# Time-Energy Uncertainty Relation for Noisy Quantum Metrology

Philippe Faist, Mischa P Woods, Victor V Albert, Joseph M Renes, Jens

Eisert, John Preskill

► To cite this version:

Philippe Faist, Mischa P Woods, Victor V Albert, Joseph M Renes, Jens Eisert, et al. Time-Energy Uncertainty Relation for Noisy Quantum Metrology. PRX Quantum, 2023, 4 (4), pp.040336. 10.1103/prxquantum.4.040336. hal-04396809

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Distributed under a Creative Commons Attribution 4.0 International License
Time-Energy Uncertainty Relation for Noisy Quantum Metrology

Philippe Faist,¹,²,³,* Mischa P. Woods,³,⁴ Victor V. Albert,⁵,²,⁶ Joseph M. Renes,³ Jens Eisert,¹ and John Preskill,²,⁶,⁷

¹Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany

²Institute for Quantum Information and Matter, Caltech, Pasadena, California, USA

³Institute for Theoretical Physics, ETH Zurich, Zurich, Switzerland

⁴University Grenoble Alpes, Inria, Grenoble 38000, France

⁵Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, Maryland, USA

⁶Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, California, USA

⁷AWS Center for Quantum Computing, Caltech, Pasadena, California, USA

<figure><img src="image_2.png" /></figure>

(Received 15 November 2022; revised 4 September 2023; accepted 16 October 2023; published 5 December 2023)

Detection of very weak forces and precise measurement of time are two of the many applications of quantum metrology to science and technology. To sense an unknown physical parameter, one prepares an initial state of a probe system, allows the probe to evolve as governed by a Hamiltonian $H$ for some time $t$, and then measures the probe. If $H$ is known, we can estimate $t$ by this method; if $t$ is known, we can estimate classical parameters on which $H$ depends. The accuracy of a quantum sensor can be limited by either intrinsic quantum noise or by noise arising from the interactions of the probe with its environment. In this work, we introduce and study a fundamental trade-off, which relates the amount by which noise reduces the accuracy of a quantum clock to the amount of information about the energy of the clock that leaks to the environment. Specifically, we consider an idealized scenario in which a party Alice prepares an initial pure state of the clock, allows the clock to evolve for a time that is not precisely known, and then transmits the clock through a noisy channel to a party Bob. Meanwhile, the environment (Eve) receives any information about the clock that is lost during transmission. We prove that Bob's loss of quantum Fisher information about the elapsed time is equal to Eve's gain of quantum Fisher information about a complementary energy parameter. We also prove a similar, but more general, trade-off that applies when Bob and Eve wish to estimate the values of parameters associated with two noncommuting observables. We derive the necessary and sufficient conditions for the accuracy of the clock to be unaffected by the noise, which form a subset of the Knill-Laflamme error-correction conditions. A state and its local time-evolution direction, if they satisfy these conditions, are said to form a metrological code. We provide a scheme to construct metrological codes in the stabilizer formalism. We show that there are metrological codes that cannot be written as a quantum error-correcting code with similar distance in which the Hamiltonian acts as a logical operator, potentially offering new schemes for constructing states that do not lose any sensitivity upon application of a noisy channel. We discuss applications of the trade-off relation to sensing using a quantum many-body probe subject to erasure or amplitude-damping noise.

DOI: 10.1103/PRXQuantum.4.040336

I. INTRODUCTION

Quantum mechanics places fundamental limits on how well we can measure a physical quantity when using

a quantum system as a probe [1]. *Quantum metrology* is an active research area addressing how physical quantities can be estimated based on observations of a probe system [2–4]. As methods for accurately controlling quantum systems steadily advance, increasingly sophisticated measurement strategies are becoming feasible [5,6], leading for example to more sensitive gravitational wave detectors [7], improved frequency standards [8], and ultra-precise quantum clocks [9]. These technological developments accentuate the need for a precise theoretical understanding of the potential of quantum

<footnote>*philippe.faist@fu-berlin.de</footnote>

<footnote>Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.</footnote>
metrology and of the ultimate limits on measurement accuracy.

Fundamental accuracy limits in quantum metrology can often be phrased in terms of uncertainty relations, wherein the accuracy of one physical quantity trades off against the accuracy of a complementary quantity. For example, a particle with a definite position has a highly uncertain momentum, and vice versa. Such trade-offs may be captured conveniently by entropic uncertainty relations [10,11]. One may envision a two-party scenario, where the entropic uncertainty relation connects the first party's ignorance about a quantity A with the second party's lack of knowledge about a complementary quantity B. Typically these quantities are values of non-commuting observables.

In this work, we focus on a related but fundamentally different type of uncertainty relation. Rather than a trade-off between the values of two observables, we consider an information-theoretic trade-off between time and energy. Specifically, we envision preparing a probe state $\rho_{init}$, which then evolves for a time $t$ as determined by some Hamiltonian $H$. By measuring the probe $\rho(t)$ at time $t$, we attempt to infer the value of $t$ [12]. The time-energy uncertainty relation relates the accuracy of our estimate of $t$ to the energy fluctuations of the probe state $\rho(t)$ [13,14]; a state with larger energy fluctuations evolves more rapidly, allowing the elapsed time to be estimated more precisely. Here, too, it is helpful to envision two parties, one attempting to measure time, the other attempting to measure energy. Indeed, such entropic time-energy uncertainty relations have recently been established [15,16].

For our purposes, a clock is a quantum system used to measure a time interval. The clock is initialized at some initial time and is measured at a later time, with the aim of the measurement being to reveal the difference in time between the initialization and the measurement. We are particularly interested in how a noise channel affects the accuracy of a clock. For that purpose we consider the following idealized scenario, involving three parties referred to as Alice, Bob, and Eve, which is amenable to precise mathematical analysis (see Fig. 1). Alice prepares a noiseless clock in the pure state vector $|\psi_{init}\rangle$, then allows that clock to evolve until some (a priori unknown) time $t$. Rather than measuring the clock herself for the purpose of estimating $t$, Alice stops the evolution of the clock and sends it to Bob through a noisy quantum channel $N_{A\to B}$. As with any noisy channel, we can represent $N_{A\to B}$ as an isometric map from Alice's system $A$ to $BE$, where $B$ is Bob's system and $E$ is the channel's environment, after which $E$ is discarded. In our scenario, Bob receives $B$ and Eve receives $E$. We wish to study the trade-off between what Bob can learn about the elapsed time by measuring $B$ and what Eve can learn about the energy of the clock by measuring $E$. Intuitively, such a trade-off is expected, because leakage to the environment of information about

<figure><img src="image_3.png" /><figcaption>FIG. 1. A noiseless clock is initialized by Alice in $|\psi_{init}\rangle$ and evolves for a time $t$ under the Hamiltonian $H$. Then Alice sends the clock through an instantaneous noisy channel $N_{A\to B}$ to Bob, who receives the state $\rho_B$, measures it, and estimates $t$. The complementary channel $\tilde{N}_{A\to E}$ describes the quantum information that leaks to the environment. Eve receives the state $\rho_E = \tilde{N}(\psi(t))$, measures, and estimates the energy parameter of $|\psi(t)\rangle$. Our main result describes the trade-off between Bob's ability to estimate the time and Eve's ability to estimate the energy.</figcaption></figure>

the clock's energy causes the clock to dephase in the energy-eigenstate basis, obscuring its evolution.

We consider the setting of local parameter estimation. This means that the value of a parameter is already approximately known, and we wish to determine it to greater accuracy. In this setting, the optimal estimate of the parameter is determined by the quantum Fisher information (QFI). For example, if $F_{Alice,t}$ denotes the QFI of Alice's state with respect to the parameter $t$, then by performing the optimal measurement on her state, Alice can estimate the value of $t$ with a mean-square error of $1/F_{Alice,t}$. For the purpose of locally estimating $t = t_0 + \Delta t$ to first order in $\Delta t$, it suffices to know the quantum state $\rho(t_0)$ and its first time derivative, and indeed the QFI is determined by just these quantities.

Bob's noisy clock, degraded by transmission through the noisy channel $N_{A\to B}$, has a reduced QFI compared to Alice's clock, and correspondingly Bob's optimal measurement yields a less accurate estimate of the time $t$ than Alice's. On the other hand, Eve receives the state of Alice's clock after transmission through the complementary noisy channel $\tilde{N}_{A\to E}$, the channel obtained if $B$ is discarded after $A$ is isometrically mapped to $BE$. We imagine that Eve wishes to learn about the energy of Alice's clock, rather than about the elapsed time. More precisely, Eve's goal is to determine an "energy parameter" denoted $\eta$ and defined in Sec. II B, which is complementary to the time $t$. Because Eve, like Bob, receives a state of the clock degraded by noise, the QFI of her state with respect to $\eta$ is in general less than Alice's. Our main result is an equality relating Bob's QFI about $t$ to Eve's QFI about $\eta$ given by

$$ \frac{F_{\text{Bob},t}}{F_{\text{Alice},t}} + \frac{F_{\text{Eve},\eta}}{F_{\text{Alice},\eta}} = 1. \qquad (1) $$
This time-energy uncertainty relation, derived in Sec. III and Appendix E using semidefinite programming duality, substantially differs from previous results [12,17,18] in that it characterizes the trade-off between Bob's and Eve's QFI, rather than the trade-off between the inherent energy variance and time uncertainty of the noiseless clock.

Figure 2 illustrates the setting of Eq. (1) in a concrete example. Alice initializes a single qubit in the pure state vector $|+\rangle = (|\uparrow\rangle + |\downarrow\rangle)/\sqrt{2}$, which evolves under the Hamiltonian $H = \omega Z/2$. Here and in the following, X, Y, Z denote the qubit Pauli-X, Y, Z operators, respectively. The qubit basis states are denoted by $|\uparrow\rangle, |\downarrow\rangle$ for consistency with which state is excited with respect to the Hamiltonian $H$, with $Z|\uparrow\rangle = |\uparrow\rangle$ and $Z|\downarrow\rangle = -|\downarrow\rangle$. Later in this work, we also use the alternative notation $|0\rangle \equiv |\uparrow\rangle$ and $|1\rangle \equiv |\downarrow\rangle$ whenever necessary to facilitate the representation of states of multiple qubits using bit strings or for consistency with the literature on quantum error-correcting codes. At time $t = t_0 + \Delta t$, the partially dephasing channel $N_p = (1-p)\text{id} + p\mathcal{D}_Z$ is applied to Alice's qubit, where $\mathcal{D}_Z(\cdot) = \langle\uparrow|\cdot|\uparrow\rangle|\uparrow\rangle\langle\uparrow| + \langle\downarrow|\cdot|\downarrow\rangle|\downarrow\rangle\langle\downarrow|$. We may describe this channel by saying that the environment (Eve) measures the qubit with probability $p$ in the energy-eigenstate basis (i.e., along the Z axis of the Bloch sphere). The partial dephasing attenuates the $t$ dependence of Bob's state $\rho_B(t)$ by the factor $1-p$, hindering his ability to estimate the time. Equation (1) captures the trade-off between Bob's information about the time (proportional to $1-p$) and Eve's information gain about the energy (proportional to $p$).

The trade-off relation Eq. (1) can be a useful tool for deriving upper bounds on QFI. The QFI for a mixed state can be tricky to characterize in cases where a diagonal representation of the state is not easily obtained. Along these lines, it is useful to note that QFI obeys a data processing inequality, which ensures that, for any state $\rho$ and any quantum channel $N$, the QFI of $N(\rho)$ is no larger than the QFI of $\rho$ [19]. We can imagine that Eve applies a channel to her state $\rho_E$, obtaining the state $\rho'_E$, which she then measures for the purpose of estimating $\eta$. Using the data processing inequality, we conclude that

$$ \frac{F_{\text{Bob},t}}{F_{\text{Alice},t}} + \frac{F'_{\text{Eve},\eta}}{F_{\text{Alice},\eta}} \le 1, \qquad (2) $$

where now $F'_{\text{Eve},\eta}$ denotes the QFI of $\rho'_E$ with respect to $\eta$. Even if the QFI of $\rho_E$ is difficult to compute, the QFI of $\rho'_E$ may be easy to compute if the channel taking $\rho_E$ to $\rho'_E$ is artfully chosen; then Eq. (2) provides a computable upper bound on $F_{\text{Bob},t}$. For example, in the case where $N_{A \to B}$ is an amplitude damping noise channel, a useful upper bound on Bob's QFI can be derived by applying a completely dephasing channel to Eve's state $\rho_E$. We apply this idea to an Ising spin chain in Sec. VIII.

<figure><img src="image_4.png" /><figcaption>FIG. 2. Illustration of Eq. (1) for a single qubit subjected to partial dephasing. Alice's clock state is initialized as $|+\rangle = (|\uparrow\rangle + |\downarrow\rangle)/\sqrt{2}$ and evolves according to the Hamiltonian $H = \omega Z/2$, where $Z$ denotes the qubit Pauli-Z operator. At time $t$, the channel $N_p(\cdot) = (1-p)\langle\uparrow|\cdot|\uparrow\rangle\langle\uparrow| + p|\downarrow\rangle\langle\downarrow|\cdot|\downarrow\rangle\langle\downarrow|$ is instantaneously applied to Alice's clock state. In effect, Eve measures the energy observable $Z$ with probability $p$, and Bob receives the partially dephased clock. Equation (1) relates Bob's reduced information about the elapsed time to Eve's information gain about the clock's energy. Unitary evolution in Eve's complementary energy variable $\eta$, generated by an optimal local time-sensing observable, rotates the state into a direction that is orthogonal to the direction of the original evolution in time $t$ (see Sec. II B).</figcaption></figure>

One consequence of Eq. (1) is a necessary and sufficient condition for the clock's sensitivity to be unaffected by transmission through the noisy channel $N_{A \to B}: F_{\text{Bob},t} = F_{\text{Alice},t}$ if and only $F_{\text{Eve},\eta} = 0$. This condition can be usefully restated in terms of the Kraus operators $\{E_k\}$ of the channel $N_{A \to B}$. Recall that we aim to estimate the time $t = t_0 + \Delta t$ in the setting of local parameter estimation, i.e., to linear order in $\Delta t$. Suppose that after evolution for time $t_0$, the state of Alice's clock is $|\psi\rangle$, and that $|\xi\rangle = (H - \langle H|\psi\rangle)|\psi\rangle = P_\psi^\perp H|\psi\rangle$ with $P_\psi^\perp = 1 - |\psi\rangle\langle\psi|$. Then the condition $F_{\text{Eve},\eta} = 0$ is equivalent to

$$ \langle\xi|E_k^\dagger E_j|\psi\rangle + \langle\psi|E_k^\dagger E_j|\xi\rangle = 0 \quad \text{for all } k,j. \qquad (3) $$

Intuitively, Eq. (3) means that the action of the channel on the clock cannot be confused with genuine time evolution.

Equation (3) may be recognized as a weakened version of the Knill-Laflamme condition for quantum error correction, the necessary and sufficient condition for the action of a noisy channel on an encoded subspace to be reversible by a suitable recovery channel [20]. This condition may be stated as $\Pi_L E_k^\dagger E_j \Pi_L \propto \Pi_L$ for all $k$ and $j$, where $\Pi_L$ is the projector onto the encoded subspace. To write Eq. (3) in a similar form, consider the two-dimensional subspace spanned by the mutually orthogonal state vectors $|\psi\rangle$ and $|\xi\rangle$; we call this two-dimensional space a “virtual qubit.” Using the notation $|+\rangle_L := |\psi\rangle, |-\rangle_L := \|\xi\rangle^{-1}|\xi\rangle$, the orthogonal projector onto the virtual qubit is $\Pi_L = |+\rangle\langle+|_L + |-\rangle\langle-|_L$,
and $Z_L = |+\rangle\langle-|_L + |-\rangle\langle+|_L$ is the logical Z Pauli operator acting on the virtual qubit. In this language, Eq. (3) becomes

$$ \mathrm{tr}(\Pi_L E_k^\dagger E_j \Pi_L Z_L) = 0 \quad \text{for all } k,j. \quad (4) $$

The condition Eq. (4) is reminiscent of a recently formulated condition for quantum coding to improve how measurement sensitivity scales with increasing sensing time [21,22]. In Sec. VII, we explain how time-covariant quantum error-correcting codes automatically fulfill Eq. (3), providing some simple examples. In particular, we consider spins on a graph with Ising or Heisenberg interactions and construct a state vector $|\psi\rangle$ that fulfills Eq. (3), where the noise model inflicts a single located erasure.

We have derived the trade-off relation Eq. (1) in a highly idealized setting, in which noiseless evolution of Alice's clock is followed by transmission to Bob through the noisy channel $N_{A\to B}$. For an actual clock, the noise acts continuously as the clock evolves, rather than after the time evolution is complete. By focusing on the idealized setting, we have been able to perform a particularly elegant analysis of the time-energy trade-off. But in Sec. VI we connect our results to the more realistic case of continuous Markovian noise described by a master equation in Lindblad form, noting that the two settings are actually equivalent, or nearly equivalent, under certain conditions. One can decompose the Lindbladian into a Hamiltonian part and a noise part that contains all the jump operators; if, for example, these two parts define commuting channels, then the Markovian evolution for time $t$ is equivalent to Hamiltonian evolution for time $t$ followed by a noise channel $N_t$. Other cases where the Lindblad evolution is compatible with a trade-off relation of the form Eq. (1) (at least to a good approximation) are identified in Sec. VI.

Although the time-energy trade-off provided the primary motivation for this work, we find that a trade-off relation similar to Eq. (1) can be derived in a more general setting. Suppose that $A$ and $B$ are Hermitian operators, and that $\psi = |\psi\rangle\langle\psi|$ is a pure quantum state. We may consider the “flow” in Hilbert space generated by $A$ or by $B$. That is, we consider a one-parameter family of pure states close to $\psi$, generated by $A$ and parameterized by $a$, and a one-parameter family generated by $B$ and parametrized by $b$, such that

$$ \partial_a \psi = -i[A, \psi], \quad \partial_b \psi = -i[B, \psi]. \quad (5) $$

In the setting of local parameter estimation, we suppose that Bob wishes to estimate the parameter $a$ and Eve wants to estimate the parameter $b$, where $a$ and $b$ are both small. Alice's QFI about $a$ is $F_{\mathrm{Alice},a}$, but Bob receives the state via the noisy channel $N_{A\to B}$, so his QFI about $a$ ($F_{\mathrm{Bob},a}$) is in general smaller than Alice's. Alice's QFI about $b$ is $F_{\mathrm{Alice},b}$, but Eve receives the state via the complementary

channel $\hat{N}_{A\to E}$, so her QFI about $b$ ($F_{\mathrm{Eve},b}$) is in general smaller than Alice's. In Sec. III we derive the trade-off relation

$$ \frac{F_{\mathrm{Bob},a}}{F_{\mathrm{Alice},a}} + \frac{F_{\mathrm{Eve},b}}{F_{\mathrm{Alice},b}} \le 1 + 2 \sqrt{1 - \frac{\langle i[A, B]\rangle_{\psi}^2}{4\sigma_A^2\sigma_B^2}}, \quad (6) $$

where $\sigma_M := [\langle M^2 \rangle_\psi - \langle M \rangle_\psi^2]^{1/2}$ denotes the standard deviation of the observable $M$. Note that, in contrast to Eq. (1), this relation is an inequality rather than an equality. It is reminiscent of the Robertson uncertainty relation, with the commutator quantifying the incompatibility of the observables $A$ and $B$.

Figure 3 summarizes the structure of this work and provides an overview of our results. In Sec. II, we introduce the setting of local parameter estimation, recall some useful properties of the QFI, define the energy parameter $\eta$, and review the concept of a complementary quantum channel. We sketch the proof of the trade-off relation Eq. (1) and its generalization Eq. (6) in Sec. III (more details can be found in Appendix E), and discuss some examples in Sec. IV. We use the trade-off relation to derive upper bounds on the QFI in Sec. V. In Sec. VI we discuss how the setting in Fig. 1 is connected with the more realistic setting of continuous Markovian noise. In Sec. VII we derive the necessary and sufficient condition Eq. (3) for the clock's sensitivity to be undiminished by transport through the noisy channel $N_{A\to B}$, and discuss some of the implications of this condition. Numerical results for our upper bound on QFI in many-body systems are reported in Sec. VIII. We summarize and comment on our results in Sec. IX. Many further details are presented in the appendices.

## II. SETTING

We review the standard setting in quantum metrology of single-parameter estimation. We then introduce our noise model and the quantities that are relevant to formulate our uncertainty relation.

### A. Quantum parameter estimation

Consider a quantum state $\rho(t)$ that depends on a single parameter $t$. The task we study is how well the parameter $t$ can be estimated by performing suitable measurements (Fig. 4). In the context of this work, the parameter $t$ is identified with physical time, although the results hold for any general real parameter that the quantum state might depend on.

We consider the setting of local sensitivity, where the goal of the quantum measurement is to refine the precision to which we determine the parameter if the value of the parameter is already known to be close to a given value $t_0$. More precisely, we seek a measurement operator $T$ with minimal variance such that the expectation value of
<figure><img src="image_5.png" /><figcaption>FIG. 3. Overview of our main results and structure of this work.</figcaption></figure>

T reveals the value of the parameter locally around $t_0$ to first order in $dt$, i.e.,

$$ \langle T \rangle_{\rho(t_0+dt)} = t_0 + dt + O(dt^2). \quad (7) $$

<figure><img src="image_6.png" /><figcaption>FIG. 4. In the setting of quantum parameter estimation, the task is to infer a parameter $t$ in a one-parameter family of states $t \mapsto \rho(t)$ through suitable measurements. For local parameter estimation, we assume the value of the parameter is already known to lie in the neighborhood of a given value $t_0$. The measurement is required to refine the parameter estimation by optimally distinguishing $\rho(t_0)$ from $\rho(t_0 + dt)$ to first order in $dt$. This setting is standard in the field of quantum metrology, and the optimal sensitivity is quantified by a quantity known as the Fisher information.</figcaption></figure>

Identifying the orders in $dt$ we see that Eq. (7) is equivalent to

$$ \langle T \rangle_{\rho(t_0)} = t_0 \quad \text{and} \quad \text{tr}(T \partial_t \rho(t_0)) = 1, \quad (8) $$

using the notation $\partial_t \rho = \partial \rho / \partial t$. (We write a partial derivative instead of a total derivative in anticipation of other variables, which will be introduced later.) In the literature, it is common to reuse the symbol $t$ for both the parameter on which $\rho$ depends as well as the reference value of the parameter $t_0$. We keep the distinction for clarity.

Here, we restricted the measurement to be projective, as described by the Hermitian observable $T$. A more general positive operator-valued measure (POVM) does not offer any more sensitivity in sensing the parameter [1,12].

A central result in quantum metrology is the quantum Cramér-Rao bound, which states that the optimal sensitivity to which one can determine the parameter $t$ locally around $t_0$ is determined by a quantity called the quantum
Fisher information [1,23,24]. The quantum Fisher information of the state $\rho(t_0)$ with respect to a direction $\partial_t\rho(t_0)$ is defined as

$$F(\rho; \partial_t \rho) = \mathrm{tr}(\rho R^2), \qquad (9)$$

where R is any Hermitian operator that solves the equation $1/2\{\rho, R\} = 1/2(\rho R + R\rho) = \partial_t\rho$, and where the quantities $\rho$ and $\partial_t\rho$ are evaluated at $t_0$. The Cramér-Rao bound can be formulated for our purposes as follows: for any observable T that satisfies Eq. (8), we must have

$$\langle (T - t_0)^2 \rangle_{\rho(t_0)} \geq \frac{1}{F(\rho(t_0); \partial_t \rho(t_0))}, \qquad (10)$$

and furthermore, the equality in Eq. (10) can always be achieved by a suitable choice of T. We refer to a choice of T, which is optimal in Eq. (10) as an *optimal local sensing observable for t*.

The operator R in Eq. (9) is called a *symmetric logarithmic derivative*. When $\rho$ and $\partial_t\rho$ commute, we can choose $R = \rho^{-1}\partial_t\rho = (\partial/\partial t)\ln\rho$. A general construction of R in terms of an eigendecomposition of $\rho$ is given as follows [25]. Consider an eigenbasis $\{|k\rangle\}$ of $\rho$ that spans the full Hilbert space, such that $\rho = \sum_k \lambda_k |k\rangle\langle k|$ and $k = 1, 2, \dots, \dim(\mathcal{H})$, then

$$R = \sum_{\substack{k, k': \\ \lambda_k + \lambda_{k'} \neq 0}} \frac{2}{\lambda_k + \lambda_{k'}} \langle k | \partial_t \rho | k' \rangle |k\rangle \langle k'|, \qquad (11)$$

where the sum ranges over all pairs of indices $k, k'$ except those for which both $\lambda_k = 0$ and $\lambda_{k'} = 0$. The expression for the Fisher information becomes $F(\rho; \partial_t\rho)$, where

$$F(\rho; \partial_t \rho) = \sum_{\substack{k, k': \\ \lambda_k + \lambda_{k'} \neq 0}} \frac{2}{\lambda_k + \lambda_{k'}} |\langle k | \partial_t \rho | k' \rangle|^2. \qquad (12)$$

The solution to the anticommutator equation $1/2\{\rho, R\} = \partial_t\rho$ is unique up to transformations of the form $R \mapsto R + P_\rho^\perp MP_\rho^\perp$, where M is an arbitrary Hermitian operator, where $P_\rho^\perp = 1 - P_\rho$, and where $P_\rho$ denotes the projector onto the support of $\rho$. In the event that $P_\rho^\perp \frac{d\rho}{dt} P_\rho^\perp \neq 0$, there is no solution for R. In such a situation, the optimal estimation variance (10) is zero and the Fisher information is not defined; such cases do not arise in the setting we consider in this work.

We review the solutions to the anticommutator equation $1/2\{\rho, R\} = \partial_t\rho$ in Appendix B. In Appendix C, the definition and elementary properties of the Fisher information are reviewed using simple techniques based on semidefinite programming. In Appendix D, we review a derivation of the Cramér-Rao bound using these methods.

Observables T that estimate the time parameter t with an accuracy that achieves the Cramér-Rao bound (10), i.e., the optimal local sensing observables, turn out to be the projective measurements with outcomes associated with the eigenspaces of a symmetric logarithmic derivative [1,23]. Specifically, any optimal local sensing observable for t is of the form

$$T = t_0 + \frac{1}{F(\rho; \frac{d\rho}{dt})} R, \qquad (13)$$

where R is as above any solution to the anticommutator equation $1/2\{\rho, R\} = d\rho/dt$ (see Appendix D for a review of the proof). Due to the freedom in the choice of R, all optimal local sensing observables for t differ by a term of the form $P_\rho^\perp MP_\rho^\perp$ where M is any Hermitian operator.

