February 1, 2008 | Karol Życzkowski*, Paweł Horodecki, Anna Sanpera and Maciej Lewenstein
The paper investigates the volume of separable states in the set of all quantum states for N-dimensional systems. A natural measure is introduced for density matrices, and it is proven that the set of separable states has a nonzero volume. Analytical bounds are derived for 2×2 and 2×3 systems, and numerical Monte Carlo simulations show that the volume of separable states decreases exponentially with the system's dimension. The study also explores the dualism between purity and separability, showing that entanglement is typical for pure states, while separability is associated with quantum mixtures. The results indicate that states with sufficiently low purity are necessarily separable. The paper concludes that the volume of separable states decreases exponentially with the system's dimension, and that the set of separable states has a nonzero volume under the proposed measure.The paper investigates the volume of separable states in the set of all quantum states for N-dimensional systems. A natural measure is introduced for density matrices, and it is proven that the set of separable states has a nonzero volume. Analytical bounds are derived for 2×2 and 2×3 systems, and numerical Monte Carlo simulations show that the volume of separable states decreases exponentially with the system's dimension. The study also explores the dualism between purity and separability, showing that entanglement is typical for pure states, while separability is associated with quantum mixtures. The results indicate that states with sufficiently low purity are necessarily separable. The paper concludes that the volume of separable states decreases exponentially with the system's dimension, and that the set of separable states has a nonzero volume under the proposed measure.