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Physical Review A

Noninformative prior in the quantum statistical model of pure states

Fuyuhiko Tanaka

Published Jun 5, 2012• 7 citations
Publisher

Abstract

In the present paper, we consider a suitable definition of a noninformative prior on the quantum statistical model of pure states. While the full pure-states model is invariant under unitary rotation and admits the Haar measure, restricted models, which we often see in quantum channel estimation and quantum process tomography, have less symmetry and no compelling rationale for any choice. We adopt a game-theoretic approach that is applicable to classical Bayesian statistics and yields a noninformative prior for a general class of probability distributions. We define the quantum detection game and show that there exist noninformative priors for a general class of a pure-states model. Theoretically, it gives one of the ways that we represent ignorance on the given quantum system with partial information. Practically, our method proposes a default distribution on the model in order to use the Bayesian technique in the quantum-state tomography with a small sample.

DOI
10.1103/physreva.85.062305

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In the present paper, we consider a suitable definition of a noninformative prior on the quantum statistical model of pure states. While the full pure-states model is invariant under unitary rotation and admits the Haar measure, restricted models, which we often see in quantum channel estimation and quantum process tomography, have less symmetry and no compelling rationale for any choice. We adopt a game-theoretic approach that is applicable to classical Bayesian statistics and yields a noninformative prior for a general class of probability distributions. We define the quantum detection game and show that there exist noninformative priors for a general class of a pure-states model. Theoretically, it gives one of the ways that we represent ignorance on the given quantum system with partial information. Practically, our method proposes a default distribution on the model in order to use the Bayesian technique in the quantum-state tomography with a small sample.
In the quantum state tomography, which is the task of estimating the unknown quantum state on a Hilbert space from measurement data, the standard Bayesian technique is often used.As is well known in Bayesian statistics, it is a nontrivial problem to define a noninformative prior in gen- eral statistical models.Although some definitions for statistical models of mixed states are given by straightforward extension of classical results, there is no counterpart for those of pure states.Recently the author proposed a conceptual extension of the famous reference prior by Bemardo and defined noninformative priors for statistical models of pure states.While the formal extension of the reference prior corresponds to the prior maximizing the von Neumann entropy, our proposed prior is shown to maximize the limit of the quantum Renyi entropy.We show some relations between two principles.
An analytical solution for the posterior estimate in Bayesian tomography of the unknown quantum state of an arbitrary quantum system (with a finite-dimensional Hilbert space) is found. First, we derive the Bayesian estimate for a pure quantum state measured by a set of arbitrary rank-1 POVMs under the uninformative (i.e. the unitary invariant or Haar) prior. The expression for the estimate involves the matrix permanents of the Gram matrices with repeated rows and columns, with the matrix elements being the scalar products of vectors giving the measurement outcomes. Second, an unknown mixed state is treated by the Hilbert-Schmidt purification. In this case, under the uninformative prior for the combined pure state, the posterior estimate of the mixed state of the system is expressed through the matrix ααα-permanents of the Gram matrices of scalar products of vectors giving the measurement outcomes. In the mixed case, there is also a free integer parameter -- the Schmidt number -- which can be used to optimise the Bayesian reconstruction (for instance, in case of Schimdt number being equal to 1, the mixed state estimates reduces to the pure state estimate). We also discuss the perspectives of approximate numerical computation and asymptotic analytical evaluation of the Bayesian estimate using the derived formula.
Abstract Quantum state estimation (or state tomography) is an indispensable task in quantum information processing. Because full state tomography that determines all elements of the density matrix is computationally demanding, one usually takes the strategy of assuming a certain model of quantum states and identifying the model parameters. However, it is difficult to make a valid assumption given little prior knowledge on a quantum state of interest, and thus we need a reasonable model selection method for quantum state estimation. Actually, in the classical statistical estimation theory, several types of information criteria have been established and widely used in practice for appropriately choosing a classical statistical model. In this study, we propose quantum information criteria for evaluating the quality of the estimated quantum state in terms of the quantum relative entropy, which is a natural quantum analogue of the classical information criterion defined in terms of Kullback–Leibler divergence. In particular, we derive two quantum information criteria depending on the type of an estimator for the quantum relative entropy; one uses the log-likelihood and the other uses the classical shadow. The general role of information criteria is to predict the performance of an estimated model for unseen data, although it is a function of only sampled data; this generalization capability of the proposed quantum information criteria is evaluated in numerical simulations.
Quantum tomography involves obtaining a full classical description of a prepared quantum state from experimental results. We propose a Langevin sampler for quantum tomography, that relies on a new formulation of Bayesian quantum tomography exploiting the Burer-Monteiro factorization of Hermitian positive-semidefinite matrices. If the rank of the target density matrix is known, this formulation allows us to define a posterior distribution that is only supported on matrices whose rank is upper-bounded by the rank of the target density matrix. Conversely, if the target rank is unknown, any upper bound on the rank can be used by our algorithm, and the rank of the resulting posterior mean estimator is further reduced by the use of a low-rank promoting prior density. This prior density is a complex extension of the one proposed in (Annales de l'Institut Henri Poincare Probability and Statistics, 56(2):1465-1483, 2020). We derive a PAC-Bayesian bound on our proposed estimator that matches the best bounds available in the literature, and we show numerically that it leads to strong scalability improvements compared to existing techniques when the rank of the density matrix is known to be small.
Using a Bayesian methodology, we introduce the maximum a posteriori probability (MAP) estimator for quantum state and process tomography. We show that the maximum likelihood, the hedged maximum likelihood, and the maximum likelihood-maximum entropy estimator, and estimators of this general type, can be viewed as special cases of the MAP estimator. The MAP, like the Bayes mean estimator, includes prior knowledge. For cases of interest to tomography MAP can take advantage of convex optimization tools, making it numerically tractable. We show how the MAP and other Bayesian quantum state estimators can be corrected for noise produced if the quantum state passes through a noisy quantum channel prior to measurement. Numerical simulations on a single qubit indicate that, on average, including these corrections significantly improves the estimate even when the measurement data are modestly large.