In the remaining part of this section, we review a few properties of the Fisher information for later use (see Appendix C for details). First is a scaling property: if $0 < \alpha \le 1$ and $\beta \in \mathbb{R}$, we have

$$F(\alpha\rho; \beta \partial_t \rho) = \frac{\beta^2}{\alpha} F(\rho; \partial_t \rho), \qquad (14)$$

where the definition (9) is formally extended to positive semidefinite operators $\rho$ that satisfy $\mathrm{tr}\rho \le 1$. Second, in case the state $\rho$ and derivative $\partial_t\rho$ commute, the Fisher information takes the simple form

$$[\rho, \partial_t \rho] = 0 \quad \Rightarrow \quad F(\rho; \partial_t \rho) = \mathrm{tr}[\rho^{-1} (\partial_t \rho)^2]. \qquad (15)$$

Finally, for general $\rho, \partial_t\rho$, we can express the Fisher information in terms of a pair of convex optimization problems [26–28] as

$$\begin{aligned} & \frac{1}{4} F(\rho; \partial_t \rho) \\ &= \max_{S=S^\dagger} \left\{ \mathrm{tr}[(\partial_t \rho) S] - \mathrm{tr}[\rho S^2] \right\} \end{aligned} \qquad (16a)$$

$$= \min_{L \text{ arb.}} \left\{ \mathrm{tr}(L^\dagger L) : \rho^{1/2} L + L^\dagger \rho^{1/2} = \partial_t \rho \right\}. \qquad (16b)$$

These two optimizations can be cast as semidefinite problems that are dual to each other. These optimizations are convenient to derive bounds on the Fisher information, as it suffices to exhibit suitable candidates in Eqs. (16a) or (16b).

## B. Time and energy parameters of the noiseless clock

Now we turn to the setup depicted in Fig. 1, in which Alice possesses a noiseless quantum clock, which she sends to Bob through a given noisy channel. In this subsection, we study Alice’s noiseless quantum clock, and in the following subsection we study the effect of the noise.
1. The noiseless clock

Suppose that Alice prepares a quantum clock in a pure state living in a finite-dimensional Hilbert space $\mathcal{H}_A$. She lets it evolve according to a Hamiltonian $H(t)$, generating a one-parameter family of state vectors $t \mapsto |\psi(t)\rangle$. The time evolution of $\psi(t) = |\psi(t)\rangle\langle\psi(t)|$ is governed by the standard Schrödinger time evolution

$$ \partial_t \psi := \frac{\partial \psi}{\partial t} = -i[H, \psi]. \qquad (17) $$

We now compute the Fisher information associated with Alice's clock locally around a time of interest $t_0$, following the definition (9). For any $t_0$, we can choose $R = 2 \partial_t \psi = -2i[H, \psi]$, because $\{\partial_t \psi, \psi\} = \partial_t (\psi^2) = \partial_t \psi$. Alice's Fisher information $F_{\text{Alice},t}$ for the evolution $|\psi(t)\rangle$ at the time of interest $t_0$ is therefore given by

$$ F_{\text{Alice},t} := F(\psi; -i[H, \psi]) = 4\sigma_H^2, \qquad (18) $$

where $\psi$ and $H$ are evaluated at time $t_0$, and where again, we denote by $\sigma_M = [(M^2)_\psi - (M)_\psi^2]^{1/2}$ the standard deviation of an observable $M$. Alternative expressions of the standard deviation are given by

$$ \sigma_M^2 = \langle (M - \langle M \rangle)^2 \rangle = -\langle [M, \psi]^2 \rangle, \qquad (19) $$

writing $\langle M \rangle := \langle M \rangle_\psi$ for brevity.

Around the point $t_0$, any optimal local sensing observable for $t$ takes the form given by Eq. (13), which we can rewrite in this context as

$$ T = t_0 - \frac{i[H, \psi]}{2\sigma_H^2} + P_\psi^\perp MP_\psi^\perp, \qquad (20) $$

where $M$ is any Hermitian operator. In the case where $H$ is time independent, then $F_{\text{Alice},t}$ does not depend on the time of interest $t_0$, but the optimal sensing observable $T$ depends on $t_0$ not only directly but also indirectly through $\psi$ and $\partial_t \psi$. In the following, we fix $t_0$ and we consider only the evolution $|\psi(t)\rangle$ locally at $t_0$. Furthermore, we use the shorthand $|\psi\rangle := |\psi(t_0)\rangle$.

2. The energy parameter

The optimal local time-sensing observable $T$ in Eq. (20), being a Hermitian operator, can be used to generate a different evolution in an alternative direction in the space of quantum states. In our setup, we define $\eta_0 = \langle H \rangle_\psi$ and we consider any family of state vectors $\eta \mapsto |\psi(\eta)\rangle$ such that $|\psi(\eta = \eta_0)\rangle = |\psi\rangle = |\psi(t = t_0)\rangle$ and such that at the point $|\psi(\eta = \eta_0)\rangle$ we have

$$ \partial_\eta \psi = i[T, \psi]. \qquad (21) $$

This evolution can be interpreted as a Schrödinger equation with the effective Hamiltonian $-T$. An example

<figure><img src="image_7.png" /><figcaption>FIG. 5. We define a parameter $\eta$ that is complementary to time evolution and that represents the energy of the state. Consider a quantum clock modeled as a pure state $\psi$ evolving according to the Schrödinger equation $\partial_t \psi = -i[H, \psi]$, where $H$ is the Hamiltonian. Locally around $t_0$, the observable $T$ that optimally distinguishes the neighboring states $\psi(t_0)$ and $\psi(t_0 + dt)$ defines an optimal local time-sensing observable. $T$ is the relevant measurement to carry out to optimally read out the information about time stored in the quantum clock. We now consider locally around $\psi(t_0)$ the direction in state space defined by $\partial_\eta \psi = i[T, \psi]$, i.e., a Schrödinger-type evolution with $-T$ playing the role of an effective Hamiltonian. It turns out that the optimal estimation procedure for the parameter $\eta$ is to measure $H$ itself. Therefore, the parameter $\eta$ represents the energy of $\psi(\eta)$. The parameters $t$ and $\eta$ are, therefore, complementary to each other in the sense that the generator associated with one parameter optimally distinguishes neighboring values of the other parameter and vice versa.</figcaption></figure>

of such an evolution is

$$ |\psi(\eta)\rangle = e^{iT(\eta-\eta_0)} |\psi\rangle. \qquad (22) $$

Interestingly, the evolution generated in this way locally around $|\psi\rangle$ turns out to be complementary to time evolution in the sense that we can derive a meaningful uncertainty relation and that the parameter $\eta$ can be identified with the average energy of the state vector $|\psi(\eta)\rangle$ (see Fig. 5).

More formally and to clarify the dependencies of $|\psi\rangle$ on $t$ and $\eta$, we consider a two-parameter family of state vectors $(t, \eta) \mapsto |\psi(t, \eta)\rangle$ with $|\psi(t_0, \eta_0)\rangle = |\psi\rangle$ and such that at the point $(t_0, \eta_0)$ we have

$$ \partial_t \psi(t_0, \eta_0) = -i[H, \psi], \quad \partial_\eta \psi(t_0, \eta_0) = i[T, \psi], \qquad (23) $$

where $T$ is given by Eq. (20). For example, we could choose

$$ |\psi(t, \eta)\rangle = \exp\{-i[(t - t_0)H - (\eta - \eta_0)T]\} |\psi\rangle. \qquad (24) $$

Unless indicated otherwise, the state vector $|\psi\rangle$ and the corresponding derivatives $\partial_t \psi, \partial_\eta \psi$ are henceforth implicitly evaluated at $(t_0, \eta_0)$. We use the shorthands $|\psi(t)\rangle := |\psi(t, \eta_0)\rangle$ and $|\psi(\eta)\rangle := |\psi(t_0, \eta)\rangle$ to denote the respective evolutions according to $t$ and $\eta$ in which the other
parameter is fixed to $\eta_0$ or $t_0$, respectively; the name of the argument ($t$ or $\eta$) determines which evolution is meant.

Let us re-express the derivative $\partial_{\eta}\psi$ of $\psi = |\psi\rangle\langle\psi|$ in terms of the Hamiltonian. Using Eq. (20), we have

$$ \partial_{\eta}\psi = i[T, \psi] = \frac{1}{2\sigma_H^2}[[H, \psi], \psi]. \quad (25) $$

A brief computation reveals that $[[H, \psi], \psi] = H\psi + \psi H - 2\langle H \rangle \psi = \{H - \langle H \rangle, \psi\}$ and therefore

$$ \partial_{\eta}\psi = \frac{1}{2\sigma_H^2}\{H - \langle H \rangle, \psi\}. \quad (26) $$

Alice's Fisher information with respect to the parameter $\eta$ is given by the same expression as Eq. (18), but with $H$ and $t$ replaced by $-T$ and $\eta$, to get

$$ F_{\text{Alice},\eta} := F(\psi; i[T, \psi]) = 4\sigma_T^2 = \frac{1}{\sigma_H^2}, \quad (27) $$

where the last equality follows from

$$ \sigma_T^2 = \langle (T - t_0)^2 \rangle = \frac{-\langle [H, \psi]^2 \rangle}{4\sigma_H^4} = \frac{1}{4\sigma_H^2}. \quad (28) $$

To justify that the parameter $\eta$ in the evolution (21) can be associated with the energy of the state vector locally around $|\psi\rangle$, we compute the optimal sensing observable for $\eta$ and show that it is the Hamiltonian $H$ itself (up to terms lying outside of the support of $\psi$). The optimal local-sensing observable that distinguishes $\psi(\eta)$ from $\psi(\eta + d\eta)$ is given by Eq. (13), but with the parameter $t$ replaced by the parameter $\eta$. Using Eq. (26), observe that the operator $R' = (H - \langle H \rangle)/\sigma_H^2$ solves the equation $\{\psi, R'\}/2 = \partial_{\eta}\psi$. From Eq. (13) and substituting $t$ by $\eta$, we see that the optimal local-sensing observable for $\eta$ is simply $\eta_0 + H - \langle H \rangle = H$. That is, the optimal measurement distinguishing $|\psi(\eta_0)\rangle$ from $|\psi(\eta_0 + d\eta)\rangle$ is the Hamiltonian $H$ itself, up to a term $P_{\psi}^{\perp}MP_{\psi}^{\perp}$ for any Hermitian $M$. [Alternatively, the same conclusion would have been reached had we started from Eq. (20) with $t, H$ replaced by $\eta, -T$. A more detailed computation is provided in Appendix D.] Therefore, the parameter $\eta$ describes an evolution along which, locally around $|\psi\rangle$, we have $\eta_0 + d\eta = \langle H \rangle_{\psi(\eta_0+d\eta)}$. In this sense, $\eta$ represents the energy of the probe $|\psi(\eta)\rangle$ locally around $\eta_0$.

To summarize, the evolution of $\psi(t) = |\psi(t)\rangle\langle\psi(t)|$ is generated by the Hamiltonian $H$; nearby states $\psi(t_0)$ and $\psi(t_0 + dt)$ are optimally distinguished by a local time-sensing observable $T$. The complementary evolution $\psi(\eta)$ is one that inverts the roles of $H$ and $T$: the evolution $\psi(\eta)$ is generated by $T$, and $H$ is the operator that optimally distinguishes neighboring states $\psi(\eta_0)$ and $\psi(\eta_0 + d\eta)$.

### 3. Single-qubit example

Consider a qubit initialized in the state vector $|\psi_{\text{init}}\rangle = |+\rangle$, where $|\pm\rangle = [[\uparrow\rangle \pm |\downarrow\rangle]/\sqrt{2}$, and let the qubit evolve according to the Hamiltonian $H = \omega Z/2$ (i.e., Alice's system in Fig. 2). The time evolution of the clock is given by $|\psi(t)\rangle = U_t|+\rangle$, where $U_t = e^{-iHt}$; we see that

$$ \begin{aligned} |\psi(t)\rangle &= \frac{1}{\sqrt{2}}[e^{-\frac{i\omega t}{2}}|\uparrow\rangle + e^{\frac{i\omega t}{2}}|\downarrow\rangle] \\ &= \cos(\frac{\omega t}{2})|+\rangle - i\sin(\frac{\omega t}{2})|-\rangle. \end{aligned} \quad (29) $$

It is also convenient to note that

$$ \psi(t) = U_t|+\rangle\langle+|U_t^\dagger = \frac{1}{2} + \frac{1}{2}U_tXU_t^\dagger \quad (30a) $$

$$ = \frac{1}{2} + \frac{1}{2}[\cos(\omega t)X + \sin(\omega t)Y], \quad (30b) $$

using the identity $|+\rangle\langle+| = [1+X]/2$ along with $e^{-iaZ}Xe^{iaZ} = \cos(2a)X + \sin(2a)Y$. The time derivative of the state is

$$ \begin{aligned} \partial_t \psi(t) &= -i[H, \psi(t)] = -\frac{i\omega}{2}[Z, U_t|+\rangle\langle+|U_t^\dagger] \\ &= -\frac{i\omega}{2}U_t[Z, \frac{1+X}{2}]U_t^\dagger \\ &= \frac{\omega}{2}U_tYU_t^\dagger \end{aligned} \quad (31a) $$

$$ = \frac{\omega}{2}[\cos(\omega t)Y - \sin(\omega t)X], \quad (31b) $$

using $e^{-iaZ}Ye^{iaZ} = \cos(2a)Y - \sin(2a)X$. The expressions (30a) and (31a) manifest the fact that the state and the derivative evolve in time by rotation around the Z axis of the Bloch sphere, whereas we can read out from the expressions (30b) and (31b) the information about the time evolution of the components of the Bloch vector. The average energy $\langle H \rangle_{\psi(t)}$ is

$$ \langle H \rangle_{\psi(t)} = \text{tr}[U_t|+\rangle\langle+|U_t^\dagger \frac{\omega Z}{2}] = 0 \quad (32) $$

for all $t$, noting that $U_t$ commutes with $Z$ and that $\langle +|Z|+\rangle = 0$. The energy's standard deviation at time $t$ is then

$$ \sigma_H^2 = \langle H^2 \rangle_{\psi(t)} = \frac{\omega^2}{4}, \quad (33) $$

noting that $Z^2 = 1$.

We now compute the time sensitivity and the optimal time-sensing observable locally around a given time $t_0$. We write $|\psi\rangle = |\psi(t_0)\rangle$ for short. The optimal time-sensing
observable is given by Eq. (20), which we can compute as (ignoring the degree of freedom $P_\psi^\perp MP_\psi^\perp$),

$$ T - t_0 = \frac{1}{2\sigma_H^2} \partial_t \psi(t_0) = \frac{1}{\omega} U_{t_0} Y U_{t_0}^\dagger \\ = \frac{1}{\omega} [\cos(\omega t_0) Y - \sin(\omega t_0) X]. \quad (34) $$

The optimal sensing observable $T$ is therefore aligned with the direction on the Bloch sphere that is tangent to the state's evolution.

We now determine the parameter $\eta$. It is generated by $T$ as per Eq. (21), and we can compute the associated derivative using Eq. (26) as

$$ \{H - \langle H \rangle, \psi\} = \frac{\omega}{2} \left\{ Z, \frac{1+X}{2} \right\} = \frac{\omega}{2} Z, \quad (35) $$

$$ \partial_\eta \psi = \frac{1}{2\sigma_H^2} \frac{\omega}{2} Z = \frac{Z}{\omega}, \quad (36) $$

recalling that the Pauli matrices along different directions anticommute. The direction associated with the $\eta$ parameter is aligned with the $Z$ axis of the Bloch sphere (Fig. 2), which is the direction in which the Hamiltonian is oriented.

We can now compute the sensitivities with respect to $t$ and $\eta$ using Eqs. (18) and (27) as

$$ F_{\text{Alice},t} = 4\sigma_H^2 = \omega^2, \quad F_{\text{Alice},\eta} = \frac{1}{\sigma_H^2} = \frac{4}{\omega^2}. \quad (37) $$

Finally, we can check that $H$ is an optimal local-sensing observable for $\eta$. First observe with $\eta_0 = \langle H \rangle_\psi = 0$ that

$$ \langle H \rangle_{\psi(t_0, \eta_0 + d\eta)} = d\eta \operatorname{tr}(H \partial_\eta \psi) = d\eta, \quad (38) $$

using Eq. (36) along with $Z^2 = 1$. Hence, $H$ satisfies the condition (7) for the parameter $\eta$. The variance of this observable was computed above as

$$ \langle (H - \langle H \rangle)^2 \rangle = \sigma_H^2 = \frac{\omega^2}{4} = \frac{1}{F_{\text{Alice},\eta}}, \quad (39) $$

and therefore $H$ also saturates the Cramér-Rao bound. It is an optimal local-sensing observable.

### C. The noisy channel and the environment

#### 1. The noisy clock

Suppose that Alice sends the clock from its noiseless environment to a receiver Bob through a noisy channel $\mathcal{N}_{A \to B}$ (Fig. 1). Bob has access to the noisy clock state

$$ \rho_B(t) = \mathcal{N}_{A \to B}(\psi(t)). \quad (40) $$

We consider the sensitivity of Bob's clock locally around $t_0$, i.e., we ask how well Bob can distinguish $\rho_B(t_0)$ from

$\rho_B(t_0 + dt)$. We assume that the noisy channel $\mathcal{N}_{A \to B}$ does not depend on $t$. This setting is nonstandard in the context of quantum metrology. Usually, one considers a quantum clock that is exposed to continuous noise as it evolves in time instead of the noise being applied separately and instantaneously after the system has evolved unitarily for a given amount of time. This alternative setting represents the situation where Alice would like to send a quantum reference frame to Bob over a noisy channel [29]. We defer the discussion of the connections between these two settings to Sec. VI.

Locally around $t_0$, Bob's optimal sensitivity is given via the Cramér-Rao bound (10) by Bob's Fisher information with respect to time,

$$ F_{\text{Bob},t} := F(\rho_B(t_0); \partial_t \rho_B(t_0)). \quad (41) $$

We may furthermore express $\rho_B$ and $\partial_t \rho_B$ as

$$ \begin{gathered} \rho_B = \mathcal{N}_{A \to B}(\psi), \\ \partial_t \rho_B = \mathcal{N}_{A \to B}(\partial_t \psi) = \mathcal{N}_{A \to B}(-i[H, \psi]). \end{gathered} \quad (42) $$

Determining $F_{\text{Bob},t}$ in principle requires the usage of a general expression of the Fisher information for mixed states such as Eq. (9) or Eq. (12), which can be significantly more cumbersome to manipulate as opposed to computing the variance of the Hamiltonian in the case of a pure-state evolution.

#### 2. The environment

Any quantum channel $\mathcal{N}_{A \to B}$ can be expressed as a unitary evolution over a larger system, where the environment is initialized in a pure state. This construction is known as a Stinespring dilation. The initial pure state of the environment can be contracted with the global unitary to give a more concise description of the Stinespring dilation in terms of an isometry $A \to BE$. More precisely, any quantum channel $\mathcal{N}_{A \to B}$ can be written as

$$ \mathcal{N}_{A \to B}(\cdot) = \operatorname{tr}_E (V_{A \to BE}(\cdot) V^\dagger), \quad (43) $$

where $E$ is a suitable environment system, and where $V_{A \to BE}$ is an isometry mapping states of $A$ into $B \otimes E$.

The system $E$, which we call Eve, represents the quantum information that is discarded by the channel $\mathcal{N}_{A \to B}$. Instead, we can consider a quantum channel that describes what Eve gets if Bob's system $B$ is discarded. By tracing out $B$ instead of $E$ in Eq. (43) we obtain the complementary channel,

$$ \widehat{\mathcal{N}}_{A \to E}(\cdot) = \operatorname{tr}_B (V_{A \to BE}(\cdot) V^\dagger). \quad (44) $$
If we write the noisy channel in an operator-sum representation with Kraus operators {$E_k$} as

$$ \mathcal{N}(\cdot) = \sum_k E_k(\cdot) E_k^{\dagger}, \qquad (45) $$

we may write a corresponding complementary channel as

$$ \hat{\mathcal{N}}(\cdot) = \sum_{k,k'} \mathrm{tr}(E_k^{\dagger} E_k(\cdot)) |k\rangle \langle k'|_E, \qquad (46) $$

for some orthonormal basis {$|k\rangle$} on the environment system $E$. The complementary channel is unique up to a partial isometry on the environment system.

Our main result involves Eve's sensitivity to the $\eta$ parameter of the state that she obtains if Alice's quantum clock is sent to $E$ via the complementary channel. Namely, we define

$$ \rho_E(\eta) = \hat{\mathcal{N}}_{A \to E}(\psi(\eta)). \qquad (47) $$

Recalling Eq. (26), we have

$$ \partial_{\eta} \rho_E = \frac{1}{2\sigma_H^2} \hat{\mathcal{N}}_{A \to E}(\{H - \langle H \rangle, \psi\}). \qquad (48) $$

As for $\psi$, $\partial_t \psi$, and $\partial_{\eta} \psi$, the states $\rho_B$, $\rho_E$ and the derivatives $\partial_t \rho_B$, $\partial_{\eta} \rho_E$ are implicitly evaluated at $(t_0, \eta_0)$ unless specified otherwise. We also abbreviate $\mathcal{N}_{A \to B}$ and $\hat{\mathcal{N}}_{A \to B}$ by $\mathcal{N}$ and $\hat{\mathcal{N}}$ for convenience and whenever it is unambiguous to do so.

### III. BIPARTITE UNCERTAINTY RELATION FOR THE FISHER INFORMATION

#### A. Equality Fisher information trade-off for time and energy and expression for sensitivity loss

Sending Alice's clock to Bob through the noisy channel $\mathcal{N}_{A \to B}$ reduces the clock's sensitivity to the time parameter $t$. On the other hand, sending the clock to Eve through the complementary channel $\hat{\mathcal{N}}_{A \to E}$ enables Eve to gain sensitivity with respect to the energy parameter $\eta$. Our main result characterizes how these two effects are related.

**Theorem 1 (Bipartite time-energy uncertainty relation).**—Suppose Alice prepares a probe in a quantum state vector $|\psi\rangle$ and consider the local parameters $t$, $\eta$ defining directions in state space generated by $H$ and $T$ and centered at $|\psi\rangle = |\psi(t_0, \eta_0)\rangle$ as in Eq. (23). Alice sends her probe to Bob through a channel $\mathcal{N}_{A \to B}$; let Eve represent the output of the corresponding complementary channel $\hat{\mathcal{N}}_{A \to E}$ (see Fig. 1). Then

$$ \frac{F_{\mathrm{Bob},t}}{F_{\mathrm{Alice},t}} + \frac{F_{\mathrm{Eve},\eta}}{F_{\mathrm{Alice},\eta}} = 1, \qquad (49) $$

provided the rank of $\mathcal{N}(\psi(t))$ does not change at $t_0$.

Recalling Eqs. (18) and (27), our uncertainty relation is equivalently stated as

$$ \frac{F_{\mathrm{Bob},t}}{4\sigma_H^2} + \sigma_H^2 F_{\mathrm{Eve},\eta} = 1. \qquad (50) $$

Using the Cramér-Rao bound (10) we can relate the optimal sensing accuracies $\langle \delta t_{\mathrm{Bob},\mathrm{est}}^2 \rangle$, $\langle \delta \eta_{\mathrm{Eve},\mathrm{est}}^2 \rangle$ associated with the parameters $t$, $\eta$ on Bob and Eve's systems,

$$ \frac{1}{4\sigma_H^2} \frac{1}{\langle \delta t_{\mathrm{Bob},\mathrm{est}}^2 \rangle} + \frac{1}{4\sigma_T^2} \frac{1}{\langle \delta \eta_{\mathrm{Eve},\mathrm{est}}^2 \rangle} \le 1, \qquad (51) $$

noting that equality can be achieved with sensing strategies that saturate the Cramér-Rao bound provided the rank of $\mathcal{N}(\psi(t))$ does not change at $t = t_0$.

A proof of Theorem 1 proceeds by writing the Fisher information on Bob's end, i.e., after the application of the noise channel, in terms of the Bures metric. The environment Eve is introduced as the purifying space over which the fidelity is computed via Uhlmann's theorem. The resulting expression is expressed as a semidefinite program as in Refs. [30,31]; suitably manipulating the corresponding dual problem yields the relation (49). The full proof is provided in Appendix E. We also provide an alternative proof using a semidefinite characterization of the Fisher information.

The condition the rank of $\mathcal{N}(\psi(t))$ does not change locally at the time $t_0$ ensures that we avoid edge cases where the correspondence between the Fisher information and the Bures metric is incomplete [32–34]. In edge cases where this condition is violated, the uncertainty relation (49) can be shown to hold as an inequality instead of an equality (see below and Appendix E). The no rank change condition is typically associated with situations where the quantum Fisher information is discontinuous. In such cases its operational relevance can be questioned; we further discuss these points below in the context of independent and identically distributed (IID) noise as well as in Appendix F.

The condition that the rank of $\mathcal{N}(\psi(t))$ does not change at $t = t_0$ is formalized by requiring that for any eigenvalue $p_k(t)$ of $\mathcal{N}(\psi(t))$ such that $p_k(t_0) = 0$ we also have $\partial_t^2 p_k(t_0) = 0$. This more precise formulation is the form of the assumption that is used in the proof. Observe that any eigenvalue $p_k(t)$ of $\mathcal{N}(\psi(t))$ that satisfies $p_k(t_0) = 0$ necessarily also satisfies $\partial_t p_k(t_0) = 0$, since the value zero is necessarily a minimum for $p_k(t)$.

