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Largest separable balls around the maximally mixed bipartite quantum state

Leonid Gurvits·Howard Barnum
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Abstract

For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral $l_p$ norms for $1 \le p \le \infty$, of separable (unentangled) matrices around the identity matrix. This implies a simple and intutively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\tr ρ^2$ not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by NMR quantum computation implementations or other experimental situations.

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Leonid Gurvits, Howard Barnum. 2002-12-19. Largest separable balls around the maximally mixed bipartite quantum state. https://doi.org/10.1103/physreva.66.062311

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