Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state, and therefore have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in terms of confidence regions-a concept originating from classical statistics. Confidence regions are subsets of the state space in which the true state lies with high probability, independently of any prior assumption on the distribution of the possible states. Our method for computing confidence regions can be applied to arbitrary measurements including fully coherent ones; it is practical and particularly well suited for tomography on systems consisting of a small number of qubits, which are currently in the focus of interest in experimental quantum information science. PACS numbers: 03.65. Wj, The state of a classical system can in principle be determined to arbitrary precision by applying a single measurement to it. Any imprecisions are due solely to inaccuracies of the measurement technique, but not of fundamental nature. This is different in quantum theory. It follows from Heisenberg's uncertainty principle that measurements generally have a random component and that individual measurement outcomes only give limited information about the state of the system-even if an ideal measurement device is used. To illustrate this difference, it is useful to take an informationtheoretic perspective. Assume, for instance, that we are presented with a two-level system about which we have no prior information except that it has been prepared in a pure state, and our task is to determine this state. If the system was classical, there are only two possible pure states, and one single bit of information is therefore sufficient for its full description. Furthermore, a single measurement of the system suffices to retrieve this bit. If the system was quantum, however, the situation becomes more interesting. A two-level quantum system (a qubit) admits a continuum of pure states that can, for example, be parameterized by a point on the Bloch sphere. To determine this point to a given accuracy ∆, at least log 2 (4/∆ 2 ) bits of information are necessary [45]. Conversely, according to Holevo's bound [1], any measurement applied to a single qubit will provide us with at most one bit of information. And even if n identically prepared copies of the qubit were measured, at most log 2 (n + 1) bits of information about their state can be obtained [46]. Hence the accuracy, ∆, to which the state can be determined always remains finite (∆ ≥ 2 √ n+1 ), necessitating the specification of error bars. The impact that randomness in measurement data has on the accuracy of estimates has been studied extensively in statistics and, in particular, estimation theory [2]. The latter is concerned with the general problem of estimating the values of parameters from data that depend probabilistically on them. The data may be obtained from measurements on a quantum system with parameter-dependent state, as considered in quantum estimation theory [3]. Quantum state tomography can be seen as a special instance of quantum estimation, where one aims to estimate a set of parameters large enough to determine the system's state completely [4][5][6][7][8][9][10]. An obvious choice of parameters are the matrix elements of a density operator representation of the state. Due to the finite accuracy, however, the individual estimates for the matrix elements do not generally correspond to a valid density operator (for instance, the matrix may have negative eigenvalues). This problem is avoided with other techniques, such as maximum likelihood estimation (MLE) [8, 11, 12], which has been widely used in experiments [13][14][15] [16] [17] [18] [19] , or Bayesian estimation [3, [20] [21] [22] [23] [24] [25] . In MLE, an estimate for the error bars can be obtained from the width of the likelihood function, which is approximated by the Fisher information matrix [10, 11, [26] [27] [28] [29] . In current experiments one also uses numerical plausibility tests known as "bootstrapping" or, more generally, "resampling" [18, 30] in order to obtain bounds on the errors. However, despite being reasonable in many practical situations, these bounds are not known to have a well-defined operational interpretation and, in the case of the resampling method, may lead to an underestimate of the errors [31] . In contrast, Bayesian methods can be used to calculate "credibility regions", i.e., subsets of the state space in which the state is found with high probability. This probability, however, depends on the choice of a "prior", corresponding to an assumption about the distribution of the states before the measurements (in particular, the assumption can not be justified by the experimental data). Furthermore, we remark that most known techniques are based on the assumption of independent and identical measurements (a notable exception is the one-qubit adaptive tomography analysis of [32] ). We refer to [24] for a further discussion of currently used approaches to quantum state tomography, including pedagogical examples illustrating their limitations. In this Letter we introduce a method to obtain confidence arXiv:1108.5329v2 [quant-ph] 7 Nov 2012 2 regions, that is, regions in state space which contain the true state with high probability. A point in the region may then serve as estimate and the maximal distance of the point to the border of the region as error bar. Our method allows to analyse data obtained from arbitrary quantum measurements, including fully coherent ones. The method does not rely on any assumptions