Another equivalent form of our uncertainty relation (49) is one that quantifies directly the difference between the sensitivity of the noiseless clock and the resulting sensitivity on Bob's end. Let us define

$$ \Delta F_{\mathrm{Bob},t} = F_{\mathrm{Alice},t} - F_{\mathrm{Bob},t} = 4\sigma_H^2 - F_{\mathrm{Bob},t}. \qquad (52) $$
A few simple algebraic manipulations of Eq. (49) lead to

$$F_{\text{Alice},t} - F_{\text{Bob},t} = \frac{F_{\text{Alice},t}}{F_{\text{Alice},\eta}} F_{\text{Eve},\eta}, \quad (53)$$

which gives us an expression for $\Delta F_{\text{Bob},t}$. We can further spell out this expression using Eqs. (18) and (27) along with simple scaling properties that follow from the definition of the Fisher information to find

$$\begin{aligned} \Delta F_{\text{Bob},t} &= (2\sigma_H^2)^2 F_{\text{Eve},\eta} = F(\rho_E; 2\sigma_H^2 \partial_\eta \rho_E) \\ &= F(\rho_E; \hat{\mathcal{N}}(\{\bar{H}, \psi\})), \end{aligned} \quad (54)$$

where we have used Eq. (48) in the last equality.

Summarizing the above argument, we obtain an alternative form of Theorem 1 as an expression for the sensitivity loss $\Delta F_{\text{Bob},t}$ in terms of the Fisher information that Eve obtains with respect to a direction associated with the anticommutator of $H$ and $\psi$.

Corollary 1 (Expression for Bob's sensitivity loss via Eve).—consider the setting of Theorem 1 and assume that the rank of $\mathcal{N}(\psi(t))$ does not change locally at $t_0$. Then

$$\Delta F_{\text{Bob},t} = F(\hat{\mathcal{N}}(\psi); \hat{\mathcal{N}}(\{\bar{H}, \psi\})), \quad (55)$$

where $\Delta F_{\text{Bob},t} = F_{\text{Alice},t} - F_{\text{Bob},t}$ and where we recall the shorthand $\bar{H} = H - \langle H \rangle$. As a consequence,

$$F_{\text{Bob},t} = 4\sigma_H^2 - F(\hat{\mathcal{N}}(\psi); \hat{\mathcal{N}}(\{\bar{H}, \psi\})). \quad (56)$$

Two extreme cases can readily be identified. One is where there is no noise and $\mathcal{N}$ = id is the identity process; in this case, the complementary channel is a channel that outputs a constant state regardless of the input, $\hat{\mathcal{N}}(\cdot) = \text{tr}(\cdot) \tau_E$ for some state $\tau_E$. In this case Eve obtains no information about the probe's energy, which can be seen in our formalism by the fact that $\hat{\mathcal{N}}(\{\bar{H}, \psi\}) = 0$ and therefore $\Delta F_{\text{Bob},t} = 0$. In the opposite extreme case, the noise destroys its input entirely and sends it to the environment, with correspondingly $\hat{\mathcal{N}}$ = id. In this case, Eve has maximal sensitivity to the $\eta$ parameter, $F_{\text{Eve},\eta} = F_{\text{Alice},\eta}$, and therefore $F_{\text{Bob},t} = 0$ and $\Delta F_{\text{Bob},t} = 4\sigma_H^2$.

## B. Trade-off relation in terms of a virtual qubit

In this section we simplify the setting required to produce the relation in Theorem 1, in an effort to identify the fundamental concepts required for our uncertainty relation to hold. It turns out that Theorem 1 can be rephrased as an uncertainty relation between Bob and Eve distinguishing states, respectively, along the Y and Z Pauli directions of a virtual qubit space, which in the setting of Theorem 1 is defined by the clock state vector $|\psi\rangle$ and its image $H|\psi\rangle$ under application of the Hamiltonian.

Consider the subspace of Alice's Hilbert space spanned by the probe state $|\psi\rangle$ and its time derivative $\propto H|\psi\rangle$. This

subspace defines a virtual qubit. We choose to identify the probe state with the $+1$ Pauli-X eigenvector. It turns out that our uncertainty relation admits a restatement as a relation between the sensitivity that Bob and Eve can achieve with respect to Pauli-Y and logical Pauli-Z directions of the virtual qubit. More precisely, we first define

$$|\xi\rangle = P_{\psi}^{\perp} H |\psi\rangle = (H - \langle H \rangle) |\psi\rangle = \bar{H} |\psi\rangle, \quad (57)$$

recalling $P_{\psi}^{\perp} = 1 - \psi$. The norm of $|\xi\rangle$ satisfies

$$\| |\xi\rangle \|^2 = \langle \xi | \xi \rangle = \sigma_H^2. \quad (58)$$

Here, we assume that $|\xi\rangle \neq 0$, otherwise the probe does not evolve in time and all the terms in our uncertainty relation are trivially zero. We can write the following derivatives in terms of $|\xi\rangle$:

$$\partial_t \psi = -i[\bar{H}, \psi] = -i(|\xi\rangle\langle\psi| - |\psi\rangle\langle\xi|), \quad (59a)$$

$$2\sigma_H^2 \partial_\eta \psi = \{\bar{H}, \psi\} = |\xi\rangle\langle\psi| + |\psi\rangle\langle\xi|. \quad (59b)$$

An orthonormal basis of the virtual qubit can be chosen as

$$|+\rangle_L = |\psi\rangle, \quad |-\rangle_L = \frac{1}{\sigma_H} |\xi\rangle. \quad (60)$$

As the logical computational basis of the virtual qubit, we choose

$$|0\rangle_L = \frac{1}{\sqrt{2}}[|+\rangle_L + |-\rangle_L], \quad |1\rangle_L = \frac{1}{\sqrt{2}}[|+\rangle_L - |-\rangle_L]. \quad (61)$$

This choice of basis is motivated to match the qubit operators of a single spin-1/2 particle prepared in the $+X$ eigen-state and evolving according to a magnetic field pointing along the Z axis.

Consider the logical Pauli-X, Y and Z operators defined as usual with respect to the basis (61). They are expressed in the $|\pm\rangle_L$ basis as

$$X_L = |+\rangle\langle+|_L - |-\rangle\langle-|_L, \quad (62a)$$

$$Y_L = -i|-\rangle\langle+|_L + i|+\rangle\langle-|_L, \quad (62b)$$

$$Z_L = |-\rangle\langle+|_L + |+\rangle\langle-|_L, \quad (62c)$$

with furthermore

$$Y_L = \frac{-i}{\sigma_H}[\bar{H}, \psi], \quad Z_L = \frac{1}{\sigma_H}\{\bar{H}, \psi\}. \quad (63)$$

We see that the logical Pauli-Y and Pauli-Z operators are parallel to the evolution, respectively, along $t$ and along $\eta$ locally at $|\psi\rangle = |\psi(t_0, \eta_0)\rangle$, as we recall Eq. (59). Our
uncertainty relation can be stated in terms of a metrological
logical qubit as follows.

Theorem 2 (Uncertainty relation for the metrological
logical qubit).—let A, B, and E be finite-dimensional quan-
tum systems. Let $\mathcal{N}_{A \to B}$ be a completely positive, trace
nonincreasing map. Let $V_{A \to BE}$ be such that $\mathcal{N}_{A \to B}(\cdot) =
\text{tr}_E(V(\cdot)V^\dagger)$ and $V^\dagger V \le 1$, i.e., $V$ is a Stinespring dilation
of $\mathcal{N}$. Consider the complementary channel $\tilde{\mathcal{N}}_{A \to E}(\cdot) =
\text{tr}_B(V(\cdot)V^\dagger)$. Let $|\pm\rangle_L$ be any two orthogonal and normal-
ized state vectors on system A, and let $X_L, Y_L, Z_L$ be defined
via Eq. (62). If $(P_{\rho_B}^\perp \otimes P_{\rho_E}^\perp) V|-\rangle_L = 0$, then

$$
F(\mathcal{N}(\psi); \mathcal{N}(Y_L)) + F(\tilde{\mathcal{N}}(\psi); \\
\tilde{\mathcal{N}}(Z_L)) = 4\langle-|\mathcal{N}^\dagger(1)|-\rangle_L. \quad (64)
$$

If $(P_{\rho_B}^{\perp} \otimes P_{\rho_E}^{\perp})V|-\rangle_L \neq 0$, then we have the inequality

$$
F(\mathcal{N}(\psi); \mathcal{N}(Y_L)) + F(\tilde{\mathcal{N}}(\psi); \\
\tilde{\mathcal{N}}(Z_L)) \le 4\langle-|\mathcal{N}^\dagger(1)|-\rangle_L. \quad (65)
$$

The above theorem provides a more formal statement
that generalizes the earlier statement Theorem 1 to trace-
non-increasing maps and to subnormalized states. The
metrological qubit construction also provides a clearer
mathematical picture of the symmetric role of Bob and
Eve in our uncertainty relation: Bob and Eve can be
interchanged (i.e., $\mathcal{N} \leftrightarrow \tilde{\mathcal{N}}$) provided we correspondingly
interchange $|\xi\rangle \leftrightarrow i|\xi\rangle$. For a state vector $|\psi\rangle$ evolving
with respect to a Hamiltonian $\tilde{H}$, the state $|\xi\rangle = \tilde{H}|\psi\rangle$ is
the derivative of $|\psi\rangle$ with respect to time, and $i|\xi\rangle$ can
be thought of the derivative of $|\psi\rangle$ with respect to imag-
inary time. The symmetry in Theorem 2 between Bob and
Eve, which involves the interchange $|\xi\rangle \leftrightarrow i|\xi\rangle$, is repro-
duced at the level of the parameters $t$ and $\eta$ by choosing
$\eta$ to parametrize the one-family parameter of state vectors
$|\psi(\eta)\rangle$ in Eq. (21) governed by the imaginary-time evo-
lution (24). The full proof of Theorem 2 is provided in
Appendix E 2.

In Theorem 2 a different condition is stated for equality
as in Theorem 1, where we require the rank of $\mathcal{N}(\psi(t))$
not to change. These conditions turn out to be equivalent,
as shown in the following proposition. We defer the proof
to Appendix E 2.

Proposition 1 (Conditions for equality in the uncertainty relation).—let {$E_k$} be a set of Kraus operators for $\mathcal{N}_{A \to B}$ and $V_{A \to BE}$ be a Stinespring dilation of $\mathcal{N}$. The following conditions are equivalent:

(a) $(P_{\rho_B}^{\perp} \otimes P_{\rho_E}^{\perp}) V|\xi\rangle = 0$;

(b) $\rho_B(t)$ does not change rank as a function of $t$ locally
at the point $t_0$;

(c) for any linear combination $E = \sum c_k E_k$ (with $c_k \in \mathbb{C}$) such that $E|\psi\rangle = 0$, then $P_{\rho_B}^{\perp} \tilde{E}|\xi\rangle = 0$.

In particular, it suffices that either $\rho_B = \mathcal{N}(\psi)$ or $\rho_E = \tilde{\mathcal{N}}(\psi)$ has full rank to ensure that these conditions are satisfied, and thereby that our uncertainty relation holds with equality [Eq. (64)].

As a consequence, the situations for which the conditions (1) do not hold, and correspondingly for which our uncertainty relation does not necessarily hold with equality, are edge cases that can be infinitesimally perturbed into situations where the corresponding conditions hold. Indeed, one can mix $\mathcal{N}$ with an infinitesimal amount of depolarizing noise to ensure that Bob's state is full rank, and therefore to ensure that equality holds in our uncertainty relation.

C. General uncertainty relation for any two
parameters

The uncertainty relation between position and momen-
tum can be generalized to any arbitrary pair of observables.
The Robertson uncertainty relation states that for any two
observables A, B, we have

$$
\sigma_A \sigma_B \ge \frac{1}{2} |\langle i[A, B] \rangle|. \quad (66)
$$

In the same spirit, we derive a generalization of (49) that
is valid for any two observables. Suppose Alice prepares a
pure state $\psi$ that can evolve along two possible directions
$\partial_a\psi$ and $\partial_b\psi$, and sends the state through the noisy channel
$\mathcal{N}$ to Bob as in Fig. 1. We assume that the directions along
$a, b$ are generated by two Hermitian operators $A, B$ acting
on $\psi = |\psi\rangle\langle\psi|$ as

$$
\partial_a \psi = -i[A, \psi], \quad \partial_b \psi = -i[B, \psi]. \quad (67)
$$

Bob is tasked with estimating a deviation locally to first
order around $\mathcal{N}(\psi)$ in the $a$ direction, whereas Eve tries
to distinguish $\tilde{\mathcal{N}}(\psi)$ from neighboring states along the $b$
direction. The parameters $a, b$ are analogous to the param-
eters $t, \eta$ considered above, but the two directions $\partial_a\psi, \partial_b\psi$
can be arbitrary.

Theorem 3 (Bipartite uncertainty relation for any two parameters).—let $\psi$ be a state vector and suppose that $A, B$ are two Hermitian operators that generate evolutions locally at $\psi$ in directions $\partial_a\psi, \partial_b\psi$ via (67). Suppose we apply a noisy channel as depicted in Fig. 1. Then

$$
\frac{F_{\text{Bob},a}}{F_{\text{Alice},a}} + \frac{F_{\text{Eve},b}}{F_{\text{Alice},b}} \le 1 + 2 \sqrt{1 - \frac{\langle i[A, B] \rangle^2}{4\sigma_A^2\sigma_B^2}}, \quad (68)
$$

where

$$
\begin{align*}
F_{\text{Alice},a} &= F(\psi; \partial_a \psi), & F_{\text{Bob},a} &= F(\mathcal{N}(\psi); \mathcal{N}(\partial_a \psi)), \\
F_{\text{Alice},b} &= F(\psi; \partial_b \psi), & F_{\text{Eve},b} &= F(\tilde{\mathcal{N}}(\psi); \tilde{\mathcal{N}}(\partial_b \psi)).
\end{align*}
$$
Furthermore, assume that $\mathcal{N}[\psi(a)]$ does not change rank locally and that

$$ \hat{\mathcal{N}}(-i[B/\sigma_B, \psi]) = \pm \hat{\mathcal{N}}(\{(A - \langle A \rangle)/\sigma_A, \psi\}). \quad (69) $$

Then

$$ \frac{F_{\text{Bob},a}}{F_{\text{Alice},a}} + \frac{F_{\text{Eve},b}}{F_{\text{Alice},b}} = 1. \quad (70) $$

The proof of this statement is presented in Appendix E 4. The argument of the square root in Eq. (68) never becomes negative, thanks to the Robertson uncertainty relation (66) for A and B. The proof we present in Appendix E 4 considers in fact a more general statement in which the two sides of Eq. (69) are proportional to one another rather than differing only by a sign.

We can identify two extreme cases of interest to gain some intuition for the relation (68). First consider A, B to be two complementary observables in the sense that they saturate the Robertson inequality (66). Consider, for instance, the Pauli-Y and the Pauli-Z operators on a qubit. In this case the right-hand side of the inequality (68) equals one. There is a trade-off between the sensitivity losses associated with Bob sensing along the $\mathcal{N}(\partial_a\psi)$ direction and Eve sensing along the $\hat{\mathcal{N}}(\partial_b\psi)$ direction, as both terms on the left-hand side of Eq. (68) cannot simultaneously be equal to one. On the other hand, we can consider two Hermitian generators A, B that commute. (Perhaps A, B act on different subsystems of Alice's noiseless clock.) In this case, the right-hand side of Eq. (68) evaluates to the constant 3. Our uncertainty relation no longer presents any obstruction to both Bob and Eve sensing along the respective directions a, b as well as Alice could, as there is room for both terms on the left-hand side of Eq. (68) to be equal to one. This is the case, for instance, if A, B act on different subsystems of Alice's clock, and the respective subsystems are sent to Bob and Eve via the noisy channel and its complementary channel.

We can recover our Theorem 1 if we consider the two generators $A = H$ and $B = -T$, with $H, T$ defined in Sec. II B, leading to $\partial_a\psi = \partial_i\psi$ and $\partial_b\psi = \partial_\eta\psi$. To see this, we first compute

$$ \begin{aligned} \langle i[H, T] \rangle &= \frac{1}{2\sigma_H^2} \langle i[H, -i[H, \psi]] \rangle \\ &= \frac{1}{2\sigma_H^2} (2\langle H^2 \rangle - 2\langle H \rangle^2) = 1. \end{aligned} \quad (71) $$

Using Eq. (27) we further see that $4\sigma_H^2\sigma_T^2 = 1$. Therefore, the square root on the right-hand side of Eq. (68) vanishes and the entire right-hand side of the inequality evaluates to the constant 1. With the identifications $F_{\text{Alice},t} = F_{\text{Alice},a}$, $F_{\text{Bob},t} = F_{\text{Bob},a}$, $F_{\text{Alice},\eta} = F_{\text{Alice},b}$, $F_{\text{Eve},\eta} = F_{\text{Eve},b}$, we recover the expression (49) with an inequality instead

of an equality. In this case, the additional condition (69) is in fact also satisfied, since $i[T, \psi] \propto \{H - \langle H \rangle, \psi\}$ [cf. Eqs. (25) and (26)]. We thus fully recover the equality statement of Theorem 1 subject to our additional condition on the absence of a rank change of the noisy state.

The strategy of the proof of Theorem 3 (Appendix E 4) is to first apply our main uncertainty relation (Theorem 2) between Bob's sensitivity to the parameter a and Eve's sensitivity to a parameter c that is complementary to a using the construction in Sec. II B and Fig. 5 identifying $t \to a, \eta \to c$. We then apply a general bound relating the quantum Fisher information with respect to two arbitrary evolution directions (Proposition 16 in Appendix C) to bound the difference between Eve's sensitivity to the parameters b and c.

One might have assumed that the equality (70) can only be achieved if the parameters a, b are complementary in the sense of Fig. 5. Yet it suffices for this property to hold on the support of the complementary channel, as seen in the condition (69). As a simple extreme example, consider $\mathcal{N} = \text{id}$ and $\hat{\mathcal{N}}(\cdot) = \text{tr}(\cdot)\tau$ is a constant channel preparing some fixed quantum state $\tau$. Then our uncertainty relation equality (70) necessarily holds for any parameters a, b, since Eve's sensitivity to any parameter b is zero and Bob's sensitivity to any parameter a is equal to Alice's. This example also illustrates how the right-hand side of Eq. (68) should necessarily be improved to depend on the channel $\mathcal{N}$ if we wanted the inequality to be tight for a fixed $\mathcal{N}$. Such an improvement can be obtained from our proof in Appendix E 4.

We furthermore provide a proof that the general uncertainty relation (68) also holds in infinite-dimensional Hilbert spaces, and even for unbounded operators. The details of this proof are given in Appendix F. The proof proceeds by considering a limiting case of the finite-dimensional setting for larger and larger system sizes, with additional care given to the definition of the Fisher information in the infinite-dimensional case and to the fact that the considered operators are not necessarily bounded.

It is expected that the bound (68) can be further tightened for observables that do not saturate the Robertson bound. For instance, consider two independent systems in a pure tensor product state, with one system evolving with a parameter t and the other with z: if we hand the first system to Bob and the second to Eve, then there is no sensitivity loss for either parties and the sum of the Fisher information ratios should be 2. But the right-hand side of our bound is 3.

## IV. A SELECTION OF EXAMPLES

We now explore some examples illustrating the application of our main results.
## A. Single qubit subject to partial dephasing

Consider the setup in Fig. 2 and described in Sec. II B, in which Alice prepares a pure qubit in the $|+\rangle$ state vector and lets it evolve according to the Hamiltonian $H = \omega Z/2$. At time $t$, the qubit is in the state $\psi(t)$ given in Eq. (30) and its derivative $\partial_t \psi$ is given by Eq. (31).

Suppose that at time $t_0$ we apply the partially dephasing noisy channel

$$ \mathcal{N}_p = (1-p) \text{id} + p \mathcal{D}_Z, \qquad (72) $$

where

$$ \mathcal{D}_Z(\cdot) = \langle \uparrow|\cdot|\uparrow \rangle \langle \uparrow| \cdot + \langle \downarrow|\cdot|\downarrow \rangle \langle \downarrow| \cdot \rangle. \qquad (73) $$

In the following, we will verify that our uncertainty relation holds in this setting, by first computing directly Bob's Fisher information with respect to $t$, and then computing Eve's Fisher information with respect to $\eta$.

### 1. Direct computation of $F_{\text{Bob},t}$

Using $\mathcal{D}_Z(X) = 0 = \mathcal{D}_Z(Y)$ we find from Eq. (30b) that Bob receives the state

$$ \rho_B(t_0) = \frac{1}{2} \begin{bmatrix} 1 & (1-p)e^{-i\omega t_0} \\ (1-p)e^{i\omega t_0} & 1 \end{bmatrix}. \qquad (74) $$

Using Eq. (30a) along with the fact that the superoperator action of $U_t = e^{-iHt}$ and $\mathcal{N}_p$ commute and that $\mathcal{N}_p(X) = (1-p)X$, we can alternatively write Bob's state as

$$ \begin{aligned} \rho_B(t_0) &= U_{t_0} \frac{1 + (1-p)X}{2} U_{t_0}^{\dagger} \\ &= \left(1 - \frac{p}{2}\right) |+_{t_0}\rangle\langle+_{t_0}| + \frac{p}{2} |-_{t_0}\rangle\langle-_{t_0}|, \end{aligned} \qquad (75) $$

defining the rotated basis state vectors $|\pm_t\rangle := U_t|\pm\rangle$. For the time derivative, using Eq. (31a) along with $\mathcal{N}_p(Y) = (1-p)Y$ we find

$$ \partial_t \rho_B (t_0) = \mathcal{N}_p (\partial_t \psi (t_0)) = \frac{\omega}{2} (1-p) U_{t_0} Y U_{t_0}^{\dagger}. \qquad (76) $$

We may compute the Fisher information with the formula (12), using the eigendecomposition of $\rho_B(t_0)$ given by Eq. (75)

$$ \begin{aligned} F_{\text{Bob},t} ={}& \frac{\omega^2}{4} (1-p)^2 \left[ \begin{aligned} & \frac{1}{1-p/2} |\langle+|Y|+\rangle|^2 \\ & + 2 |\langle+|Y|-\rangle|^2 + 2 |\langle-|Y|+\rangle|^2 \\ & + \frac{1}{p/2} |\langle-|Y|-\rangle|^2 \end{aligned} \right] \\ ={}& \omega^2 (1-p)^2, \end{aligned} \qquad (77) $$

using $\langle+|Y|-\rangle = \langle+|YZ|+\rangle = i\langle+|X|+\rangle = i$ and $\langle+|Y|+\rangle = 0 = \langle-|Y|-\rangle$.

Recalling Eq. (37), the ratio of the Fisher information of the noisy versus the noiseless clock is

$$ \frac{F_{\text{Bob},t}}{F_{\text{Alice},t}} = (1-p)^2. \qquad (78) $$

### 2. Computation of $F_{\text{Eve},\eta}$

Now we turn to Eve's picture. We start with computing a complementary channel to $\mathcal{N}_p$. We can use Eq. (46) for this effect from any Kraus representation of $\mathcal{N}_p$. It is useful to choose a representation with the fewest possible Kraus operators to simplify our computation of $F_{\text{Eve},\eta}$. From Eq. (72), and using $\mathcal{D}_Z(\cdot) = 1(\cdot)1/2 + Z(\cdot)Z/2$ we can read off a representation of $\mathcal{N}_p$ with the two Kraus operators

$$ E_0^{(p)} = \sqrt{1 - \frac{p}{2}} 1, \quad E_1^{(p)} = \sqrt{\frac{p}{2}} Z. \qquad (79) $$

The complementary channel constructed via Eq. (46) takes the form

$$ \hat{\mathcal{N}}_p(\cdot) = \begin{bmatrix} (1 - \frac{p}{2}) \text{tr}(\cdot) & \sqrt{\frac{p}{2}(1 - \frac{p}{2})} \text{tr}[Z(\cdot)] \\ \sqrt{\frac{p}{2}(1 - \frac{p}{2})} \text{tr}[Z(\cdot)] & \frac{p}{2} \text{tr}(\cdot) \end{bmatrix}. \qquad (80) $$

Hence Eve's state is

$$ \rho_E(t_0) = \hat{\mathcal{N}}_p(\psi(t_0)) = \begin{bmatrix} 1 - \frac{p}{2} & 0 \\ 0 & \frac{p}{2} \end{bmatrix}. \qquad (81) $$

The derivative in the $\eta$ direction is given by the image of Eq. (36) under $\hat{\mathcal{N}}_p$, namely

$$ \partial_{\eta} \rho_E (t_0) = \hat{\mathcal{N}}_p (\partial_{\eta} \psi (t_0)) = \frac{2}{\omega} \sqrt{\frac{p}{2} (1 - \frac{p}{2})} X. \qquad (82) $$

We may now directly compute $F_{\text{Eve},\eta}$ using Eq. (12),

$$ \begin{aligned} F_{\text{Eve},\eta} ={}& \frac{4}{\omega^2} \frac{p}{2} \left(1 - \frac{p}{2}\right) \left[0 + 2 + 2 + 0\right] \\ ={}& \frac{4}{\omega^2} (2p - p^2). \end{aligned} \qquad (83) $$

Using Eq. (37) we find that the ratio of Eve's Fisher information to Alice's Fisher information with respect to $\eta$ is

$$ \frac{F_{\text{Eve},\eta}}{F_{\text{Alice},\eta}} = 2p - p^2 = 1 - (1 - p)^2. \qquad (84) $$

The fact that Eqs. (78) and (84) sum to unity is a manifestation of Theorem 1 in the present setting.
Consider now our Fisher information loss formula (55). Using Eq. (35) and $\tilde{H} = H - \langle H \rangle = H$ we have

$$ \hat{\mathcal{N}}_p(\{\tilde{H}, \psi\}) = \frac{\omega}{2} \hat{\mathcal{N}}_p(Z) = \omega \sqrt{\frac{p}{2}(1-\frac{p}{2})} X. \quad (85) $$

Then we can compute $F(\rho_E; \hat{\mathcal{N}}_p(\{\tilde{H}, \psi\}))$ using Eq. (12) as

$$ F(\rho_E; \hat{\mathcal{N}}_p(\{\tilde{H}, \psi\})) = \omega^2 (2p - p^2). \quad (86) $$

We can then verify that the difference in Fisher information between the noiseless clock and the noisy clock is indeed

$$ \begin{aligned} \Delta F_{\text{Bob},t} &= F_{\text{Alice},t} - F_{\text{Bob},t} = \omega^2(1 - (1-p)^2) \\ &= F(\rho_E; \hat{\mathcal{N}}_p(\{\tilde{H}, \psi\})). \end{aligned} \quad (87) $$

## B. Single qubit subject to complete dephasing along a transversal axis

Now we consider a variant of the above single-qubit example: we replace the noisy channel by a complete dephasing along the X axis (Fig. 6). The qubit is initialized in the state vector $|\psi_{\text{init}}\rangle = |+\rangle$, where $|\pm\rangle = [|\uparrow\rangle \pm |\downarrow\rangle]/\sqrt{2}$, with a Hamiltonian $H = \omega/(2)Z$. After a time $t$, the state is given by Eq. (29) and at all times we have $\langle H \rangle_{\psi(t)} = 0$ and $\sigma_H^2 = \omega^2/4$. At time $t \approx t_0$ the clock is completely dephased in the X basis, as described by the noisy channel

$$ \mathcal{D}_X(\cdot) = \langle +|\cdot|+\rangle |+\rangle \langle +| + \langle -|\cdot|-\rangle |-\rangle \langle -| \quad (88) $$

This completely dephasing map acts on the Pauli operator basis as $\mathcal{D}_X(1) = 1$, $\mathcal{D}_X(X) = X$, and $\mathcal{D}_X(Y) = 0 = \mathcal{D}_X(Z)$. Bob receives the density matrix

$$ \rho_B = \cos^2\left(\frac{\omega t}{2}\right) |+\rangle\langle +| + \sin^2\left(\frac{\omega t}{2}\right) |-\rangle\langle -| \quad (89) $$

Now the complementary channel of $\mathcal{D}_X$ is again $\hat{\mathcal{D}}_X = \mathcal{D}_X$, and so Eve gets the same density matrix as Bob.

