Precision bounds for characterising quantum measurements

Domain: physics
Published: 2026-01-20
Editorial outcome: accept

Abstract
Quantum measurements, alongside quantum states and processes, form a cornerstone of quantum information processing. However, unlike states and processes, their efficient characterisation remains relatively unexplored. We resolve this asymmetry by introducing a comprehensive framework for efficient detector estimation that reveals the fundamental limits to extractable parameter information and errors arising in detector analysis-the detector quantum Fisher information. Our development eliminates the need to optimise for the best probe state, while highlighting aspects of detector analysis that fundamentally differ from quantum state estimation. Through proofs, examples and experimental validation, we demonstrate the relevance and robustness of our proposal for current quantum detector technologies. By formalising a dual perspective to state estimation, our framework completes and connects the triad of efficient state, process, and detector tomography, advancing quantum information theory with broader implications for emerging technologies reliant on precisely calibrated measurements.

Introduction
Measurements hold a special place in quantum mechanics, bridging quantum states and real-world classical observations. This probabilistic transition from the quantum to the classical, originally postulated by the Born rule1, typically washes out key quantum features like superposition and entanglement, thereby limiting the information that can be extracted from a quantum system2. The modern theory of quantum measurements has evolved well beyond the observe-and-collapse paradigm3–5, encompassing generalised measurements such as weak measurements that blur the line between observation and interaction5–9. Experimental practice has lent further credence to the utility of these generalised measurements5,8–10, for instance, by showing the advantage of entangling measurements for the precise characterisation of quantum states11–15.

The measurement of a quantum state constitutes one part of the triad of quantum states, processes and detectors that forms the basis of any quantum information protocol16. From quantum estimation theory, the Quantum Fisher Information (QFI) provides a metric for the distinguishability of parametrised quantum states and processes, leading to bounds on estimation errors known as quantum Cramér-Rao bounds (CRBs)17,18. Strikingly, despite the fundamental and practical importance of quantum detectors, whether similar bounds exist for general quantum detectors remains an open question. In this paper, we answer this question. More precisely, we ask whether information-theoretic precision bounds exist for the accurate estimation of detector parameters.

The aforementioned disparity is particularly surprising given the dual nature of states and measurements in quantum theory2—a symmetry suggesting their informational properties should be balanced. However, while provably optimal state estimation protocols are known, existing approaches to detector estimation often overlook the efficiency of the process19,20, and thus fail to inform optimal estimation strategies. Optimal detector estimation strategies are essential to extract maximal benefit from future quantum devices. For example, precise characterisations of state preparation and measurement (SPAM) errors on quantum computing platforms hinge on reliable detector reconstructions21. Large uncertainties in reconstructed detectors propagate into SPAM error estimates22 that could limit platform performance and impede effective error mitigation. Similarly, photonic experiments rely on photodetectors23–27 whose precise calibration must be known a priori. Although coherent state probes are one option for photodetector tomography16,24,25, it remains unclear to what extent quantum properties of light such as entanglement and squeezing, could improve this precision28. These questions underscore the need for a general framework for efficient and high-precision reconstruction of the measurement operators.

In this work, we formulate and prove the maximum information extractable from probing unknown quantum measurements and call it the detector quantum Fisher information (DQFI). The DQFI leads to fundamental limits, that is, the quantum Cramér-Rao bounds (QCRBs), on the uncertainties in locally estimating parameters of measurement operators19,20,29–32, thereby setting a performance benchmark for past and future detector tomography experiments. In addition to providing the first fundamental bound for detector estimation, the DQFI enables us to draw the following physical insights: 1) We draw a direct analogy between the DQFI and the well-known state quantum Fisher information (SQFI), finding some similarities and some surprising differences. 2) We apply the DQFI to noisy on-off single-photon detectors16,24,25 and implement the first provably-optimal detector estimation experiment using the IBM platform. 3) We show that the proposed DQFI framework provides an alternative approach for optimal sensing of a quantum process18, which requires both optimal probe states and optimal measurements (Fig. 1(c)). In practice, our alternative approach could lead to more appropriate benchmarks in settings where probe preparation capabilities by far exceed control over the measurement device.

Overall, our work reveals surprising connections between the theories of state and detector tomography while highlighting the unique aspects of detector analysis. These findings complete the triad of optimal state, process, and detector tomography (Fig. 1)16,24, paving the way for a more balanced role of states and measurements in precision quantum metrology.

Results
Introduction to the framework
Before introducing precision bounds for the estimation of quantum measurements, we briefly recap the concept of efficiency and how it leads to precision bounds in quantum state estimation17,29–34. The task here (Fig. 1(a)) is to estimate parameters θ of an unknown state ρ θ, by first measuring the state via a m-outcome POVM, Π ≡ {π 1,…,π m}, and then processing the measured probabilities, p θ (j)=Tr(ρ θ π j ), to get the parameter estimates17. The efficiency or effectiveness of the measurement Π in extracting parameter information depends on how sensitive its measured statistics are to changes in the parameter value. In the local setting, where the unknown parameters are close to some known true values17,30, efficiency is quantified by the classical Fisher information (CFI) of the distribution17,35–37,

F θ [ρ θ,Π] := ∑ j∈[m] (∂ θ p θ (j))2 / p θ (j) = E p θ (∂ θ log p θ )2 , (1)

capturing the variance of the parameter-derivative ∂ θ p θ/p θ of the distribution (abbreviating ∂ θ := ∂/∂ θ and [m]:={1,…,m}). The minimum mean-squared error (MSE) of parameter estimates achievable (on average per trial) using a measurement Π is given by 1/F θ, as per the classical CRB (CCRB)31. Accordingly, optimal measurements are those that maximise the CFI in Eq. (1), thus minimising the CCRB and yielding the most precise estimates17,38,39. However, maximising this non-linear quantity is challenging in practice, even more so in the multi-parameter case38,39, so the state QFI (SQFI) I θ of state ρ θ was introduced as an upper bound to the maximum CFI over all measurements29–31. The SQFI is defined as

I θ [ρ θ]:=Tr(ρ θ L2 θ )=E ρ θ [L2 θ ], and satisfies

max Π F θ [ρ θ,Π] ≤ I θ [ρ θ] (2)

for any valid ρ θ, with equality holding in several cases17,34. The symmetric logarithmic derivative (SLD) operator L θ appearing in the definition generalises the classical logarithmic derivative ∂ θ log p θ in Eq. (1) implicitly via L θ ρ θ+ρ θ L θ=2∂ θ ρ θ. Following Eq. (2), the QCRB given by 1/I θ lower-bounds the MSE of estimates for any measurement Π, thus setting a precision bound31. In the following, we introduce a general framework for bounding estimation precision for parameters of quantum measurements. To start off, we consider detector models with a single unknown parameter; these are relevant for practical single-photon detectors and noisy qubit detectors. Then, we generalise the framework to multi-parameter scenarios, addressing general quantum detector tomography.

Single-parameter detector estimation
The task here (Fig. 1(b)) is to characterise an unknown m-outcome measurement Π θ ≡ {π jθ } j∈[m] by first probing it with a known quantum state ρ in, and then processing the measured probabilities, p θ (j)=Tr(ρ in π jθ ), to obtain parameter estimates. The efficiency of the probe state ρ in in extracting parameter information from Π θ is determined by the CFI, F θ [ρ in , Π θ], which leads to the CCRB, 1/F θ, lower-bounding MSE when using this probe. Thus, the optimal or most precise probe state, ρopt, is the one maximising the CFI, i.e.,

ρopt[Π θ] := argmax ρ in F θ [ρ in , Π θ], (3)

F θmax [Π θ] := max ρ in F θ [ρ in , Π θ].

The maximisation in Eq. (3) has no known analytical solution, though numerical techniques from channel literature may be applicable40. Specifically, by translating the detector POVM to Kraus operators for quantum-classical maps (particular choice of Kraus set shown in Eq. (27)), the maximisation for F θmax can, in principle, be converted to a maximisation over all equivalent Kraus operators through a numerical semi-definite program40. In practice, however, this approach is computationally demanding due to the large number of Kraus operators for a measurement channel. The question of the optimal probe state is thus a significant hurdle for detector tomography, although it has been addressed for specific measurement channels41,42. The lack of efficiently computable precision bounds for the minimum MSE, 1/F θmax , for detector estimation reveals a gap in our understanding of the information content of measurements and resulting fundamental limits on estimation errors.

We now clarify how the domain of maximisation in Eq. (3) corresponds to various probing strategies for detector estimation. Let H d denote the d-dimensional Hilbert space of probe states and D(H d ) the space of density matrices on H d. The simplest probing strategy is to pick a single state, ρ in , from D(H d ) to probe the detector, repeatedly and independently. Alternatively, one may use an ensemble of p different probe states, ρ k ∈ D(H d ) for k=1,…,p, each with probability q k (∑ k q k=1). Which strategy is ultimately favoured? As we show in Lemma 1 in Methods, from the convexity of the CFI37, a single quantum state is sufficient to be optimal.

Central to the CFI and the SQFI is the logarithmic derivative quantity: both information measures capture the variance of this quantity, as in Eq. (1). To quantify it, we introduce the SLD operators for each measurement operator. For each outcome j ∈ [m] of Π θ, we define an SLD operator L jθ via

L jθ π jθ +π jθ L jθ := 2∂ θ π jθ : (4)

This implicit definition has known solutions in terms of the measurement operators and their parameter derivatives (Eqs. (18), (19) in Methods)43. More importantly, the SLD operators let us express the measurement probabilities p θ (j|ρ)=Tr(ρπ jθ ) and their derivatives ∂ θ p θ (j|ρ)=Tr(ρ∂ θ π jθ )=Re Tr(π jθ ρL jθ ) as linear operations on the state and the measurement (where Re[⋅] denotes real part)17,34. These two expressions, when substituted into the CFI in Eq. (1), open the door to operator inequalities that can gauge F θmax without actual optimisation17.

DQFI definition
To upper-bound F θ by an expression independent of ρ, we borrow from state estimation a chain of inequalities17,34 that, when applied to the CFI expressed in terms of π jθ , L jθ , and ρ, yields

F θ [ρ, Π θ] ≤ ∑ j∈[m] |Tr(√π jθ ρ L jθ )|2 / Tr(ρπ jθ ) ≤ Tr(Q θ ρ), (5)

where Q θ := ∑ j∈[m] L jθ π jθ L jθ is a positive semi-definite operator. The first inequality above relies on Re[z]2 ≤ |z|2 for complex number z, whereas the second is the operator Cauchy-Schwarz inequality. In state estimation we obtain the measurement-independent SQFI I θ at this point, but here the trace quantity upper-bounding F θ still depends on ρ. Our final step settles this by maximising both sides of Eq. (5) over the probe space D(H d ), resulting in F θmax ≤ max ρ∈D(H d) Tr(Q θ ρ)= ||Q θ ||2 sp , where ||⋅||2 sp denotes the largest eigenvalue (or spectral radius) of Q θ. Moreover, the probe state maximising Tr(Q θ ρ) is the eigenvector of Q θ corresponding to its largest eigenvalue—this state is thus always pure.

These results lay the ground for defining the DQFI: an upper bound to the maximum CFI that leads to precision bounds for detector estimation. We propose two definitions for the DQFI.

Definition 1. (Spectral DQFI). Define the spectral DQFI J jj,θ of a measurement Π θ ≡ {π jθ } j∈[m] as

J jj,θ [Π θ] := || ∑ j∈[m] L jθ π jθ L jθ ||2 sp , (6)

where ||X||2 sp denotes the spectral radius or largest eigenvalue of operator X.

A simpler but less tight upper bound on F θmax worth consideration is Tr Q θ. In fact, Tr Q θ= ∑ j∈[m] Tr(π jθ L j 2 θ ) resembles the SQFI I θ. Assembling the measurement operators and their SLD counterparts into md × md block-diagonal operators Π θ,bd := ⊕ j∈[m] π jθ and L θ,bd := ⊕ j∈[m] L jθ reveals Tr Q θ=Tr(Π θ,bd L2 θ,bd )=I θ (Π θ,bd ). This approach is justified in that Π θ,bd is trace-constant (though not unit-trace) like a state and L θ,bd is indeed its SLD operator when treated as such. This weaker upper bound results from treating the measurement as an unnormalised, higher-dimensional state.

Definition 2. (Trace DQFI). Define the trace DQFI J Tr,θ of a measurement Π θ ≡ {π jθ } j∈[m] as

J Tr,θ [Π θ]:= ∑ j∈[m] Tr(L jθ π jθ L jθ ). (7)

Having proposed two definitions for the DQFI, we must compare the two to understand their maximum disagreement and cases where they are equivalent. It is evident from the positivity of Q θ that J jj,θ ≤ J Tr,θ. On the other hand, the largest eigenvalue of Q θ cannot be smaller than the average of its eigenvalues, so that J Tr,θ ≤ d J jj,θ. This leads to the following ordering of the DQFIs,

1/d J Tr,θ ≤ J jj,θ ≤ J Tr,θ ≤ d J jj,θ : (8)

The cause of the disagreement clarifies why the spectral DQFI is tighter—it respects that quantum states are normalised, so that probing along multiple directions requires sacrificing some probability of detection, and hence parameter information, along each direction. Nonetheless, the trace DQFI serves as a simpler upper bound that is more amenable to analytical evaluation.

To consolidate the two DQFI definitions, we propose Theorem 1 in the following.

Theorem 1. (DQFI upper-bounds maximum CFI). For estimating quantum detectors, the spectral DQFI J jj,θ and the trace DQFI J Tr,θ upper-bound the maximum CFI over probe states, F θmax , as F θmax ≤ J jj,θ ≤ J Tr,θ.

Proof. The proof follows from Eqs. (5) and (8) and is provided in Methods.

Theorem 1 leads to the trace QCRB 1/J Tr,θ and the spectral QCRB 1/J jj,θ, which lower-bound the minimum MSE of estimates as 1/J Tr,θ ≤ 1/J jj,θ ≤ 1/F θmax , thereby setting precision bounds for detector estimation.

