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

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 $\gamma =0.5(\ln 3∕\ln 2-1)\approx 0.29248125$ 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 $\sqrt{3^{m+1}∕(3^m+3)}\times 2^{-(1+\gamma )m}\equiv \sqrt{3^{m+1}∕(3^m+3)}\times 6^{-m∕2}$ (note that $\kappa =\sqrt{3}$), compared to the previous $2\times 2^{-3m∕2}$. This implies that with parameters realistic for current experiments, nuclear magnetic resonance (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 $d_0\to \infty$. DOI: http://dx.doi.org/10.1103/PhysRevA.72.032322 © 2005 The American Physical Society