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.
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.
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.
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.
We consider several definitions of the quantum Wasserstein distance based on an optimization over general bipartite quantum states with given marginals. Then, we examine the quantities obtained after the optimization is carried out over bipartite separable states instead. We prove that several of these quantities are equal to each other. Thus, we connect several approaches in the literature. We prove the triangle inequality for some of these quantities for the case of one of the three states being pure. As a byproduct, we show that the square root of the Uhlmann-Jozsa quantum fidelity can also be written as an optimization over separable states with given marginals. We use this to prove that some of these quantities equal the Uhlmann-Jozsa quantum fidelity for qubits. We also find relations with the superfidelity.
We prove dimension-independent Hyers--Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here ε≥0 is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most 9ε/2 in the trace metric and 22ε in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to 3ε even under the hypothesis of δ-surjectivity, independently of δ. In infinite dimensions, we construct spectrum-preserving homeomorphisms of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as ε→0, but no such modulus can be chosen independently of dimension.
When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as the complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sjöqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we explicitly illustrate the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial state and the final mixed state when calculated with the two metrics. Interestingly, in analogy with what happens when studying the topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of a finite distance between mixed quantum states is not preserved when comparing distances determined with the Bures and the Sjöqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of the complexity and volume of mixed quantum states.