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arXiv:1503.04146

Generalized Fubini-Study Metric and Fisher Information Metric

Mondal, Debasis

Published Mar 13, 2015• 3 citations
arXivHTMLPDF

Abstract

We provide an experimentally measurable local gauge $U(1)$ invariant Fubini-Study (FS) metric for mixed states. Like the FS metric for pure states, it also captures only the quantum part of the uncertainty in the evolution Hamiltonian. We show that this satisfies the quantum Cramer-Rao bound and thus arrive at a more general and measurable bound. Upon imposing the monotonicity condition, it reduces to the square-root derivative quantum Fisher Information. We show that on the Fisher information metric space dynamical phase is zero. A relation between square root derivative and logarithmic derivative is formulated such that both give the same Fisher information. We generalize the Fubini-Study metric for mixed states further and arrive at a set of Fubini-Study metric---called $α$ metric. This newly defined $α$ metric also satisfies the Cramer-Rao bound. Again by imposing the monotonicity condition on this metric, we derive the monotone $α$ metric. It reduces to the Fisher information metric for $α=1$.

DOI
10.48550/arxiv.1503.04146

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We provide an experimentally measurable local gauge U(1)U(1)U(1) invariant Fubini-Study (FS) metric for mixed states. Like the FS metric for pure states, it also captures only the quantum part of the uncertainty in the evolution Hamiltonian. We show that this satisfies the quantum Cramer-Rao bound and thus arrive at a more general and measurable bound. Upon imposing the monotonicity condition, it reduces to the square-root derivative quantum Fisher Information. We show that on the Fisher information metric space dynamical phase is zero. A relation between square root derivative and logarithmic derivative is formulated such that both give the same Fisher information. We generalize the Fubini-Study metric for mixed states further and arrive at a set of Fubini-Study metric---called ααα metric. This newly defined ααα metric also satisfies the Cramer-Rao bound. Again by imposing the monotonicity condition on this metric, we derive the monotone ααα metric. It reduces to the Fisher information metric for α=1α=1α=1.
A fundamental starting point in quantum information theory is the consideration of the von Neumann entropy and its generalization to relative quantum entropies. A particular feature of quantum relative entropies is their relation via their Hessian to monotonic Riemannian metrics on the dense set of invertible mixed quantum states (Lesniewski and Ruskai 1999). These metrics are also known as quantum Fisher information metrics and provide a direct link to quantum estimation theory (Helstrom 1969). Quantum Fisher information metrics which are extendable to pure states coincide all with the Fubini Study metric of the projective Hilbert space of complex rays. This theses outlines possible advantages of an inverse approach to quantum information theory, by starting with the Fubini Study metric rather then with the von Neumann entropy. This is done in a first step by associating to the Fubini Study metric a covariant and a contra-variant structure on the punctured Hilbert space as being available in the geometric formulation of quantum mechanics. While the contra-variant structure leads to a quantum version of the Cramér-Rao inequality for general 1-dimensional submanifolds of pure states, the covariant structure provides alternative entanglement monotones by identifying an inner product on the pullback tensor fields on local unitary group orbits of quantum states. It is shown in the case of two qubits that these monotones yield a more efficient estimation of entanglement than standard measures from the literature as those associated with the linearization of the von Neumann entropy.
The quantum analog of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cramér–Rao-type inequalities or the generalization of the Berry phase to mixed states. They are shown to be stochastically monotone here. All stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them. A class of metrics can be extended to pure states and a constant multiple of the Fubini–Study metric appears in the extension.
After a review of the pure state case, we discuss from a geometrical point of view the meaning of the quantum Fisher metric in the case of mixed states for a two-level system, i.e. for a q-bit, by examining the structure of the fiber bundle of states, whose base space can be identified with a co-adjoint orbit of the unitary group. We show that the Fisher information metric coincides with the one induced by the metric of the manifold of , i.e. the three-dimensional sphere S 3 , when the mixing coefficients are varied. We define the notion of Fisher tensor and show that its anti-symmetric part is intrinsically related to the Kostant–Kirillov–Souriau symplectic form that is naturally defined on co-adjoint orbits, while the symmetric part is non-trivially again represented by the Fubini–Study metric on the space of mixed states, weighted by the mixing coefficients.
In classical statistics,the Fisher information is unique in the sense that it is essentially the only monotone Riemannian metric on the space of probability densities.In quantum theory,this uniqueness breaks down,and there are many natural quantum analogues of the Fisher information,among which two particular versions distinguish themselves by their intuitive and informational significance:The first has its origin in the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurement,and is defined via the square root of the density operator. The second arises from Helstrom's study of quantum detection in 1967,and is defined via the symmetric logarithmic derivative.The aim of this paper is to compare these two versions of quantum Fisher information,and to establish two informational inequalities relating them.
In classical statistics, the Fisher information is unique in the sense that it is essentially the only monotone Riemannian metric on the space of probability densities. In quantum theory, this uniqueness breaks down, and there are many natural quantum analogues of the Fisher information, among which two particular versions distinguish themselves by their intuitive and informational significance: The first has its origin in the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurement, and is defined via the square root of the density operator. The second arises from Helstrom's study of quantum detection in 1967, and is defined via the symmetric logarithmic derivative. The aim of this paper is to compare these two versions of quantum Fisher information, and to establish two informational inequalities relating them.
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.