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Infinite Dimensional Analysis Quantum Probability and Related Topics

Metrics induced by certain Hilbert-Schmidt fidelities on positive semi-definite matrices

Ho Minh Toan, Vu The Khoi

Published Apr 23, 2023• 0 citations
Publisher

Abstract

Motivated by measuring the degree of similarity of a pair of quantum states (density matrices), we consider the metric property of the modified Bures angles and modified Bures distances of symmetric functions which are extensions of some fidelity measures on the spaces [Formula: see text] of nonzero positive semi-definite matrices. We use the positive semi-definiteness of the Gram-type matrices to characterize the metric property of the modified Bures angles. As a consequence, we can show that the modified Bures angles induced by the geometric mean, harmonic mean, minimum and maximum of two positive numbers are metrics on [Formula: see text]. In addition, we can also show that the metric property of the modified Bures angles is stronger than that of the modified Bures distances.

DOI
10.1142/s0219025723500078

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Motivated by measuring the degree of similarity of a pair of quantum states (density matrices), we consider the metric property of the modified Bures angles and modified Bures distances of symmetric functions which are extensions of some fidelity measures on the spaces [Formula: see text] of nonzero positive semi-definite matrices. We use the positive semi-definiteness of the Gram-type matrices to characterize the metric property of the modified Bures angles. As a consequence, we can show that the modified Bures angles induced by the geometric mean, harmonic mean, minimum and maximum of two positive numbers are metrics on [Formula: see text]. In addition, we can also show that the metric property of the modified Bures angles is stronger than that of the modified Bures distances.
Density matrices are fundamental mathematical objects for describing quantum states, and their set can be viewed as a subset of the manifold of symmetric positive definite (SPD) matrices. Based on the Riemannian geometric structure on the SPD manifold, this paper analyzes geometric paths and distance functions on the space of density matrices. We adopt the affine-invariant Riemannian metric and review the associated geodesics and Riemannian distance, comparing them with linear interpolation and the Frobenius distance defined in the ambient Euclidean space. Numerical experiments are conducted to visualize geodesic paths between density matrices and to quantitatively compare several commonly used matrix distances, including the Frobenius, Riemannian, and Bures distances. The results demonstrate that distances induced by different geometric structures exhibit fundamentally different behaviors in characterizing discrepancies between density matrices, reflecting the intrinsic non-Euclidean geometry of the space. These analyses and numerical results provide a coherent geometric perspective for understanding the structure and similarity of density matrices.
On the space of mixed quantum states, several alternative fidelities have been proposed besides the standard Uhlmann-Jozsa fidelity. It has been known that several properties of the Uhlmann-Jozsa fidelity still hold true for these alternative fidelities. The aim of this paper is to give positive answers to some questions about the metric properties of functionals of alternative quantum fidelities raised by Y. C. Liang {\it et al.} in \cite{LYM}. Our method is to use the non-negativity of the Gram determinant of three vectors constructed from the quantum states to prove the triangle inequality for the modified Bures angle.
We propose three alternative measures for non-Gaussianity of quantum states: sine distance, Bures angle, and Bures distance, which are based on quantum fidelity introduced by Wang [Phys. Lett. A 373 58 (2008)]. Using them, we evaluate the non-Gaussianity of some relevant single-mode and two-mode non-Gaussian states and find a good consistency of the three examined measures. In addition, we show that such metrics can exactly quantify the degree of Gaussianity of even Schrödinger-cat-like states of small amplitudes that can not be measured by other known non-Gaussianity measures such as the Hilbert–Schmidt metric and the relative entropy metric. We make a comparative study between all existing non-Gaussianity measures according to the metric axioms and point out that the sine distance is the best candidate among them.
The notion of fidelity for quantum states is a measure of how much two states overlap. In the matrix formalism of quantum mechanics, states are represented by density operators, i.e., positive semi-definite matrices with trace equal to 1 in a complex Euclidean space Mn(C)M_n(\mathbb{C})Mn​(C). Felipe-Sosa and Felipe (2022) introduced the notion of quantum states on certain Krein spaces with indefinite metric induced by a fundamental symmetry JJJ, calling these JJJ-states. We define an analogous notion of measurement for JJJ-states to the regular quantum theory and use it to show that a notion of fidelity holds in the Krein setting. We also show that an analogous result to the Fuchs-Caves measurement holds in this setting. Following the developments of Bag, Rohilla, and Trivedi (2024), we then extend this definition of fidelity to UUU-quantum states on SSS-spaces. We demonstrate that the analogous geometric motivation holds in the Krein and SSS-space setting, as holds for quantum fidelity and geometric means of operators.
Uhlmann's fidelity function is one of the most widely used similarity measures in quantum theory. One definition of this function is that it is the minimum classical fidelity associated with a quantum-to-classical measurement procedure of two quantum states. In 2010, Matsumoto introduced another fidelity function which is dual to Uhlmann's in the sense that it is the maximimum classical fidelity associated with a classical-to-quantum preparation procedure for two quantum states. Matsumoto's fidelity can also be defined using the well-established notion definition of the matrix geometric mean. In this work, we examine Matsumoto's fidelity through the lens of semidefinite programming to give simple proofs that it possesses many desirable properties for a similarity measure, including monotonicity under quantum channels, joint concavity, and unitary invariance. Finally, we provide a geometric interpretation of this fidelity in terms of the Riemannian space of positive definite matrices, and show how this picture can be useful in understanding some of its peculiar properties.
The Bures distance holds a special place among various distance measures due to its several distinguishing features and has applications in diverse problems in quantum information theory. It is related to fidelity, and among other things, it serves as a bona fide measure for quantifying the separability of quantum states. In this work, we calculate exact analytical results for the mean root fidelity and mean-square Bures distance between a fixed density matrix and a random density matrix and also between two random density matrices. In the course of derivation, we also obtain the spectral density for the product of the above pairs of density matrices. We corroborate our analytical results using Monte Carlo simulations. Moreover, we compare these results with the mean-square Bures distance between reduced density matrices generated using coupled kicked tops and find very good agreement.