### 1. Computation of $F_{\text{Eve},\eta}$

Recalling Eq. (35), we find

$$ \hat{\mathcal{D}}_X(\{H - \langle H \rangle, \psi\}) = \frac{\omega}{2} \hat{\mathcal{D}}_X(Z) = 0, \quad (90) $$

because $\mathcal{D}_X$ maps the Pauli-Y and Pauli-Z operators to zero. Therefore, Eve obtains zero information about $\eta$, i.e., $F_{\text{Eve},\eta} = 0$. Therefore, there is no sensitivity loss for Bob regardless of the time $t \approx t_0$ at which the noisy channel is applied, as long as the rank of $\rho_B(t_0)$ does not change locally at $t_0$. The state $\rho_B$ changes rank whenever either term of Eq. (89) vanishes, i.e., when $t_0$ is a multiple of

<figure><img src="image_8.png" /><figcaption>FIG. 6. Single-qubit probe evolving according to the Hamiltonian $H = \omega Z/2$ and subject to complete dephasing along the X direction at time close to $t_0$. For almost every $t_0$, the noisy probe remains maximally sensitive to time to first order around $t_0$. This property might sound surprising, because Bob's state can be very mixed. In the purified picture, Eve is given the outcome of a measurement of Alice's state along the X axis. Observe that in contrast to the setting in Fig. 2, this information does not reveal any information about the energy of Alice's state.</figcaption></figure>

$\pi/\omega$. At those discrete points, we hit the edge cases where our main uncertainty relation does not hold with equality and we cannot deduce that Bob has maximal sensitivity at those points. However, at all other points $t_0$ the clock does not lose any sensitivity when sent to Bob.

The same conclusions apply for any noisy channel that is a complete dephasing operation along an axis that lies in the equatorial plane, by rotational symmetry of the problem around the Z axis. (Any axis in the equatorial plane can be described as a rotation of the X axis that is equivalent to a time evolution of the system for some given time $t_*$.) Because the Fisher information is invariant under unitary transformations, the calculation of Bob's Fisher information of this qubit after complete dephasing along that given axis at time $t_0$ is equivalent to calculating the Fisher information after a complete dephasing along the X axis at the time $t_0 \to t_0 - t_*$.)

### 2. Check by direct computation of $F_{\text{Bob},t}$

We now compute $F_{\text{Bob},t} = F(\rho_B; \mathcal{D}_Z(\partial_t\psi))$ directly, by using the definition of the Fisher information. From Eq. (31b) we find

$$ \mathcal{D}_X(\partial_t\psi) = -\frac{\omega}{2} \sin(\omega t_0) X. \quad (91) $$

If $\sin(\omega t_0) = 0$, which happens when $t_0$ is a multiple of $\pi/\omega$, we find that Bob's state is locally stationary and Bob has no sensitivity to first order in $t$. (For this discrete set of points one could argue that the Fisher information no longer represents the relevant sensitivity for Bob, since the evolution should be considered to its leading order—here the second order—and no longer only to first order.)
We now compute $F_{\text{Bob},t}$ for all times $t_0$ where $\sin(\omega t_0) \neq 0$. Observe that $\rho_B$ and $D_X(\partial_t \psi)$ commute. Using Eq. (15) and $X^2 = 1$, we find

$$F_{\text{Bob},t} = \frac{\omega^2}{4} \sin^2(\omega t_0) \text{tr}(\rho_B^{-1}) = \omega^2 \left[ \sin\left(\frac{\omega t_0}{2}\right) \cos\left(\frac{\omega t_0}{2}\right) \right]^2 \\ \times \left[ \frac{1}{\cos^2\left(\frac{\omega t_0}{2}\right)} + \frac{1}{\sin^2\left(\frac{\omega t_0}{2}\right)} \right] \\ = \omega^2, \qquad (92)$$

using $\sin(\omega t_0) = 2 \sin(\omega t_0/2) \cos(\omega t_0/2)$ in the second equality.

Overall, we see that Bob still has maximal sensitivity even after application of the completely dephasing channel along the transversal X axis, for all times except for the discrete set of times $t_0$ where the rank of $\rho_B$ changes. This conclusion matches our earlier conclusions obtained via considerations from Eve's perspective (except for a discrete set of times $t_0$).

It might appear counterintuitive that Bob's state still has as high a sensitivity as Alice's noiseless state for almost all $t_0$, especially as Bob's state can get arbitrarily mixed. Indeed, $\rho_B$ coincides with the maximally mixed state for times $t_0$ that are midpoints between the multiples of $\pi/\omega$. However, we see that $\rho_B(t)$ still varies with $t$ sufficiently to enable optimal discrimination of nearby states to first order around $t_0$.

### C. Probe in a GHZ state with one partial erasure

Consider as initial state an n-party GHZ state vector,

$$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}} [ |\uparrow \cdots \uparrow\rangle + |\downarrow \cdots \downarrow\rangle ], \qquad (93) $$

and let the system evolve according to the local Hamiltonian $H = \sum_i (\omega/2)Z^{(i)}$ where $Z^{(i)}$ denotes the Pauli-Z operator acting on the $i$th site. Suppose that the first qubit is lost with probability $p$. This is represented by the noisy channel

$$\mathcal{N}(\cdot) = p |\phi_{\perp}\rangle\langle\phi_{\perp}| \otimes \text{tr}_1(\cdot) + (1-p) \langle\cdot|\phi_{\perp}\rangle\langle\phi_{\perp}|. \qquad (94)$$

where $\text{tr}_1$ traces out the first qubit and where $|\phi_{\perp}\rangle$ is a state vector in a new, orthogonal dimension that has no overlap with the input state. A Stinespring dilation of the first term in $\mathcal{N}$ is described as giving the first qubit of Alice's system to Eve, and the remaining qubits to Bob; any missing qubits on either Bob or Eve's side is replaced by $|\phi_{\perp}\rangle$. The complementary channel can thus be computed as

$$\hat{\mathcal{N}}(\cdot) = p \text{tr}_{2...n}(\cdot) + (1-p) \text{tr}(\cdot) |\phi_{\perp}\rangle\langle\phi_{\perp}|. \qquad (95)$$

We compute the sensitivity loss associated with the noise according to Eq. (55). We have

$$H|\psi\rangle = \frac{n\omega}{2\sqrt{2}} [ |\uparrow \cdots \uparrow\rangle - |\downarrow \cdots \downarrow\rangle ] = P_{\psi}^{\perp} H|\psi\rangle, \qquad (96)$$

noting that $H|\psi\rangle$ is already orthogonal to $|\psi\rangle$ since $\langle H|\psi\rangle = 0$. The optimal noiseless sensitivity is

$$F_{\text{Alice},t} = 4\sigma_H^2 = 4\langle\psi|H^2|\psi\rangle = n^2\omega^2, \qquad (97)$$

exhibiting the expected Heisenberg scaling for optimally entangled probe states. We write $\{\psi, \bar{H}\} = P_{\psi}^{\perp}H\psi + \text{h.c.} = \sigma_H Z_L$, with $Z_L$ defined in Eq. (63). The local reduced operator of $\{\psi, \bar{H}\}$ on a single site is

$$\text{tr}_{\setminus i}(\{\psi, \bar{H}\}) = \frac{n\omega}{4} |\uparrow\rangle\langle\uparrow| - \frac{n\omega}{4} |\downarrow\rangle\langle\downarrow| + \text{h.c.} = \frac{n\omega}{2} Z^{(i)}, \qquad (98)$$

where $\text{tr}_{\setminus i}$ denotes the partial trace over all subsystems except the $i$th subsystem. Noting that $\text{tr}(\{\psi, \bar{H}\}) = 0$, we obtain

$$\hat{\mathcal{N}}(\{\psi, \bar{H}\}) = p \frac{n\omega}{2} Z. \qquad (99)$$

On the other hand, the reduced state of $\psi$ on a single site is simply the maximally mixed state $1_2/2$ and thus

$$\rho_E = \hat{\mathcal{N}}(\psi) = p \frac{1_2}{2} + (1-p) |\phi_{\perp}\rangle\langle\phi_{\perp}|. \qquad (100)$$

As $\rho_E$ and $\hat{\mathcal{N}}(\{\psi, \bar{H}\})$ commute, we can use Eq. (15) to see that

$$\Delta F_{\text{Bob},t} = \text{tr} \left[ \frac{2}{p} \left( p \frac{n\omega}{2} Z \right)^2 \right] = p n^2 \omega^2. \qquad (101)$$

If $p=1$, Eve is maximally disturbing and completely blocks Bob's ability to measure time, if $p=0$ there is no sensitivity loss. Any value in between interpolates between these two cases.

Note that while it might appear here that Heisenberg scaling ($F_{\text{Bob},t} \propto n^2$) is achieved for $p > 0$, this is an artifact of the lack of scaling in $n$ of our choice of noisy channel and does not contradict the findings of, e.g., Ref. [22,31].

### D. Estimating a signal Hamiltonian term

In this subsection, we briefly comment on the case where the parameter to estimate is not time $t$ itself, but a parameter $f$ in the Hamiltonian that influences time evolution. In other words, we now account for possible other terms in the Hamiltonian that contribute to time evolution but that reveal nothing about the parameter of interest.
We assume that the noiseless probe evolves according to a Hamiltonian

$$H_f = H_0 + fG, \quad (102)$$

where $H_0$ does not depend on $f$, and where $H_0$ and $G$ are time independent. References [35,36] have determined that the Fisher information with respect to $f$ that one achieves by initializing the system in some initial state vector $|\psi_0\rangle$ and letting the system evolve according to $H_f$ for some fixed time $T$. Let $U_f(T) = e^{-iH_f T}$ be the time-evolution operator, and define $|\psi_f\rangle = U_f(T)|\psi_0\rangle$. The question is, how much sensitivity does the family of state vectors $f \mapsto |\psi_f\rangle$ offer with respect to $f$? The derivative relevant for the Fisher information is given by [36]

$$\partial_f \psi_f = -i[K_f, \psi_f], \quad (103)$$

where

$$K_f = -iU_f^{-1} \frac{dU_f}{df} = T \sum_{k=0}^{\infty} \frac{(-iT)^k}{(k+1)!} \mathrm{ad}_{H_f}^k(G), \quad (104)$$

where $\mathrm{ad}_M(G) := [M, G]$ and

$$\mathrm{ad}_M^k(G) := [M, [M, \dots, [M, G]] \quad (105)$$

is the $k$th commutator of $M$ with $G$. The operator $K_f$ can be thought of as an effective “Hamiltonian” for the parameter $f$, driving an “evolution” in $|\psi_f\rangle$ with respect to $f$ according to Eq. (103).

If we send this probe state through a noisy channel following the setting in Fig. 1, then our uncertainty relation can be applied, where the complementary parameter evolution is generated by the operator $L = -i[K_f, \psi_f]/(2\sigma_{K_f}^2)$. That is, Bob’s sensitivity to $f$ trades off with Eve’s sensitivity to the parameter generated by $L$.

### E. Symmetric codes against erasures via superpositions of Dicke states

Based on the relevance of Dicke states for metrology [37–41] and for quantum error correction [42–44], we can ask whether our uncertainty relation can guide a search for good clock states. To ensure good sensitivity even in the noiseless setting, we seek probe states with a large spread over energy eigenstates. So we consider a general superposition of Dicke states corresponding to different numbers of excitations. We note an important class of permutation-invariant codes are those developed in Refs. [41,42].

Consider the $n$-spin noninteracting Hamiltonian $H = \sum_{i=1}^n (\omega/2) Z^{(i)}$. A Dicke state is an eigenstate of $H$ that

is symmetric under permutations of the sites. Consider the Dicke state

$$|h_q^n\rangle := \binom{n}{q}^{-1/2} \sum_{\substack{s_i=\pm 1 \\ \sum s_i = n-2q}} |s_1 \dots s_n\rangle, \quad (106)$$

where $s_i = \pm 1$ represents the eigenstates of $Z$ and where $q = 0, \dots, n$. We construct our probe states as a superposition of Dicke states of different values of $q$. In general, such a state vector can be written as

$$|\psi\rangle = \sum_{q=0}^{n} \psi_q |h_q^n\rangle, \quad (107)$$

for some arbitrary complex amplitudes $\{\psi_q\}$ that satisfy $\sum_q |\psi_q|^2 = 1$.

As a noise model, we assume that $k$ systems chosen at random are entirely erased. Because the probe state is completely symmetric, it does not matter which subsystems are erased; we may assume that the first $k$ sites are erased. The complementary channel to the erasure of $k$ subsystems is a channel that provides those lost subsystems to Eve,

$$\hat{\mathcal{N}}(\cdot) = \mathrm{tr}_{k+1\dots n}(\cdot), \quad (108)$$

where $\mathrm{tr}_{k+1\dots n}$ denotes the partial trace over sites $k+1$ to $n$.

We compute numerical values for the quantities $\Delta F_{\mathrm{Bob},t}$ and $4\sigma_H^2$, enabling us to infer $F_{\mathrm{Bob},t}$. Consider the probe state vector consisting of an even superposition of two Dicke states with associated parameters $q_1, q_2$

$$|\psi\rangle = [|h_{q_1}^n\rangle + |h_{q_2}^n\rangle]/\sqrt{2}. \quad (109)$$

The sensitivity of this probe state for $n = 100$ and subject to $k = 9$ erasures is plotted as a function of $q_1, q_2$ in Fig. 7 (with $\omega/2 = 1$). The sensitivity $F_{\mathrm{Bob},t}$ is obtained by computing $\Delta F_{\mathrm{Bob},t}$ and $\sigma_H^2$ via Eq. (52). On the one hand, our trade-off relation facilitates the calculation of the remaining Fisher information after the erasures. On the other hand, the trade-off relation explains that the high sensitivity loss experienced for states with a broad spread in energy ($q_1 \to 0$ and $q_n \to n$) is directly related to the fact that the environment can well infer the energy of the state from few-site reduced states.

Because the noise is local, numerical computations only have to take place on a smaller system representing the local degrees of freedom. Because of permutation symmetry globally and also locally (the reduced state also lives in the local symmetric subspace), our computations run on $k+1$ dimensions and not on the full $(n+1)$-dimensional symmetric subspace. We will return to the example of permutation-invariant states on $n$ spins in Sec. VIII, where we consider an IID amplitude-damping noise model instead of erasures.
<figure><img src="image_9.png" /><figcaption>FIG. 7. Fisher information of an even superposition of two Dicke states of magnetizations $m_1 = n - 2q_1$ and $m_2 = n - 2q_2$ on a n-site noninteracting spin chain with local terms $H_i = (\omega/2)Z$. A good probe state has $m_1, m_2$ far from one another (for a large energy spread), but also far from the edges $-n$ and $n$ (to avoid decoherence caused by the erasures). Here we set $n = 100$ total spins, $\omega/2 = 1$, and $k = 9$ spins are lost to the environment. Our trade-off relation facilitates the calculation of the Fisher information plotted above. It also gives an interpretation of the loss in sensitivity with respect to the noiseless case (where the GHZ state $m_1 = -m_2 = \pm n$ would be optimal; leftmost and rightmost edges of the plot) as the sensitivity that Eve gains with respect to the energy of the state.</figcaption></figure>

## V. BOUNDS ON THE FISHER INFORMATION

Because it might not always be simple to compute the Fisher information trade-off quantity $\Delta F_{\text{Bob},t}$ in Eq. (55), we provide a few bounds that might be applicable to different settings, and that avoid the calculation of the symmetric logarithmic derivative on Eve's system.

### A. Upper bound on Bob's sensitivity by postprocessing Eve's system

A useful bound for the Fisher information is the data processing inequality [19]. The inequality states that for any $\rho(t)$, and for any $t$-independent completely positive, trace-non-increasing map $\mathcal{E}$, the sensitivity after application of the channel can only decrease:

$$F(\rho(t)) \geq F(\mathcal{E}(\rho(t))). \quad (110)$$

A trace-non-increasing map can be used to describe only a subspace of interest of a larger Hilbert space while accounting for leakage outside of that subspace.

Consider our setup with Alice, Bob, and Eve as in Fig. 1. Suppose now that Eve sends her state to another agent, Eve', through a trace-non-increasing, completely positive map $\mathcal{N}'$ as depicted in Fig. 8(a). The data-processing inequality ensures that $F_{\text{Eve},\eta} \geq F_{\text{Eve}',\eta}$. Combining this

<figure><img src="image_10.png" /><figcaption>FIG. 8. Combining our uncertainty relation with the data processing inequality for the Fisher information yields new bounds for the Fisher information. (a) Suppose Eve applies a suitably chosen map $\mathcal{N}'$ to her system, resulting in a system we denote by Eve', on which the sensitivity to energy might be significantly easier to compute. Eve' can only have a worse sensitivity to energy than Eve, so our uncertainty relation gives an upper bound to Bob's sensitivity to time. (b) Suppose that Eve's output can be written as a composition of two maps $\hat{\mathcal{N}}_0$ and $\mathcal{N}'$ via an intermediate system $\text{Eve}_0$. Then we obtain a lower bound on Bob's sensitivity to time by computing $\text{Eve}_0$'s sensitivity to energy.</figcaption></figure>

with our uncertainty relation (49) yields

$$\frac{F_{\text{Bob},t}}{F_{\text{Alice},t}} + \frac{F_{\text{Eve}',\eta}}{F_{\text{Alice},\eta}} \leq 1. \quad (111)$$

We can also obtain this inequality by starting from the quantum Fisher information loss on Bob's end, Eq. (55),

$$\begin{aligned} \Delta F_{\text{Bob},t} &= F(\rho_E; \hat{\mathcal{N}}(\{\bar{H}, \psi\})) \\ &\geq F(\mathcal{N}'(\rho_E); \mathcal{N}'(\hat{\mathcal{N}}(\{\bar{H}, \psi\}))), \end{aligned} \quad (112)$$

which in turn provides an upper bound on Bob's Fisher information via Eq. (52) as

$$F_{\text{Bob},t} \leq 4\sigma_H^2 - F(\mathcal{N}'(\rho_E); \mathcal{N}'(\hat{\mathcal{N}}(\{\bar{H}, \psi\}))). \quad (113)$$

By choosing the map $\mathcal{N}'$ suitably, one can potentially significantly simplify the computation of the Fisher information. For instance, $\mathcal{N}'$ can be a dephasing map that ensures that $\mathcal{N}'(\rho_E)$ and $\mathcal{N}'(\hat{\mathcal{N}}(\{\bar{H}, \psi\}))$ commute, therefore enabling the use of Eq. (15) and removing the necessity of computing the symmetric logarithmic derivative. Alternatively $\mathcal{N}'$ can be chosen to enforce some symmetry that might be convenient for the computation of the Fisher information.

The bound (113) can be spelled out in the case of IID noise on a many-body probe state. Consider a single-site noisy channel $\mathcal{N}_1$ with Kraus operators $\{E_x\}$ for $x =$
0, ..., m − 1. The full noisy channel is N = N ⊗ $^{n}$. Its Kraus operators are E$_{x}$ , where x = ( x$_{1}$, . . . , x$_{n}$ ) is a collec- tion of indices x$_{i}$ = 0 , . . . , m − 1 indicating which Kraus operator is applied on the i th site

$$
E_{\mathbf{x}} = \bigotimes_{i=1}^{n} E_{x_i}. \qquad (114)
$$

The complementary channel ̂ N can then be written in
terms of the Kraus operators of N as

$$
\hat{\mathcal{N}}(\cdot) = \sum_{\mathbf{x},\mathbf{x}'} \mathrm{tr}(E_{\mathbf{x}}^{\dagger} E_{\mathbf{x}}(\cdot)) |\mathbf{x}\rangle \langle \mathbf{x}'|, \quad (115)
$$

where {|**x**} is a basis of the Hilbert space of E.

Computing the Fisher information analytically on the output of either N or ̂N might not be straightforward if the state and its derivative are mapped to operators whose eigenbases are not aligned in any obvious way, which would complicate the calculation of the symmetric logarithmic derivative when computing the expression (55). Here, we see that by completely dephasing the output of ̂N in the computational basis, and projecting onto the subspace of the environment associated with low-weight Kraus operators of N , we obtain a lower bound on Δ F$_{Bob,t}$ which translates into a upper bound on F$_{Bob,t}$ that is easy to compute. Here, we assume that the first Kraus operator E$_{0}$ is close to the identity and that the other Kraus operators represent “jump terms.” We mean by “weight” the number of Kraus operators that are jump terms.

We now choose a suitable completely positive, trace-
non-increasing map $\mathcal{N}'$ in order to use Eq. (113) to obtain
an upper bound on the Fisher information at Bob's end.
In the following, we assume that $m = 2$, but the argu-
ment generalizes straightforwardly to noisy channels that
have more Kraus operators. We design the map such
that it (i) completely dephases the environment system in
the computational basis, and (ii) projects its input onto
the subspace associated with basis vectors $|\mathbf{x}\rangle$ with small
Hamming weight $|\mathbf{x}|$. Fix $k > 0$ and let

$$
\mathcal{N}'(\cdot) = \sum_{\mathbf{x}: |\mathbf{x}| \le k} |\mathbf{x}\rangle \langle \mathbf{x}|(\cdot) |\mathbf{x}\rangle \langle \mathbf{x}|. \quad (116)
$$

Then we can see that

$$
(\mathcal{N}' \circ \hat{\mathcal{N}})(\cdot) = \sum_{\mathbf{x}: |\mathbf{x}| \le k} \mathrm{tr}(E_{\mathbf{x}}^{\dagger} E_{\mathbf{x}}(\cdot)) |\mathbf{x}\rangle \langle \mathbf{x}|. \quad (117)
$$

The upper bound on Bob’s Fisher information with respect
to time comes from Eq. (113). Starting from Eq. (112)
and since the two arguments of the Fisher information

commute, we can use Eq. (15) to find

$$
\begin{align}
\Delta F_{\text{Bob},t} &\ge \operatorname{tr}\left\{\left[(\mathcal{N}' \circ \hat{\mathcal{N}})(\psi)\right]^{-1}\left[(\mathcal{N}' \circ \hat{\mathcal{N}})\left(\{\psi, \bar{H}\}\right)\right]^2\right\} \nonumber \\
&= \sum_{\mathbf{x}: |\mathbf{x}| \le k} \frac{\left[2 \operatorname{Re}\langle\psi|\bar{H} E_{\mathbf{x}}^{\dagger} E_{\mathbf{x}}|\psi\rangle\right]^2}{\operatorname{tr}(E_{\mathbf{x}}^{\dagger} E_{\mathbf{x}} \psi)}, \tag{118}
\end{align}
$$

where we have used the fact that the output of N' ◦ ̂N
is diagonal in the computational basis. The completely
dephasing channel ensures that the expression (118) is a
classical Fisher information, which is easier to compute
than the quantum Fisher information in which the state and
the derivative do not commute.

The number of terms in the above sum, which cor-
responds to the dimension of the subspace associated
with basis vectors | **x**〉 satisfying | **x**| ≤ *k*, is given by
⟨**n**<sub>*k*</sub> + ⟨**n**<sub>*k*−1</sub>⟩ + ... + ⟨**n**<sub>0</sub>⟩ = *O*(*n*<sup>*k*</sup>). For fixed *k*, this number
scales polynomially in *n*. The complexity of computing
the numerator and denominator in Eq. (118) also scales
only polynomially in *n* as long as | *ψ*⟩ and *H*| *ψ*⟩ can
be expressed using a representation that enables efficient
computation of local expectation values, such as a superpo-
sition of a constant number of computational basis vectors,
or alternatively as matrix-product states [45]. We discuss
below the case of IID amplitude damping noise, where
numerical evidence indicates that for small values of *p*
(say *p* ≲ 0.1), even for *n* = 50 it can suffice to set *k* = 4 to
obtain meaningful bounds (see Sec. VIII).

B. Lower bound on Bob’s Fisher information by
preprocessing Eve’s system

Let us return to the original setting with Alice, Bob, and
Eve as in Fig. 1. Suppose now that we can find a com-
pletely positive, trace-preserving map ̂N₀ and a completely
positive, trace-non-increasing N' such that ̂N = N' ◦ ̂N₀.
That is, we suppose that Eve gets her state through an
intermediary, which we call Eve₀ as shown in Fig. 8(b).
The data-processing inequality now tells us that F_Eve,η ≤
F_Eve₀,η. Combining this with our uncertainty relation gives
us

$$
\frac{F_{\text{Bob},t}}{F_{\text{Alice},t}} + \frac{F_{\text{Eve}_{0,\eta}}}{F_{\text{Alice},\eta}} \ge 1. \qquad (119)
$$

A more explicit bound on $F_{\text{Bob},t}$ can be obtained starting
from Eq. (55) and writing

$$
\begin{align}
\Delta F_{\text{Bob},t} &= F(\rho_E; \hat{\mathcal{N}}(\{\bar{H}, \psi\})) \nonumber \\
&= F(\mathcal{N}'(\hat{\mathcal{N}}_0(\psi)); \hat{\mathcal{N}}'(\hat{\mathcal{N}}_0(\{\bar{H}, \psi\}))) \nonumber \\
&\le F(\hat{\mathcal{N}}_0(\psi); \hat{\mathcal{N}}_0(\{\bar{H}, \psi\})). \tag{120}
\end{align}
$$
We present two simple example uses of this type of bound. The first example applies to permutation-invariant systems. The second example applies to the setting where Eve's state is reasonably close to being diagonal.

