The (Fisher) information matrix (I2p) for a p×p density matrix (ρ) is taken to be the (2p2+p)-dimensional information matrix of a 2p-variate real normal distribution with an a priori zero mean vector and a covariance matrix (Σ2p), the two diagonal blocks of which are formed from the symmetric (real) component of ρ, and the two off-diagonal blocks from the skew-symmetric (imaginary) part of ρ. The inverse square root ‖nI2p‖−1/2 (where n is the number of observations) serves as a unitary invariant prior distribution over (the precision matrices) Σ2p−1 and ‖nI2p‖1/2 over Σ2p—and consequently over the p×p (mixed and pure) density matrices. The availability of these priors should facilitate the process of state determination. An uncertainty principle for the estimation of ρ is formulated. The determinant of Σ2p proves for integral p≥1 to equal (2−p‖ρ‖)2.
The (Fisher) information matrix (I2p) for a p×p density matrix (ρ) is taken to be the (2p2+p)-dimensional information matrix of a 2p-variate real normal distribution with an a priori zero mean vector and a covariance matrix (Σ2p), the two diagonal blocks of which are formed from the symmetric (real) component of ρ, and the two off-diagonal blocks from the skew-symmetric (imaginary) part of ρ. The inverse square root ‖nI2p‖−1/2 (where n is the number of observations) serves as a unitary invariant prior distribution over (the precision matrices) Σ2p−1 and ‖nI2p‖1/2 over Σ2p—and consequently over the p×p (mixed and pure) density matrices. The availability of these priors should facilitate the process of state determination. An uncertainty principle for the estimation of ρ is formulated. The determinant of Σ2p proves for integral p≥1 to equal (2−p‖ρ‖)2.
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.
Given a pure state vector | x ⟩ and a density matrix , the function defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher–Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.
The classical Bayes rule computes the posterior model probability from the prior probability and the data likelihood. We generalize this rule to the case when the prior is a density matrix (symmetric positive definite and trace one) and the data likelihood a covariance matrix. The classical Bayes rule is retained as the special case when the matrices are diagonal. In the classical setting, the calculation of the probability of the data is an expected likelihood, where the expectation is over the prior distribution. In the generalized setting, this is replaced by an expected variance calculation where the variance is computed along the eigenvectors of the prior density matrix and the expectation is over the eigenvalues of the density matrix (which form a probability vector). The variances along any direction is determined by the covariance matrix. Curiously enough this expected variance calculation is a quantum measurement where the co-variance matrix specifies the instrument and the prior density matrix the mixture state of the particle. We motivate both the classical and the generalized Bayes rule with a minimum relative entropy principle, where the Kullbach-Leibler version gives the classical Bayes rule and Umegaki’s quantum relative entropy the new Bayes rule for density matrices. 1
For a quantum state and a set of observables, we can construct an associated covariance matrix and a natural quantum Fisher information matrix. These two matrices characterize the uncertainty and information content of the observables in the relevant state. An inequality between these two matrices is established. This inequality may be interpreted as a general quantification of the Heisenberg uncertainty principle from a statistical estimation perspective. In particular, it implies a new uncertainty relation which refines the celebrated Schrödinger uncertainty relation.
Classical measurement of four-vectors In the preceding chapter, we found that the accuracy in an estimate of a single parameter θ is determined by an information I that has some useful physical properties. It provides new definitions of disorder, time and temperature, and a variational approach to finding a single-component PDF law p ( x ) of a single variable x. However, many physical phenomena are describable only by multiple- component PDFs, as in quantum mechanics, and for vector variables x since worldviews are usually four-dimensional (as required by covariance). Our aim in this chapter, then, is to form a new, scalar information I that is appropriate to this multi-component, vector scenario . The information should be intrinsic to the phenomenon under measurement and not depend, e.g., upon exterior effects such as the noise of the measuring device. The ‘intrinsic’ measurement scenario In Bayesian statistics, a prior scenario is often used to define an otherwise unknown prior probability law; see, e.g., Good (1976), Jaynes (1985) or Frieden (1991). This is a model scenario that permits the prior probability law to be computed on the basis of some ideal conditions, such as independence of data, and/or ‘maximum ignorance’ (see below), etc. We will use the concept to define our unknown information expression.