about the prior distribution of the states to be measured. This makes it highly robust so that it can, for instance, be applied in the context of quantum cryptography, where the states to be estimated are chosen adversarially. The remainder of this Letter is organized as follows. We first describe a very general setup for tomography of quantum states prepared in a sequence of experiments, where we do not make the typical assumption that the states are independent and identically distributed (i.i.d.). We then show that, nevertheless, properties of the states can be inferred reliably using a suitable tomographic data analysis procedure (Theorem 1). In motivation and spirit, this result relates to recent research efforts on quantum de Finetti representations. We then specialise our setup to the case where, in principle, the experiments may be run an arbitrary number of times (while still only finitely many runs are used to generate data). This special case is (by the quantum de Finetti theorem) equivalent to an i.i.d. preparation of the states, thereby justifying the common i.i.d. assumption in data analysis. The theorem, applied to this special case, then results in a construction for confidence regions for quantum state tomography (Corollary 1). General Scenario.-Consider a collection S 1 , . . . , S n+k of finite-dimensional quantum systems with associated Hilbert space H, as depicted in Fig. 1 (see also [33] and [34] , where a similar setup is considered). We denote by d the dimension of H. For example, one may think of n + k particles prepared in a series of experiments, where H could correspond to the spin degree of freedom. From this collection, a sample consisting of n systems is selected at random and measured according to an (arbitrary) Positive Operator Valued Measure (POVM) {B n }, a family of positive semi-definite operators B n on H ⊗n such that B n B n = 1 ⊗n H . That is, each POVM element B n corresponds to a possible sequence of outcomes resulting from (not necessarily independent) measurements on the n systems. The goal of quantum state tomography is to infer the state of the remaining k systems, using the outcomes of these measurements. Note that the k extra systems are not measured during data acquisition. Nevertheless, they play a role in the above scenario, as they are used to define operationally what state we are inferring. (In the special case of i.i.d. states, the extra systems are simply copies of the measured systems-see below.) We also remark that, instead of measuring a sample of n systems chosen at random, one may equivalently permute the initial collection of n + k systems at random and then measure the first n of them, i.e., S 1 , . . . , S n . We will use this alternative description for our theoretical analysis. In order to describe our main results, we imagine that the measurement outcomes B n are processed by a data analysis routine that outputs a probability distribution µ B n on the set Measurements are applied to a sample S1, . . . , Sn consisting of n systems, chosen at random from a collection of n + k systems. The outcomes of the measurements are collected and given as input, B n , to a data analysis procedure (orange). The aim of quantum state tomography is to make reliable predictions about the state of the remaining k (non-measured) systems Sn+1, . . . , S n+k . To model such predictions, we consider hypothetical tests, which output "success" whenever their input has a desired property (blue). Given only the output of the data analysis procedure, µBn , it is possible to characterize the tests Tµ B n that are passed with high probability-independently of the initial state of the n + k systems (Theorem 1). of mixed states, defined by Fig. 2 for an illustration). Here dσ denotes the HilbertSchmidt measure with dσ = 1. Furthermore, is a normalisation constant, where K ∼ = H ∼ = C d and where 1 Sym n (H⊗K) is the projector onto the symmetric subspace of (H ⊗ K) ⊗n . Note that, in Bayesian statistics, µ B n (σ)dσ corresponds to the a posteriori distribution when updating a Hilbert-Schmidt prior dσ. Furthermore, in MLE, σ → tr[σ ⊗n B n ] is known as the likelihood function. Since our work is not based on either of these approaches, however, we will not use this terminology and simply refer to µ B n . Reliable Predictions.-We now show that µ B n contains all information that is necessary in order to make reliable predictions about the state of the remaining systems S n+1 , . . . , S n+k . To specify these predictions, we consider hypothetical tests, a quantum version of a similar concept used in classical statistics. Any such test acts on the joint system consisting of S n+1 , . . . , S n+k (see Fig. 1 ). Mathematically, a 3 FIG. 2: Illustration of µBn . The graphs show µBn for measurements performed on n = 20 and n = 240 qubits (for illustration purposes, we only depict the density µBn on the surface of the Bloch ball). Half of the qubits have been measured in the z direction and half in the y direction with relative frequencies of (0.2, 0.8) and (0.7, 0.3), respectively. One observes a rapid decrease in the size of the bright regions (which are connected by a bright tube inside the Bloch ball), which correspond to large values of µBn .
Beyond its use as point estimator, the maximum-likelihood estimator is used as a basis for inference around the point estimate, via confidence intervals REF and credible regions.
Reliable Quantum State Tomography
Physical Review Letters 109, 120403 (2012)
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