## 1. Permutation-invariant system

Consider a permutation-invariant clock state $\psi$ and Hamiltonian $H$. If the noise $\mathcal{N}$ acts only locally on at most $k$ known sites (or $k/2$ unknown sites), then $\tilde{\mathcal{N}}$ can be written as the composition of a channel that traces out all but $k$ sites, and another channel that completes the implementation of $\tilde{\mathcal{N}}$. To see this, observe that we can write $\tilde{\mathcal{N}}(\cdot) = \sum_{j',j} \text{tr}(E_j^\dagger E_j(\cdot)) |j'\rangle\langle j|$, where $\{E_j\}$ are the Kraus operators of $\mathcal{N}$. By assumption, $E_j^\dagger E_j$ acts nontrivially on at most $k$ sites. Therefore, the expression $\text{tr}(E_j^\dagger E_j(\cdot))$ depends only on the $k$-site reduced state of the input. The full complementary channel can be written as the composition of a channel that traces all but $k$ sites, and the channel $\rho_k \mapsto \sum_{j',j} \text{tr}(E_j^\dagger E_j \rho_k) |j'\rangle\langle j|$ (where here we reuse the notation $E_j^\dagger E_j$ to denote the action of those operators on only the $k$ sites where either operator acts nontrivially). Therefore, the sensitivity loss $\Delta F_{\text{Bob},t}$ can be upper bounded, for any noisy channel consisting of Kraus operators of weight at most $k/2$, by the sensitivity loss associated with $k$ located erasures.

## 2. If Eve's state is nearly diagonal

Computing useful expressions of the Fisher information when a diagonal representation of the state is not known can be tricky. The idea if $\rho_E$ is reasonably close to being diagonal is to hope that one can essentially neglect the off-diagonal elements of $\rho_E$ and still obtain a good approximation of the Fisher information via the formula (12).

Suppose we find an invertible matrix $A$ (with hopefully $A \approx 1$) and a diagonal matrix $\tau = \text{diag}(\tau_0, \dots, \tau_{d_E}) \ge 0$ such that

$$ \rho_E = A \tau A^\dagger. \qquad (121) $$

Such a matrix is given, for instance, by the LDLT and Cholesky decomposition of $\rho_E$. [The eigendecomposition of $\rho_E$ also gives such a matrix $A$, but if we can compute an eigendecomposition one might as well use Eq. (12) to compute the Fisher information directly.] Now we decompose $\tilde{\mathcal{N}}$ by including a scaling factor $\alpha$ as

$$ \alpha \tilde{\mathcal{N}} = \mathcal{N}' \circ \tilde{\mathcal{N}}_0, \qquad (122) $$

with $\alpha = \|A\|^{-2}\|A^{-1}\|^{-2}$ and with the two completely positive, trace-non-increasing maps

$$ \tilde{\mathcal{N}}_0(\cdot) = \frac{1}{\|A^{-1}\|^2} A^{-1} \tilde{\mathcal{N}}(\cdot) A^{-\dagger}, \qquad (123) $$

$$ \mathcal{N}'(\cdot) = \frac{1}{\|A\|^2} A(\cdot) A^{\dagger}. \qquad (124) $$

If $A$ is close to $1$ then we have $\alpha \approx 1$. Recalling the scaling property (14) of the quantum Fisher information, we find

$$ \begin{aligned} \Delta F_{\text{Bob},t} &= F(\rho_E; \tilde{\mathcal{N}}(\{\bar{H}, \psi\})) \\ &= \frac{1}{\alpha} F(\alpha \tilde{\mathcal{N}}(\psi); \alpha \tilde{\mathcal{N}}(\{\bar{H}, \psi\})) \\ &\le \frac{1}{\alpha} F(\tilde{\mathcal{N}}_0(\psi); \tilde{\mathcal{N}}_0(\{\bar{H}, \psi\})) \\ &= \frac{1}{\alpha} F\left(\frac{1}{\|A^{-1}\|^2} \tau; \frac{1}{\|A^{-1}\|^2} A^{-1} \tilde{\mathcal{N}}(\{\bar{H}, \psi\}) (A^{-1})^{\dagger}\right) \\ &= \|A\|^2 F(\tau; A^{-1} \tilde{\mathcal{N}}(\{\bar{H}, \psi\}) (A^{-1})^{\dagger}). \end{aligned} \qquad (125) $$

In the last expression, the Fisher information is evaluated on a state that is diagonal, so one can directly use Eq. (12). Furthermore, if $A$ is determined by a LDLT and Cholesky decomposition then it is lower triangular and its inverse can be computed efficiently (matrix multiplication of the inverse with another matrix can be done by forward substitution).

## C. Bound in terms of Eve's access to the probe's energy

In this section, a further bound on Bob's sensitivity to time is presented, which is given in terms of how well Eve can approximate a measurement of energy on the noiseless clock state. The properties that Eve can measure on the noiseless probe are given by the adjoint of the complementary channel: Eve applying an operator $W$ on her system can equivalently be described as the operator $\tilde{\mathcal{N}}^\dagger(W)$ being applied onto Alice's system, because $\text{tr}(\tilde{\mathcal{N}}(\psi)W) = \text{tr}(\psi \tilde{\mathcal{N}}^\dagger(W))$. One measure of how well Eve can approximate a measurement of the Hamiltonian around $|\psi\rangle$ with an observable $S$ on her system is the minimum root-mean-squared error $\min_{S=S^\dagger} [((\tilde{\mathcal{N}}^\dagger(S) - H)^2]_\psi]^{1/2}$. It turns out that the minimum square of this quantity is a lower bound to Bob's Fisher information to time

$$ F_{\text{Bob},t} \ge \min_{S=S^{\dagger}} 4 \left\langle (\tilde{\mathcal{N}}^{\dagger}(S) - H)^2 \right\rangle_{\psi}. \qquad (126) $$

While this bound is aesthetically interesting, finding the optimal $S$ in this expression is not significantly easier than directly solving the semidefinite program (16a). Furthermore, a candidate for $S$ in Eq. (16a) immediately provides an upper bound on $F_{\text{Bob}}$, whereas a candidate in Eq. (126) does not provide any useful bound on $F_{\text{Bob}}$ because of the direction of the inequality.
The bound (126) is proven as follows. Starting from Eq. (56) and using Eq. (16a),

$$ 
\begin{aligned}
& \frac{1}{4} F_{\text{Bob},t} \\
&= \min_{S=S^{\dagger}} [\langle \bar{H}^2 \rangle - \text{tr}(\psi \{\bar{H}, \hat{\mathcal{N}}^{\dagger}(S)\} + \langle \hat{\mathcal{N}}^{\dagger}(S^2) \rangle)] \\
&\geq \min_{S=S^{\dagger}} [\langle \bar{H}^2 \rangle - \langle \{\bar{H}, \hat{\mathcal{N}}^{\dagger}(S)\} \rangle + \langle [\hat{\mathcal{N}}^{\dagger}(S)]^2 \rangle] \\
&= \min_{S=S^{\dagger}} \langle (\bar{H} - \hat{\mathcal{N}}^{\dagger}(S))^2 \rangle,
\end{aligned} 
\quad (127) $$

where we have used $\hat{\mathcal{N}}^{\dagger}(S^2) \geq [\hat{\mathcal{N}}^{\dagger}(S)]^2$ (see Corollary 2 in Appendix A). Finally, we can replace $\bar{H}$ by $H$ in Eq. (127) because any shifts of $\bar{H}$ by the identity can be canceled out by corresponding shifts of $S$ by the identity.

### D. If Eve can measure the probe's energy almost perfectly

If Eve has (approximate) access to the energy of the probe state, then this (approximately) kills sensitivity on Bob's end. Suppose we can find an observable $S$ on Eve's system such that $\|\hat{\mathcal{N}}^{\dagger}(S) - \bar{H}\| \leq \|\bar{H}\| \delta$ and $\|\hat{\mathcal{N}}^{\dagger}(S^2) - \bar{H}^2\| \leq \|\bar{H}\|^2 \delta$. Then

$$ F_{\text{Bob},t} \leq 12\delta \|\bar{H}\|^2. \quad (128) $$

To show this inequality, we first write $\Delta = \hat{\mathcal{N}}^{\dagger}(S) - \bar{H}$ and $\Delta' = \hat{\mathcal{N}}^{\dagger}(S^2) - \bar{H}^2$, with $\|\Delta\| \leq \|\bar{H}\|\delta$ and $\|\Delta'\| \leq \|\bar{H}\|^2\delta$. Then, from Eq. (56) and using Eq. (16a) we obtain

$$ 
\begin{aligned}
& \frac{1}{4} F_{\text{Bob},t} \\
&= \min_{S=S^{\dagger}} [\langle \bar{H}^2 \rangle - \text{tr}(\{\psi, \bar{H}\} \hat{\mathcal{N}}^{\dagger}(S) + \langle \hat{\mathcal{N}}^{\dagger}(S^2) \rangle)] \\
&\leq \min_{S=S^{\dagger}} \{-\text{tr}(\{\psi, \bar{H}\} \Delta) + \text{tr}(\psi \Delta')\} \\
&\leq 2 \|\bar{H}\| \|\Delta\| + \|\Delta'\| \leq 3\delta \|\bar{H}\|^2.
\end{aligned} 
\quad (129) $$

### E. Clock sensitivity loss for weak IID noise

Here, we consider an $n$-site system subject to weak IID noise, where each site is affected by a noisy channel $\mathcal{N}_{\epsilon}$ such that $\mathcal{N}_{\epsilon} \to \text{id}$ if $\epsilon \to 0$. Clearly for $\epsilon = 0$ there is no sensitivity loss. For a given clock state and Hamiltonian, we develop a set of tools to understand and determine to which order $m$ in $\epsilon$ the Fisher information loss is suppressed, $\Delta F_{\text{Bob},t} = O(\epsilon^m)$.

The question is partly motivated by a similar question in the context of quantum error correction. A quantum error-correcting code of distance $d$ can correct any $(d-1)/2$ arbitrary single-site errors. In the case of a weak IID noisy channel $\mathcal{N}_{\epsilon}^{\otimes n}$ affecting the $n$ sites, a weight-[(d-1)/2] error happens with probability of order $O(\epsilon^{(d-1)/2})$ if we

assume that a single-site error happens with probability $O(\epsilon)$. This means that the chance of an uncorrectable error occurring is upper bounded by $O(\epsilon^{(d-1)/2})$. In this scenario, we see that the higher the distance of the code, the better robustness is achieved against weak IID noise. In the context of quantum metrology, we ask the following analogous question: can we determine the robustness of the sensitivity of the clock to time when affected by a weak IID noisy channel, a function of a certain feature (analogous to the code distance) of the clock state, the Hamiltonian, and the noisy channel?

There does not appear to be any obvious property of the setup (analogous to the code distance) that immediately determines the order $m$ in the Fisher information loss $\Delta F_{\text{Bob},t} = O(\epsilon^m)$. Instead, we explain a general procedure for how to obtain a bound on $m$ when given a weak IID noisy channel, a clock state and a Hamiltonian.

The simplest case presents itself if the complementary channel $\hat{\mathcal{N}}_{\epsilon}^{\otimes n}$ maps the clock state $\psi$ onto a full-rank state $\rho_E = \sum_{\mathbf{x}} p_{\mathbf{x}} |\mathbf{x}\rangle\langle\mathbf{x}|_E$ that is diagonal in the tensor product computational basis on $E$. (This is equivalent to all vectors $\{E_{\mathbf{x}}|\psi\}\}_{\mathbf{x}}$ being orthogonal on Bob's system.) In such a case we can use Eq. (12) to express the Fisher information loss as

$$ 
\begin{aligned}
\Delta F_{\text{Bob},t} &= \sum_{\mathbf{x},\mathbf{x}'} \frac{2}{p_{\mathbf{x}} + p_{\mathbf{x}'}} |\langle \mathbf{x} | \hat{\mathcal{N}}_{\epsilon}^{\otimes n} (\{\bar{H}, \psi\}) | \mathbf{x}' \rangle|^2 \\
&= \sum_{\mathbf{x},\mathbf{x}'} \frac{O(\epsilon^{2q_{\mathbf{x},\mathbf{x}'}})}{\Omega(\epsilon^{\min(r_{\mathbf{x}},r_{\mathbf{x}'})})} \\
&= O(\epsilon^m)
\end{aligned} 
\quad (130) $$

defining $r_{\mathbf{x}}$ and $q_{\mathbf{x},\mathbf{x}'}$ via $p_{\mathbf{x}} = \Omega(\epsilon^{r_{\mathbf{x}}})$ and $|\langle \mathbf{x} | \hat{\mathcal{N}}_{\epsilon}^{\otimes n} (\{\bar{H}, \psi\}) | \mathbf{x}' \rangle| = O(\epsilon^{q_{\mathbf{x},\mathbf{x}'}})$, and with

$$ m = \min_{\substack{\mathbf{x},\mathbf{x}': \\ r_{\mathbf{x}} \leq r_{\mathbf{x}'}}} \{2q_{\mathbf{x},\mathbf{x}'} - r_{\mathbf{x}}\}. \quad (131) $$

As we can see above, it is not obvious which $\mathbf{x}, \mathbf{x}'$ minimizes the expression in the exponent above. One might have expected that events $\mathbf{x}$ whose probability of occurring vanish faster than other events (large $r_{\mathbf{x}}$ compared to other $r_{\mathbf{x}'}$) are less relevant and would not contribute significantly to the Fisher information loss. However, this is not the case; terms with high $r_{\mathbf{x}}, r_{\mathbf{x}'}$ can contribute to leading order to the sensitivity loss if the corresponding term $q_{\mathbf{x},\mathbf{x}'}$ is sufficiently small. If the state $\rho_E$ is not diagonal, then it is unclear whether or not one can easily determine the order of the Fisher information loss.
## VI. CLOCK SENSITIVITY IN THE PRESENCE OF CONTINUOUS NOISE

The setting presented in Fig. 1 is nonstandard in metrology, because in typical settings the noise and the signal both get imprinted on the state in the same physical time-evolution process. It is more common to consider, for instance, a Lindbladian master equation that governs the time evolution of the clock state, with terms that encode any noise processes via jump operators.

Here we consider the situation where the noise is described by a Lindbladian master equation. Under suitable conditions, we can decompose the time evolution into a pure unitary evolution followed by some effective noisy channel, and the time dependence of the effective noisy channel can be neglected. In this case our Theorem 1 can be applied to compute the sensitivity loss after some time $t_0$.

One can follow a similar procedure in the setting where the goal is to determine an unknown parameter in the Hamiltonian when the overall evolution is governed by a Lindbladian master equation. The full derivation is presented in Appendix G. We can carry out a similar decomposition in the case of a clock sensing an unknown parameter in the Hamiltonian, while subject to continuous noise described by a Lindblad evolution.

### A. Decomposing a Lindbladian evolution of a clock into a pure unitary time evolution and an instantaneous noisy channel

Consider a clock initialized at time $t=0$ in the state vector $|\psi_{init}\rangle$. Suppose that the dynamics $\rho(t)$ of the clock are given by the Lindblad master equation

$$ \partial_t \rho = \mathcal{L}_{\text{tot}}[\rho], \qquad (132a) $$

where

$$ \mathcal{L}_{\text{tot}} = \mathcal{L}_0 + \mathcal{L}_1, \quad \mathcal{L}_0(\rho) = -i[H, \rho], \qquad (132b) $$

$$ \mathcal{L}_1(\rho) = \sum_j \left[ L_j \rho L_j^\dagger - \frac{1}{2} \{ L_j^\dagger L_j, \rho \} \right]. \qquad (132c) $$

Here we assume that the operators $H$ and $L_j$ are time independent. The evolution up to a time $t$ is given by the completely positive, trace-preserving map

$$ \mathcal{E}_t = e^{i(\mathcal{L}_0 + \mathcal{L}_1)}. \qquad (133) $$

The evolution driven by the Hamiltonian part $\mathcal{L}_0$ of the dynamics can be written as $e^{i\mathcal{L}_0(\cdot)} = e^{-iHt}(\cdot) e^{iHt}$.

We would like to compute the sensitivity of the clock at a given time $t_0$, meaning that the relevant quantity to

compute is the Fisher information

$$ F_{\text{clock},t}(t_0) = F(\rho(t_0); \partial_t \rho(t_0)). \qquad (134) $$

We can decompose the evolution $\mathcal{E}_t$ as first a unitary evolution according to $H$ for a time $t$ followed by the instantaneous application of an effective noisy channel $\mathcal{N}_t$. Define

$$ \mathcal{N}_t = \mathcal{E}_t e^{-t\mathcal{L}_0} = e^{i(\mathcal{L}_0 + \mathcal{L}_1)} e^{-t\mathcal{L}_0}. \qquad (135) $$

Here, $e^{-t\mathcal{L}_0}$ is the inverse of the unitary evolution $e^{t\mathcal{L}_0}$. By construction, if we apply $\mathcal{N}_t$ after applying $e^{t\mathcal{L}_0}$, then the overall effect is the same as letting the system evolve for time $t$ under the full Lindbladian dynamics $\mathcal{L}_0 + \mathcal{L}_1$:

$$ \mathcal{E}_t = \mathcal{N}_t e^{i\mathcal{L}_0}. \qquad (136) $$

An alternative expression for $\mathcal{N}_t$ is obtained from Eq. (135) using the Baker-Campbell-Hausdorff formula,

$$ \mathcal{N}_t = e^{i\mathcal{L}_1 - \frac{t}{2}[\mathcal{L}_1, \mathcal{L}_0] + \dots}. \qquad (137) $$

Observe that if $[\mathcal{L}_1, \mathcal{L}_0] = 0$, then we simply have $\mathcal{N}_t = e^{i\mathcal{L}_1}$. This situation is known as phase-covariant dynamics (cf., e.g., Refs. [46,47]). This is the case if $[L_j, H] = 0$ for all jump operators $L_j$. In other cases, the map can be determined from Eq. (135) directly if the superoperator $\mathcal{E}_t$ can be computed.

Let us introduce the family of states $\psi(t) = e^{-iHt} \psi_{\text{init}} e^{iHt}$ associated with the (fictitious) pure unitary evolution of $\psi_{\text{init}}$ if we artificially turn off the noise terms.

The derivative of the quantum state $\rho(t) = \mathcal{E}_t(\psi_{\text{init}})$ can then be written as

$$ \partial_t \rho(t) = \partial_t \left[ \mathcal{N}_t(\psi(t)) \right] = \mathcal{N}_t(\partial_t \psi(t)) + (\partial_t \mathcal{N}_t)(\psi(t)). \qquad (138) $$

Therefore, the derivative of the noisy state can be decomposed into a sum of two terms, the first associated with the unitary dynamics $\psi(t)$, and the other associated with the time dependence of the effective noisy channel $\mathcal{N}_t$. Plugging into Eq. (134), this gives us

$$ F_{\text{clock},t} = F(\mathcal{N}(\psi); \mathcal{N}(\partial_t \psi) + (\partial_t \mathcal{N})(\psi)), \qquad (139) $$

where now $F_{\text{clock},t}$, $\mathcal{N}$, $\partial_t \mathcal{N}$, $\psi$, and $\partial_t \psi$ are all implicitly evaluated at $t_0$.

In the following, we consider settings where the local time dependence of the state due to the time dependence of the effective noisy channel terms can be neglected when computing $F_{\text{clock},t}$. (We will study in greater depth below
when exactly this situation arises.) In other words, for now we assume that

$$F_{\text{clock},t} \approx F(\mathcal{N}(\psi); \mathcal{N}(\partial_t \psi)) =: F_{\text{clock},U,t}. \quad (140)$$

Expanding $\partial_t \psi$, we obtain

$$F_{\text{clock},U,t} = F(\mathcal{N}(\psi); \mathcal{N}(-i[H, \psi))). \quad (141)$$

This quantity is what we defined as $F_{\text{Bob},t}$ in the context of our main uncertainty relation.

The complementary channel $\tilde{\mathcal{N}}_{t_0}$ is directly determined by the complementary channel of the overall evolution up to that time $\tilde{\mathcal{E}}_{t_0}$, since the two channels differ only by a unitary evolution $e^{-t_0 \mathcal{L}_0}$ on their input:

$$\tilde{\mathcal{N}}_t = \tilde{\mathcal{E}}_t e^{-t_0 \mathcal{L}_0}. \quad (142)$$

This means that the Fisher information on Eve's end with respect to the complementary direction can be expressed entirely in terms of the complementary channel $\tilde{\mathcal{E}}_{t_0}$ to the entire evolution up to time $t_0$:

$$\Delta F_{\text{clock},U,t} = F(\tilde{\mathcal{E}}_{t_0}(\psi_{\text{init}}); \tilde{\mathcal{E}}_{t_0}(\{\bar{H}, \psi_{\text{init}}\})), \quad (143)$$

with $\bar{H} = H - \langle H \rangle_{\psi(t_0)}$, and Theorem 1 states that

$$F_{\text{clock},U,t} = 4\sigma_H^2 - \Delta F_{\text{clock},U,t}. \quad (144)$$

Now we turn to discussing when the approximation (140) is a reasonable assumption, by characterizing the error induced on the Fisher information. First of all, the approximation is exact in the case of phase-covariant dynamics, where $[\mathcal{L}_1, \mathcal{L}_0] = 0$ (e.g., Refs. [46,47]). In other settings, we can use a continuity bound of the Fisher information in its second argument (Proposition 17 in Appendix C) to try to get a handle on the error terms involved in the approximation (140). Denote by $\delta$ the error in the approximation (140),

$$\delta = F_{\text{clock},t} - F_{\text{clock},U,t}, \quad (145)$$

then we have

$$|\delta| \le F(\rho; (\partial_t \mathcal{N})(\psi)) + [F(\rho; (\partial_t \mathcal{N})(\psi)) F_{\text{clock},U,t}]^{1/2}. \quad (146)$$

That is, the relative error in the approximation (140) is demonstrably small if $F(\rho; (\partial_t \mathcal{N})(\psi))$ is much smaller than $F_{\text{clock},U,t}$. We can rewrite this term using (138) as

$$(\partial_t \mathcal{N})(\psi) = \partial_t \rho(t) - \mathcal{N}_t(-i[H, \psi(t)]) = \mathcal{L}_{\text{tot}}[\rho(t)] - \mathcal{E}_t(-i[H, \psi_0]). \quad (147)$$

The above expression is given in terms of the Lindbladian map and the overall evolution map, and can aid in

determining an analytical or numerical upper bound to the quantity $F(\rho; (\partial_t \mathcal{N})(\psi))$. In Appendix G, we study two single-qubit examples that are subject to continuous dephasing along various axes in order to illustrate the connections between the Lindbladian setting and the setting in Fig. 1.

## VII. ERROR-CORRECTION CONDITIONS FOR ZERO SENSITIVITY LOSS

The uncertainty relation (55) enables us to provide a characterization of when the noise reduces a probe's sensitivity to time. In this section, we study the situation where the sensitivity loss $\Delta F_{\text{Bob},t}$ introduced in Eq. (52) is equal to zero. This is a situation where the probe is chosen cleverly enough such that the noise has no effect on sensitivity. The main contribution of this section is a set of necessary and sufficient conditions for $\Delta F_{\text{Bob},t} = 0$, which bear resemblance to the Knill-Laflamme conditions for quantum error correction [20] and which are closely related to the Hamiltonian-not-in-Lindblad-span condition of Refs. [21,22].

### A. Conditions for zero sensitivity leakage

In the following, we suppose that our uncertainty relation holds with equality, i.e., that the conditions given in Proposition 1 hold. Recall the expression for the Fisher information loss on Bob's end (55), and consider the expression (16b) for the Fisher information. If $\Delta F_{\text{Bob},t} = 0$, then there exists an operator $L$ such that $\text{tr}(L^\dagger L) = 0$ and $\rho^{1/2}L + L^\dagger\rho^{1/2} = \tilde{\mathcal{N}}(\{\psi, \bar{H}\})$; the former condition implies $L = 0$ and thus the latter implies $\tilde{\mathcal{N}}(\{\psi, \bar{H}\}) = 0$. Therefore, we see that $\Delta F_{\text{Bob},t} = 0$ if and only if

$$\tilde{\mathcal{N}}(\{\psi, \bar{H}\}) = 0, \quad (148)$$

i.e., $\{\psi, \bar{H}\}$ must lie in the kernel of the superoperator $\tilde{\mathcal{N}}$. It is instructive to rewrite this condition in terms of the “virtual qubit” introduced in Sec. III B. With $Z_L$ defined in Eq. (62), then Eq. (148) becomes

$$\tilde{\mathcal{N}}(Z_L) = 0. \quad (149)$$

Alternatively, the above condition is equivalent to requiring that for all operators $O$,

$$\text{tr}[\tilde{\mathcal{N}}^\dagger(O) Z_L] = 0, \quad (150)$$

meaning that error operations of the form $\tilde{\mathcal{N}}^\dagger(O)$ should not have any overlap with the “logical” $Z_L$ operator on the qubit subspace.

So the task of finding probe states that perfectly counter the noisy channel $\mathcal{N}$ can be formulated as ensuring the logical Z Pauli operator in the logical qubit subspace spanned
by $|+\rangle = |\psi\rangle$ and $|−\rangle \propto |\xi\rangle = P_\psi^\dagger H |\psi\rangle$ is in the kernel of the complementary channel to the noisy channel.

Note that simply looking for zero sensitivity loss is not sufficient to find the best probe states; we still need to make sure that $|\psi\rangle$ has as large energy variance as possible to ensure good sensitivity.