Example 1. We now present a simple example to illustrate the importance of the DQFI. Consider a qubit Z-basis detector with inherent bit-flip noise4 of unknown strength p (0 ≤ p ≤ 1), corresponding to the POVM Π p ≡ {π 1p ,π 2p } with

π 1p = (1-p 0; 0 p), π 2p = (p 0; 0 1-p). (9)

Figure 2(a) illustrates the Fisher information in estimating p for both non-optimal and optimal probe states. We find that the spectral DQFI (dashed black curve) is tight, as it equals the maximum CFI F θmax . The Fisher information for probe states notated as |θ,ϕ⟩ := cos(θ/2)|0⟩+exp(iϕ)sin(θ/2)|1⟩ is depicted as circular markers (orange, purple and blue) on dashed lines for non-optimal probes, and as star-shaped markers (golden) for the optimal probes. The DQFI J jj,p is 4/(1/〈Z〉2-(1-2p)2) where 〈Z〉=Tr(ρZ) is the Pauli Z expectation of ρ. The CFI is maximised at 〈Z〉=±1, corresponding to |0⟩ or |1⟩, attaining a maximum of F pmax =1/p(1-p) in either case (golden points). As Q p=1/p(1-p) 1 2, its eigenstates are also |0⟩ and |1⟩. Thus, the maximum CFI over probe states equals J jj,p and is attained by probes |0⟩ or |1⟩, in agreement with J jj,p. Here we have J Tr,p=2F pmax =2J jj,p, displaying the maximum disagreement possible for d=2.

Attainability of DQFI
In the above example, the spectral DQFI is attainable, i.e., J jj,θ=F θmax , and the optimal probe states are among the basis states. These two statements are generally true for diagonal or phase-insensitive measurements, because inequality (5) is saturated, as we state in the following theorem.

Theorem 2. (Attainability for diagonal measurements). For estimating a phase-insensitive detector represented by a diagonal POVM Π θ, the DQFI J jj,θ is attainable, i.e., J jj,θ = F θmax , and an optimal probe exists within the family of basis states.

Proof. The proof is presented in Supplementary Mat. I.

Phase-insensitive measurements feature prominently in detector characterisation experiments, in part because they do not require phase stabilisation of probes16,24. Indeed, avalanche photodiodes (APDs)16,24, multiplexed photon-number-resolving detectors16,24,25, and superconducting nanowire single-photon detectors26,27,44 have been experimentally characterised in this setting. Given the extensive usage of these detectors across quantum technologies, understanding the information-theoretic bounds for estimating them has practical implications for benchmarking existing approaches to detector characterisation and designing more effective strategies. Our result in Theorem 2 means that the spectral QCRB sets the ultimate precision limit here.

The proof of Theorem 2 also covers measurements that are simultaneously diagonalisable independent of the parameters (see Supplementary Mat. I). This category includes qubit detectors with phase-flip noise (Fig. 2(b)) and bit-phase-flip noise (Fig. 2(c))4, so the spectral DQFI is tight for all single-Pauli error qubit detectors. More generally, we find a necessary condition for attainability of the spectral DQFI to be that the SLD operators commute, [L jθ , L kθ ]=0, akin to compatible multi-parameter state estimation14,17,36,45,46 (see Theorem 3 in Methods for the complete set of necessary and sufficient attainability criteria). However, if the SLD operators do not commute, the spectral DQFI could be unattainable even in the single-parameter regime, which is not the case in state estimation. Interestingly, in this case, the attainable bound can be computed through a semi-definite program (Supplementary Mat. II) that effectively incorporates non-Hermitian components into the SLD operator definition in Eq. (4) (see Methods).

Platform demonstration
We now consider a simple, but practically relevant and experimentally feasible example, of estimating a qubit detector affected by dephasing noise4. Given that dephasing is particularly prominent on current quantum computing platforms47,48, its precise characterisation is crucial for effective noise control and suppression. Other noise models4 are deferred to Supplementary Mat. III.

Example 2. Consider a generic dephased qubit projective measurement, corresponding to the POVM Π p ≡ {π 1p ,π 2p } with elements

π 1p = 1/2 (1 + (1-2p)cosθ sinθ e-iϕ; (1-2p)sinθ eiϕ 1-cosθ), π 2p = 1/2 (1 - (1-2p)cosθ sinθ e-iϕ; -(1-2p)sinθ eiϕ 1+cosθ), (10)

where p denotes the unknown dephasing strength (0 ≤ p ≤ 1/2). Here, the noiseless measurement Π ideal corresponding to p=0 is assumed known. It corresponds to projections along polar and azimuthal angles θ and ϕ on the Bloch sphere (double arrows in Fig. 3(b)), via two orthogonal projectors, |± θ,ϕ⟩, given by the nontrivial eigenvectors of π 1p |p=0. The spectral and trace DQFI can be calculated as

J k,p = sin2θ / p(1-p) & J Tr,p = 2sin2θ / p(1-p) : (11)

The two optimal probe states, ρ± opt, are both pure and have phase ϕ, thus lying in the same vertical plane as measurement projectors |± θ,ϕ⟩, as shown in Fig. 3(a). This observation, magnified in Fig. 3(b), provides an intuitive explanation of the optimal probes (black dots in Fig. 3(b)): they are the pure states that align perfectly with the measurement direction upon dephasing. The polar angles θ ± opt of these optimal probes, given by tanθ ± opt = ±tanθ/(1-2p), can be deduced geometrically from Fig. 3(b).

We now experimentally implement Example 2 on an IBM Eagle r3 quantum computer as follows. First, we introduce dephasing noise of fixed strength p=0.2 by interacting the probe qubit with an ancilla qubit (Fig. 3(c)). Then we estimate the noise strength using a range of probe states, including both optimal and non-optimal ones (see Methods for details). Critically, the noiseless measurement is fixed at (θ, ϕ) = (π/8, 0), differentiating the problem from state or process estimation. In Fig. 3(d), we compare the experimental MSEs for each probe state (blue dots) to the CCRB 1/F p (black curve) that gives the theoretical minimum MSE20 for this probe state and to the MSEs from noiseless simulation (green dots). The spectral QCRB 1/J k,p (horizontal blue dashed line) lower-bounds the MSE for any probe state. The theory-optimal (vertical black dashed line), simulated-optimal and experimentally-inferred optimal probes agree within confidence intervals (green and blue regions), consistent with platform noise.

In contrast to previous efforts to characterise the platform’s SPAM errors through detector tomography22, which do not use optimal quantum states, our work provides the first instance of provably optimal detector tomography.

Multi-parameter detector tomography
In quantum state estimation, there are many physically motivated scenarios where we will want to estimate multiple parameters simultaneously13,14,49–54. Similarly, the example is full detector tomography, where the maximal number of unknown parameters is simultaneously inferred (see below and Refs. 16, 24 for more examples). For state estimation, it is known that measuring the parameters simultaneously can provide greater sensitivity compared to measuring them sequentially55. Analogously, multiple parameters of a detector can be successively estimated using the single-parameter framework56, but achieving the ultimate precision limit requires the simultaneous estimation of all parameters. In this setting, the CFI is a matrix reflecting the probe’s sensitivity to each parameter individually as well as correlations in sensitivities to multiple parameters (defined in Eq. (SM.VII.1), Supplementary Mat. VII). The corresponding QCRBs minimise a weighted-sum of the different parameter variances and covariances18, but an ensemble of probe states is generally required (see discussion in Methods). For simplicity, here, we set the weight to identity and minimise total MSE of the parameters, though our techniques also apply to arbitrary weight matrices (see Supplementary Mat. VII). Below, we propose two QCRBs that lower-bound the attainable minimum total MSE, which is defined as the minimum CCRB over all probing ensembles,

CCCRB := min Tr(F θ -1) : (12)

We first extend the trace DQFI to the multi-parameter setting. For estimating n real parameters, θ:=(θ 1,…,θ n), the trace DQFI from Def. 2 symmetrically extends to the n×n real matrix J Tr,θ with elements

(J Tr,θ ) jk := 1/2 ∑ l∈[m] Tr(L l θ j π l L l θ k +L l θ k π l L l θ j ), (13)

where L l θ j is the SLD operator for the lth outcome and the jth parameter. This matrix upper-bounds the CFI matrix for any probing strategy as J Tr,θ ≽ F θ (Theorem 5 in Supplementary Mat. VII), where A ≽ 0 denotes A is positive semi-definite. The trace QCRB, CQCRB Tr := Tr J Tr,θ -1, thus lower-bounds CCCRB in Eq. (12). However, due to the limitation discussed below Eq. (8), the trace QCRB is generally unattainable, as in Example 3.

Next we consider the more complex problem of extending the spectral DQFI to the spectral QCRB, denoted CQCRB k. This is done through an efficient semi-definite program (in Supplementary Mat. VII), where we define a linear objective that is maximised, symmetrically extending Tr(Q θ ρ) from Eq. (5) to multiple parameters. This results in a bound tighter than the trace QCRB, see Supplementary Mat. VII for details. Notably, the spectral QCRB does not require considering an ensemble of probes, unlike the minimisation in Eq. (12), thus offering a practical computational advantage.

Example 3 below compares the two QCRBs to CCCRB, showing the spectral bound to be tight for an experimentally relevant problem16,24,25.

Example 3. Consider the tomography of a phase-insensitive, on-off qubit detector. The POVM Π θ ≡ {π 1θ ,π 2θ } representing such a measurement may be parametrised as

π 1θ = (θ/2 0; 0 1-θ/2), π 2θ = (1-θ/2 0; 0 θ/2). (14)

The trace DQFI for simultaneously estimating θ≡{θ 1,θ 2} is

J Tr,θ = diag(θ 1(1-θ 1), θ 2(1-θ 2)),

and the resulting QCRB is CQCRB Tr = ∑ j=1 2 1/θ j(1-θ j). The spectral QCRB for this problem is

CQCRB k = (∑ j=1 2 1/√(θ j(1-θ j)))2 = Tr J Tr,θ -1 = CQCRB Tr + 2/θ(1-θ)θ 2(1-θ 2).

The spectral QCRB is attainable, i.e., CCCRB=CQCRB k. The optimal probing strategy requires an ensemble of two states: |0⟩ and |1⟩ prepared with probability q and 1-q in the proportion q/(1-q) = √(θ 1(1-θ 1))/√(θ 2(1-θ 2)).

Probeincompatibility effect
An interesting feature of Example 3 is that the optimal multi-parameter probing strategy is a convex mixture of the single-parameter optimal probes, namely, |0⟩ and |1⟩ for θ 1 and θ 2, respectively. Although this may not hold for generic multi-parameter models, it underscores the fact that when the optimal single-parameter probes for different parameters are incompatible, termed probe incompatibility in Ref. 56, there is an uncertainty trade-off in simultaneous multi-parameter estimation. Naturally, we can expect the optimal simultaneous strategy to generally outperform the optimal strategy that utilizes a fraction of detector uses to estimate each parameter separately. This latter strategy, which we call sequential multi-parameter estimation, is connected to the total QFI metric56, and we compare our bounds to this metric in Supplementary Mat. VIII. While the comparison shows the spectral technique to produce a tighter bound than the total QFI metric56, the former approach may still be viewed as a special case of a particular choice of channel purification and computation of the corresponding QFI matrix.

Application to photodetector tomography
Example 3 generalises to phase-insensitive measurements of higher-dimensional states, such as the APDs experimentally tomographed in Refs. 16, 24, 25. These detectors correspond to diagonal two-outcome measurements in the photon number basis for the continuous-variable state space, truncated at some large dimension d. The spectral QCRB here, CQCRB k = (∑ j=1 d 1/√(θ j(1-θ j)))2, is tight and sets the precision limit for APD tomography. Attaining this precision requires preparing an ensemble of photon-number states {|j⟩}, each with probability q j ∝ √(θ j(1-θ j)), and assigning a higher q j to states |j⟩ that produce more uniformly distributed detection probabilities. Interestingly, the spectral QCRB in Eq. (15) mirrors the Gill-Massar QCRB for state estimation57, with similar behaviour seen in Fig. 2(d) for the simultaneous estimation of bit-flip and phase-flip rates (Example 4 in Supplementary Mat. VII).

Methods
Probing strategies for detector estimation
Local unbiased estimation. In the local estimation setting, where unknown parameters θ are close to known true values θ*, the Fisher information framework requires locally-unbiased estimators, meaning estimators with zero bias at the true parameter values17,35,70. Effectively, this constrains how the probabilistic detector outcomes are mapped to corresponding predicted values for the unknown parameters20. In linear estimation20,28, this is done through a classical estimator function represented by an m×n real matrix Ξ jk that, given a detector outcome j∈[m], predicts the estimate Ξ jk for θ k(k∈[n]). The overall estimate produced for each θ k is then ∑ j p θ (j)Ξ jk. The local-unbiasedness condition is then ∑ j p θ (j)Ξ jk |θ=θ*=θ k. The maximum-likelihood estimator is asymptotically unbiased, and therefore can saturate the CCRB in that limit20.

Single-parameter detector estimation. We first present Lemma 1, establishing single-probe strategies to be sufficient for optimal single-parameter detector estimation.

Lemma 1. For single-parameter detector estimation, a single-probe strategy is optimal and at least one optimal state ρopt ∈ D(H d ) exists that attains F θmax in Eq. (3).

Proof of Lemma 1. Suppose, to estimate a single parameter θ of a measurement Π θ, we probe with two distinct states, ρ 1 ≠ ρ 2, ρ 1, ρ 2 ∈ D(H d ), with probabilities q 1 and q 2 = 1-q 1. The joint list of probabilities,

plist={q 1 p θ (1|ρ 1), ..., q 1 p θ (m|ρ 1), q 2 p θ (1|ρ 2), ..., q 2 p θ (m|ρ 2)},

forms a valid distribution and, as q j are (by definition) independent of θ, we see from Eq. (1) that the CFI of plist,

F θ [plist]=q 1 F θ [ρ 1, Π θ]+(1-q 1)F θ [ρ 2, Π θ], (16)

is a convex combination of the CFIs of each probe ρ k individually. Unless the two individual CFIs are equal, the combined CFI F θ [plist] is smaller than the maximum individual CFI. Even if the two individual CFIs are equal, combining them with the only probe ρ 1 or only ρ 2 is sufficient for proving optimality. Thus, for single-parameter detector estimation, a one-state probing strategy is optimal.

Lemma 1 says that for single parameters, the maximisation in Eq. (3) can be performed over ρ in ∈ D(H d ).

Multi-parameter detector estimation. For multi-parameter problems, Lemma 1 does not hold and an ensemble of probes is generally required for optimality. This is because multiple unknown parameters cannot be uniquely determined without sufficiently many linearly-independent outcome probabilities. The multi-parameter CFI matrix F θ for an ensemble {q k, ρ k} k∈[p] comprising p probe states is a convex sum of the individual CFI matrices F θ [ρ k, Π θ] = ∑ k q k F θ [ρ k, Π θ], (17)

similar to Eq. (16). Here, the difference from the single-parameter setting is that the CCRB Tr(F θ -1) requires matrix inversion of F θ. We note that even if the individual CFI matrices are equal, the combined CFI matrix F θ [ρ k, Π θ] = F θ [ρ k, Π θ] is not necessarily equal to the individual matrices, and the inverse of the sum is not the sum of the inverses. In general, the minimum number of probes required for estimating n parameters is ⌈n/(m-1)⌉, where m is the number of outcomes. For n parameters, the optimal strategy is an ensemble of probes {q k, ρ k} k∈[p] that can be used to estimate each parameter simultaneously. However, this minimum number of probes merely guarantees the feasibility of multi-parameter estimation, whereas optimality may require even larger ensembles.

