Better bound on the exponent of the radius of the multipartite separable ball
Leonid I Gurvits, Howard Barnum · Physical Review A · 2005
We show that for an $m$-qubit quantum system, there is a ball of radius asymptotically approaching $\ensuremath{\kappa}{2}^{\ensuremath{-}\ensuremath{\gamma}m}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent $\ensuremath{\gamma}=0.5(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}3∕\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2\ensuremath{-}1)\ensuremath{\approx}0.292\phantom{\rule{0.2em}{0ex}}481\phantom{\rule{0.2em}{0ex}}25$ 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 $\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}(1+\ensuremath{\gamma})m}\ensuremath{\equiv}\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{6}^{\ensuremath{-}m∕2}$ (note that $\ensuremath{\kappa}=\sqrt{3}$), compared to the previous $2\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}3m∕2}$. 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 ${d}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$.