Abstract In the estimation for a parametric family of quantum state on a Hilbert space H , the Gill and Massar bound is known as a lower bound of weighted traces of covariances of unbiased estimators. The Gill and Massar bound is derived by considering the convexity of the set of classical Fisher information matrices, and the bound is locally achievable by using randomized strategies when H = C 2 . In this paper, we show that estimation for a parametric SU(2) unitary channel model has a similar convex structure as qubit state model, and a Gill and Massar type lower bound of weighted traces of covariances of unbiased estimators can be derived for any weight matrix. We show that the Gill and Massar type lower bound is achievable by using randomized strategies when certain conditions are satisfied. To derive a convex structure of the set of classical Fisher information matrices, we introduce a Fisher information matrix J (U) for a SU(2) unitary channel model, and we show a upper bound of inverse J (U) weighted trace of classical Fisher information matrix. The optimal randomized strategy we construct in this paper does not require ancilla systems in many cases.
Abstract In the estimation for a parametric family of quantum state on a Hilbert space H , the Gill and Massar bound is known as a lower bound of weighted traces of covariances of unbiased estimators. The Gill and Massar bound is derived by considering the convexity of the set of classical Fisher information matrices, and the bound is locally achievable by using randomized strategies when H = C 2 . In this paper, we show that estimation for a parametric SU(2) unitary channel model has a similar convex structure as qubit state model, and a Gill and Massar type lower bound of weighted traces of covariances of unbiased estimators can be derived for any weight matrix. We show that the Gill and Massar type lower bound is achievable by using randomized strategies when certain conditions are satisfied. To derive a convex structure of the set of classical Fisher information matrices, we introduce a Fisher information matrix J (U) for a SU(2) unitary channel model, and we show a upper bound of inverse J (U) weighted trace of classical Fisher information matrix. The optimal randomized strategy we construct in this paper does not require ancilla systems in many cases.
We study lower bounds for Bayesian quantum parameter estimation, with a particular focus on qubit models. While several lower bounds on the Bayes risk have been proposed, including the Bayesian symmetric logarithmic derivative (B-SLD) type bound and the Bayesian Nagaoka-Hayashi (B-NH) bound, there is no definite proof available to show they are attainable except for special cases. Thus, identifying attainable bounds together with their corresponding optimal measurement strategies remains a central open problem in Bayesian quantum estimation. In this work, we introduce a new Bayesian lower bound, referred to as the Bayesian Gill-Massar (B-GM) bound, inspired by the logic of Gill-Massar bound in point estimation. We derive an analytical closed-form expression of the bound and show that it is attainable for any qubit model. In particular, we prove that the optimal Bayesian strategy can be realized by a projection-valued measure associated with a single effective direction determined by the weight matrix and the B-SLD-type Fisher information matrix. We provide numerical comparisons between the B-GM, B-NH, and B-SLD-type bounds in higher-dimensional models. Our results show that the B-GM bound has a limitation in high-dimensional models with few parameters, since it can be negative.
We determine the exact local precision limits for single-copy estimation of smooth multiparameter quantum statistical models contained in spin factors, a class of matrix Jordan algebras whose state spaces generalize the qubit Bloch ball. At any parameter point where the symmetric logarithmic derivative (SLD) Fisher information is positive definite, we characterize the entire attainable classical Fisher-information region over all finite-outcome positive-operator-valued measurements. After SLD normalization, this region consists exactly of the real symmetric positive semidefinite matrices with trace at most one, independently of the ambient Hilbert-space dimension. This yields a sharp weighted covariance bound for every positive definite weight, attained by an explicit locally unbiased estimator based on randomized spectral measurements of SLD directions. The proof combines a statistics-preserving positive projection onto the spin factor with the two-eigenvalue structure of its effects, revealing the Jordan-algebraic origin of the tradeoff. The result extends the qubit information tradeoff to models with more than three parameters. Since the optimal measurement depends on the unknown parameter, we simulate an adaptive scheme for a five-parameter model on a four-dimensional Hilbert space and observe performance close to the optimal local benchmark.
We address the `major open problem' of evaluating how much increased efficiency in estimation is possible using non-separable —as opposed to separable—measurements of N copies of m -level quantum systems. First, we study the six cases m = 2, N = 2, ..., 7 by computing the 3 × 3 Fisher information matrices for the corresponding optimal measurements recently devised by Vidal et al . We obtain simple polynomial expressions for the (`Gill–Massar') traces of the products of the inverse of the quantum Helstrom information matrix and these Fisher information matrices. The six traces all have minima of 2 N − 1 in the pure state limit—while for separable measurements, the traces can equal N , but not exceed it. Then, the result of an analysis for m = 3, N = 2 leads us to conjecture that for optimal measurements for all m and N , the Gill–Massar trace achieves a minimum of (2 N − 1)( m − 1) in the pure state limit.
We consider the problem of determining the weights of a quantum ensemble. That is to say, given a quantum system that is in a set of possible known states according to an unknown probability law, we give strategies to estimate the individual probabilities, weights, or mixing proportions. Such strategies can be used to estimate the frequencies at which different independent signals are emitted by a source. They can also be used to estimate the weights of particular terms in a canonical decomposition of a quantum channel. The quality of these strategies is quantified by a covariance-type error matrix. According with this cost function, we give optimal strategies in both the single-shot and multiple-copy scenarios. The latter is also analyzed in the asymptotic limit of large number of copies. We give closed expressions of the error matrix for two-component quantum mixtures of qubit systems. The Fisher information plays an unusual role in the problem at hand, providing exact expressions of the minimum covariance matrix for any number of copies.
We show that the sample complexity required in quantum learning theory within a general parametric framework is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error, with high success probability under maximum-likelihood estimation. Notably, both the upper and lower bounds are determined by the supremum of the maximum diagonal entry of the inverse Fisher information matrix. We then apply the general bounds to Pauli channel learning and Pauli expectation value learning, which serve as representative tasks in quantum channel and state learning, respectively, in the asymptotic small-error regime. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value learning without quantum memory by comparing the quantum Fisher information matrix and the classical Fisher information matrix. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators, deriving the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli channel learning with entangled probes. We highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, the learning sample complexity is governed by the inverse Fisher information matrix, which is the central quantity in quantum metrology that determines the ultimate achievable mean squared error.
We address the "major open problem" of evaluating how much increased efficiency in estimation is possible using non-separable measurements, rather than separable ones, of N copies of two-level quantum systems. To proceed, we compute the the 3 x 3 Fisher information matrices for the optimal measurements recently devised by Vidal et al (quant-ph/9812068) for N = 2,...,7. We obtain simple polynomial expressions for the traces of the products of the inverse of the quantum Helstrom information matrix and the computed Fisher information matrices. These traces all have minima of 2 N - 1 in the pure state limit, while for separable measurements, as established by Gill and Massar (quant-ph/9902063), they are bound above by N. This indicates both that there is a large discontinuity at the pure states (where the computed Fisher information matrices are indeterminate) and that the gain in using joint measurements over separate measurements is substantial for all impure states. In seeking to extend our results to N > 7, we investigate possible relationships between the Fisher information matrices generated here and the class of monotone Riemannian metrics, studied by Petz and Sudar as well as others.