An alternative representation of the zero sensitivity loss condition can be obtained if we consider an operator-sum representation of the noisy channel in terms of Kraus operators {$E_k$} as in Eq. (45). The condition (148) is then equivalent to the condition

$$ \langle\psi| E_{k'}^{\dagger} E_k |\xi\rangle + \langle\xi| E_{k'}^{\dagger} E_k |\psi\rangle = 0 \quad \text{for all } k, k'. \quad (151) $$

These may be interpreted as Knill-Laflamme-like conditions for optimal sensitivity. Whereas for a traditional quantum error-correcting code, we require any two code words $|\psi_i\rangle, |\psi_j\rangle$ to satisfy $\langle\psi_i|E_{k'}^\dagger E_k|\psi_j\rangle \propto \delta_{i,j}$, here we require that the error operator $E_k^\dagger E_{k'}$ cannot map the state $|\psi\rangle$ onto the vector $|\xi\rangle$, or at least not in a way that is not suitably antisymmetric. The weird antisymmetrization in Eq. (151) can be expressed in a more elegant form if we switch back to the picture of the logical qubit spanned by $|\psi\rangle$ and $|\xi\rangle$. Analogously to Eq. (149), we may rewrite the condition (151) as

$$ \mathrm{tr}[Z_L \Pi_L E_{k'} E_k \Pi_L] = 0, \quad (152) $$

where $\Pi_L = |+\rangle\langle+|_L + |−\rangle\langle−|_L$ is the projector onto the virtual qubit subspace spanned by $|\psi\rangle$ and $|\xi\rangle$. The full Knill-Laflamme conditions applied to the subspace $\Pi_L$ would require $\Pi_L E_{k'} E_k \Pi_L \propto \Pi_L$. The condition (152) is simply a weaker condition where only the corresponding projection onto the logical Pauli operator $Z_L$ is considered and where the projection onto the other Pauli operators is unconstrained.

The form (152) also helps clarify that for zero sensitivity loss, the terms in Eq. (151) need not vanish individually. Indeed, only the Hilbert-Schmidt projection of $\Pi_L E_{k'} E_k \Pi_L$ onto $Z_L$ is required to vanish, and not in principle on $Y_L$ or $X_L$. An example below in Sec. VII H 1, consisting of a single-qubit subject to transversal noise, will illustrate this point.

The conditions (152) are reminiscent of quantum error correction for operator algebras, where we require a code to preserve the outcomes of any operator in a given algebra [48–50]. In fact, if the algebra associated with any choice of optimal sensing operator of the form (20) is preserved, then our conditions (152) are satisfied. Indeed, suppose that $[T, \hat{\mathcal{N}}^\dagger(W)] = 0$ for any operator $W$ on Eve and for a fixed choice of $M$ in Eq. (20), meaning that the Abelian algebra generated by $T$ is correctable [48–50]. Then taking the expectation value $\langle\cdot|\psi$ of this commutator we find $0 = \langle[T, \hat{\mathcal{N}}^\dagger(W)]\rangle = \mathrm{tr}([\psi, T]\hat{\mathcal{N}}^\dagger(W)) \propto \mathrm{tr}(Z_L \hat{\mathcal{N}}^\dagger(W))$, using

Eq. (25), which holds for all $W$, and therefore our Knill-Laflamme-like condition (150) holds. The converse implication is unclear, in part because the optimal sensing operator is not unique and different choices can generate different algebras.

The conditions (151) are actually tightly related to the Hamiltonian-not-in-Kraus-span condition of Refs. [21,22,30,31,51–53]. There, it was shown that there exists a clock state vector $|\psi\rangle$ that achieves Heisenberg scaling in the presence of noise using quantum error correction if and only if the Hamiltonian signal term is not in the linear span of the Lindblad noise operators. Here we argue that the Hamiltonian-not-in-Kraus-span condition is in fact equivalent to the existence of a state $|\psi\rangle$ that satisfies our zero sensitivity-loss conditions (151). (In our setting, the clock state vector $|\psi\rangle$ is a given fixed state.) As we have a discrete noisy channel, we consider the Kraus operators {$E_k$} of the noisy channel instead of Lindblad operators. If $H = \sum \alpha_{k',k} E_{k'}^\dagger E_k$, and supposing the conditions (151) are satisfied for some $|\psi\rangle$, then by taking a linear combination $\sum \alpha_{k',k}$ of the conditions (151) we obtain $0 = 2\langle\psi|H P_\psi^\dagger H|\psi\rangle = 2\sigma_H^2$; therefore the conditions (151) cannot be satisfied by any $\psi$ that has nonzero energy variance. Conversely, we know (see, e.g., Refs. [21,22,52]) that if the Hamiltonian is not in the span of the noisy channel’s Kraus operators, then there is a code space $\Pi$, possibly involving an ancilla system, with $\Pi E_{k'}^\dagger E_k \Pi = c_{k',k} \Pi$ such that $[\Pi, H] = 0$ (i.e., $\Pi$ is spanned by a subset of energy eigenvectors) and such that $\Pi$ contains a state vector $|\psi\rangle$ with nonzero energy variance; then for any $k, k'$ we have $\langle\psi|E_{k'}^\dagger E_k P_\psi^\dagger H|\psi\rangle = \langle\psi|\Pi E_{k'}^\dagger E_k P_\psi^\dagger H \Pi|\psi\rangle = \langle\psi|\Pi E_{k'}^\dagger E_k \Pi P_\psi^\dagger H|\psi\rangle = c_{k',k}\langle\psi|P_\psi^\dagger H|\psi\rangle = 0$ using the fact that $[P_\psi^\dagger, \Pi] = [H, \Pi] = 0$, so the conditions (151) are satisfied. Therefore, if the Hamiltonian is not in the span of the Kraus operators, then there exists a clock state vector $|\psi\rangle$ that suffers no sensitivity loss after being exposed to the noise locally at $t_0$. This state is constructed in the above mentioned references using a quantum error-correcting code.

We can ask whether there is a relation between our conditions for no sensitivity loss and when the sensitivity can achieve Heisenberg scaling in the system size [2]. The Heisenberg scaling refers to situations where $F_{\mathrm{Bob},t}$ scales like $n^2$, where $n$ is the number of systems that are jointly prepared in the clock state vector $|\psi\rangle_n$. (If no entanglement is present between the $n$ systems, the best scaling that can be achieved is $F_{\mathrm{Bob},t} \propto n$.) We assume that the clock state vector $|\psi\rangle_n$ has a variance that scales quadratically in $n$, i.e., $[\sigma_H(\psi_n)]^2 \propto n^2$, as otherwise even the noiseless clock does not achieve Heisenberg scaling. Suppose the conditions (151) are satisfied: then $F_{\mathrm{Bob},t} = 4[\sigma_H(\psi_n)]^2 \propto n^2$ as there is no sensitivity loss, and the Fisher information displays Heisenberg scaling. On the other hand, even if there is some loss of sensitivity due to the noise, the
Heisenberg scaling might survive. Suppose, for example, that we consider two independent one-dimensional spin chains, each consisting of $n/2$ sites that are prepared in a GHZ state and that evolve according to an on-site Z Hamiltonian. Both spin chains are independent probes whose sensitivity each scales as approximately $n^2$, and therefore the overall probe state exhibits Heisenberg scaling. Now consider the noisy channel that erases one of the spin chains. Half the sensitivity is lost; because there is sensitivity loss our Knill-Laflamme-like conditions cannot be satisfied. However, the single spin chain that is left for Bob still exhibits Heisenberg scaling. This shows that Heisenberg scaling is guaranteed if the environment has zero sensitivity to energy (and the noiseless probe itself has Heisenberg scaling), but that there are also situations where the environment induces sensitivity loss without hindering the Heisenberg scaling of the probe. In the language of Refs. [21,22], this corresponds to a Hamiltonian that might have both a parallel component to the signal as well as a perpendicular component that can be exploited to achieve Heisenberg scaling. We see that zero sensitivity loss implies Heisenberg scaling for a family of state vectors $|\psi\rangle$ that are sufficiently entangled. But there are states that achieve the Heisenberg scaling even if some sensitivity is lost due to the noise.

When the zero sensitivity loss conditions (148) hold, then by definition there must exist a sensing observable for Bob to estimate the parameter $t$, whose sensitivity matches that of Alice. We can extract this optimal sensing observable from our technical analysis using semidefinite programming (see Appendix E2). Namely, in Appendix E3 we show that if the zero sensitivity-loss conditions hold, then the operator $i\rho\mathcal{N}(|\xi\rangle\langle\psi|)$ is Hermitian. Furthermore, the operator

$$R_B = -2i\mathcal{N}(|\xi\rangle\langle\psi|)\rho^{-1} + 2i\rho^{-1}\mathcal{N}(|\psi\rangle\langle\xi|)P_{\rho}^{\perp} \quad (153)$$

is also Hermitian and satisfies $1/2\{R_B, \rho_B\} = \mathcal{N}(Y_L)$, i.e., we obtain an explicit expression of the symmetric logarithmic derivative on Bob's end. The optimal sensing observable on Bob's system is then given via Eq. (13) as $T_b = [F_{\text{Bob,}}]^{-1}R_B + t_0$. That is, when a clock state and associated Hamiltonian fulfill the metrological code conditions for a given noise channel, we obtain an explicit expression for the optimal measurement on Bob's end.

## B. Metrological codes and metrological distance

We now introduce the concept of a metrological code. The idea is to study the qubit space spanned by the vectors $|\psi\rangle$ and $|\xi\rangle = \tilde{H}|\psi\rangle = (H - \langle H \rangle)|\psi\rangle$. If the state loses no sensitivity upon the action of a noisy channel, one could expect these states to span some kind of quantum error-correcting code space. We can see that they do not necessarily form a full error-correcting code as follows. Consider the single-qubit state $|\psi\rangle = |+\rangle = [|0\rangle + |1\rangle]/\sqrt{2}$

evolving under the Hamiltonian $H = \omega\sigma_Z/2$, which we expose to an error channel whose Kraus operators are proportional to $1$ and $X$. We see that the condition (151) is satisfied, given that $|\xi\rangle = |-\rangle = [|0\rangle - |1\rangle]/\sqrt{2}$ is orthogonal to $|+\rangle$ and that $|+\rangle$ is an eigenstate of both $1$ and $X$. Yet a quantum state stored on this qubit would be corrupted by the noise, as the bit flips would be uncorrectable. We identify a concept that is weaker than a full error-correcting code, which applies precisely to states that satisfy the condition (151). Here, we assume that the setting is specified as a pair of orthogonal states $|\psi\rangle, |\xi\rangle$, whereby $|\xi\rangle$ is presumably obtained from a Hamiltonian $H$ as $|\xi\rangle = (H - \langle H \rangle)|\psi\rangle$. Specifying the full Hamiltonian is not necessary as the relevant quantum Fisher information quantities can be fully expressed only in terms of $|\psi\rangle, |\xi\rangle$.

**Metrological code.** Let $\mathcal{E}$ be any set of operators. We say that the state vectors $|\psi\rangle$ and $|\xi\rangle$ form a metrological code against the errors $\mathcal{E}$ if for all $E, E' \in \mathcal{E}$, we have

$$\mathrm{tr}[E^{\dagger}E (|\xi\rangle\langle\psi| + |\psi\rangle\langle\xi|)] = 0. \quad (154)$$

As a consequence of the zero sensitivity loss condition (151), a metrological code prevents sensitivity loss against any noise channel whose Kraus operators are linear combinations of elements in $\mathcal{E}$ (as long as the conditions of Proposition 1 are satisfied).

A natural class of errors to consider is the set of all operators that act on only a subset of $n$ components of a composite quantum system $A = A_1 \otimes A_2 \otimes \cdots \otimes A_n$. The weight $\mathrm{wgt}(O)$ of an operator $O$ acting on the $n$ systems is defined as the number of systems on which $O$ acts nontrivially. Specifically, if $O$ is expanded in the Pauli operator basis (or in any tensor basis using a single-site operator basis that includes the identity matrix), all nonidentity elements in tensor products of basis operators that appear in the decomposition of $O$ must be supported on a fixed set of $\mathrm{wgt}(O)$ sites. Equivalently, the expectation value of $O$ on any state can be computed exactly even after tracing out all but a given set of $\mathrm{wgt}(O)$ sites.

We say that the pair of state vectors $|\psi\rangle$ and $|\xi\rangle$ form a metrological code of distance $d_m$ if it is a metrological code against all operators of weight at most $d_m - 1$; in other words, for all operators $O$ satisfying $\mathrm{wgt}(O) < d_m$, we have

$$\mathrm{tr}[O(|\xi\rangle\langle\psi| + |\psi\rangle\langle\xi|)] = 0. \quad (155)$$

Metrological codes of distance $d_m$ have the property that for any noise channel $\mathcal{N}$ whose Kraus operators $\{E_k\}$ are such that $\mathrm{wgt}(E_k^\dagger E_k) < d_m$ for all $k', k$, the associated sensitivity loss is zero (as long as Proposition 1 is satisfied).

Metrological codes are, roughly speaking, in between classical and quantum codes. On one hand, they are not full-blown classical codes because condition (155)
requires protection against both X- and Z-type physi-
cal noise. Because of this, the pair |ψ⟩ ∝ |0⟩ⁿ + |1⟩ⁿ and
|ξ⟩ ∝ |0⟩ⁿ − |1⟩ⁿ of GHZ states *is not* a metrological code
of nontrivial distance because single-qubit Z errors cause
a logical-X error, thereby violating Eq. (155). On the
other hand, metrological codes are not full-blown quantum
codes because the sensitivity conditions say nothing about
other types of logical noise. In other words, noise can cause
logical-Y and logical-Z errors for a metrological code, but
not for a bona-fide error-correcting code.

C. Uncertainty relation equality and conditions for metrological codes

In order to deduce from Eve's lack of sensitivity to
energy that Bob loses no sensitivity to time, it is necessary
to ensure that the conditions of Proposition 1 hold. When
we presented Proposition 1, we already noted that the situa-
tions where these conditions are not satisfied are edge cases
that can be perturbed away. Here, we strengthen this state-
ment for metrological codes: if a metrological code for a
given noise channel happens not to satisfy the conditions of
Proposition 1, then the noise channel can be infinitesimally
perturbed to obtain a situation for which these conditions
hold, and furthermore, the zero sensitivity loss conditions
(148) are preserved.

Proposition 2 (Perturbation bound for noise chan-
nels consistent with a metrological code).—Let $V_{A→BE}$
be an isometry, let $|\psi\rangle_A, |\xi\rangle_A$ with $\langle\psi|\xi\rangle_A = 0$ and let
$\mathcal{N}(\cdot) = \text{tr}_E(V(\cdot) V^\dagger)$, $\tilde{\mathcal{N}}(\cdot) = \text{tr}_B(V(\cdot) V^\dagger)$. Suppose that
$\tilde{\mathcal{N}}(|\xi\rangle\langle\psi| + |\psi\rangle\langle\xi|) = 0$. We furthermore assume that
there exists a unitary operator $G_B$ acting on the system
$B$ with the properties that $0 = P_{\rho_B} G_B P_{\rho_B} = P_{\zeta_B} G_B P_{\zeta_B} =$
$P_{\rho_B} G_B P_{\zeta_B} = P_{\zeta_B} G_B P_{\rho_B}$, where $\zeta_B = \mathcal{N}(|\xi\rangle\langle\xi|)$. Then,
for any $\epsilon > 0$, there exists an isometry $V'_{A→BE}$ with
$\|V' - V\| \le \epsilon$ such that

$$
(P_{\rho'_B}^{\perp} \otimes P_{\rho'_E}^{\perp}) V |\xi\rangle = 0; \quad \text{and} \tag{156a}
$$

$$
\tilde{\mathcal{N}}'(|\xi\rangle\langle\psi| + |\psi\rangle\langle\xi|) = 0, \qquad (156\text{b})
$$

where $\rho'_B = \text{tr}_E\{V'\psi V'^\dagger\}$, $\rho'_E = \text{tr}_B\{V'\psi V'^\dagger\}$, and $\tilde{\mathcal{N}}'(\cdot) = \text{tr}_B\{V'(\cdot)V'^\dagger\}$.

The proof is presented as Proposition 22 in Appendix H.
Note that the existence of such an operator $G_B$ can always
be ensured by augmenting the $B$ system to include a qubit,
which $\mathcal{N}$ prepares in a fixed pure state vector $|0\rangle$ for all
inputs. The operator $G_B$ can be chosen to flip the qubit to
$|1\rangle$. The additional qubit can represent an additional “fail-
ure” flag such as, for instance, an additional photon that is
emitted at the output of the noise process.

D. Sensitivity loss of metrological codes under weak IID noise

1. Sensitivity loss under weak IID noise

If we encode a logical quantum state using a quantum
error-correcting code of a distance *d*, and each site has
a small probability *O*(*ε*) of incurring an error, then we
know that the errors that the code cannot correct occur with
probability at most *O*(*ε*<sup>*d*</sup>/<sup>2</sup>). In turn, this implies that the
infidelity of recovery of the logical information also scales
as *O*(*ε*<sup>*cd*</sup>) with a constant *c* depending on which conven-
tion for the infidelity measure we choose. It is then natural
to conjecture that if |ψ⟩ and |ξ⟩ form a metrological code
of metrological distance *d*<sub>*m*</sub>, then the loss in Fisher infor-
mation must similarly be upper bounded by *O*(*ε*<sup>*cd*<sub>*m*</sub></sup>), for
some universal constant *c*.

Interestingly, the order of the Fisher information loss in
ε is not directly related to the metrological distance of a
metrological code. In fact, there are examples of metrolog-
ical codes with large metrological distance, but for which
the Fisher information loss is always of order ε. This
behavior appears to contradict the expectation that events
of vanishing probability should not significantly influence
observable properties of the system (such as its sensitiv-
ity to time). An explanation stems from the fact that the
operational interpretation of the Fisher information via the
Cramér-Rao bound involves an implicit averaging of the
error over infinitely many samples. It might turn out in
the present case that events with vanishing probability can
contribute non-negligibly to the quantum Fisher informa-
tion. To remedy this issue, it would be desirable to consider
a measure of sensitivity that accounts for finite data acqui-
sition. One such measure has been put forward in Ref. [54].
We refer to Appendix I for a more detailed discussion.

E. Clock states from time-covariant quantum error-correcting codes

Here, we explore a simple method to construct states
that satisfy the zero sensitivity-loss condition, using time-
covariant quantum error-correcting codes. A code is said to
be time-covariant code with respect to a given Hamiltonian
*H* if *H* (and hence also time evolution generated by *H*) is
a nontrivial logical operator. In the following, Pauli oper-
ators *X*, *Y*, *Z* carry an index indicating the qubit on which
the operator acts. This strategy is the one pursued by, e.g.,
Refs. [22,51–53].

1. *Four nearest-neighbor interacting qubits in a square pattern*

As a warm-up example, we first consider how to lever-
age the [[4, 2, 2]] code for quantum metrology with a
Hamiltonian on four qubits with ZZ interactions arranged
in a square pattern. Consider four qubits arranged in a
square as depicted in Fig. 9. The Hamiltonian is defined
<figure><img src="image_11.png" /><figcaption>FIG. 9. Metrology with interacting qubits. (a) Consider four qubits in a square with nearest-neighbor ZZ Ising interactions (alternatively with additional XX and YY interactions). A clock state with maximal sensitivity and zero sensitivity loss under a single located erasure can be obtained via the time-covariant [[4, 2, 2]] code. (b) We can extend the construction based on the [[4, 2, 2]] code to any number of qubits interacting with respect to any graph of ZZ interactions (alternatively with additional XX and YY interactions), while offering protection against a single located erasure.</figcaption></figure>

by placing a ZZ interaction on each side of the square,

$$H = \omega(Z_1Z_2 + Z_1Z_3 + Z_2Z_4 + Z_3Z_4). \qquad (157)$$

The [[4, 2, 2]] code [55,56] has stabilizers $X_1X_2X_3X_4$ and $Z_1Z_2Z_3Z_4$. The logical operators $X_1, Z_1$ and $X_2, Z_2$ for the first and second logical qubits are $\overline{X}_1 = X_1X_3, \overline{X}_2 = X_1X_2, \overline{Z}_1 = Z_1Z_2$, and $\overline{Z}_2 = Z_1Z_3$.

Observe that the Hamiltonian is a logical operator: the second and fourth terms in Eq. (157) have the same action on the code space as the first and third terms, respectively, because they differ only by the stabilizer $Z_1Z_2Z_3Z_4$. When acting on the code space, we have

$$H \Pi = 2\omega(\overline{Z}_1 + \overline{Z}_2)\Pi. \qquad (158)$$

Let us choose the clock state as a logical state with the largest possible energy spread under this Hamiltonian,

$$|\psi\rangle = \frac{1}{\sqrt{2}}[|\overline{00}\rangle + |\overline{11}\rangle], \qquad (159)$$

where $|\overline{00}\rangle$ and $|\overline{11}\rangle$ refer to logical state vectors with the first and second logical qubits in the given logical computational basis states.

Now we check our Knill-Laflamme-like condition. Having distance 2, the code can correct a single erasure at a known location. Crucially, the operator $|\xi\rangle = (H - \langle H \rangle)|\psi\rangle = 2\omega[|\overline{00}\rangle - |\overline{11}\rangle]/\sqrt{2}$ is still in the code space because $H$ is a logical operator. Then from the Knill-Laflamme conditions we know that $\langle \xi | O_i | \psi \rangle = 0$ for any single-site operator $O_i$, because $|\xi\rangle$ and $|\psi\rangle$ are orthogonal vectors in the code space, and hence our conditions (151) are satisfied for single located errors.

If we have some freedom in engineering our Hamiltonian, there are other choices of logical operators to use in the Hamiltonian that would achieve a similar sensitivity

while also offering protection against single located erasures. For instance, we could ignore the second logical qubit (or treat it as a gauge qubit) and the Hamiltonian could be chosen to act only on sites 1 and 2 as $H = 2\omega\overline{Z}_1 = 2\omega Z_1Z_2$.

We see that the probe state (159) does not lose any sensitivity to time if a system is erased at a known location. The variance of $|\psi\rangle$ is given by

$$\sigma_H^2 = \langle \psi | H^2 | \psi \rangle = 16\omega^2. \qquad (160)$$

Because we have not specified how this model scales with $n$, we cannot talk yet about achieving Heisenberg scaling.

In this example, the sensitivity is in fact as good as you can get without any noise at all, for any probe state: the state (159) is a superposition between two states that have extremal eigenvalues with respect to $H$, which is optimal in the absence of noise. What is special about the state vector $|\psi\rangle$ is that it retains its sensitivity even after a single located error, which is not in general the case of other probe states that would be optimal in the noiseless setting. For instance, the state vector $[|0000\rangle + |0110\rangle]/\sqrt{2}$ has the same sensitivity as $|\psi\rangle$ if no noise is applied, but it does not satisfy our conditions (151) and so is subject to sensitivity loss under single-site errors.

The above construction can also be applied if we include XX and YY interactions between the neighboring qubits on top of the existing ZZ interactions (enabling us to model, e.g., Heisenberg interactions):

$$H = \omega \sum_{\langle i,j \rangle} [s_x X_i X_j + s_y Y_i Y_j + Z_i Z_j], \qquad (161)$$

with the additional coupling constants $s_x, s_y$ allowing for some anisotropy in the interaction strengths. In this case, the interaction terms are again all logical operators, which can be seen from the fact that $X_1X_2X_3X_4$ and $Y_1Y_2Y_3Y_4 = (X_1X_2X_3X_4)(Z_1Z_2Z_3Z_4)$ are stabilizers. Our zero sensitivity-loss conditions are therefore still satisfied. To compute the variance of $|\psi\rangle$ under this new Hamiltonian, we need to determine the action of the additional terms on $|\psi\rangle$. The $X$ terms give us again $\overline{X}_1 + \overline{X}_2$ when acting on the code space following the same argument as for the $Z$ terms. Now $|\psi\rangle$ is a maximally entangled state vector between the two logical qubits, satisfying $(\overline{A}_1 \otimes \overline{1})|\psi\rangle = (\overline{1} \otimes \overline{A}_2^T)|\psi\rangle$ where $(\cdot)^T$ denotes the matrix transpose in the (logical) computational basis, and where $\overline{A}_i$ is a logical operator acting on the $i$th logical qubit. For the $Y$ terms, we then find

$$\begin{aligned}
Y_1 Y_2 |\psi\rangle &= -\overline{X}_2 \overline{Z}_1 |\psi\rangle = -\overline{Z}_1 \overline{X}_1 |\psi\rangle = i \overline{Y}_1 |\psi\rangle, \\
Y_1 Y_3 |\psi\rangle &= -\overline{X}_1 \overline{Z}_2 |\psi\rangle = -\overline{X}_1 \overline{Z}_1 |\psi\rangle = -i \overline{Y}_1 |\psi\rangle, \\
Y_2 Y_4 |\psi\rangle &= Y_1 Y_3 |\psi\rangle = -i \overline{Y}_1 |\psi\rangle, \\
Y_3 Y_4 |\psi\rangle &= Y_1 Y_2 |\psi\rangle = i \overline{Y}_1 |\psi\rangle.
\end{aligned} \qquad (162)$$
Thus the sum of all four Y interacting terms vanishes when applied onto $|\psi\rangle$. The variance of H is hence given by

$$H|\psi\rangle = 2\omega[\bar{Z}_1 + \bar{Z}_2 + s_x(\bar{X}_1 + \bar{X}_2)]|\psi\rangle \\ = 4\omega[\bar{Z}_1 + s_x\bar{X}_1]|\psi\rangle, \qquad (163)$$

using the fact that $|\psi\rangle$ is a maximally entangled state vector between the two logical qubits, and

$$\sigma_H^2 = \langle\psi|H^2|\psi\rangle = 4\omega^2(1 + s_x^2). \qquad (164)$$

The increase in the variance $\sigma_H^2$ when we switch on transversal interactions can be simply associated with the increased norm of the Hamiltonian. Had we defined the clock state (159) with a $-1$ relative phase, then the YY terms would contribute instead of the XX terms and we would get $\sigma_H^2 = 4\omega^2(1 + s_y^2)$.

## 2. Time-covariant codes lead to states with no sensitivity loss

The construction above based on the [[4, 2, 2]] code exploited a key property of that code with respect to the Hamiltonian, namely time covariance [44,57–60]. A time-covariant code with respect to a given Hamiltonian $H$ is a code for which the time evolution generated by $H$ is a (nontrivial) logical operator. If we can find a time-covariant code with respect to the system’s Hamiltonian, then the clock state can be chosen to lie within the code space, so that errors that affect it can be corrected, all while evolving nontrivially in time and thus serving as a clock.

