Braunstein and Caves have recently demonstrated that the Bures metric on the mixed quantum states is equivalent - up to a proportionality factor of four - to the statistical distinguishability or quantum Fisher information metric. The volume element of these metrics can then - adapting a fundamental Bayesian principle of Jeffreys to the quantum context - serve as a reparametrization-invariant prior measure over the quantum states. The implications of this line of reasoning for the two-level systems, in general, and an embedding of them into a certain set of three-level systems are investigated.
Braunstein and Caves have recently demonstrated that the Bures metric on the mixed quantum states is equivalent - up to a proportionality factor of four - to the statistical distinguishability or quantum Fisher information metric. The volume element of these metrics can then - adapting a fundamental Bayesian principle of Jeffreys to the quantum context - serve as a reparametrization-invariant prior measure over the quantum states. The implications of this line of reasoning for the two-level systems, in general, and an embedding of them into a certain set of three-level systems are investigated.
Clarke and Barron have recently established that the (classical) Jeffreys' prior yields universal codes - ones which do relatively well (in the sense of relative entropy) no matter what the true state. Twamley has recently computed the Bures metric - proportional to the quantum Fisher information (statistical distinguishability) metric, as shown by Braunstein and Caves - for squeezed thermal states. The volume elements of these metrics - that is, the quantum Jeffreys' prior - are found to be simply the product of a function of the squeeze factor and a function of the inverse temperature, the phase being irrelevant in this regard. A computational strategy to find a universal coding using the quantum Jeffreys' prior - previously implemented for the two-level quantum systems - is then discussed.
We find, in an analysis involving four prior probabilities (p's), that the information-theoretic-based comparative noninformativity test devised by Clarke, and applied by Slater in a quantum setting, yields a ranking (p_{F_{q=1}} > p_{B} > p_{B_{q=1}trunc} >p_{F}) fully consistent with Srednicki's recently-stated criterion for priors of ``biasedness to pure states''. Two of the priors are formed by extending certain metrics of quantum-theoretic interest from three- to four-dimensions -- by incorporating the q-parameter (nonextensivity/Tsallis index/escort parameter). The three-dimensional metrics are the Bures (minimal monotone) metric over the two-level quantum systems and the Fisher information metric over the corresponding family of Husimi distributions. The priors p_{B} and p_{F} are the (independent-of-q) normalized volume elements of these metrics, and p_{F_{q=1}} is the normalized volume element of the q-extended Fisher information metric, with q set to 1. While originally intended to similarly be the q-extension of the Bures metric, with q then set to 1, the prior p_{B_{q=1}trunc}, actually entails the truncation of the only off-diagonal entry of the extended Bures metric tensor. Without this truncation, the q-extended Bures volume element is null, as is also the case in two other quantum scenarios we examine.
In the Bayesian theory of statistical inference, as first suggested by Harold Jeffreys in highly influential work, one can employ the square root of the determinant of an n×n Fisher information matrix as a reparametrization-invariant prior (generally unnormalized) measure over an n-dimensional family (Riemannian manifold) of probability distributions. Jeffreys’ ansatz is adopted here to the quantum context, that is, with regard to density matrices rather than probability distributions, by computing the quantum Fisher information matrices (associated with Helstrom and Holevo) for the three-, five-, and eight-dimensional convex sets of two-level complex, two-level quaternionic, and three-level complex systems, respectively. In both the two-level cases, the priors have been normalized to probability distributions over the 2×2 density matrices, while, in the much more computationally demanding three-level situation, no such normalization has been accomplished. An argument is made for the general form, in terms of eigenvalues, that the (unnormalized) prior should assume over the (n2−1)-dimensional convex set of n×n density matrices.
We study the interrelationships between the Fisher information metric recently introduced, on the basis of maximum entropy considerations, by Brody and Hughston (quant-ph/9906085) and the monotone metrics, as explicated by Petz and Sudar. This new metric turns out to be not strictly monotone in nature, and to yield (via its normalized volume element) a prior probability distribution over the Bloch ball of two-level quantum systems that is less noninformative than those obtained from any of the monotone metrics, even the minimal monotone (Bures) metric. We best approximate the additional information contained in the Brody-Hughston prior over that contained in the Bures prior by constructing a certain Bures posterior probability distribution. This is proportional to the product of the Bures prior and a likelihood function based on four pairs of spin measurements oriented along the diagonal axes of an inscribed cube.
Braunstein and Caves (Braunstein S L and Caves C M 1994 Phys. Rev. Lett. 72 3439-43) proposed to use Helstrom's quantum information number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing other than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on n identical copies of the given quantum system. We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two-stage adaptive measurement procedure. When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of `distinguishability of close-by states' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.
Recently, there is growing interest in studying quantum mechanics from the information geometry perspective, where a quantum state is depicted by a point in the projective Hilbert space (PHS). However, the absence of high-dimensional measures limits information geometry in the study of multiparameter systems. In this paper, we propose a measure of the intrinsic density of quantum states (IDQS) in the PHS with the volume element of quantum Fisher information (QFI). Theoretically, the IDQS is a measure to define the (over)completeness relation of a class of quantum states. As an application, the IDQS is used to study quantum measurement and multiparameter estimation. We find that the density of distinguishable states (DDS) for a set of efficient estimators is measured by the invariant volume element of the classical Fisher information, which is the classical counterpart of the QFI and serves as the metric of statistical manifolds. The ability to infer the IDQS via quantum measurement is studied with a determinant-form quantum Cram\'er-Rao inequality. As a result, we find a gap between the IDQS and the maximal DDS over the measurements. The gap has tight connections with the uncertainty relationship. Exemplified by the three-level system with two parameters, we find that the Berry curvature characterizes the square gap between the IDQS and the maximal attainable DDS. Specific to vertex measurements, the square gap is proportional to the square of the Berry curvature.