Whole and Half Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities (original) (raw)
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Generalized two-qubit whole and half Hilbert–Schmidt separability probabilities
Journal of Geometry and Physics, 2015
Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state (ρ) is separable/disentangled is 8 33 (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal 4F 3-hypergeometric moment formula (Appendix A), we reach, via densityapproximation (inverse) procedures, the conclusion that one-half (4 33) of this probability arises when the determinantal inequality |ρ P T | > |ρ|, where P T denotes the partial transpose, is satisfied, and, the other half, when |ρ| > |ρ P T |. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of 4 × 4 density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-"re[al]bit" and two"quater[nionic]bit" separability probabilities of 29 64 and 26 323 , respectively. The new determinantal 4F 3-hypergeometric moment formula is, then, adjusted (Appendices B and C) to the boundary case of minimally degenerate states (|ρ| = 0), and its consistency manifestedalso using density-approximation-with an important theorem of Szarek, Bengtsson andŻyczkowski (arXiv:quant-ph/0509008). This theorem states that the Hilbert-Schmidt separability probabilities of generic minimally degenerate two-qubit states are (again) one-half those of the corresponding generic nondegenerate states.
Advances in Mathematical Physics
Previously, a formula, incorporating a5F4hypergeometric function, for the Hilbert-Schmidt-averaged determinantal momentsρPTnρk/ρkof4×4density-matrices (ρ) and their partial transposes (|ρPT|), was applied withk=0to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as we note here, as an averaging over “induced measures in the space of mixed quantum states.” The associated induced-measure separability probabilities (k=1,2,…) are found—viaa high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit (α=1/2), (standard) two-qubit (α=1), and two-quater[nionic]bit (α=2) cases. We deduce rather simple companion (rebit, qubit, quaterbit, …) formulas that successfully reproduce the rational values assumed forgeneral k. These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.
Random Bures mixed states and the distribution of their purity
Journal of Physics A: Mathematical and Theoretical, 2010
Ensembles of random density matrices determined by various probability measures are analysed. A simple and efficient algorithm to generate at random density matrices distributed according to the Bures measure is proposed. This procedure may serve as an initial step in performing Bayesian approach to quantum state estimation based on the Bures prior. We study the distribution of purity of random mixed states. The moments of the distribution of purity are determined for quantum states generated with respect to the Bures measure. This calculation serves as an exemplary application of the "deform-and-study" approach based on ideas of integrability theory. It is shown that Painlevé equation appeared as a part of the presented theory.
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Purity distribution for generalized random Bures mixed states
Journal of Physics A: Mathematical and Theoretical, 2012
We compute the distribution of the purity for random density matrices (i.e.random mixed states) in a large quantum system, distributed according to the Bures measure. The full distribution of the purity is computed using a mapping to random matrix theory and then a Coulomb gas method. We find three regimes that correspond to two phase transitions in the associated Coulomb gas. The first transition is characterized by an explosion of the third derivative on the left of the transition point. The second transition is of first order, it is characterized by the detachement of a single charge of the Coulomb gas. A key remark in this paper is that the random Bures states are closely related to the O(n) model for n = 1. This actually led us to study "generalized Bures states" by keeping n general instead of specializing to n = 1.
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A quantum version of the Monge-Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states ρ AB , such that both of its reduced density matrices ρ A and ρ B of dimension N are fixed. We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between ρ A and ρ B , which is bounded from below by the rescaled Bures distance and from above by the root infidelity. In the single qubit case we provide a semi-analytic expression for the optimal transport cost between any two states and prove that its square root satisfies the triangle inequality and yields an analogue of the Wasserstein distance of order two on the set of density matrices. We introduce an associated measure of proximity of quantum states, called SWAP-fidelity, and discuss its properties and applications in quantum machine learning.