On the structure of the body of states with positive partial transpose

Abstract

We show that the convex set of separable mixed states of the 2 x 2 system is a body of constant height. This fact is used to prove that the probability to find a random state to be separable equals 2 times the probability to find a random boundary state to be separable, provided the random states are generated uniformly with respect to the Hilbert-Schmidt (Euclidean) distance. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system.

arXiv comment10 pages, 1 figure; ver. 2 - minor changes, new proof of lemma 3

References in corpus (17)

• Separability Criterion for Density Matrices
• A computable measure of entanglement
• Separability of Mixed States: Necessary and Sufficient Conditions
• On the volume of the set of mixed entangled states
• Largest separable balls around the maximally mixed bipartite quantum state
• On the volume of the set of mixed entangled states II
• Geometry of entangled states
• Bures volume of the set of mixed quantum states
• Hilbert--Schmidt volume of the set of mixed quantum states
• Separable balls around the maximally mixed multipartite quantum states
• Geometry of quantum systems: density states and entanglement
• The volume of separable states is super-doubly-exponentially small
• A priori Probabilities of Separable Quantum States
• Relativity of pure states entanglement
• Better bound on the exponent of the radius of the multipartite separable ball
• Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems
• Qubit-Qutrit Separability-Probability Ratios

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• Approximating the Set of Separable States Using the Positive Partial Transpose Test
• How often is a random quantum state k-entangled?
• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities
• A Concise Formula for Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities
• Eigenvalues, Separability and Absolute Separability of Two-Qubit States
• The absolute positive partial transpose property for random induced states
• Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities
• Two-Qubit Separability Probabilities and Beta Functions
• Bound entangled states with extremal properties
• Subnormalized states and trace-nonincreasing maps
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities
• Convex set of quantum states with positive partial transpose analysed by hit and run algorithm
• Ratios of maximal concurrence-parameterized separability functions, and generalized Peres-Horodecki conditions
• Invariance of Bipartite Separability and PPT-Probabilities over Casimir Invariants of Reduced States
• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
• Scalar leptoquark and vector-like quark extended models as the explanation of the muon $g-2$ anomaly: bottom partner chiral enhancement case
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems
• Two-Qubit Separability Probabilities as Joint Functions of the Bloch Radii of the Qubit Subsystems
• Formulas for Generalized Two-Qubit Separability Probabilities