On the volume of the set of mixed entangled states

On the volume of the set of mixed entangled states

(February 1, 2008) | Karol Życzkowski, Paweł Horodecki, Anna Sanpera, Maciej Lewenstein
The paper explores the volume of the set of separable states in the space of all quantum states, focusing on mixed entangled states. The authors propose a natural measure on the space of density matrices and prove that the set of separable states has a non-zero volume. Analytical lower and upper bounds for the volume are derived for 2x2 and 2x3 systems. Numerical Monte Carlo simulations suggest that the volume of separable states decreases exponentially with the dimension of the composite system. The study also examines the relationship between purity and separability, finding that entanglement is more common in pure states, while separability is associated with quantum mixtures. The results highlight the dualism between purity and separability in composite systems.The paper explores the volume of the set of separable states in the space of all quantum states, focusing on mixed entangled states. The authors propose a natural measure on the space of density matrices and prove that the set of separable states has a non-zero volume. Analytical lower and upper bounds for the volume are derived for 2x2 and 2x3 systems. Numerical Monte Carlo simulations suggest that the volume of separable states decreases exponentially with the dimension of the composite system. The study also examines the relationship between purity and separability, finding that entanglement is more common in pure states, while separability is associated with quantum mixtures. The results highlight the dualism between purity and separability in composite systems.
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