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Better bound on the exponent of the radius of the multipartite separable ball

Leonid Gurvits·Howard Barnum
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Abstract

We show that for an m-qubit quantum system, there is a ball of radius asymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is much smaller in magnitude than the best previously known exponent, from our earlier work, of 1/2. For normalized m-qubit states, we get a separable ball of radius sqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + γ)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\times 6^{-m/2} (note that κ= \sqrt{3}), compared to the previous 2 * 2^{-3m/2}. This implies that with parameters realistic for current experiments, NMR with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for m-partite systems of fixed local dimension d_0, although approaching our earlier exponent as d_0 approaches infinity.

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Leonid Gurvits, Howard Barnum. 2005-06-13. Better bound on the exponent of the radius of the multipartite separable ball. https://doi.org/10.1103/physreva.72.032322

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