Notably, an ensemble of p different d-dimensional probe states can be realised through access to an ancillary system, A, that can be measured perfectly. This involves preparing a single bipartite-entangled state in D(H d ⊗ HA p ), where HA p denotes the p-dimensional Hilbert space corresponding to system A. For example, the ensemble {q k, ρ k} can be realised by preparing the state ∑ k √q k |ρ k⟩ ⊗ |k⟩A and measuring the ancilla perfectly in the {|k⟩A} basis28. In this sense, Lemma 1 extends to multi-parameter problems, with the optimisation domain becoming ρ in ∈ D(H d ⊗ HA p ).

The equivalence between bipartite-entangled states and ensemble probes means that unless the probe and ancilla are measured jointly, the convexity of the CFI (Lemma 1) rules out any entanglement advantage in precision in the single-parameter setting. However, for the most general estimation scenario, ancilla measurements conditioned on the detector outcomes should be considered. This estimation strategy was applied to detector tomography in Ref. 28 and extended to imperfect but pre-characterised ancilla measurements. In Supplementary Mat. II, using Theorem 4, we show that for single-parameter detector estimation, even conditional measurements cannot enable an entanglement advantage, and the maximum precision can be attained using a single separable pure-state probe. At the same time, the advantages of ancilla-assisted probe states for multi-parameter detector estimation have been addressed in recent experimental demonstrations71,72, though the extent of precision enhancement remains to be explored.

Single-parameter DQFI derivation
The Lyapunov equation (4) can be explicitly solved by utilising either the eigenspectrum of π jθ = ∑ k∈[d] λ jk |λ jk⟩⟨λ jk|17,29,30,

L jθ = 2 ∑ m,n∈[d] ⟨λ jm|∂ θ π jθ |λ jn⟩ / (λ jm+λ jn) |λ jm⟩⟨λ jn|, (18)

or the vectorisation approach43,

vec(L jθ ) = 2(π j * θ ⊗ 1 d+1 d ⊗ π jθ ) -1 vec(∂ θ π jθ ), (19)

where (⋅)* denotes complex conjugation and the pseudo-inverse can be used instead of the inverse, if required. Below, we first prove Eq. (5), and then prove Theorem 1.

Proof of Eq. (5). To upper-bound F θ by an expression independent of ρ, we employ the chain of inequalities17,34

F θ [ρ, Π θ] = ∑ j∈[m] (Re Tr(π jθ ρ L jθ ))2 / Tr(ρπ jθ ) ≤ ∑ j∈[m] |Tr(√π jθ ρ L jθ )|2 / Tr(ρπ jθ ) (20)

= ∑ j∈[m] |Tr(√ρ √π jθ L jθ )|2 / Tr(ρπ jθ ) ≤ ∑ j∈[m] Tr(ρ L jθ π jθ L jθ ) = Tr(Q θ ρ) :

The first inequality above follows from Re[z]2 ≤ |z|2 for any complex number z, and the second inequality is the operator Cauchy-Schwarz inequality, Tr(A B) ≤ Tr(A A) Tr(B B). As ρ ∈ D(H d ) satisfies ρ ≼ 1/d, it follows that Tr(Q θ ρ) ≤ Tr(Q θ ) always. Consolidating Eq. (20) and its maximisation, max ρ∈D(H d) Tr(Q θ ρ)= ||Q θ ||2 sp , into

F θ [ρ, Π θ] ≤ Tr(Q θ ρ) ≤ ||Q θ ||2 sp ≤ Tr(Q θ ) (21)

leads to

F θmax [Π θ] ≤ J jj,θ [Π θ] ≤ J Tr,θ [Π θ], (22)

as claimed in Theorem 1.

Attainability criteria of the DQFI
The attainability criteria for the spectral DQFI J jj,θ are threefold. The first two of these conditions ensure that the two inequalities in Eq. (5) are saturated, implying that F θmax =Tr(Q θ ρopt), for ρopt the CFI-optimal probe state from Eq. (3). The third condition, unique to detector estimation, ensures that Tr(Q θ ρopt)= ||Q||2 sp =J jj,θ. Taken together, these criteria require the existence of a common eigenstate to each SLD operator, that is also common to Q θ. This common eigenstate, if it exists, corresponds to the optimal probe state. We state this result as Theorem 3 below. Main-text Theorem 2 is recovered as a special case of Theorem 3, where basis or number states comprise the common eigenstates.

Theorem 3. For estimating a detector, the DQFI J jj,θ is attainable if and only if the SLD operators {L jθ } of the POVM and Q θ all share a common eigenvector, which constitutes an optimal probe state.

Proof of Theorem 3 (Backward Direction). It is clear from Eq. (21) that J jj,θ is attained whenever the first two inequalities in Eq. (21) are saturated. The first inequality in Eq. (20) is saturated if and only if Tr(π jθ ρ L jθ ) is real for all j ∈ [m]. This is satisfied if ρ is a pure state |ψ⟩⟨ψ| and L jθ is Hermitian, which is true for any measurement. The second inequality in Eq. (21) is saturated if and only if ρ is a projector |ψ⟩⟨ψ| onto the eigenvector |ψ⟩ of Q θ corresponding to its largest eigenvalue, i.e.,

3. Q θ ρ=λmaxρ where λmax=maxeig Q θ

Condition (2), for saturating the Cauchy-Schwarz inequality, is equivalent to

ρπ jθ / Tr(ρπ jθ ) = ρL jθ π jθ / Tr(ρL jθ π jθ ), (23)

for all j ∈ [m]. This is satisfied if ρL jθ is a projector onto the same subspace as π jθ , which can be ensured if |ψ⟩ is a simultaneous eigenstate of every L jθ for j ∈ [m] (or a combination of projectors onto multiple degenerate simultaneous eigenstates). Criterion (2) is thus equivalent to all the L jθ sharing at least one common eigenvector, whereas criterion (3) requires this eigenvector to also be the largest-eigenvalue eigenvector of Q θ.

In the case where the largest eigenvalue of Q θ is degenerate among multiple eigenvectors, the definition for J jj,θ still holds, but the DQFI-optimal probe state can be some superposition of these degenerate eigenvectors. The optimal superposition state is prescribed by criterion (2) above to be a simultaneous eigenstate of every L jθ , if one exists. If no simultaneous eigenstate exists, DQFI J jj,θ only upper-bounds F θmax but fails to reveal the optimal probe state. Below, we prove Theorem 3 in the forward direction, i.e., that probe states ρ= |ψ⟩⟨ψ| satisfying criteria (1) through (3) attain a CFI F θ [ρ, Π θ]=J jj,θ [Π θ].

Proof of Theorem 3 (Forward Direction). Let us assume pure probe state ρ= |ψ⟩⟨ψ| satisfies the attainability criteria enumerated 1 to 3. As |ψ⟩ is an SLD-simultaneous eigenstate, L jθ |ψ⟩ = λ jθ |ψ⟩ for real eigenvalues λ jθ . Then, Tr(π jθ ρ L jθ )=λ j * θ Tr(π jθ ρ)=λ jθ p jθ . The CFI from Eq. (20) then equals

F θ [ρ, Π θ]= ∑ j∈[m] λ j 2 θ p jθ : (24)

As |ψ⟩ is also an eigenvector of Q θ= ∑ j∈[m] L jθ π jθ L jθ , and corresponds to its largest eigenvalue, ||Q θ ||2 sp ,

J jj,θ [Π θ] = Tr(Q θ ρ)= ⟨ψ| ∑ j∈[m] L jθ π jθ L jθ |ψ⟩ = ∑ j∈[m] λ j 2 θ ⟨ψ|π jθ |ψ⟩ = ∑ j∈[m] λ j 2 θ p jθ , (25)

so that F θ [ρ, Π θ]=J jj,θ [Π θ], thus proving the claim.

Tight DQFI for single-parameter estimation
The spectral DQFI (Def. 1) is tight for phase-insensitive measurements but not for the general phase-sensitive case. However, a minor modification to the spectral technique, specifically to the defining equation (4), can produce the tight bound, J Ext,θ, for general single-parameter detector models40,63,67,68,73. By considering non-Hermitian SLD (nSLD) operators L0 jθ 74 that satisfy

L0 j y θ π jθ +π jθ L0 jθ = 2∂ θ π jθ (26)

instead of Eq. (4), we show in Supplementary Mat. II that the tight bound J Ext,θ may be formulated as40,63,67

J Ext,θ := min {L0 jθ } || ∑ j L0 j y θ π jθ L0 jθ ||2 sp | Eq.(26)holds.

This minimisation can be efficiently solved by an SDP in terms of the POVM and its derivative40, which we present in Supplementary Mat. II. Lemma 2 therein proves the validity of the upper bound F θmax ≤ J Ext,θ, directly extending main-text Theorem 1 and its proof in Eq. (20) of Methods. Then, Theorem 4 proves the tightness of the bound by showing F θmax = J Ext,θ, extending main-text Theorem 2 to general single-parameter models. While a closed-form analytical solution remains unknown, the proof reveals the optimal nSLD operators to satisfy all three attainability criteria listed in the previous subsection.

Notably, the tight bound is based on a channel estimation bound that is not always tight for the original channel but is tight for the extended channel67. Here, the extended channel refers to augmenting the POVM Π θ with Kraus operators

K j,k= |j⟩⟨k| √π jθ for j∈[m], k∈[d] (27)

for a quantum-classical channel40. Unlike general quantum channels40,63, however, for single-parameter detector estimation, ancilla-entangled probes do not offer any advantage over separable probes (Theorem 4 and Corollary 1 in Supplementary Mat. II).

Experimental parameters
The quantum circuit implementing the estimation experiment is shown in Fig. 3(c). The first qubit, representing the probe, is prepared in a pure state at polar angle θ in and azimuthal angle ϕ=0 (we choose ϕ=0 for the ideal measurement Π ideal so only the ϕ=0 cross-section of the Bloch surface is relevant). This probe undergoes dephasing noise of strength p, implemented via interaction with an ancilla qubit, before the final projection measurement along polar angle θ and azimuthal angle ϕ=0. By scanning probe angle θ in over the interval [0.2, 1] while fixing detector angle θ=π/8, we estimate parameter p. By repeating the process N=105 times for each θ in, we report the empirical MSE of estimates (green dots) in Fig. 3(d) (with error bars in grey).