However, there are constraints on the possibility of constructing time-covariant codes. Consider a Hamiltonian that is a sum of terms of weight at most $k$, which we call a k-local Hamiltonian. Any code that can correct up to $k$ arbitrary errors at known locations cannot be time covariant with respect to a k-local Hamiltonian, because the Hamiltonian would be a sum of correctable terms that cannot have a nontrivial action that preserves the code space. On the other hand, physical systems like spin chains and the anti-de Sitter–conformal field theory (AdS-CFT) correspondence as a model for quantum gravity offer natural examples of time-covariant codes that can approximately correct against low-weight errors [43]. The above example using the [[4, 2, 2]] code is a concrete case of a time-covariant code with respect to a 2-local Hamiltonian and which can correct a single erasure at a known location.

We can see that whenever we can find a time-covariant code with respect to a given Hamiltonian, then we can construct from the code a clock state with zero sensitivity loss. Consider a code space $\Pi$ and suppose that the Hamiltonian $H$ is a nontrivial logical operator. We can choose $|\psi\rangle$ to be any logical state vector that has nonzero variance with respect to $H$. Let $|\xi\rangle = (H - \langle H \rangle)|\psi\rangle$, noting that $|\xi\rangle$

lies in the code space. Denoting by $\{E_k\}$ the Kraus operators of $\mathcal{N}$, we see that $\langle\psi|E_k^\dagger E_k|\xi\rangle \propto \langle\psi|\xi\rangle = 0$ from the Knill-Laflamme conditions of the code, and therefore the conditions (151) are satisfied. Therefore, the following can be observed.

**Observation 1 (Clock state from a time-covariant code).**—Let $\Pi$ be the projector onto a code space that corrects errors of the error channel $\mathcal{N}$. Assume that the code is time covariant with respect to the Hamiltonian $H$. Then any logical state vector $|\psi\rangle$ and associated $|\xi\rangle = (H - \langle H \rangle)|\psi\rangle$ satisfy the conditions (151). Furthermore, if $\Pi$ defines a [[n, 1, d]] quantum code, then $|\psi\rangle$ and $P_{\psi}^{\perp}H|\psi\rangle$ define a metrological code of metrological distance $d$.

That is, any logical state of the code satisfies our Knill-Laflamme-like conditions for zero sensitivity loss. The sensitivity is maximized by picking the state with the largest energy variance.

If we are given an $\epsilon$-approximate quantum error-correcting code that is time-covariant, that is, if the error-correction procedure is allowed to fail with some probability $\epsilon > 0$, then we can still use a state lying in the code space to construct a clock state with little sensitivity loss. Approximate quantum error-correcting codes can be characterized by the fact that the channel that maps the code space to the environment, $\hat{\mathcal{N}}(\Pi(\cdot)\Pi)$, is close to a constant channel that always outputs a fixed state [61,62]. Specifically, $\hat{\mathcal{N}}(\Pi(X)\Pi) \approx \text{tr}(X) \tau_E$ for all $X$, for some fixed state $\tau_E$. If we pick a logical state vector $|\psi\rangle$ with nonzero energy variance, then we have that $\{\psi, \hat{H}\} = \Pi\{\psi, \hat{H}\}\Pi$ is a logical operator and therefore $\hat{\mathcal{N}}(\{\psi, \hat{H}\}) = \hat{\mathcal{N}}(\Pi\{\psi, \hat{H}\}\Pi) \approx \text{tr}(\{\psi, \hat{H}\}) \tau_E = 0$ since $\langle \hat{H} \rangle = 0$. Therefore, $\Delta F_{\text{Bob},t}$ in Eq. (55) satisfies $\Delta F_{\text{Bob},t} \approx 0$, and $F_{\text{Bob},t} \approx F_{\text{Alice},t} = 4\sigma_H^2$. This choice of a clock state is hence expected to lose little sensitivity under action of the noisy channel. Deriving a universal quantitative bound on $F_{\text{Bob},t}$ in this scenario in terms of $\epsilon$ does not appear easy. In such a scenario, a direct use of our uncertainty relation (49) [or of a corresponding bound such as Eq. (120)] seems likely to be the most straightforward way to obtain useful quantitative expressions for $F_{\text{Bob},t}$ in the case where the clock state is prepared using an approximate error-correcting code.

## F. Clock state for interacting many-body systems

Consider now an arbitrary interaction graph, where each vertex is associated with a single qubit [Fig. 9(b)] and consider the Hamiltonian

$$H = \frac{J}{2} \sum_{\langle i,j \rangle} (Z_i Z_j + s_x X_i X_j + s_y Y_i Y_j), \qquad (165)$$

where the sum ranges over all graph vertices $i, j$ that are connected by an edge, and where $s_x, s_y$ are arbitrary real coefficients. (In fact, the coefficients $s_x, s_y$ may also vary
for each pair of sites $i,j$, though we omit the dependence here for clarity.) We recover the Ising model with $s_x = s_y = 0$ and the Heisenberg model with $s_x = s_y = 1$. We denote by $m$ the number of edges in the graph, which is also the number of terms in the sum.

We define the clock-state vector $|\psi\rangle$ as follows. Denote by $|0^n\rangle$ and $|1^n\rangle$ the all-zero and the all-one state. Choose any bit string $\mathbf{x}$ and let $|\mathbf{x}\rangle$ be the corresponding spin configuration, where each bit corresponds to one of the qubit basis vectors on the corresponding vertex. We assume that $\mathbf{x}$ violates a number $c$ out of the $m$ possible ZZ-interaction terms, i.e., we denote by $c$ the number of pairs of bits in $\mathbf{x}$ that differ and that are connected by an edge in the graph. (It might not be possible to violate all the interaction terms simultaneously, as the graph might be frustrated.) An assumption we will need later is that the bit strings $0^n, 1^n$, and $\mathbf{x}$ all differ on at least four sites. Now define

$$|\psi\rangle = \frac{1}{2} [ |0^n\rangle + |1^n\rangle + |\mathbf{x}\rangle + |\tilde{\mathbf{x}}\rangle ], \quad (166) $$

where the bit string $\tilde{\mathbf{x}}$ is obtained by flipping all the bits of $\mathbf{x}$. We then have

$$H|\psi\rangle = \frac{J}{4}[m|0^n\rangle + m|1^n\rangle + (m-2c)|\mathbf{x}\rangle + (m-2c)|\tilde{\mathbf{x}}\rangle] \\ + \frac{J}{2}\sum_{\langle i,j \rangle} (s_x X_i X_j + s_y Y_i Y_j) |\psi\rangle. \quad (167) $$

The XX and YY operators applied on $|\psi\rangle$ generate terms associated with new bit strings where, each time, two bits are flipped and a possible phase is acquired. These new configurations are all orthogonal to $|0^n\rangle, |1^n\rangle, |\mathbf{x}\rangle$, and $|\tilde{\mathbf{x}}\rangle$ thanks to our assumption that the chosen configurations differ on at least four sites. So we have

$$\langle H \rangle_{\psi} = \frac{J}{8} [2 \times m + 2 \times (m - 2c)] = \frac{J}{2}(m - c). \quad (168)$$

With $\tilde{H} = H - J/2(m-c)$ and $|\xi\rangle = \tilde{H}|\psi\rangle$, we see that

$$|\xi\rangle = \frac{J}{4}[c|0^n\rangle + c|1^n\rangle - c|\mathbf{x}\rangle - c|\tilde{\mathbf{x}}\rangle] \\ + \frac{J}{2}\sum_{\langle i,j \rangle} (s_x X_i X_j + s_y Y_i Y_j) |\psi\rangle. \quad (169) $$

To check the zero sensitivity-loss conditions (148), we compute the following expression for any single-site operator $O_i$,

$$\begin{aligned} \langle \psi | O_i | \xi \rangle &= \frac{J}{8} [c \langle 0 | O_i | 0 \rangle + c \langle 1 | O_i | 1 \rangle \\ &\quad - c \langle x_i | O_i | x_i \rangle - c \langle \tilde{x}_i | O_i | \tilde{x}_i \rangle] \\ &\quad + \frac{J}{2} \sum_{\langle i,j \rangle} \langle \psi | O_i (s_x X_i X_j + s_y Y_i Y_j) | \psi \rangle \\ &= \frac{J}{8} [c \operatorname{tr}(O_i) - c \operatorname{tr}(O_j)] + 0 = 0, \quad (170) \end{aligned} $$

where $x_i$ (respectively $\tilde{x}_i$) denote the value of the $i$th bit in $\mathbf{x}$ (respectively, $\tilde{\mathbf{x}}$). The terms corresponding to XX and YY interactions vanish because all configurations $0^n, 1^n, \mathbf{x}$, and $\tilde{\mathbf{x}}$ differ on at least four sites, and XX and YY flip two bits of the basis vector state on which they are applied (with a possible phase). Therefore, the zero sensitivity-loss conditions (151) are satisfied, and the clock state can suffer a single located erasure while retaining full sensitivity.

The energy variance of the probe state is given by

$$\begin{aligned} \sigma_H^2 &= \langle \psi | (H - \langle H \rangle)^2 | \psi \rangle = \langle \xi | \xi \rangle \\ &= \frac{1}{4} J^2 c^2 + (\text{contrib. from XX/YY terms}). \end{aligned} \quad (171) $$

The contribution from XX and YY terms is zero if the configurations $0^n, 1^n, \mathbf{x}, \tilde{\mathbf{x}}$ all differ on at least five sites (or in the case of Ising interactions with $s_x = s_y = 0$).

The question of whether this achieves $n^2$ scaling depends on how we choose the graph and the string $\mathbf{x}$ to grow with $n$. In the case of a square lattice with nearest-neighbor interactions, we have that the number of edges scales like the number of vertices ($m \sim 2n$) and we can simultaneously violate all ZZ interaction terms by choosing an alternating configuration of 0's and 1's. In this case $\sigma_H^2 \sim J^2 n^2$, achieving Heisenberg scaling. For other graphs, the question of whether $\sigma_H^2 \sim n^2$ is determined by how the number of edges scales with the number of vertices in the graph, and how many of those ZZ-interaction terms can be simultaneously violated. If there is a linear relationship between these quantities then Heisenberg scaling is achieved, noting that only a single error at a known location can be incurred without sensitivity loss.

## G. Metrological codes from stabilizer codes

In this section, we present a general scheme to construct metrological codes based on the stabilizer formalism [63] and study some simple examples. We show that our construction is strictly more general than constructing time-covariant error-correcting codes. Our aim is to study and illustrate our general construction; the Hamiltonians in
our examples are not intended as practical schemes to be
engineered with near-term technology.

Consider the Pauli group $\mathcal{G}_n$ on $n$ qubits, defined as
comprising all tensor product operators on $n$ qubits of
single-site Pauli operators and the identity operator, with
all possible prefactors $\pm 1$ and $\pm i$ [63]. Consider a sub-
group $S \subset \mathcal{G}_n$ presented as $S = \langle S_1, \dots, S_\ell \rangle$ with indepen-
dent commuting generators $S_1, \dots, S_\ell$ such that $-1 \notin S$.
The normalizer of $S$ in $\mathcal{G}_n$ is $N(S) = \{E \in \mathcal{G}_n : \forall g \in S, E g E^{-1} \in S\}$. A state is said to be stabilized by $S$ if it
lies in the simultaneous $+1$ eigenspace of all $S \in S$; the
elements of $S$ are called stabilizers. The code space asso-
ciated with the Pauli stabilizer group $S$ is the subspace
spanned by all states that are stabilized by $S$. If $\mathcal{E} \subset \mathcal{G}_n$
is a set of error operators such that for all $E, E' \in \mathcal{E}$ either
$E'^\dagger E \notin N(S)$ or $E'^\dagger E$ lies in $S$ up to a phase, then a fun-
damental theorem of quantum error-correction states that
the subspace of all common $+1$ eigenstates of the opera-
tors $\{S_i\}$ forms a code space that can correct any error in
$\mathcal{E}$. One defines the distance $d$ of the code as the minimal
weight of an element in $N(S) \setminus S$, i.e., of a nontrivial log-
ical operation. Then, the code can correct any $t$ errors at
unknown locations as long as $2t + 1 \le d$.

As a simple example, consider the *n*-qubit GHZ state
vector $|\psi\rangle = [|\uparrow^n\rangle + |\downarrow^n\rangle]/\sqrt{2}$ and the Hamiltonian $H =$
$\sum_{j=1}^n Z_j$. We have $|\xi\rangle \propto [|\uparrow^n\rangle - |\downarrow^n\rangle]/\sqrt{2}$. Suppose our
error model consists of an arbitrary number of *X* errors.
From Eq. (151), since acting with *X* operators on $|\psi\rangle$ can
never generate any overlap with $|\xi\rangle$, we see that $|\psi\rangle$, $|\xi\rangle$
form a metrological code against any number of *X* errors.
We now present an overview of our procedure using this
example. In our procedure, we first find a set $\{S_i\}$ of inde-
pendent commuting Pauli operators that stabilize $|\psi\rangle$. We
fix a set of error operators $\mathcal{E}$, which we choose in our
example to consist of all *n*-qubit Pauli operators that are
a product of only 1's and *X*'s. Suppose that we are given
an operator *H* with the following property: for any opera-
tors *E*, *E'* ∈ *E*, there exists a *S* ∈ *S* such that {*H*, *S*} = 0
and [*E*†*E*, *S*] = 0. The state vector $|\psi\rangle$ is stabilized by
the choice of commuting Pauli operators $Z_1Z_2, Z_2Z_3, \dots$,
$Z_{n-1}Z_n, X^{\otimes n}$. Multiplying all but the last stabilizer by $X^{\otimes n}$,
we obtain the following choice of independent stabilizer
generators:

$$
\begin{equation}
\begin{split}
& -Y_1 Y_2 X_3 X_4 \dots X_n, \quad -X_1 Y_2 Y_3 X_4 \dots X_n, \quad \dots, \\
& \qquad \qquad \qquad \qquad -X_1 \dots X_{n-2} Y_{n-1} Y_n, \quad X^{\otimes n}.
\end{split}
\tag{172}
\end{equation}
$$

For any site *j*, the operator $Z_j$ anticommutes with all the
above stabilizer generators. Our structural constraint turns
out to apply in this case; it will be detailed later. Our
construction then implies that the pair $(|\psi\rangle, H|\psi\rangle)$ is a
metrological code. Here, $|\xi\rangle = H|\psi\rangle$ is in fact the state
vector that is stabilized by all the operators $\{-S_i\}$.

1. Statement of the construction

Our construction is given by the following theorem.

*Theorem 4 (Metrological codes from stabilizer states).*—
let $S \subset \mathcal{G}_n$ be an abelian subgroup of the Pauli group with
$-1 \notin S$, and let $|\psi\rangle$ be stabilized by $S$. Let $H$ be any Her-
mitian operator such that $H|\psi\rangle \neq 0$ and let $\mathcal{E} \subset \mathcal{G}_n$ be any
set of Pauli error operators. Assume that for all $E, E' \in \mathcal{E}$,
there exists $S \in \mathcal{S}$ such that $\{H, S\} = 0$ and $[E^\dagger E, S] = 0$.
Then $|\psi\rangle, H|\psi\rangle$ form a metrological code against $\mathcal{E}$.

We recall the definition of a metrological code as satisfying the condition (154). On the other hand, a defining property of a time-covariant code (recall definition in Sec. VII E) is that the Hamiltonian $H$ must be a logical operator, and thus, for a stabilizer code, must commute with all the stabilizers of the code. This is not in contradiction with Theorem 4 since the stabilizer group $S$ in the theorem is not necessarily that of the code for which $H$ is a logical operator. Below, we present examples of metrological codes; some are error-correcting time-covariant codes in disguise, yet others cannot be written as a time-covariant error-correcting code with similar distance as the metrological code.

*Proof*.—First, let $S_0 \in S$ with $\{H, S_0\} = 0$; such a stabilizer must exist from our assumption. We then have

$$
\langle \psi | H | \psi \rangle = \langle \psi | H S_0 | \psi \rangle = - \langle \psi | S_0 H | \psi \rangle = - \langle \psi | H | \psi \rangle \tag{173}
$$

and thus $\langle H \rangle_{\psi} = 0$. Let

$$
|\xi\rangle = H|\psi\rangle, \tag{174}
$$

with $\langle \xi | \psi \rangle = 0$ automatically satisfied. Let $E, E' \in \mathcal{E}$. We need to show that Eq. (154) holds. From our assumption there exists an $S \in S$ with $\{H, S\} = 0$ and $[E'^{\dagger} E, S] = 0$. We have

$$
\begin{align}
\langle \xi | E^{\dagger} E | \psi \rangle &= \langle \psi | H E^{\dagger} E S | \psi \rangle \nonumber \\
&= -\langle \psi | S H E^{\dagger} E | \psi \rangle \nonumber \\
&= -\langle \xi | E^{\dagger} E | \psi \rangle, \tag{175}
\end{align}
$$

and thus $\langle \xi | E^{\dagger} E | \psi \rangle = 0$, confirming that Eq. (154) holds and that $|\psi\rangle$, $|\xi\rangle$ indeed constitute a metrological code against $\mathcal{E}$.

In the remainder of this section we review some exam-
bles of codes resulting from the construction of Theorem
4. We begin by connecting our construction with error-
correcting codes in which the Hamiltonian is a nontriv-
ial logical operator, i.e., time-covariant error-correcting
codes. We present an example of a time-covariant code
based on the seven-qubit Steane code, and we then show
that all time-covariant error-correcting codes are special
cases of Theorem 4. We then show that there are metro-
logical codes that cannot be formulated in terms of a cor-
responding time-covariant error-correcting code; in other
words, there are schemes that enable the communication of a clock state through a noisy channel that achieve zero sensitivity loss without having to construct a full quantum error-correcting code.

## 2. Example based on the [[7, 1, 3]] Steane code

As an example of a time-covariant code, we consider an example deriving from the Steane stabilizer code [64]. The latter is given by the following generators and logical $\overline{X}$, $\overline{Z}$ operators:

$$ 
\begin{aligned}
\hat{S}_1 &= X_4 X_5 X_6 X_7, \\
\hat{S}_2 &= X_2 X_3 X_6 X_7, \\
\hat{S}_3 &= X_1 X_3 X_5 X_7, \\
\hat{S}_4 &= Z_4 Z_5 Z_6 Z_7, \\
\hat{S}_5 &= Z_2 Z_3 Z_6 Z_7, \\
\hat{S}_6 &= Z_1 Z_3 Z_5 Z_7, \\
\overline{X} &= X_1 X_2 X_3 X_4 X_5 X_6 X_7, \\
\overline{Z} &= Z_1 Z_2 Z_3 Z_4 Z_5 Z_6 Z_7.
\end{aligned} 
\quad (176) 
$$

Let $|\psi\rangle = |\mp\rangle$ be the state vector in the logical space associated with the +1 logical eigenspace of the $\overline{X}$ operator, and consider the Hamiltonian

$$ H = \overline{Z}\hat{S}_4 = Z_1 Z_2 Z_3. \quad (177) $$

The Hamiltonian is a logical operator, being stabilizer-equivalent to the logical $\overline{Z}$ operator, and rotates the state vector $|\psi\rangle$ to $|\xi\rangle = H|\psi\rangle = |\mp\rangle$. (The above choice of $H$ was preferred to the choice $H = \overline{Z}$ because it has lower weight.) The code is therefore time covariant with respect to the action of $H$, and we for this reason already know that it is a metrological code of metrological distance 3. To illustrate our construction, we explain how the same conclusion can be reached by applying Theorem 4. We now define a set of stabilizer generators that serve to define the state vector $|\psi\rangle$ of the resulting metrological code. The new stabilizer generators $\{S_i\}$ are obtained by multiplying each of the $\hat{S}_i$ by $\overline{X}$, all while including $\overline{X}$ itself, as

$$ 
\begin{aligned}
S_1 &= \overline{X}\hat{S}_1 = X_1 X_2 X_3, \\
S_2 &= \overline{X}\hat{S}_2 = X_1 X_4 X_5, \\
S_3 &= \overline{X}\hat{S}_3 = X_2 X_4 X_6, \\
S_4 &= \overline{X}\hat{S}_4 = X_1 X_2 X_3 Y_4 Y_5 Y_6 Y_7, \\
S_5 &= \overline{X}\hat{S}_5 = X_1 Y_2 Y_3 X_4 X_5 Y_6 Y_7, \\
S_6 &= \overline{X}\hat{S}_6 = Y_1 X_2 Y_3 X_4 Y_5 X_6 Y_7, \\
S_7 &= \overline{X} = X_1 X_2 X_3 X_4 X_5 X_6 X_7.
\end{aligned} 
\quad (178) 
$$

One can verify that $H$ anticommutes with each $S_i$ listed above. (For the application of Theorem 4, it is convenient to use a choice of stabilizer generators that anticommute with $H$.) Let $\mathcal{E}$ be the set of all single-site operators. For any $E, E' \in \mathcal{E}$, we will show that there is a $S \in \mathcal{S}$ with $\{H, S\} = 0$ and $[E^\dagger E, S] = 0$. If one of the $S_i$ has support outside of that of $E^\dagger E$, it will do the job. Alternatively, any product of an odd number of the $S_i$ will also do, for instance,

$$ 
\begin{aligned}
S_1 S_2 S_3 &= X_3 X_5 X_6, \\
S_1 S_2 S_7 &= X_1 X_6 X_7, \\
S_2 S_3 S_7 &= X_3 X_4 X_7.
\end{aligned} 
\quad (179) 
$$

One can verify that for any two among the seven sites, at least one operator among $S_1, S_2, S_3, S_1 S_2 S_3, S_1 S_2 S_7, S_2 S_3 S_7$ has its support outside of those two sites. Since these operators all anticommute with $H$, we have that for all $E, E' \in \mathcal{E}$, there is a $S \in \{S_1, \dots, S_7\}$ such that $[E^\dagger E, S] = 0$ and $\{S, H\} = 0$. From Theorem 4, we see that $|\psi\rangle$ and $|\xi\rangle = H|\psi\rangle$ must form a metrological code against $\mathcal{E}$, and is therefore a metrological code with metrological distance 3.

## 3. Time-covariant codes

In this paragraph, we show that the assumptions of Theorem 4 are in fact always satisfied for time-covariant stabilizer codes like the seven-qubit Steane code example above.

Let $\hat{S} = \langle \hat{S}_1, \dots, \hat{S}_\ell \rangle$ be a stabilizer code with a non-trivial logical operator $\overline{Z}$. Let $\overline{X}$ be a logical operator that anticommutes with $\overline{Z}$, and define the stabilizer group $S = \langle \overline{X}\hat{S}_1, \overline{X}\hat{S}_2, \dots, \overline{X}\hat{S}_\ell, \overline{X}\rangle$. Observe that $\overline{Z}$ anticommutes with all the chosen generators for $S$. (Such an operator $\overline{X}$ must always exist, cf. e.g., [63, Proposition 10.4].)

We show the following: for any Pauli operator $A \notin N(\hat{S}) \setminus \hat{S}$, there exists $S \in \mathcal{S}$ such that $[S, A] = 0$ and $\{S, \overline{Z}\} = 0$. This property implies that for a given set of errors $\mathcal{E}$ that are correctable for $\hat{S}$, i.e., if we have $E^\dagger E \notin N(\hat{S}) \setminus \hat{S}$ for all $E, E' \in \mathcal{E}$, then the conditions of Theorem 4 are satisfied, where the Hamiltonian is $H = \overline{Z}$.

Suppose first that $A \in \hat{S} \subset \mathcal{S}$. Then $A$ commutes with all stabilizers in $\mathcal{S}$, including $\overline{X}$, which anticommutes with $\overline{Z}$. Now suppose that $A \in \mathcal{G}_n$ and $A \notin N(\hat{S})$, i.e., there is a $\hat{S} \in \hat{S}$ with $\{A, \hat{S}\} = 0$. If $[A, \overline{X}] = 0$, then the choice $S = \overline{X} \in \mathcal{S}$ satisfies $[S, A] = 0$ and $\{S, \overline{Z}\} = 0$. If, instead, we have $\{A, \overline{X}\} = 0$, we can set $S = \overline{X}\hat{S}$ to find $S\overline{Z} = \overline{X}\hat{S}\overline{Z} = \overline{X}\overline{Z}\hat{S} = -\overline{Z}\overline{X}\hat{S} = -\overline{Z}\hat{S}$ and $AS = A\overline{X}\hat{S} = -\overline{X}A\hat{S} = \overline{X}\hat{S}A = SA$, and thus $\{S, \overline{Z}\} = 0$ and $[S, A] = 0$ as required.
### 4. Example: a [[4, 2, 2]] code state with an auxiliary qubit

Whereas in the earlier seven-qubit Steane code example the state vectors $|\psi\rangle$ and $|\xi\rangle$ both lie within a subspace of a distance $d = 3$ code, the following example illustrates a situation in which $|\psi\rangle$ and $|\xi\rangle$ cannot be contained in a code space that can correct the same errors against which the states form a metrological code. In other words, Theorem 4 can be used to construct metrological codes that cannot be formulated as time-covariant quantum error-correcting codes with respect to the same errors. Consider the five-qubit Pauli operators

$$ 
\begin{aligned}
S_1 &= X_1 X_2, \\
S_2 &= X_3 X_4, \\
S_3 &= X_1 X_3, \\
S_4 &= X_5, \\
S_5 &= Z_1 Z_2 Z_3 Z_4.
\end{aligned} 
\quad (180) 
$$

We can see that the stabilizer group $S = \langle S_1, \dots, S_5 \rangle$ is generated by

(i) the stabilizers for the [[4, 2, 2]] code on the first four qubits ($Z_1Z_2Z_3Z_4 = S_5$ and $X_1X_2X_3X_4 = S_1S_2$);

(ii) the logical $X$ operators of the first and second logical qubits of that [[4, 2, 2]] code ($X_1X_3 = S_3$ and $X_1X_2 = S_1$); and

(iii) an independent stabilizer fixing the state of the fifth qubit ($X_5 = S_4$).

We choose the Hamiltonian

$$ H = Y_1 Z_4 Y_5. \quad (181) $$

The Hamiltonian can be written as a product of three terms: a logical $Z$ operator on both logical qubits of the [[4, 2, 2]] code ($Z_1Z_4$), a $Y$ operation on the fifth physical qubit, and a single $X_1$ on the first physical qubit. The Hamiltonian is not a logical operator of the [[4, 2, 2]] code. Also, a suitable permutation of the qubits would make $H$ geometrically local, should this property be desired.

We can verify that $H$ anticommutes with each of the stabilizers $S_1, \dots, S_5$. Furthermore, for any two sites $i, j$, one of the $S_k$ acts as the identity on the sites $i, j$; therefore, for any two-site operator $A$, there always exists a stabilizer $S$ with $[S, A] = 0$ and $\{S, H\} = 0$. We can apply Theorem 4 to deduce that $|\psi\rangle, |\xi\rangle$ define a distance-3 metrological code.