The empirical MSEs agree well with theoretically-expected MSEs (black curve) (for true value of p chosen to be 0.2), and both are lower-bounded by 1/J k,p=1.093 from Eq. (11), in accordance with the spectral QCRB (blue dashed line). Theoretically, the optimal probe state angle is arctan(tanθ/(1-2p))=0.6042 rad (grey dashed line). From the simulation data points, which form a grid of step size 0.006 rad within [0.5, 0.7] rad, the optimal inferred angle is 0.6162 rad, with 90% confidence interval [0.6121, 0.6202] rad (green shaded region). From experimental data points, which form a grid of step size 0.014 rad within [0.5, 0.7] rad, the optimal inferred angle is 0.6121 rad, with 90% confidence interval [0.5394, 0.6848] rad (blue shaded region).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper introduces new tools to determine the ultimate precision limits of quantum detectors. Detector characterization involves determining the associated POVMs, a task addressed by quantum detector tomography (QDT). In QDT, an unknown element \\Pi_k is characterized through a set of known measurements using known input probe states. By performing many repetitions on multiple copies of each probe state, one can estimate \\Pi_k, thereby allowing the inverse problem to be solved.\nFollowing the pioneering work of Ian Walmsley's group, QDT has been performed for a wide range of detectors, including SPADs, time-multiplexed detectors, transition-edge sensors, superconducting nanowire detectors, and even the human eye. Due to the complexity of the task, various advanced methods have been developed, such as the Fitting of Data Patterns, self-guided tomography, compressed sensing tomography, and techniques based on neural networks. To the best of my knowledge, all of these approaches rely on fidelity (or variants thereof) for benchmarking.\nThe current work introduces a novel tool: the detector quantum Fisher information (DQFI). The concept is elegant and the theoretical development is clear. The authors propose two definitions of DQFI—one tighter than the other. While this is insightful, I would appreciate a clearer operational preference or justification for using one over the other.\nThe authors draw an analogy with the standard quantum Fisher information (QFI). But we know that for pure states, QFI reduces to the variance of the generator. In this sense, QFI provides a lower bound on the mean square error (MSE), which can be related to fidelity. From a benchmarking standpoint, I question whether this represents a significant advance over existing approaches in QDT.\nMore importantly, it is well known that a key requirement for QDT is that the set of probe states must be experimentally feasible. This means the states must be well-characterized themselves. In standard tomography once the QFI is known and the quantum Cramér-Rao bound (QCRB) is shown to be saturable, one can advantageously identify the optimal probe states for characterization. The authors focus on the single-parameter case, where the situation is relatively simple. Can this be done for QDT?\nHowever, a complete QDT requires a multiparameter scenario, where the QCRB is generally not saturable. In standard QDT, the set of reference states must be tomographically complete; i.e., they must span the space of POVMs to enable inversion. In photodetector measurements, this is relatively straightforward. Coherent states of an optical beam serve as ideal candidates: they are overcomplete and can form a tomographically complete set by varying amplitude (e.g., via attenuation with a beam splitter) and phase (e.g., with a delay line). Crucially, such states are easy to generate using lasers.\nA question that remains is whether the proposed DQFI can be used to identify optimal probe states for full QDT. In the standard multiparameter QFI framework, one can convert the QCRB matrix into a scalar bound (e.g., via a trace with a weighting matrix) and thereby derive optimal states. Would a similar approach be possible or meaningful in the DQFI context?\nIn summary, this is a well-written and rigorous work introducing new concepts that may prove important in refining the limits of QDT. I recommend it for publication. However, I believe that addressing the concerns mentioned above would enhance the readability and practical impact of the paper.", "reviewer_2": "The authors investigate the precision achievable in estimation of parameters of general quantum measurements. The authors introduce detector quantum Fisher information, investigate its properties and present applications of the established formalism. The present study is comprehensive and it nicely complements the earlier results established of optimal estimation of parameters encoded in quantum states. As pointed out by the authors in the Supplemental material, there are close connections between quantum measurements and quantum processes. The quantum measurement can be seen as a specific instance of a quantum process (or quantum channel), where the output state is one of orthogonal states associated with the measurement outcomes, c.f. Eq. (SM8.3) when traced over the second degree of freedom. Nevertheless, the quantum measurements play so important role that it is fully justified to treat them separately and establish a theory which provides bounds on precision of estimation of parameters encoded in the measurements. Given the central role that sophisticated quantum measurements play in modern quantum technologies and quantum computing architectures the potential impact and importance of the present work are high enough to in principle deserve publication in Nature Communications.\nThere are several points that should be considered by the authors before the manuscript could be published:\n1) In the Supplemental Material 7 the authors consider a scenario where two copies of the measurement are available. In this case, entangled probe states may be optimal and may provide advantage. What about using entangled probes when probing a single copy of the measurement. Suppose a pure bipartite entangled states of systems A and B is prepared and the measurement is applied to system A. Subsequently, also system B is measured and the measurement on B can depend on the outcome of the measurement performed on system A. Could such scenario provide any advantage with respect to using only a single system A to probe the characterized quantum measurement? Note that the entanglement of A and B need not be maximal.\n2) The authors base their work on the widely utilized concept of Fisher information. To saturate the Cramer-Rao bound, an unbiased estimator is usually required. The authors should clarify in more details which estimators do they have in mind in general, whether they assume the maximum likelihood estimator in the asymptotic limit of very large number of samples.\n3) In the Supplemental Material 3, the authors provide an example of a linear estimator of parameter p, Eq. (SM.3.2). What are the signs of the coefficients alpha and beta in this estimator? It seems that one of the coefficients is negative. In principle, the estimator can yield a negative value of the estimated parameter, which would be unphysical. This may happen in case of low number of samples and if the true p is close to 0. While such situations may be rare in occurrence, the authors should comment on this, because e.g. changing the negative p to p=0 could introduce a bias in the estimator.\n4) The authors should check Eq. (SM.8.16).\n5) In Figure 1 in the Supplemental Material, symbol theta denotes both the angle of a probe quantum state and also the unknown parameter to be estimated. The authors should check the notation used in this figure.", "reviewer_3": "The authors of the paper, propose the concept of detector quantum fisher information (DQFI), which in their words, feels the gap in the triad “State, Process and Measurements”. They claim that while State and Process estimation are well investigated fields, the Measurement estimation problem is not. They derive the upper bound on the achievable classical information in terms of DQFI, which they show to be saturable for a subclass of diagonal povms. They propose also a generalization to multi-parameter case, and discuss an experimental using the IBM quantum computing platform.\nThe paper is in my opinion correct, and also highlights in an interesting way the relevance of measurement estimation problem. There is a serious issue with the paper, however, which authors are either not aware or intentionally avoid discussion of.\nAny quantum measurement is a special case of a quantum channel, more concretely a quantum-classical channel! Given any generalized measurement, with measurement operators \\pi_j, I can construct a corresponding channel with Kraus operators K_j = |j> \\sqrt{\\pi_j}, where outputs represent measurement outcomes.\nI can now use all the results from quantum channel estimation theory to obtain channel QFI – which in this case will correspond to DQFI. In particular, in J. Phys. A: Math. Theor. , 41 p. 255304 (2008) , it is shown that channel QFI is upper bounded by 4 | \\sum_i \\dot{K}_i^\\dagger \\dot{K}_i|, where the norm is the spectral norm, and \\dot{K}_i are derivatives of Kraus operators. In our case since, we can plug in operators K_j = |j> \\sqrt{\\pi_j}, and use identity \\dot{(K_i^\\dagger K_i)} = \\dot{K_i}^\\dagger K_i + K_i^\\dagger \\dot{K_i}. to effectively get the result (6) obtained by the authors.\nMoreover, the above result from quantum channel estimation theory is much stronger than the result presented by the authors, as one can prove its general saturability provided one minimizes the channel over different Kraus representations. Furthermore, this problem may be implemented as a simple SDP programme as was shown in Nat. Commun. , Vol. 3 p. 1063 (2012). Hence, these tools are much more powerful than the ones proposed by the authors.\nRegarding the multi-parameter case, it should be stressed here, that the non-trivial issue here is the ‘probe-incomplatibilty’ , which has been also investigated thoroughly in the quantum channel estimation literature, see e.g. Phys. Rev. X , Vol. 12 p. 011039 (2022). In particular in this reference, a generalization of Fujiwara bound to multiparameter case is given. I presume that this bound again would allow for a more efficient numerical analysis than the bound proposed by the authors, as it again may be solved using the SPD. This at least should at least be tested and compared.\nIn the light of these results, the novely of the presented work is not as high as might seem at a first sight. Still, despite this harsh criticism, I would not formulae here a negative recommendation yet. Provided the Authors are able to show that they are able to propose an improved tools compared to the ones I mentioned above, making use of the particular structure of the quantum-claasical channel that represents the measurement process and showing that they are superior to the all-purpose channel estimation methods.\nIn particular, if authors were able to show that minimization over different Kraus representations, that is required to find the channel QFI in the general case, can be simplified for the detector QFI case – ideally solved analytically!- and show some insight behind it I would regard this as an important contribution to the field. In the same spirit If the authors show that their multi-parameter approach is superior to existing channel estimation approaches, this would be an argument in favour of this paper to be published in Nat Comm. Otherwise, the paper may at most be regarded as a nice highlight of an important special case of a problem which has a known solution, and as such does not meet the standards of Nat. Comm.\nApart from this main criticism, I have some minor remarks:\n- Above Eq.(2) the authors write: “. However, this maximisation is not tractable in general [38, 39],” This is only true for multi-parameter case. For single parameter this problem is completely solved, and hence this sentence will be misleading for the readers, as immediately below the authors present single parameter formulas\n- Authors do not mention the possibility of using an ancilla with which the probe state is entangled. I guess in this case this would not help, as the quantum channel we consider is quantum-classical, but It would be nice at least to comment this.\n- Below Eq.3 Authors write “The maximization is not analytically tractable”. This is true, but in the light of what I wrote, above it is also slightly misleading, as there are efficient methods to find channel QFI with a simple SDP programme."}, "author_response": "We thank the reviewers for their insightful comments and constructive feedback, which have significantly improved our manuscript. Below we address the reviewers’ comments in detail.\nResponses to Reviewer 1\nWe thank the reviewer for their constructive comments and detailed suggestions. We have implemented their suggestions in our revised manuscript that has improved both its clarity and overall impact.\nThis paper introduces new tools to determine the ultimate precision limits of quantum detectors. Detector characterization involves determining the associated POVMs, a task addressed by quantum detector tomography (QDT). In QDT, an unknown element \\Pi k is characterized through a set of known measurements using known input probe states. By performing many repetitions on multiple copies of each probe state, one can estimate \\Pi k, thereby allowing the inverse problem to be solved.\nFollowing the pioneering work of Ian Walmsley’s group, QDT has been performed for a widerangeofdetectors,includingSPADs,time-multiplexeddetectors,transition-edgesen-sors,superconductingnanowiredetectors,andeventhehumaneye. Duetothecomplexity of the task, various advanced methods have been developed, such as the Fitting of Data Patterns, self-guided tomography, compressed sensing tomography, and techniques based on neural networks. To the best of my knowledge, all of these approaches rely on fidelity (or variants thereof) for benchmarking.\nWe appreciate the reviewer’s clear summary of the detector tomography problem that we address in ourwork. WeagreethatQDTiscurrentlyanactivetopicofresearchandseveralinnovativetechniques have been developed to address its various aspects.\nThe current work introduces a novel tool: the detector quantum Fisher information (DQFI). The concept is elegant and the theoretical development is clear. The authors propose two definitions of DQFI–one tighter than the other. While this is insightful, I would appreciate a clearer operational preference or justification for using one over the other.\nWe thank the reviewer for their positive evaluation of our work and appreciate the reviewer’s constructive feedback on the two alternative DQFI definitions. Our original manuscript compared these two definitions mainly on the aspect of tightness. While tightness is an important factor that guides the choice of the definition to be employed in practice, our revised manuscript now clarifies the operational preference in more direct terms.\nOn pg. 4, comparing the two definitions, we now include the following (above Theorem 1).\n“Nonetheless, the trace DQFI serves as a simpler upper bound that is more amenable to analytical evaluation.”\nFurther,inourrevisedConclusionandOutlooksection(pg.8),wenowexplicitlyincludethefollowing operational distinction between the alternative DQFI definitions.\n1\n“Inpractice,thismakesthespectralboundthedefaultbenchmarkbecauseitcloselytracks theattainableboundinseveralaspects(seenumericalcomparisonsinFigs.1&2inSupp. Mat. II), whereas the trace bound serves as a more convenient analytical approximation but can overestimate by a factor of the dimensions of the system in question.”\nOurrevisedmanuscriptnowclearlycommunicatesanoperationalbasisforchoosingbetweenthetwo DQFI definitions: one definition is tighter but more complex to compute, while the other is less tight but can be evaluated analytically.\nTheauthorsdrawananalogywiththestandardquantumFisherinformation(QFI).Butwe know that for pure states, QFI reduces to the variance of the generator. In this sense, QFI provides a lower bound on the mean square error (MSE), which can be related to fidelity. Fromabenchmarkingstandpoint,Iquestionwhetherthisrepresentsasignificantadvance over existing approaches in QDT.\nWe agree with the reviewer that for parameters encoded unitarily onto pure states, the variance of thegeneratorcanbeusedtolower-boundtheminimummean-squarederror(MSE).However,unitary dynamics cannot directly describe most noisy measurement processes, because the parameter dependencehereistypicallylessstructured. Forinstance,forthedephaseddetectorconsideredinExample2 (Eq.(10),maintext,pg.5),theunitary-basedreasoningfailsproducealowerboundtotheMSE.The sameistrueforthetomographyprobleminExample3(Eq.(14),maintext,pg.6),wheretheparam- eterdependencecannotbeconvertedintoaunitary. Insuchcases,theQFIcanonlyproduceafidelity bound for a specific input state; there is no notion of optimality or a universal bound.\nMore generally, for characterising measurements subjected to noisy channels, as well as for most measurement tomography problems, the variance of the generator does not produce a lower bound on MSE and the optimal probe states cannot be identified by this method. In contrast, our DQFI technique can produce tight precision bounds for such problems and identify the optimal probestates. In practical settings, the parameters of interest would often correspond to either lossy effects or noisy interactionswiththeenvironment(e.g.,darkcountsinsinglephotondetectors),neitherofwhichcan be accounted for unitarily. By producing informative precision bounds for these non-unitary settings, the DQFI makes a major advance over existing approaches.\nWhile the lower bound on MSE can be related to reconstruction fidelity, it is important to note that the Fisher information is not concerned with fidelity itself, but rather with the scaling of fidelity with the number of samples, N. This is because, from the quantum Cram´er-Rao bound, the minimum (unscaled) MSE decreases inversely proportionally with increasing N, meaning that fidelity increases with increasing N, ideally converging to 100% in the asymptotic limit. Effectively, our tools quantify the maximal rate of this convergence, thereby benchmarking how efficiently the task of QDT has been performed, which is not captured in the fidelity metric alone. When applied to experimental setups where fidelity is used as a metric, our tools can lead to higher reconstruction fidelities using reduced numbers of samples, via the optimal choice of probe states. More broadly, our work enables benchmarkingexistingapproachesinQDT,andcomparingalternativeproposals(e.g.,ancilla-assisted strategies, as proposed in Phys. Rev. Lett. 93, 250407 (2004) and demonstrated in Phys. Rev. Lett. 108, 253601 (2012)) to better understand their merits and limitations. In these ways, our work significantly advances the topic of QDT.\nForthemulti-parametersetting,ourproposedquantumCram´er-Raoboundalsoservesasafinermetric than fidelity, for instance allowing the prioritisation of parameters through a weight matrix. Notably, 2\nour proposed DQFI has a more natural connection to local fidelity, the differential version of reconstruction fidelity, that we address in Supplemental Material V. In summary, the variance of the generator cannot address the characterisation of general quantum detectors, and the quantum Cram´er-Rao boundandreconstructionfidelityaretwometricsthataddresstwodifferent,albeitrelated,aspectsof precision estimation.\nMore importantly, it is well known that a key requirement for QDT is that the set of probe states must be experimentally feasible. This means the states must be well-characterized themselves. In standard tomography once the QFI is known and the quantum Cram´er- Rao bound (QCRB) is shown to be saturable, one can advantageously identify the optimal probe states for characterization. The authors focus on the single-parameter case, where the situation is relatively simple. Can this be done for QDT?\nWe appreciate the reviewer’s thoughtful question. Our proposed DQFI can identify the optimal probe states for QDT, although the probing strategies here are more involved than in the single-parameter case, typically requiring a collection of different input states for completeness. As an example, we address the QDT of on-off detectors in Example 3 (main text pg. 6)), where we identify the experimentally-feasible optimal probe states, |0⟩ and |1⟩, using our framework and demonstrate that the estimation errors saturate our proposed quantum Cram´er-Rao bound.\nSimilar to standard state tomography, our developed tools can identify the optimal probe states for general QDT settings, once the QCRB is shown to be attainable. In such cases, the optimal pure stateprobesmaybederivedbyinspectingthecommoneigenvectorsacrosstheSLDoperatorsforeach parameter. Further, being single-qubit pure states, these optimal probes can be precisely prepared on several platforms, therefore being experimentally feasible. For instance, on currently available superconducting platforms, such states can be prepared with errors less than 0.01%, whereas the measurement imperfections typically characterised are of the order of 1% (as noted previously on pg. 14 of Supplemental Material IV).\nHowever, a complete QDT requires a multiparameter scenario, where the QCRB is generally not saturable. In standard QDT, the set of reference states must be tomographically complete; i.e., they must span the space of POVMs to enable inversion. In photodetector measurements, this is relatively straightforward. Coherent states of an optical beam serve as ideal candidates: they are overcomplete and can form a tomographically complete set by varying amplitude (e.g., via attenuation with a beam splitter) and phase (e.g., with a delay line). Crucially, such states are easy to generate using lasers.\nA question that remains is whether the proposed DQFI can be used to identify optimal probestatesforfullQDT.InthestandardmultiparameterQFIframework,onecanconvert theQCRBmatrixintoascalarbound(e.g.,viaatracewithaweightingmatrix)andthereby derive optimal states. Would a similar approach be possible or meaningful in the DQFI context?\nThe reviewer is correct that in multi-parameter QDT, the set of reference states must be tomographically complete, which is also a key requirement of our approach (see Methods, section Multi- parameter detector estimation, pg. 9). As coherent states are experimentally accessible in most optical setups, we agree that they constitute a convenient probing strategy for complete QDT of photodetectors. 3\nHowever, the precision achievable by coherent probes is limited by their quadrature uncertainties. More importantly, it has remained unclear how such performance compares with the ultimate precision allowed by quantum mechanics, as well as what the corresponding optimal probe states are that could guide future practice. Our new DQFI framework addresses this challenge, which holds both fundamental and practical significance – particularly in light of the rapid advances in generating high-quality quantum light sources such as squeezed states, whose reduced quadrature uncertainties could lead to higher sensitivity to detector parameters. We have added the following sentences to the Discussion section to acknowledge the importance of this issue and to outline future directions.\n“Whereas the Fock basis probes that are generally optimal for phase-insensitive measurementscanbegeneratedusingheraldedmeasurements,lowstategenerationfidelitiescould limit the actual enhancement in practical setups. The exploration of more accessible continuous-variableinputprobes, including displacedsqueezedprobeensembles, remains the scope of future work.”\nWe agree that in standard multi-parameter state estimation, the QCRB matrix can be converted to scalar bounds via a trace with a weight matrix. This can then be used to derive the optimal measurements,givenbytheeigenvectorsofdifferentSLDoperators. Similarly,boththetraceDQFImatrixand the spectral QCRB can accommodate a weight matrix and lower-bound the corresponding weighted sum of parameter variances and covariances. We revise Supplemental Material VII to explicitly include these weighted bounds (trace bound in Eq. (SM.VII.15) and spectral bound in Eqs. (SM.VII.17) & (SM.VII.19), pg. 20), and we clarify this point in the revised manuscript as follows (under Multi- parameter detector estimation and tomography on pg. 5).\n“For simplicity, here, we set the weight to identity and minimise total MSE of the parameters, though our techniques also apply to arbitrary weight matrices (see Supp. Mat. VII).”\nIn terms of the optimal probes, we stress that the proposed DQFI framework can be used to identify optimal probe states for multi-parameter QDT if the chosen DQFI bound is tight for the particular problem. For highly symmetric or commuting problems, the spectral bound can be attainable and the optimal pure probes are the common eigenstates of the SLD operators, considered one by one parameter-wise. One such case is phase-insensitive tomography, presented in main-text Example 3 (pg. 6), where the spectral DQFI explicitly identifies the optimal ensemble over basis states with analytically determined mixing. This physically relevant example has been the subject of extensive investigation in previous QDT works (including Nat. Phys. 5, 27 (2009) & New J. Phys. 11, 093038 (2009)). However,wenotethatforthemostgeneralfullQDTsetting,theDQFIboundisnotexpected tobetight,andtheoptimalprobestatesremainunknown. Thisisnotsurprising,giventhattheDQFI’s counterpartinstateestimation,themulti-parameterSLDQFI,isalsonotgenerallytightandtherefore doesnotidentifytheoptimalmeasurementswhenunattainable. OurrevisedConclusionandOutlook section clarifies this point (pg. 8).\n“Apart from tighter information measures for multi-parameter measurement models that could help identify the optimal probe states for full tomography, future works could probe the quantum-geometric properties of measurements by employing other information metricsfromthePetzfamily,oraddressthepracticallimitationsofthelocalestimationregime through a Bayesian approach.”\nNonetheless, our current bounds are tight for single-parameter models and some multi-parameter models, and therefore reveal useful optimal probe information about these settings. 4\nIn summary, this is a well-written and rigorous work introducing new concepts that may prove important in refining the limits of QDT. I recommend it for publication. However, I believe that addressing the concerns mentioned above would enhance the readability and practical impact of the paper.\nWe appreciate the reviewer’s positive evaluation of our work. We are grateful for their recommendations and have accordingly revised our manuscript to achieve better readability and practical impact. WehaveclarifiedanoperationalpreferencebetweenthevariousDQFIdefinitions,discussedtheimplications for high-fidelity POVM reconstruction, and highlighted how our framework can help identify experimentally-feasible optimal probe states in both the single- and the multi-parameter settings. 5\nResponses to Reviewer 2\nWe thank the reviewer for their positive and helpful comments. The reviewer suggests several interesting and fundamental points, and we have performed new theoretical analysis in order to address theircomments. Webelievethattheserevisions,promptedbythereviewer’sfeedback,haveimproved the clarity and quality of our work. We detail our responses below.\nThe authors investigate the precision achievable in estimation of parameters of general quantum measurements. The authors introduce detector quantum Fisher information, investigate its properties and present applications of the established formalism. The present studyiscomprehensiveanditnicelycomplementstheearlierresultsestablishedofoptimal estimation of parameters encoded in quantum states. As pointed out by the authors in the Supplemental material, there are close connections between quantum measurements and quantum processes. The quantum measurement can be seen as a specific instance of a quantumprocess(orquantumchannel),wheretheoutputstateisoneoforthogonalstates associatedwiththemeasurementoutcomes,c.f. Eq. (SM8.3)whentracedoverthesecond degree of freedom. Nevertheless, the quantum measurements play so important role that it is fully justified to treat them separately and establish a theory which provides bounds on precision of estimation of parameters encoded in the measurements. Given the central role that sophisticated quantum measurements play in modern quantum technologies and quantumcomputingarchitecturesthepotentialimpactandimportanceofthepresentwork are high enough to in principle deserve publication in Nature Communications.\nWe thank the reviewer for their concise but insightful summary of our study and its practical impact. We appreciate the recognition of detector quantum Fisher information as a comprehensive and complementary framework to state-parameter estimation. In particular, the observation that quantum measurements can be framed as channels while still warranting a dedicated precision theory aligns closely with our motivation and perspective. We are grateful for the reviewer’s assessment that the scope and relevance of our results meet the high bar set by Nature Communications, given their potential impact for modern quantum technologies where measurement sophistication is important.\nThere are several points that should be considered by the authors before the manuscript could be published:\n1)IntheSupplementalMaterial7theauthorsconsiderascenariowheretwocopiesofthe measurement are available. In this case, entangled probe states may be optimal and may provide advantage. What about using entangled probes when probing a single copy of the measurement. Suppose a pure bipartite entangled states of systems A and B is prepared andthemeasurementisappliedtosystemA.Subsequently,alsosystemBismeasuredand the measurement on B can depend on the outcome of the measurement performed on system A. Could such scenario provide any advantage with respect to using only a single system A to probe the characterized quantum measurement? Note that the entanglement of A and B need not be maximal.\nWe thank the reviewer for raising this interesting question. While our original manuscript studied the role of entanglement in two-copy detector estimation and showed the possibility of a quantum advantage, we did not sufficiently address the situation where entangled probes are used to probe a 6\nsingle copy of the measurement. In the revised manuscript, we analyse this interesting setting from twoperspectivesanddiscussthepossibilityofanadvantagearisingfromtheresourceofentanglement.\nFirstly, we consider the scenario where the probe and ancilla are measured independently—that is, no conditional measurement is performed. This setting had been partially discussed in Methods in the original manuscript (pg. 9), where we proved the equivalence between entangled probes and ensemble probes, indicating that without joint measurements on the probe and ancilla, there is no entanglement advantage in detector estimation. We further clarify this in the revised manuscript as follows (Methods, pg. 10).\n“The equivalence between bipartite-entangled states and ensemble probes means that unless the probe and ancilla are measured jointly, the convexity of the CFI (Lemma 1) rules outanyentanglementadvantageinprecisioninthesingle-parametersetting. However,for the most general estimation scenario, ancilla measurements conditioned on the detector outcome should be considered.”\nSecondly,intherevisedmanuscript,weextendthisresulttogeneralancilla-assistedprobingstrategies where conditional measurements are performed on the ancilla subsystem, as illustrated in Fig. 1(b) below. We present these results in a new section (II) in the Supplemental Material. More concretely, we consider the measurement Π ⊗1 and call it the ‘extended detector’. Here 1 represents the θ A A identitychannelontheancillaspaceA,andtheoutputsoftheextendeddetectorarejointlymeasured. Based on previous literature (J. Phys. A: Math. Theor. 41, 255304 (2008)), we know that for characterising general quantum channels, there is no further processing required on the ancilla arm beyondmeasurement. Developingourmethodfordetectorestimation,wederivean‘extendedbound’ thatisknowntobetightfortheextendeddetector. Finally,weprovethatthisextendedbound,which capturestheperformanceofancilla-assistedprobingtechniques,canbeattainedbyasingleseparable probe state, thereby ruling out any advantage from entangled probe states in the single-parameter setting. To summarise these new results, we add the subsection Tight DQFI for Single-Parameter Estimation to the Methods section (pg. 11).\n(a) (b)\nDetector Extended Detector\nρ j j\nin\nρ\nin\nk\nA\nj\nFigure1: Comparisonofdetectorestimation(a)andextendeddetectorestimation(b). Estimatingthe extended detector requires an ancillary system (A) and conditional ancilla measurements (Πj) [12].\nItis,however,worthwhiletonotethatthesituationformulti-parameterestimationcouldbedifferent. Infact,themulti-parametersettingtypicallyrequiresacollectionofseparableprobestates,sousinga single entangled probe state instead could provide an operational simplification (Phys. Rev. Lett. 90, 193601 (2003)). Further, experimental works using non-classical probe states for multi-parameter measurement calibration have demonstrated improvements in convergence and robustness for this setting (Phys. Rev. Lett. 108, 253601 (2012)). We summarise these points in the revised Methods section (subsection Multi-parameter detector estimation, pg. 10). The role of entangled probes in multi-parameter estimation is an interesting question that merits thorough investigation, which we leave for future work. 7\n2) The authors base their work on the widely utilized concept of Fisher information. To saturate the Cramer-Rao bound, an unbiased estimator is usually required. The authors should clarify in more details which estimators do they have in mind in general, whether they assume the maximum likelihood estimator in the asymptotic limit of very large number of samples.