The state vector $|\psi\rangle$ can be expressed in terms of the logical $+1$ $X$ eigenvectors $|++\rangle$ of the [[4, 2, 2]] code, and

in terms of the $|\pm\rangle_i$ physical state vectors, as

$$ 
\begin{aligned}
|\psi\rangle &= |\overline{++}\rangle_{1234} \otimes |+\rangle_5 \\
&= \frac{1}{\sqrt{2}} (|++\overline{++}\rangle + |\overline{---}++\rangle).
\end{aligned} 
\quad (182) 
$$

Recalling $Y|\pm\rangle = \mp i|\mp\rangle$ and $Z|\pm\rangle = |\mp\rangle$, we find

$$ 
\begin{aligned}
|\xi\rangle = H|\psi\rangle &= \frac{1}{\sqrt{2}} (-|-++--\rangle + |+---+-\rangle) \\
&= \frac{1}{\sqrt{2}} (|+---+\rangle - |-++-\rangle) \otimes |-\rangle_5.
\end{aligned} 
\quad (183) 
$$

We see that

$$ \langle\psi|X_5|\psi\rangle = 1, \quad \langle\xi|X_5|\xi\rangle = -1, \quad (184) $$

so it is not possible for $|\psi\rangle, |\xi\rangle$ to lie in the code space of a distance $d = 3$ quantum error-correcting code.

This example shows that metrological codes are a class of codes that is broader than traditional error-correction codes as there are certain errors that the former does not have to completely correct. Metrological codes might therefore offer additional possibilities to find noise-resilient schemes for communicating clock states across a noise channel.

### 5. Metrological toric code

A further example application of Theorem 4 is based on Kitaev's toric code [65,66]. We consider a two-dimensional square lattice of dimension $L \times L$ that wraps around a torus. We define star operators $A_x$ and plaquette operators $B_x$ as depicted in Fig. 10(a), where $x$ ranges over all pairs of the lattice coordinates.

First, we can always use the toric code to form a time-covariant code, by choosing a state vector $|\psi\rangle$ in the code space (for instance, $|\overline{++}\rangle$), and choosing the Hamiltonian to be a logical operator (for instance, $\overline{Z_1 + Z_2}$). This code being by construction time covariant, it is necessarily a metrological code with distance equal to the lattice side length $L$.

For the sake of the example, we construct here a metrological code from the toric code that cannot be written as a time-covariant error-correcting code of similar distance. Our example is meant to (i) illustrate our construction as combining states that lie either in the simultaneous $+1$ or simultaneous $-1$ eigenspaces of all the stabilizer generators of some given stabilizer code, (ii) furnish another example of a metrological code that cannot be phrased in terms of a time-covariant error-correcting code with similar distance, and (iii) illustrate how a metrological code can be a terrible quantum error-correcting code—any single
<figure><img src="image_12.png" /></figure>

FIG. 10. Metrological code based on the toric code. (a) Star ($A_x$) and plaquette ($B_x$) operators generate the stabilizer group of the toric code, where $x$ is a pair of integer coordinates on the two-dimensional lattice. Two encoded logical qubits have logical Pauli operators $\bar{X}_1, \bar{Z}_1, \bar{X}_2, \bar{Z}_2$ corresponding to strings of physical Pauli $X$ or $Z$ operators that wrap around the torus. (b) In our metrological code example based on the toric code, we map a state from the toric code to a state of a related code, which we call the *antitoric code*. The antitoric code is the subspace stabilized by all the operators $\{-A_x\}$ and $\{-B_x\}$. Assuming the lattice side length is even, the depicted operator $H$ anticommutees with all star and plaquette operators, meaning that it maps a logical state of the toric code to a logical state of the antitoric code. Picking $|\psi\rangle$ in the code space of the toric code and choosing the depicted operator $H$ as the corresponding Hamiltonian yields an example of a distance-$\Omega(L^2)$ metrological code. Interestingly, this metrological code cannot be phrased in terms of a time-covariant error-correcting code of similar distance, since $|\psi\rangle$ and $H|\psi\rangle$ can be distinguished by measuring a single star or plaquette operator. This example shows that there are additional possible schemes for sending a clock state through a noisy channel without any sensitivity loss, without resorting to a time-covariant quantum error-correcting code.

plaque or star operator acts nontrivially on the subspace spanned by the state and its time-evolution state. Our example is more of a conceptual illustration than a practical proposal, as it requires a Hamiltonian that is highly nonlocal.

To better explain our example, we first define the *antitoric code* as the code whose code space is stabilized by all negative star $-A_x$ and negative plaquette operators $-B_x$. Being equivalent to the standard toric code, the antitoric code also has distance $L$ and we can see it also has the logical operators $\bar{X}_1, \bar{X}_2, \bar{Z}_1, \bar{Z}_2$ defined as for the toric code.

As the state vector $|\psi\rangle$ of our metrological code, we simply choose a logical state vector of the toric code; we can conventionally fix it to be $|00\rangle_{\text{toric}}$ stabilized by $\bar{Z}_1, \bar{Z}_2$ along with all the toric code stabilizers $\{A_x\}$ and $\{B_x\}$. For the Hamiltonian we choose an operator $H$ that anticommutees with all star and all plaquette operators. Such an operator is depicted in Fig. 10(b); we assume for convenience that $L$ is even. The operator $H$ has the property that

it maps a code word of the toric code (i.e., a state vector $|\psi\rangle$ satisfying $A_x|\psi\rangle = |\psi\rangle = B_x|\psi\rangle$) to a code word of the *antitoric code* (we have $A_x H|\psi\rangle = -H A_x|\psi\rangle = -|\psi\rangle$ and similarly for $B_x$). We can verify that the assumptions of Theorem 4 are satisfied. The operator $H$ anticommutees with our choice of stabilizer generators for $|\psi\rangle$. Also, for any operator $O$ of weight $< L^2/4$, there must be a star or plaquette operator that has disjoint support with, and therefore commutes with, $O$. [Indeed, there are $(L/2)^2$ disjoint plaquette operators that cover all qubits; an operator that has overlapping support with all plaquette operators must therefore have support on one qubit in each plaquette. The bound can presumably be improved by accounting for the star operators as well.] As a consequence of Theorem 4, the state vectors $(|\psi\rangle, |\xi\rangle = H|\psi\rangle)$ form a metrological code of distance $L^2/4$.

Is the space spanned by $(|\psi\rangle, |\xi\rangle)$ secretly a code space of a similar-distance code in which $H$ acts as a logical operator? We can rule out this possibility because the state vectors $|\psi\rangle$ and $|\xi\rangle$ can easily be distinguished by measuring any single star or plaquette operator, recalling that $A_x|\psi\rangle = B_x|\psi\rangle = |\psi\rangle$ but that $A_x|\xi\rangle = B_x|\xi\rangle = -|\xi\rangle$. The environment only has to measure a weight-4 operator to determine whether $|\psi\rangle$ or $|\xi\rangle$ was encoded.

## 6. Simultaneous +1 eigenspace and simultaneous −1 eigenspace of stabilizers

The intuition behind the construction in Theorem 4 is that if we can choose $|\psi\rangle$ to be stabilized by $S = \langle S_1, \dots, S_\ell \rangle$, then we might want to pick $|\xi\rangle$ to be stabilized by the closely related stabilizer group $S' = \langle -S_1, \dots, -S_\ell \rangle$. This idea was already illustrated by the example above based on the toric code, where the Hamiltonian maps a code word of the toric code to a code word of the antitoric code. We now show in general that such a construction is a special case of Theorem 4.

Let $S = \langle S_1, \dots, S_\ell \rangle$ be a subgroup of the Pauli group with $-1 \notin S$, where $S_1, \dots, S_\ell$ are a choice of independent commuting stabilizer generators. Let $S' = \langle -S_1, \dots, -S_\ell \rangle$. Let $\mathcal{E}$ denote any set of Pauli operators with the following property: for any $E, E' \in \mathcal{E}$, there exists $S \in S$ such that $-S \in S'$ and such that $[E^\dagger E, S] = 0$.

The two stabilizer groups $S, S'$ share many stabilizers, including $S_1S_2, S_1S_3, \dots, S_1S_\ell$. We can pick a Pauli operator $H$ such that $H$ anticommutees with $S_1$ and such that $H$ commutes with each of the operators $S_1S_2, S_1S_3, \dots, S_1S_\ell$ (see, e.g., Ref. [63, Proposition 10.4]). Observe that for all $i = 2, \dots, \ell$, we have

$$HS_i = HS_1^2S_i = -S_1HS_1S_i = -S_1^2S_iH = -S_iH \quad (185)$$

and thus we have that $\{H, S_i\} = 0$ for all $i = 1, \dots, \ell$. Suppose $|\psi\rangle$ is stabilized by $S$. Then $H|\psi\rangle$ is stabilized by $S'$,
since

$$S_i H |\psi\rangle = -HS_i |\psi\rangle = -H |\psi\rangle. \quad (186)$$

Furthermore, supposing $E, E' \in \mathcal{E}$, by assumption we have $S \in \mathcal{S}$ such that $-S \in \mathcal{S}'$ and such that $[E^{\dagger}E, S] = 0$. We can write $S = S_{i_1}S_{i_2}\cdots S_{i_m}$ in terms of our choice of independent generators $S_i$ above. Writing $-S = (-1)^{m+1}(-S_{i_1})(-S_{i_2})\cdots(-S_{i_m})$, we see that $m$ must be odd, as otherwise, we would have

$$-S = (-1)(-S_{i_1})(-S_{i_2})\cdots(-S_{i_m}) \notin \mathcal{S}'. \quad (187)$$

This observation implies that $\{H, S\} = 0$, because we can anticommute $H$ through the product of an odd number of $S_i$'s. Therefore there exists $S \in \mathcal{S}$ such that $[E^{\dagger}E, S] = 0$ and $\{H, S\} = 0$. At this point, all the assumptions of Theorem 4 are satisfied, implying that $|\psi\rangle, H|\psi\rangle$ form a metrological code that can protect against the error set $\mathcal{E}$.

## H. Further examples of metrological codes

We now present two additional examples of state vectors $|\psi\rangle, |\xi\rangle$ that satisfy the zero sensitivity loss conditions (152). These metrological codes serve to illustrate the sense in which the conditions (152) are weaker than the conditions for quantum error correction.

### 1. Single-qubit subject to complete X/Y dephasing

Consider the qubit example studied in Sec. IV B, where the clock state vector $|+\rangle$ evolves according to $H = \omega Z/2$ and is exposed to complete dephasing along the X axis around a given time $t_0$. From Eq. (90) we immediately see that the zero sensitivity loss condition (148) is satisfied for all $t_0$. In this setting, the clock state loses no sensitivity after complete dephasing in the X axis for any $t_0$, with the exception of possible discrete points where the rank of $\rho_B(t)$ changes (see Sec. IV B).

Alternatively one could also check the form (151) of the zero sensitivity-loss conditions. For any $t_0$, we have from Eq. (29) that

$$\begin{aligned} |\xi(t_0)\rangle &= H|\psi(t_0)\rangle \\ &= \frac{\omega}{2} \left[ \cos\left(\frac{\omega t_0}{2}\right) |-\rangle - i \sin\left(\frac{\omega t_0}{2}\right) |+\rangle \right]. \end{aligned} \quad (188)$$

At this point, we can compute

$$\begin{aligned} \langle\psi|\cdots|+\rangle\langle+|\xi\rangle + \langle\xi|\cdots|+\rangle\langle+|\psi\rangle &= \frac{\omega}{2} \left[ -i \cos\left(\frac{\omega t_0}{2}\right) \sin\left(\frac{\omega t_0}{2}\right) + i \sin\left(\frac{\omega t_0}{2}\right) \cos\left(\frac{\omega t_0}{2}\right) \right] \\ &= 0, \end{aligned} \quad (189)$$

and similarly for $\langle\psi||-\rangle\langle-||\xi\rangle + \langle\xi||-\rangle\langle-||\psi\rangle = 0$, showing that Eq. (151) are satisfied for all $t_0$.

An interesting aspect of this example is that there exists no recovery operation that can restore the noiseless clock state vector $|\psi(t_0 + dt)\rangle$ accurately to first order in $dt$. Let us consider for simplicity the point $t_0 = \pi/(2\omega)$. Using Eqs. (29), (31), and (89), we have at that point

$$\begin{aligned} \psi(t_0) &= |+i\rangle\langle+i|, & \rho_B(t_0) &= \frac{1}{2}, \\ \partial_t \psi(t_0) &= -\frac{\omega}{2}X, & D_X(\partial_t \psi(t_0)) &= -\frac{\omega}{2}X, \end{aligned} \quad (190)$$

where $|+i\rangle := [|\uparrow\rangle + i|\downarrow\rangle]/\sqrt{2}$. We seek a completely positive, trace-preserving map $\text{Rec}$ such that $\text{Rec}(\rho_B(t_0 + dt)) = \psi(t_0 + dt) + O(dt^2)$, which means that

$$\text{Rec}\left(\frac{1}{2}\right) = |+i\rangle\langle+i|, \quad \text{Rec}(X) = X. \quad (191)$$

There is no completely positive map that satisfies these constraints. If there was such a $\text{Rec}$ map, then we would have $\text{Rec}(|+\rangle\langle+|) = \text{Rec}(1/2) + \text{Rec}(X/2) = |+i\rangle\langle+i| + X/2 = 1/2 + Y/2 + X/2$. One can easily check that the final expression has a negative eigenvalue, contradicting the requirement that $\text{Rec}$ be completely positive. We conclude that in general, a metrological code does not necessarily come with a recovery operation that enables an agent to recover the noiseless clock state, even if the agent can sense the parameter to the same precision as before the application of the noise.

### 2. A superposition of a simple state and a generic pure state

Consider a one-dimensional chain of $n$ qubits. Consider a generic pure state vector $|\chi\rangle$, chosen, for instance, randomly from the Haar measure on the $n$-qubit system. For a given $d_m > 0$, let us perturb the state vector $|\chi\rangle$ to $|\tilde{\chi}\rangle$ by projecting it onto the subspace of all computational basis states that do not contain fewer than a number $d_m$ of 1's,

$$|\tilde{\chi}\rangle = \tilde{\Pi}|\chi\rangle, \quad \tilde{\Pi} = \prod_{|\mathbf{x}| \ge d_m} |\mathbf{x}\rangle\langle\mathbf{x}|. \quad (192)$$

If $|\chi\rangle$ is generic in some suitable sense (e.g., chosen Haar randomly), then $|\tilde{\chi}\rangle \approx |\chi\rangle$ and $|||\tilde{\chi}\rangle|| \approx 1$. The present example metrological code is constructed by picking $|\psi\rangle = |0^n\rangle$, which is the computational basis all-zero state, and $|\xi\rangle = |||\tilde{\chi}\rangle||^{-1} |\tilde{\chi}\rangle \approx |\chi\rangle$.

We proceed to check that the zero sensitivity loss conditions (148) are satisfied as long as operators of the form $E_k^\dagger E_k$ have weight at most $d_m - 1$. If $\mu \subset \{1, \dots, n\}$ denotes a subset of at most $|\mu| = d_m - 1$ systems, then the reduced operator of $|\tilde{\chi}\rangle\langle 0^n|$ on the sites labeled by $\mu$ can
be written as

$$ \mathrm{tr}_{\setminus \mu} [ |\tilde{\chi}\rangle \langle 0^n| ] = (1_{d_m} \otimes \langle 0^{n-d_m} |) |\tilde{\chi}\rangle \langle 0^{d_m} | = 0, \quad (193) $$

because $|\tilde{\chi}\rangle$ has no overlap with bit strings that have $(n-d_m)$ or more zeros. Hence,

$$ \mathrm{tr}_{\setminus \mu} (|\psi\rangle\langle\xi| + |\xi\rangle\langle\psi|) = 0. \quad (194) $$

For any operator $O$ of weight $\mathrm{wtg}(O) < d_m$, the condition (155) is thus satisfied and $|\psi\rangle, |\xi\rangle$ form a metrological code of distance $d_m$.

An interesting observation is that this code does not form a quantum error-correcting code in the usual sense. The reason is that the environment, by receiving a few sites, can tell the difference between whether the state vector $|\psi\rangle$ or the state vector $|\xi\rangle$ was prepared. More precisely, the environment can test whether the received qubits are all in the state vector $|0\rangle$. If this is the case, it is much more likely that the original state vector was $|\psi\rangle$ and not $|\xi\rangle$, as long as $\mathrm{tr}_{\setminus \nu}(\tilde{\chi})$ is sufficiently distinct from $|0\rangle\langle 0|_{\nu}$ (which is the case for a Haar-random state).

For some choices of $|\chi\rangle$, the metrological code can be interpreted as a quantum error-correcting code that protects only against certain types of errors. For instance, we can choose $|\chi\rangle = |1\rangle^{\otimes n}$ in the above and our conclusions still hold; this choice corresponds to a classical repetition code that can correct bit flips but which is vulnerable to phase flips.

Yet, there are choices of $|\chi\rangle$ for which this interpretation appears more problematic. Consider for instance the choice $|\chi\rangle = |+\rangle^{\otimes n}$. From the above argument we have that $|\psi\rangle = |0\rangle^{\otimes n}$ and $|\xi\rangle \approx |+\rangle^{\otimes n}$ form again a metrological code. Again, the environment can distinguish $|\psi\rangle$ from $|\xi\rangle$ with access only to a few sites. Here, the environment can use either an $X$ or a $Z$ measurement to (imperfectly) distinguish between the two state vectors $|0\rangle$ and $|+\rangle$. It is hence not obvious how to interpret this code as an error-correcting code that is tailored to biased noise.

While this example might illustrate the conceptual differences between quantum error-correcting codes and metrological codes, we expect this construction of a metrological code to be of limited practical use as it would require a Hamiltonian that is extremely nonlocal.

## VIII. MANY-BODY SYSTEM SUBJECT TO IID AMPLITUDE DAMPING NOISE

In this section we consider a system consisting of $n$ spin-1/2 particles evolving under a many-body Hamiltonian $H$ that is either noninteracting or that has Ising interaction terms. The system is exposed to IID amplitude damping noise. First, we consider a noninteracting Hamiltonian with an on-site magnetic field, and in the second part of this section we consider a Hamiltonian with Ising interactions.

We consider an IID amplitude damping noise model, meaning that each site is independently exposed to the noisy channel

$$ \begin{aligned} \mathcal{N}_{\mathrm{AD}}^{(p)}(\cdot) &= E_0^{(p)}(\cdot) E_0^{(p)\dagger} + E_1^{(p)}(\cdot) E_1^{(p)\dagger}, \\ E_0^{(p)} &= \begin{pmatrix} \sqrt{1-p} & 0 \\ 0 & 1 \end{pmatrix}, \quad E_1^{(p)} = \begin{pmatrix} 0 & 0 \\ \sqrt{p} & 0 \end{pmatrix}, \end{aligned} \quad (195) $$

sticking to the convention that the first basis vector is $|\uparrow\rangle$ and the second one is $|\downarrow\rangle$. The amplitude-damping noise is often also called the spontaneous emission channel.

As in Fig. 1, the system is initialized in a state $|\psi_{\mathrm{init}}\rangle$ and evolves according to $H$; at time $t_0$ we apply the noisy channel $[\mathcal{N}_{\mathrm{AD}}^{(p)}]^{\otimes n}$ to obtain Bob's state. We seek to characterize the Fisher information of Bob's state with respect to time.

In this section, we present simple numerical computations of the upper bound (118) for IID amplitude damping noise for different clock states. In the first part of this section, we suppose the spins are exposed to a uniform external magnetic field aligned along the $Z$ axis. We present numerical calculations of Bob's Fisher information and our lower bound (118) for a choice of clock states, and we numerically optimize the initial state to achieve better output sensitivity. In the second part of this section, we place the spins on a 1D chain with strong Ising interactions. We present numerical calculations of Bob's Fisher information and our lower bound (118) for a choice of clock states; we numerically show that the sensitivity loss for the metrological code state given in Eq. (166) is suppressed to first order in the amplitude damping parameter.

### A. Noninteracting Hamiltonians

The system of $n$ spins is assumed to evolve under the Hamiltonian

$$ H = \sum_{i=1}^{n} \frac{\omega}{2} Z_i. \quad (196) $$

We compute Bob's Fisher information with respect to time of a selection of states after exposure to the channel $\mathcal{N} = [\mathcal{N}_{\mathrm{AD}}^{(p)}]^{\otimes n}$. First we consider the GHZ state, which has the optimal sensitivity if no noise is present:

$$ |\psi_{\mathrm{GHZ}}\rangle = \frac{1}{\sqrt{2}} [ |\uparrow\uparrow\cdots\uparrow\rangle + |\downarrow\downarrow\cdots\downarrow\rangle ]. \quad (197) $$

The GHZ state satisfies

$$ F_{\mathrm{Alice},t}[\psi_{\mathrm{GHZ}}] = 4\langle H^2\rangle_{\mathrm{GHZ}} = n^2\omega^2. \quad (198) $$

We can also consider the product state vector of all spins pointing in the $+X$ direction,

$$ |\psi_{+}\rangle = |+\rangle^{\otimes n} = \frac{1}{\sqrt{2}^n} [ |\uparrow\rangle + |\downarrow\rangle ]^{\otimes n}. \quad (199) $$
Then

$$F_{\text{Alice},t}[\psi_+] = 4\langle H^2 \rangle_+ = n\omega^2. \qquad (200)$$

Our upper bound (118) on Bob's Fisher information for these states is presented for $n = 12$ and for $n = 50$ in Fig. 11. In Figs. 11(a) and 11(b) are also depicted an ad hoc lower bound for the state vectors $|\psi_{\text{GHZ}}\rangle$ and $|+^n\rangle$ for the same values of $n$ and $\omega$. We can see that for our choice of the amplitude damping noise model and at least for our choice of states, the upper bound on $F_{\text{Bob},t}$ provided by Eq. (118) is reasonably tight for $k = n$. The inset of Fig. 11(a) depicts the same bound for different choices of the value $k = 1, 4, 8, 12$. Recall that the bound includes a projection onto Eve's subspace associated with error operators of weight at most $k$. The probability that this projection fails is the total probability of observing an error with weight greater than $k$; this probability is of the order of $p^{k+1}$. In the inset of Fig. 11(a), for $k = 1, 4, 8$ we display gray lines identifying the values of $p$ for which $p^{k+1} = 10^{-3}$. Values of $p$ beyond the corresponding gray line represent

situations in which the projection is expected to fail with probability greater than the order of approximately $10^{-3}$. Here, we see that our bound is indeed reasonably tight up until the corresponding value of $p$. In the inset of Fig. 11(b), we determine more precisely the total weight of the events neglected by ignoring Kraus operators of weight greater than $k$. Namely, for $k = 1, 2, 5, 10$, we compute the smallest value of $p$ for which $\sum_{|x|>k} \text{tr}(E_x^\dagger E_x |\psi_{\text{GHZ}}\rangle\langle\psi_{\text{GHZ}}|) > 10^{-3}$. We see that these values of $p$ correspond approximately to where our upper bound (118) fails to accurately predict the value of the quantum Fisher information on the state vector $|\psi_{\text{GHZ}}\rangle$.

We can ask, which state vector $|\psi\rangle$ has the best sensitivity after application of the noisy channel for a given value of $p$? Here we use the understanding brought by our main Fisher information trade-off relation. In this case, Eve receives any photons emitted by spontaneous emission, which tell her exactly which sites suffered a decay. As a consequence, if Eve observes a number $k$ of photons, then she can safely guess that the energy of Alice's state must have been at least the energy corresponding to $k$ excitations. Eve has therefore obtained information about

<figure><img src="image_13.png" /><figcaption>FIG. 11. Quantum Fisher information of a system of $n$ spin-1/2 particles with the Hamiltonian $H = \sum_i \omega Z_i / 2$ (with $\omega = 2$) after the application of an amplitude damping channel of parameter $p$ on all sites. (a) Here $n = 12$. Solid lines depict our upper bound (118) with $k = n = 12$ for the state vectors $|\psi_{\text{GHZ}}\rangle$ (red), $|+^n\rangle$ (blue), and $|\psi_{\text{unif}}\rangle$ (green), which are defined in the main text, as well as for the family of states corresponding to an even superposition of the most excited state and a symmetric state (Dicke state) $|h_j^n\rangle$ with a fixed number $n-j$ of excitations (shades of orange) with $j = 3, 6, 9$. Dotted lines are corresponding ad hoc lower bounds for the state vectors $|\psi_{\text{GHZ}}\rangle$ and $|+^n\rangle$ (see main text). Dash-dotted lines are the corresponding exact values of the quantum Fisher information, which can still be directly computed for $n = 12$. The curves corresponding to the superpositions of pairs of Dicke states illustrate situations where the upper bound is not tight. The inset depicts our upper bound (118) for different values of $k = 1, 4, 8, 12$ for the state vectors $|\psi_{\text{GHZ}}\rangle$, $|+^n\rangle$ and $|\psi_{\text{unif}}\rangle$. The value of $k$ corresponds to a projection onto the subspace on Eve's system associated with errors of weight less than or equal to $k$, which is included in our bound (118). The three vertical gray lines indicate values of $p$ for which $p^{k+1} = 10^{-3}$ for $k = 1, 4, 8$; these lines roughly indicate the values of $p$ beyond which the this projection is expected to fail with a probability exceeding approximately $10^{-3}$. Indeed, for our choice of noisy channel and states, our bound (118) for each $k$ is reasonably tight up until values of $p$ for which $p^{k+1}$ is no longer negligibly small. (b) The same computations are repeated for $n = 50$ and $\omega = 2$. Solid lines depict our upper bound with $k = n$ for the same states as in (a), and dotted lines depict an ad hoc lower bound for $|+^n\rangle$ and $|\psi_{\text{GHZ}}\rangle$. The red vertical lines in the inset depict values for $p$ for which the total weight of the IID amplitude-damping Kraus operators $E_x$ with $|x| > k$ exceeds $10^{-3}$ for $k = 1, 2, 5, 10$ when applied onto the GHZ input state vector $|\psi_{\text{GHZ}}\rangle$. These values of $p$ are where our upper bound (118) for the corresponding $k$ value is expected to no longer be accurate.</figcaption></figure>