\nWe thank the reviewer for raising this important question. The reviewer is correct that the concept of Fisherinformationappliestounbiasedestimation. Inourmanuscript,weconsiderthelocalestimation setting, where the unknown parameters are close to known true values, and assume locally-unbiased estimators. However, we do not specifically assume the maximum likelihood estimator, which is a widely-used classical estimator that is asymptotically-unbiased and therefore a valid choice in that limit. We would like to point out that the choice of classical estimator for detector estimation has been explored in depth in previous literature (cited in the revised manuscript). Our current work, by contrast,focusesonthequantumaspectofdetectorestimationandidentifiestheoptimalprobestates.\nWe clarify these points in the revised manuscript (Methods, subsection Local Unbiased Estimation, pg. 9) as follows:\n“Local Unbiased Estimation. In the local estimation setting, where unknown parameters θ are close to known true values θ∗ [17, 30], the Fisher information framework requires locally-unbiased estimators, meaning estimators with zero bias at the true parametervalues[17,35,69]. Effectively,thisconstrainshowtheprobabilisticdetectoroutcomes are mapped to corresponding predicted values for the unknown parameters [20]. In linear estimation [20, 28], this is done through a classical estimator function represented by anm×nrealmatrixΞ that, givenadetectoroutcomej ∈ [m], predictstheestimateΞ jk jk (cid:80) for θ (k ∈ [n]). The overall estimate produced for each θ is then p Ξ . The local- k k j jθ jk unbiasedness constraint on the estimator [69] now translates to ( (cid:80) j p jθ Ξ jk )| θ=θ∗ = θ k ∗. The widely-used maximum-likelihood estimator [20, 24] is a suitable example, because it is asymptotically unbiased, and therefore can saturate the CCRB in that limit [20].”\n3) In the Supplemental Material 3, the authors provide an example of a linear estimator of parameter p, Eq. (SM.3.2). What are the signs of the coefficients alpha and beta in this estimator? It seems that one of the coefficients is negative. In principle, the estimator can yield a negative value of the estimated parameter, which would be unphysical. This may happen in case of low number of samples and if the true p is close to 0. While such situations may be rare in occurrence, the authors should comment on this, because e.g. changing the negative p to p=0 could introduce a bias in the estimator.\nWe thank the reviewer for their insightful comment. The reviewer is correct that one of the coefficients in the linear estimator is negative. In fact, an increase in dephasing level shrinks the Bloch ball inwards, thereby moving the two outcome probabilities closer to each other for any projective measurement. This means that the dephasing strength is positively correlated with one probability and negatively correlated with the other probability, so the negative sign is physically justified.\nHowever,therevieweriscorrectthatthislinearestimatorcouldinprincipleyieldanegativeestimate. This could happen for true values close to 0 and for a low number of samples, because otherwise, with increasing samples, the shrinking variance would eventually guarantee a positive estimate for any interior-point true value. In this setting, the cause of the unphysical estimate would be that the physical parameter is bounded but our estimation does not enforce this constraint. While our 8\ndesign of the dephasing experiment was based on prior knowledge of true values (being 0.2 for p), in the above hypothetical setting, one has a few alternative options to avoid the unphysical estimate. For true values close to the boundary, one could use constrained maximum-likelihood estimation, explicitly enforcing 0 < p < 1, so as to obtain unbiased, physical estimates. Alternatively, because the Cram´er-Raoboundcanoftenbeunattainableinthefinite-sampleregion,onecouldfurtherrepeatthe experimentwithoutdiscardingormodifyinganyempiricaldata,terminatingwhenaphysicalestimate is achieved. If the number of samples cannot be increased, one could employ alternative bounds like Bhattacharyya bounds or Barankin bounds, which are better suited to finite-sample statistics, or shift from point estimation to region estimation, which could provide more accurate error bars.\nMore broadly, quantum estimation in the low-sample number regime is an active area of research concerned with developing effective approaches to precision bounds that circumvent the limitations of the Cram´er-Rao bound in this setting.\nWe clarify these points in the revised manuscript (Supplemental Material IV, pg. 14) as follows.\n“Notably, the linear estimator pˆcould, in principle, yield unphysical estimates if the true value p∗ were close to 0 or 1 and the number of samples was insufficient. For large numbers of samples or true values inside the interior, this is unlikely to occur due to the the- ory of large deviations [21]. Nonetheless, for ensuring physical estimates at low sample numbers, alternative methods like constrained maximum-likelihood could be employed forunbiasedestimation,orregionestimationcouldbeusedinsteadofpointestimation,or Bhattacharyyabounds[22]orBarankinbounds[23]couldbeusedinplaceofCram´er-Rao bounds.”\nIn any case, local estimation is typically infeasible at the boundary points, so additional care must be taken if the true values are close to either boundary. Our revised discussion of the linear estimator forthedephasingexperimentnowaddressestheseissues,andsuggestspossibleworkaroundstoavoid them.\n4) The authors should check Eq. (SM.8.16).\nWe thank the reviewer for pointing out this typo. We have corrected the equation in question (now Eq. (SM.X.16) in the revised Supplemental Material X on pg. 29).\n5)InFigure1intheSupplementalMaterial,symbolthetadenotesboththeangleofaprobe quantumstateandalsotheunknownparametertobeestimated. Theauthorsshouldcheck the notation used in this figure.\nWe thank the reviewer for spotting this confusing notation. We have revised Fig. 1 (now Fig. 4 in Supplemental Material III, pg. 10) and the notation used in the accompanying discussion so as to consistently use p for the unknown parameter in this section.\nIn closing, we once again thank the reviewer for their comments. The reviewer raised several interesting points, and we have addressed these comprehensively in the revised manuscript. Our revisions incorporatenewtheoreticalresultsanddiscussionsintotwonewsubsectionswithintheMethodssec- tion, and a new section in the Supplemental Material. We hope that the reviewer finds our revised manuscript suitable for publication in Nature Communications. 9\nResponses to Reviewer 3\nWe thank the reviewer for their time assessing our manuscript and for their detailed and constructive feedback. The reviewer raises several fundamentally important and thought-provoking points, and we have carried out major revisions including both new theoretical analyses and new numerical experiments in addressing their comments. Our revisions reveal measurement-specific physical insight and favourable comparisons against multi-parameter channel estimation techniques. We present our detailed responses below.\nThe authors of the paper, propose the concept of detector quantum fisher information (DQFI), which in their words, feels the gap in the triad “State, Process and Measurements”. TheyclaimthatwhileStateandProcessestimationarewellinvestigatedfields,the Measurement estimation problem is not. They derive the upper bound on the achievable classical information in terms of DQFI, which they show to be saturable for a subclass of diagonal povms. They propose also a generalization to multi-parameter case, and discuss an experimental using the IBM quantum computing platform.\nWe thank the reviewer for their comprehensive evaluation of our work. We kindly point out that our derivedupperbound,thespectralDQFI,issaturableforastrictlylargerclassofPOVMsthandiagonal POVMs, as evidenced, e.g., by Example 2 in the main text.\nThepaperisinmyopinioncorrect,andalsohighlightsinaninterestingwaytherelevance of measurement estimation problem. There is a serious issue with the paper, however, which authors are either not aware or intentionally avoid discussion of.\nWeappreciatethereviewer’srecognitionthatourworkiscorrectandinterestinginsomeaspects. The reviewer raises a valid concern that we address in depth below and in the revised manuscript.\nAny quantum measurement is a special case of a quantum channel, more concretely a quantum-classical channel! Given any generalized measurement, with measurement operators \\pi j, I can construct a corresponding channel with Kraus operators K j = |j⟩ \\sqrt{\\pi j}, where outputs represent measurement outcomes.\nIcannowusealltheresultsfromquantumchannelestimationtheorytoobtainchannelQFI – which in this case will correspond to DQFI. In particular, in J. Phys. A: Math. Theor., 41 p. 255304(2008),itisshownthatchannelQFIisupperboundedby4|\\sum i\\dot{K} i† \\dot{K} i |, where the norm is the spectral norm, and \\dot{K} i are derivatives of Kraus operators. In our case since, we can plug in operators K j = |j⟩ \\sqrt{\\pi j}, and use identity\\dot{(K i† K i)}=\\dot{K i}† K i+K i† \\dot{K i}. toeffectivelygettheresult(6) obtained by the authors.\nThe reviewer is correct that a measurement process can be represented via Kraus operators for a quantum-classical channel. In case of detectors, because only the classical measurement outcome is retained and not the output quantum state, the appropriate Kraus operators are K = |x⟩⟨j| π , x,j x where index j ∈ [d] labels a basis {|j⟩} for the Hilbert space H and index x ∈ [m] labels the clas- d sical outcomes. In total, there are md number of Kraus operators for an m-outcome detector in d dimensions. 10\nWe agree with the reviewer that, in principle, the channel QFI method from J. Phys. A: Math. Theor., 41 p. 255304 (2008) could be applied to the Kraus operators K = |x⟩⟨j| π representing the x,j x measurement process to derive precision bounds for detector estimation. However, there are a few points to be clarified.\n√\nFirst, inserting the Kraus operators K = |x⟩⟨j| π into the channel QFI upper x,j x bound 4∥ (cid:80) K ˙† K ˙ ∥2 (with ∥ · ∥2 denoting the spectral norm) produces an upper bound to x,j x,j x,j sp sp theDQFIbutdoesnotingeneralcorrespondtothespectralDQFIdefinedinEq.(6)ofthemanuscript. The identity mentioned by the reviewer, ∂ (K † K ) = ∂ (K )†K + K † ∂ (K ), reduces θ x,j x,j θ x,j x,j x,j θ x,j to ∂ ( π ) π + π ∂ ( π ) = ∂ (π ) for detector estimation. Although this identity resembles θ x x x θ x θ x the defining equation L π +π L = 2∂ (π ) for the symmetric logarithmic derivative (SLD) opera- x x x x θ x√ √ tors, they are not equivalent. This is because π L is not necessarily Hermitian, whereas ∂ ( π ) is x x θ x Hermitian. In fact, the non-Hermitian component of this derivative-like quantity plays an important role in regard to tightness of the resulting bound, and therefore should not be neglected. As a result, √ √ we cannot equate π L = 2∂ ( π ), and therefore, the channel QFI upper bound obtained thus x x θ x does not generally equal the DQFI definition in Eq. (6).\nIn addition, we note that the bound obtained by following the suggested channel-based transfor- √ mation, which is 4∥ (cid:80) ∂ ( π )2∥2 , is generally weaker than the bound ∥ (cid:80) L π L ∥2 defined in x θ x sp x x x x sp Eq. (6). As an example, for the dephased detector estimation problem (Example 2 in the main text), the bound obtained by directly using the channel QFI upper bound is not tight (without additional optimisation) whereas we demonstrated our bound introduced in Eq. (6) to be tight. This example is discussed further below.\nFurthermore, there is a significant practical advantage of our DQFI framework over existing channel- based techniques in the context of detector estimation. In the latter, one must consider md Kraus operators and their derivatives. For measurement channels that involve many Kraus operators, such as for qudits or for truncated continuous variable systems, alternative representations that are more compact and convenient could be desirable for both computational efficiency and analytical tractabil- ity. Thisisparticularlythecaseforanalysingphotonicdetectors,thecharacterisationofwhichwasone of our main motivations, and where the system dimension d is typically much larger than the number ofmeasurementoutcomesm(d ≫ m). Theon-offdetectortomographedexperimentallyinNat. Phys. 5, 27 (2009) would require 400 Kraus operators (d = 200, m = 2) and the photon-number-resolving detector tomographed there would require 540 Kraus operators (d = 60, m = 9). This is exactly the regime in which we envision the DQFI as being most powerful – as the m SLD operators are defined implicitly, we have a fundamental limit that can be easily evaluated even for large systems.\nMoreover,theaboveresultfromquantumchannelestimationtheoryismuchstrongerthan the result presented by the authors, as one can prove its general saturability provided one minimizes the channel over different Kraus representations. Furthermore, this problem may be implemented as a simple SDP programme as was shown in Nat. Commun. , Vol. 3 p. 1063 (2012). Hence, these tools are much more powerful than the ones proposed by the authors.\nWe agree with the reviewer that when an additional minimisation over all equivalent Kraus operators isperformed,theminimumchannelQFIupperboundshouldbesaturable,i.e.,shouldequalthemaxi- mumCFIoverprobestatesforsingle-parameterdetectorestimation. Inprinciple,theSDPformulated in Nat. Commun., 3, 1063 (2012) is applicable to this problem, by relaxing the β = 0 constraint K˜ and setting N = 1. At the same time, we would like to note that with additional minimisation via 11\nan SDP program, our spectral DQFI also produces the tight bound (elaborated in a subsequent re- sponse point on pg. 16 of this letter), as we prove in our revised work (Theorem 4, Supplemental Material IIF, pg. 8). Accordingly, in the revised manuscript, we compare our DQFI bounds (analytical trace and spectral DQFIs, and numerical spectral DQFI with SDP) against the two single-parameter channel QFI-based bounds: (i) the analytical channel QFI upper bound (J. Phys. A: Math. Theor., 41 p. 255304 (2008)) and (ii) the numerical bound obtained via its SDP minimisation (Nat. Commun., 3, 1063 (2012)). In light of these comparisons, we emphasize that our DQFI framework makes three key advances compared to existing channel QFI-based approaches, as we summarise below. We dis- cusstheseaspectsinmoredetailinlatersectionsoftheresponseletter(pg.16)andinanewly-added section (II) in the Supplemental Material.\nAdvance 1: Tight DQFI for Single-Parameter Setting\nIn the revised manuscript, we address the tight bound for single-parameter detector estimation (Methods subsection Tight DQFI for Single-Parameter Estimation (pg. 11) & Supplemental Mate- rialII(pg.4)). Weestablishthisboundusingaminimisationofthespectralboundovernon-Hermitian derivative operators (Lemma 2 in Supplemental Material IIE, pg. 6). We also show an alternative derivation of this bound based on quantum channel estimation theory, inspired by the reviewer (Sup- plemental Material IIB & C, pg. 4). While the proof of tightness requires a combination of the two approaches (Theorem 4, Supplemental Material IIF, pg. 8), our framework leads to a simplified SDP formulation of the tight bound that can be more efficient than the Kraus-formalism SDP and more di- rectly applied to detector estimation problems. More importantly, our simplified SDP admits a partial analytical solution that explains the optimal non-Hermitian derivatives that solve this minimisation (Lemma 3 in Supplemental Material IIF, pg. 7). This explanation provides physical insight into the attainable precision limit for the most general detector models for estimating a single parameter.\nAdvance 2: Direct Identification of Optimal Probes\nAn important question in detector estimation is that of the optimal probe state, which is not directly addressed within the framework of channel estimation theory. While numerical techniques could po- tentially brute-force the search for optimal probe states, more physical insight such as the connection between the optimal probe state and the optimal Kraus derivatives (or optimal measurement deriva- tives) is currently lacking. Remarkably, our DQFI framework goes beyond existing techniques by directly identifying the optimal probe state and providing its general physical characterisation – as a simultaneous eigenstate of the SLD operators defined in our approach (Theorem 3 for spectral bound (Methods section, pg. 10) and Lemma 3 for tight bound (Supplemental Material II, pg. 7)). This characterisation applies to all single-parameter detector models and, therefore, significantly advances our general understanding of a physically important aspect of detector estimation.\nAdvance 3: Computational Efficiency and Analytical Bounds\nAsmentionedinthepreviousresponsepoint,adirectapplicationoftheKraus-formalismSDPtodetec- torestimationischallengingbecauseofthelargenumberofKrausoperatorsinvolved. Ourdeveloped tools address exactly this challenge – providing analytical expressions involving the POVM and its derivative that can be tight under a well-characterised set of conditions. Our spectral bound repre- sentsageneralanalyticalsolutiontothenon-trivialminimisationoverKrausoperatorsforalldetector problems satisfying the attainability criteria (as discussed in Supplemental Material IID, pg. 5). For general single-parameter detector estimation problems, our analytical bound can outperform the an- alytical channel QFI-based upper bound (as discussed in detail in the previous response point) and provides a good approximation to the tight bound, which typically needs to be evaluated numerically. We demonstrate the efficacy of this approximation by performing new benchmarking experiments on randomly-sampledqubitdetectormodels. Wereporttheseobservationsinmoredetailinasubsequent response point (Figs. 5–6 on pg. 17–18 of this response) and in a new subsection (IID) in the Supple- 12\nmental Material. This analytical development could enable broader physical and geometrical insights into general features of detector estimation.\nIn summary, we have shown that our analytical spectral bound can be more informative than the analytical channel QFI-based bound, while directly identifying the optimal probe states. Both the spectral bound and the channel QFI-based upper bound can produce the tight bound via additional numerical minimisation (SDP); more importantly, our perspective provides valuable physical under- standingoftheoptimalprobestatesachievingthisattainableprecisionlimit. Therefore,webelieveour fundamentally-motivatedformalismmakessignificantadvancesonthemeasurementcharacterisation problem that would be of independent interest.\nRegarding the multi-parameter case, it should be stressed here, that the non-trivial issue here is the ‘probe-incomplatibilty’ , which has been also investigated thoroughly in the quantum channel estimation literature, see e.g. Phys. Rev. X , Vol. 12 p. 011039 (2022). Inparticularinthisreference,ageneralizationofFujiwaraboundtomultiparametercaseis given. Ipresumethatthisboundagainwouldallowforamoreefficientnumericalanalysis than the bound proposed by the authors, as it again may be solved using the SPD. This at least should at least be tested and compared.\nWethankthereviewerfortheircomment. Weagreethatprobeincompatibilityplaysanimportantrole inmulti-parameterdetectorestimation. Inresponse,wediscussprobeincompatibilityinSupplemental Material VIII and add the following discussion on pg. 7 of the main text to highlight this matter.\n“Probe incompatibility effect. An interesting feature of Example 3 is that the optimal multi-parameter probing strategy is a convex mixture of the single-parameter optimal probes, namely, |0⟩ and |1⟩ for θ and θ , respectively. Although this may not hold for 1 2 generic multi-parameter models, it underscores the fact that when the optimal single- parameterprobesfordifferentparametersareincompatible,termedprobeincompatibility inRef.[57],thereisanuncertaintytrade-offinsimultaneousmulti-parameterestimation.”\nTherevieweriscorrectthatthemulti-parameterboundfromPhys. Rev. X,12011039(2022)(Single- channel use (N = 1) bound in Appendix F therein) can be applied to detector estimation. We refer to this bound as the total QFI QCRB, and compare it against our multi-parameter bounds in the re- vised manuscript. More specifically, we numerically test the two approaches by performing random detector-sampling experiments—generating 10,000 random multi-parameter qubit detector estima- tion problems—and report the results in a newly added section (VIII) in the Supplemental Material. Thecomparison,shownbelowinFigs.2and3,showsthatourmulti-parameterspectralQCRBisgen- erally, albeit not always, tighter than the total QFI QCRB (Fig. 3(a)). Further, as seen in Fig. 3(b), the trace QCRB can also be tighter than the total QFI QCRB, which is notable given that the former is analytically solved while the latter typically requires numerical evaluation of an SDP. 13\n100\n80\n60\n40\n20\n0\n0 1000 2000 3000 4000 5000\nSorted Runs\nsdnuoBrewoLnoisicerP\nMulti-parameter Qubit Detector Estimation Multi-parameter QubitDetector Estimation\nTraceQCRB(Closed-form) 25\nTotal QFI QCRB(SDP)\n20\n15\n10\n5\n0\n0 1000 2000 3000 4000 5000\nSorted Runs\nsdnuoBrewoLnoisicerP\n(a) (b)\nSpectralQCRB(SDP)\nTraceQCRB(Closed-form)\nTotal QFI QCRB(SDP)\nFigure 2: Comparison of the total QFI QCRB and the DQFI QCRBs for two-parameter, on-off, qubit detector estimation. (a) For two-parameter estimation across 10,000 randomly-generated qubit mea- surements, we find that the spectral QCRB (red) is typically tighter than the total QFI QCRB (light blue), but not always. (b) Zoomed-in version of (a). The scatter points are sorted in the increasing order of the spectral QCRB.\n35\n30\n25\n20\n15\n10\n5\ny = x\n0\n2 4 6 8 10 12 14\nBRCQ\nlartcepS\nBRCQ\necarT\n(a) (b)\n20\n15\n10\n5\ny = x\n0\n0 2 4 6 8 10\nTotal QFI QCRB Total QFI QCRB\nFigure 3: One-to-one comparison of the spectral QCRB (a) and the trace QCRB (b) versus the total QFI QCRB for two-parameter estimation from 10,000 random, two-outcome, qubit detectors. (a) The spectral QCRB is often tighter than the total QFI QCRB, evidenced by a majority of the points lying above the y = x line. (b) The trace QCRB (closed-form) can also be more informative than the total QFI QCRB (SDP).\nIn addition, our comparison reveals that our spectral DQFI approach is substantially more computa- tionally efficient than the total QFI approach for detector estimation, because the latter has to con- sider multi-parameter derivatives of md Kraus operators, as opposed to multi-parameter derivatives of the m POVM elements required for our spectral bound. Here m denotes the number of mea- surement outcomes and d denotes the dimensions of the quantum system being considered. Resul- (cid:112) tantly, the total QFI SDP has a worst-case complexity of order O( d(1+nm2)log(1/ε)) (number of time-steps for desired accuracy ε) in contrast to the spectral QCRB’s worst-case complexity of or- (cid:112) der O( 2(n+d)log(1/ε)) (where n denotes the number of parameters). Moreover, our proposed multi-parameter trace QCRB is solved in closed-form, and therefore can be readily applied to any multi-parameter problem with no need for numerical optimisation; this bound, therefore, is the most straight-forward to evaluate out of the three considered bounds.\nWhileprobeincompatibilityisanimportantaspectofdetectorestimationthatwehavenowaddressed, there remain other sources of incompatibility in the multi-parameter setting that also merit further 14\ninvestigation. Here, we only provide an illustrative example demonstrating these other effects: a four-outcome, two-parameter example where probe incompatibility can be neglected (Example 5 in Supplemental Material VIII, pg. 22). We compare our proposed detector QCRBs with the total QFI QCRBtofindthatinthiscase,thespectralQCRBisalwaystighterthanthetotalQFIQCRB(asshown in Fig. 4(b) below). Notably, we find that the total QFI QCRB can be off by more than a factor of 2, suggesting the presence of additional incompatibility effects beyond measurement and probe incompatibility in multi-parameter detector estimation.\n0.25\n0.20\n0.15\n0.10\n0.05\n0.00\n0 1000 2000 3000 4000 5000\nSortedRuns\nsdnuoB\nrewoL\nnoisicerP\nTwo-ParameterQubitDetectorEstimation\n0.25\nTotalQFIQCRB (SDP)\nSpectralQCRB (SDP)\nTraceQCRB (Closed-form) 0.20\n0.15\n0.10\n0.05\n0.00\n0.00 0.05 0.10 0.15 0.20 0.25\nTotalQFIQCRB\nBRCQlartcepS\n(a) (b) Two-ParameterQubitDetectorEstimation\ny = x\nFigure4: ComparisonofthetotalQFIQCRBandtheDQFIQCRBsforatwo-parameterdetectormodel without probe incompatibility (Example 5). (a) For estimating two parameters 0 < θ < 1/2, we find j that the spectral QCRB (black) is always tight and equals twice the trace QCRB (red). The total QFI QCRB (light brown) is generally less tight than the spectral QCRB, and sometimes less tight than the trace QCRB as well. (b) A direct comparison of the spectral QCRB versus the total QFI QCRB shows the former to be generally tighter (all the points lie above the y = x line), whereas the latter could be close to zero even if simultaneous estimation is feasible. In (a), the scatter points are sorted in increasing order of the spectral QCRB.\nIn conclusion, we find that our newly-introduced precision bounds can be more informative and ef- ficient than previous techniques for multi-parameter detector estimation. Although we believe there remains more to be studied to fully understand the attainable multi-parameter precision limit, we leave this to future work. 15\nIn the light of these results, the novely of the presented work is not as high as might seem at a first sight. Still, despite this harsh criticism, I would not formulae here a negative recommendation yet. Provided the Authors are able to show that they are able to propose an improved tools compared to the ones I mentioned above, making use of the particular structure of the quantum-claasical channel that represents the measurement process and showing that they are superior to the all-purpose channel estimation methods.\nIn particular, if authors were able to show that minimization over different Kraus representations, that is required to find the channel QFI in the general case, can be simplified for the detector QFI case – ideally solved analytically! – and show some insight behind it I would regard this as an important contribution to the field.\nWe appreciate the reviewer’s perspective and thank them for their insightful comments. We believe that while certain channel-based techniques can indeed be adapted to detector estimation, this does notdiminishthenoveltyofourresults. Onthecontrary,thenoveltyofourcontributionarisesprecisely fromdevelopingtoolstailoredtodetectorestimation. Aswehaveemphasisedabove,thereareseveral key advances enabled by the DQFI, namely: 1) Tight DQFI for Single-Parameter Setting, 2) Direct Identification of Optimal Probes, and 3) Computational Efficiency and Analytical Bounds.\nFurthermore, the multi-parameter comparisons discussed above (and in Supplemental Material VIII) have shown that our proposed multi-parameter bounds can be more powerful than existing channel- based techniques. For both generic random models (Figs. 2 & 3 above) and specific hand-crafted examples (Fig. 4 above and Example 5 in Supplemental Material VIII, pg. 22), our spectral QCRB can produce tighter precision bounds and does so using a more efficient numerical program. Moreover, thesetoolsuncovermulti-parameterestimationphenomenathatcannotbedirectlyaddressedusinges- tablished techniques. For instance, we show that there exist scenarios where simultaneous estimation can outperform individual estimation by more than a factor of 2 in precision, and where simultane- ous multi-parameter estimation may be feasible though individual estimation is not (Supplemental Material VIII, pg. 23).\nRemarkably,ournewtechniquescanalsooutperformexistingtechniquesatthesingle-parameterlevel. As we show in Supplemental Material II, our fresh perspective on detector estimation leads to a good analytical approximation, and sometimes even an analytical solution, to the detector extension-based boundfromNat. Commun.,3,1063(2012),whichotherwiseneedstobenumericallyevaluatedusing an SDP. In doing so, our results uncover an interesting connection between the optimal derivative operators, the optimal probe state, and the SLD operators defined in main-text Eq. (4). These SLD operatorsareuniquetoourframeworkanddistinguishitfrompreviousworks. Theyexplicitlyexploit theparticularstructureofthequantum-classicalchannelthatrepresentsthemeasurementprocess,and their significance is underscored by the observation that they represent a Hermitian approximation of the optimal logarithmic derivatives that minimise the detector extension bound. Importantly, this Hermitian approximation offers several physical insights into detector estimation that also generally hold true for the tight bound for this task. For instance, we show in Supplemental Material IIF that the optimal probe state is a common eigenvector of the optimal non-Hermitian SLDs (Lemma 3, pg. 7), which is the same relationship that was obtained for the spectral bound and Hermitian SLDs in main-text Theorem 3. For the practically-relevant class of phase-insensitive measurements, our development exactly reproduces the attainable precision limit and therefore can be readily applied to enhanceestimationabilitiesinpracticalsetups. Theseresultssignifythemajorimprovementsenabled bytheDQFIovercurrentlyavailabletoolsandadvanceourunderstandingofattainableprecisionlimits and optimal probing strategies for detector estimation. 16\nLastly, inspired by the reviewer’s comments, we perform a simplification of the minimisation over Krausrepresentationsfordetectorestimationandshowthatthesimplifiedboundagreeswiththetight bound obtained by numerically minimising the spectral DQFI bound. We summarise the key steps of the simplification below and include more details in Supplemental Material II. We first reformulate the extension bound from Nat. Commun., 3, 1063 (2012) for detector estimation.\n“This expression can be further simplified to remove the square-root derivatives to get\n(cid:13) (cid:13)2\nJ Ext,θ := m {S i j n } (cid:13) (cid:13) (cid:13) (cid:88) j (∂ θ π jθ −S j )π j − θ 1(∂ θ π jθ +S j ) (cid:13) (cid:13) (cid:13) sp , (SM.II.9)\n√ √ √ √\nwhere S = 2 π ∂ π − ∂ π − 2i π h π are skew-Hermitian matrices (S † =\nj jθ θ jθ θ jθ jθ j jθ j\n−S ) for j ∈ [m]. We call the bound F ≤ J the detector extension bound, and\nj θmax Ext,θ\nthe quantity J the extended DQFI. The minimisation in Eq. (SM.II.9) can be readily\next,θ\nformulated as the following SDP,\n t1 (∂ π −S )π −1/2 ... (∂ π −S )π −1/2\nd θ 1θ 1 1θ θ mθ m mθ\nJ Ext,θ = (cid:110) t m ,{S in j} t (cid:12) (cid:12) (cid:12)    \n\nπ 1 − θ 1/2 (∂ θ . . . π 1θ +S 1 ) 1 0 d . 0 .. 0 0    \n\n≽ 0 (cid:111) ,\nπ −1/2 (∂ π +S ) 0 0 1\nmθ θ mθ m d\n(SM.II.10)\nwhich simplifies to\n t1 ∂ π −S ... ∂ π −S \nd θ 1θ 1 θ mθ m\n(cid:110) (cid:12) ∂ θ π 1θ +S 1 π 1θ 0 0  (cid:111)\nJ Ext,θ =\nt,\nm\n{S\nin\nj}\nt (cid:12) (cid:12)  \n\n. .\n. 0\n...\n0\n \n\n≽ 0 . (SM.II.11)\n∂ π +S 0 0 π\nθ mθ m mθ\nNote that the choice S = 0 reverts to the unoptimised bound in Eq. (SM.II.4). On the\nj\nother hand, the optimal skew Hermitian matrices S , which are related to the optimal h\nj j\ninEq.(SM.II.8)ortheoptimalU inEq.(SM.II.6),arenotknownintermsofΠ and∂ Π .”\nθ θ θ θ\nNotably, the unoptimised bound is inferior to the spectral DQFI, as discussed previously. Nonetheless, theminimisationinEq.(SM.II.9)canbeefficientlyevaluatedusingtheSDP,whichshould,inprinciple, yield the tight bound for single-parameter detector estimation.\nWe then show that the SLD operators defined in main-text Eq. (4) can produce a good approximation to the optimal derivatives, thus recovering the spectral DQFI from this choice.\n“Remarkably, the SLD operators L defined in main-text Eq. (4) yield a good ansatz\njθ\nfor the optimal S . Consider the choice S = π L − ∂ π , where skew-Hermicity,\nj j jθ jθ θ jθ\ni.e. S + S † = 0, follows from the SLD definition (Eq. (4)). It is easy to verify that us-\nj j\ning this ansatz in Eq. (SM.II.9) produces the spectral DQFI (Definition 1 in main text),\nwhichimpliesJ ≤ J . Moreover,wheneverthespectralDQFIistight,i.e.,whenever\nExt,θ ∥,θ\nthe attainability criteria (1)–(3) (Methods, Section Attainability Criteria of the DQFI)\nare satisfied, this ansatz is exactly the optimal choice for S . For instance, in Example 2\nj\nof the main-text characterising a dephased PVM, the spectral DQFI is tight, and there-\nfore S = π L − ∂ π is the optimal solution for the minimisation in Eq. (SM.II.9).\nj jθ jθ θ jθ\nMore generally, whenever the attainability criteria are satisfied, this ansatz serves as an\nanalytical solution to the minimisation in Eq. (SM.II.9).”\n“In fact, even when the attainability criteria are not satisfied, this ansatz produces a good\napproximation to the tight bound. To demonstrate this, we numerically compare the tight\nboundandthespectralDQFIforrandomly-generateddetectorestimationproblems. Specif-\nically, we sample 10,000 random qubit (d = 2) POVMs and their derivatives with two\noutcomes (m = 2). Our results, shown below in Figs. 5 & 6, reveal that the gap between\nthe spectral DQFI J and the detector extension bound J can be small, averaging\n∥,θ Ext,θ\nbetween 2-3% over 10,000 random models.”\nSingle-parameter QubitDetector Estimation Single-parameter Qubit Detector Estimation\nSpectralDQFI (Closed-form) 1.5 SpectralDQFI (Closed-form)\nExtended DQFI (SDP) Extended DQFI (SDP)\n3\n1.0\n2\n0.5\n1\n0 0.0\n0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000\nSorted Runs\nnoitamrofnI\nrehsiF\nmutnauQ\nnoitamrofnI\nrehsiF\nmutnauQ\n(a) (b)\nSorted Runs\nFigure 5: Comparison of detector quantum Fisher information (DQFI) measures for single-parameter estimation from randomly-generated qubit measurement models. (a) Across 10,000 random models, the extended DQFI SDP (red) provides a tighter bound for single-parameter estimation than the trace DQFI (light orange) and the spectral DQFI (black). (b) A zoomed-in version of (a) shows that the trace DQFI can be far from the extended DQFI but the spectral DQFI is typically close. In (a) and (b), the scatter points are sorted in the increasing order of the extended DQFI.\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0\n0.0 0.5 1.0 1.5 2.0 2.5\nExtended DQFI Extended DQFI\nIFQD\nlartcepS\n3.5\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\ny = x\n0.0\n0.0 0.5 1.0 1.5 2.0 2.5\nIFQD\necarT\n(a) (b)\ny = x\ny = 2x\nFigure 6: One-to-one comparison of the spectral DQFI (a) and the trace DQFI (b) with the extended DQFI, over 10,000 randomly-generated, single-parameter, qubit measurement models. (a) The spec- tral DQFI is not always tight but is generally close to the extended DQFI. (b) The trace DQFI and the extended DQFI can disagree by up to a factor of two.\n“These results suggest a further simplification of Eq. (SM.II.9) that can help us better\nunderstand the attainable precision limit in single-parameter detector estimation. Consider\nthe substitution S = π L −∂ π +S′ in Eq. (SM.II.9), where S′ are skew-Hermitian\nj jθ jθ θ jθ j j\nmatrices. This results in the following bound:\n(cid:13) (cid:13)\nJ Ext,θ = m S i ′ n (cid:13) (cid:13) (cid:13) (cid:88) (L jθ +π j − θ 1S j ′)†π jθ (L jθ +π j − θ 1S j ′) (cid:13) (cid:13) (cid:13) . (SM.II.10)\nj j\nIf we identify L′ := L + π −1S′ as the modified SLD operators, the objective\njθ jθ jθ j\nabove becomes ∥ (cid:80) L′† π L′ ∥, resembling the spectral DQFI. This is justified because,\nj jθ jθ jθ\nwhereasL isHermitianandsatisfiestheequationL π +π L = 2∂ π ,themodified\njθ jθ jθ jθ jθ θ jθ\nSLD operators are non-Hermitian and satisfy\nL′† π +π L′ = 2∂ π , (SM.II.11)\njθ jθ jθ jθ θ jθ\nfor any choice of S′. In fact, any general solution to Eq. (SM.II.13) is of the form L′ =\nj jθ\nL +π −1S′ forsomeskew-HermitianS′. Therefore,thetightboundmaybereformulated\njθ jθ jθ j j\nas follows:\n(cid:26) (cid:12) (cid:27)\nJ Ext,θ = m L′ in (cid:13) (cid:13) (cid:88) L′ j † θ π jθ L′ jθ (cid:13) (cid:13) (cid:12) (cid:12) (cid:12) L′ j † θ π jθ +π jθ L′ jθ = 2∂ θ π jθ . (SM.II.12)\njθ j\nThisformulationrevealsthatwhiletheSLDapproach,asusedtoderivethespectralDQFI,\ndoes not always result in a tight bound, including a non-Hermitian component in the SLD\noperator and optimising this component produces the tight bound. Therefore, a minor\nmodification to the SLD approach still yields the tight bound for detector estimation, un-\nderscoring the versatility of this technique.”\nLastly, in Supplemental Material IIF, we provide a proof of tightness for the extended bound (Theo- rem 4, pg. 8). In proving this theorem, we present an analytical solution for the optimal derivatives (via the optimal skew-symmetric operators S ), in terms of the optimal pure state probe ρ = |ψ⟩⟨ψ|\nj\n(Lemma 3, pg. 7).\nScand = ⟨ψ|∂ θ π jθ |ψ⟩ (cid:2) π , ρ (cid:3) − (cid:2) ∂ π , ρ (cid:3) (SM.II.17)\nj ⟨ψ|π |ψ⟩ jθ θ jθ\njθ\nThis solution for S can be connected to the optimal Hermitian generators for the channel extension\nj\nminimisation over Kraus operators (detailed in Supplemental Material IIB), and therefore represents a major simplification of the minimization over different Kraus representations, as suggested by the reviewer. Further,thissolutionshedslightonhowthephase-insensitivecasebenefitsfromaddedsym-\nmetry: as the POVM and its derivative are both diagonal, we can always find an optimal probe ρ that\nsimultaneously commutes with both π and ∂ π , making S = 0 the optimal choice. This makes\njθ θ jθ j\n19\nthe optimal non-Hermitian SLD operators Hermitian—precisely the Hermitian SLD operators defined in main-text Eq. (4), thereby explaining why the spectral bound is tight for phase-insensitive mea- surements. At the same time, the proof of Lemma 3 shows that even for generic detector estimation problems, the optimal pure-state probes satisfy the same common-eigenvector property as identified previously in Theorem 3 (Methods section, pg. 10). These results illustrate that the spectral DQFI, despite being an approximation of the tight bound, accurately reflects many properties of the tight bound, thereby serving as a good analytical substitute for it.\nOverall, these developments, including the simplified SDP for the detector extension bound, the ana- lyticalcharacterisationoftheoptimalderivatives,andtheconnectionbetweenthenon-HermitianSLD\noperators and the optimal probe state, offer valuable physical insight into the measurement estima- tion problem. For instance, they directly show that for detector estimation, unlike generic quantum channels, ancilla entanglement offers no additional enhancement for a single parameter (Corollary 1 in Supplemental Material IIF, pg. 8). We believe these findings make important contributions to the topic of detector estimation and significantly advance the state-of-the-art on this topic.\nIn the same spirit If the authors show that their multi-parameter approach is superior to existingchannelestimationapproaches,thiswouldbeanargumentinfavourofthispaper to be published in Nat Comm.\nOtherwise, the paper may at most be regarded as a nice highlight of an important special case ofaproblemwhichhasaknownsolution,andassuchdoesnotmeetthestandardsof Nat. Comm.\nWe appreciate the reviewer’s suggestion. We believe that our revised manuscript emphasises the valuable advances made by the DQFI framework beyond existing channel estimation methods. These advances underscore the effectiveness of our measurement-focused approach. We summarise our key advances in the following:\n1. Forthemulti-parametersetting,wehavedemonstratedthatourproposedframeworkcanbesu- periortoexistingchannelestimationtechniquesbyprovidingtighterprecisionboundsandmore efficient numerical programs, as well as illuminating fundamental aspects of multi-parameter detector estimation that merit further study. As detailed in our response to the reviewer’s sug- gestion for numerical comparisons, our multi-parameter trace QCRB is directly computable in most settings and can outperform existing channel-based techniques, which may need numeri- cal optimisation (as in Fig. 3(b)). Our multi-parameter spectral bound SDP goes a step further, offeringatightboundforasub-categoryofdetectormodelsincludingaphysically-relevantpho- todetectormodel(Example3inmaintext)andafundamentally-interestingqubitdetectormodel for studying incompatibility effects (Example 5 in Supplemental Material VIII). We believe that these improvements justify the novelty and significance of our results compared to existing approaches.\n2. For the single-parameter setting, we report the following advances: (i) identification of the optimal probe states and an analytical expression for the attainable bound (spectral bound) for phase-insensitivedetectorsthatareofpracticalinterest,(ii)anefficientanalyticalapproximation totheattainableboundforgeneraldetectormodels,whereanalyticalresultscanenablebroader physical and geometrical insight into general features beyond specific models, (iii) physical un- derstandingoftheconnectionbetweentheoptimalprobestateandtheSLDoperatorsnewlyin- troducedinourwork(Theorem3forspectralbound(Methods,pg.10),Lemma3forattainable 20\nbound (Supplemental Material II, pg. 7)), and (iv) formulation, proof of tightness, and partial analytical solution of the attainable bound for detector estimation. In light of these points, we believe our revised framework provides an advanced tool in detector estimation over existing channel-based approaches.\nWe hope that the revisions are satisfactory and the reviewer finds our work suitable for Nature Communications.\nApart from this main criticism, I have some minor remarks: - Above Eq.(2) the authors write: “. However, this maximisation is not tractable in general [38, 39],” This is only true formulti-parametercase. Forsingleparameterthisproblemiscompletelysolved,and hence this sentence will be misleading for the readers, as immediately below the authors present single parameter formulas\nThe reviewer is correct, and we have modified the sentence above Eq. (2) as follows.\n“However, maximising this non-linear quantity is challenging in practice, even more so in themulti-parametercase[38,39], sothestateQFI(SQFI)I ofstateρ wasintroducedas θ θ an upper bound to the maximum CFI over all measurements [29–31].”\n- Authors do not mention the possibility of using an ancilla with which the probe state is entangled. I guess in this case this would not help, as the quantum channel we consider is quantum-classical, but It would be nice at least to comment this.\nWe thank the reviewer for their comment. We note that in the Methods section of our original manuscript (under Probing Strategies for Detector Estimation on pg. 9), we had already clarified the role of ancillary systems:\n“Notably, an ensemble of p different d-dimensional probe states can be realised through access to an ancillary system, A, that can be measured perfectly. This involves preparing a single bipartite-entangled state in D(H ⊗ HA), where HA denotes the p-dimensional d p p Hilbert space corresponding to system A. For example, the ensemble comprising states |0⟩ √ and|1⟩withprobabilitiesq and1−q isrealisedbyprobingΠ withthestate q|0⟩|0 ⟩+ θ A 1−q|1⟩|1 ⟩ and measuring the ancilla perfectly in the |0 ⟩,|1 ⟩ basis [28].” A A A\nFrom the above equivalence, we had concluded that independently measuring the ancilla and probe systemscannothelpinsingle-parameterdetectorestimation,astheeffectivestrategyisthatofprobing with an ensemble, whereas the convexity of the CFI demands that the optimal probe state be pure (Lemma 1).\nIn the revised manuscript, we also consider ancillary measurements conditioned on the detector out- comes (Supplemental Material II), and show that at least in the single-parameter case, this does not help in enhancing estimation precision. Notably, in the multi-parameter setting, ancilla-assisted en- tangled probes could offer an operational simplification over ensembles of probes, but the question of an entanglement advantage in precision remains open. This is an interesting question that merits further investigations, which we leave to the scope of future work. 21\n- Below Eq.3 Authors write “The maximization is not analytically tractable”. This is true, but in the light of what I wrote, above it is also slightly misleading, as there are efficient methods to find channel QFI with a simple SDP programme.\nWe thank the reviewer for their suggestion and have modified the above sentence in the revised manuscript, along with a suitable citation.\n“The maximisation in Eq. (3) has no known analytical solution, though numerical tech- niques from channel literature may be applicable [40].”\nIn closing, we thank the reviewer for their in-depth and insightful comments. Their comments have enabledsignificantimprovementsinthequalityandimpactofourwork. Aspartofourmajorrevisions, we have incorporated new theoretical and numerical results and discussions, including a new subsectionwithintheMethodssection,andtwonewsectionsintheSupplementalMaterial(II&VIII).With these revisions, which uncover measurement-specific physical insight into the tight single-parameter bound and compare favourably against multi-parameter channel estimation-based techniques, we hope that the reviewer now finds our work suitable for Nature Communications. 22\nWe would like to thank all the reviewers once again for their insightful comments and constructive feedback, which have significantly improved our manuscript. Below we address the reviewers’ comments in detail.\nResponses to Reviewer 1\nTheauthorshavesubstantiallyrevisedthemanuscript,addressinginaconvincingmanner the numerous concerns raised by all reviewers. In my view, the paper has considerably improved. I have no hesitation in recommending publication.\nWe thank the reviewer for their previous constructive feedback and detailed suggestions. We are glad that the reviewer deems our revised submission now suitable for publication in Nature Communications.\nResponses to Reviewer 2\nTheauthorshaveconvincinglyandcomprehensivelyrespondedtomycommentsandques- tionsaswellastothecommentsoftheotherReviewers. Iverymuchappreciatethenewly added material and discussion and in particular the additional clarification of the relation ofthepresentresultstothepreviousresultsoncharacterizationofquantumchannels. The revised manuscript is suitable for publication in its present form.\nWethankthereviewerfortheirpositiveassessmentofournewlyaddedmaterial. Wearegladthatthe reviewer finds our revised manuscript suitable for publication in Nature Communications.\nResponses to Reviewer 3\nIamingeneralsatisfiedwiththeAuthorsextensiverevisionandalotofefforttoshowthe relationoftheirapproachtoknowngeneralresultsofquanutumchannelestimationtheory. Iamnowconvincedthatthepaperoffersalevelofnoveltythatjutsifiedthepublicationin Nat. Comm., despte the fact, that the original presentation (and also present version, see below) was a bit missleading in terms of the novelty of the problem considered.\nWe thank the reviewer for their time assessing our manuscript and for their previous detailed and constructive comments. We are glad that the reviewer is satisfied by our revised manuscript and finds that our novelty justifies publication in Nature Communications. We respond to the reviewer’s final comment in detail below.\nIndeed, this is the one remaining issue I would like the authors to ammend in their pa- per. While the authors clearly point out that their problem may be viewed as a quantum- classical channel estimation problem in the Supplementary material, they avoid mention- ing this explicitly in the main text. I think this is crucial to point out this important fact explicitly in the main text as well, including the explicit form of the Kraus operators that 1\nrepresent the quantum-classical channel. Only then the Authors can argue that despite the fact that this is a special case of a problem for which general numerical optimization techniques are known (at least for single parameter case), they offer a deeper insight into the problem thanks to its ”quantum-classical” nature. This would give the reader a proper perspective on the problem.\nWe appreciate the reviewer’s suggestion to incorporate the quantum-classical channel estimation per- spective into the main text. Accordingly, we have revised our main text to explicitly state this connec- tion as well as the form of the appropriate Kraus operators, as suggested.\nSpecifically, in the revised section Single-parameter detector estimation, we now clarify the following immediately below the defining Eq. (3):\n“The maximisation in Eq. (3) has no known analytical solution, though numerical tech- niques from channel literature may be applicable [40]. Specifically, by translating the detector POVM to Kraus operators for quantum-classical maps (particular choice of Kraus set shown in Eq. (27)), the maximisation for F can, in principle, be converted to a θmax maximisation over all equivalent Kraus operators through a numerical semi-definite pro- gram [40]. In practice, however, this approach is computationally demanding due to the largenumberofKrausoperatorsforameasurementchannel,andleavesthequestionofthe optimal probe ρopt unanswered. The lack of efficiently computable precision bounds for theminimumMSE,1/F ,fordetectorestimationrevealsagapinourunderstandingof θmax the information content of measurements and resulting fundamental limits on estimation errors.”\nSimilarly, in the revised Discussion section, we now highlight comparisons with channel estimation andexplicitlystatethatdetectorestimationmaybeaddressedusingexistingchannelestimation-based techniques, as suggested by the reviewer:\n“These unique features re-emphasize the fact that while detector estimation may be tech- nically considered as a special case of channel estimation, generic estimation tools that do not utilise the quantum-classical nature of the measurement process may fail to capture the unique aspects of detector estimation.”\nIn addition, we explicitly include the form of the Kraus operators for a quantum-classical channel in Eq. (27), as suggested by the reviewer.\n“The tight bound for the extended detector is evaluated by identifying the POVM Π with θ Kraus operators\n√\nK = |j⟩⟨k| π for j ∈ [m], k ∈ [d] (27)\nj,k jθ\nfor a quantum-classical channel [40].”\nThese revisions improve the clarity of our presentation of the novelty of the problem considered. We hope the reviewer finds these changes satisfactory.\nApart from that I am also confused with the paragraph in the conlcusion section:\n2\n“While our findings suggest that the DQFI can grow quadratically with number of copies (Example 6), the precise conditions enabling a Heisenberg scaling are yet un- known [40, 64, 65]”\nwhile the references are exactly the papers where the condtions for possiblity of quadratic scaling of QFI are given! And could be applied to the DQFI problem, as well, using the Kraus operators for the quantum-classical channel. So I am not sure what the authors want to communicate here.\nWe thank the reviewer for noticing this issue. We have revised the sentence in question to be clearer and more specific.\n“While our findings suggest that the DQFI can grow quadratically with number of copies (Example 6), the precise strategies including error-correcting techniques that can pre- serve this Heisenberg scaling for realistic noisy detector estimation require further explo- ration [40, 61, 62].”\nIn closing, we thank the reviewer for their detailed and insightful comments. We believe their com- ments have significantly improved the quality of our work. We hope that the reviewer finds our improved presentation satisfactory.\n3"}]

Decision letter