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Size of the separable neighborhood of the maximally mixed bipartite quantum state

Leonid Gurvits, Howard Barnum

Published Apr 26, 2002• 4 citations
OpenAlex

Abstract

For finite-dimensional bipartite quantum systems, we find the size of the largest balls, in spectral $l_p$ norms for $1 \\le p \\le \\infty$, of separable (unentangled) matrices around the identity matrix. We discuss corollaries and applications to density matrices and bulk quantum information processing such as high-temperature nuclear magnetic resonance.

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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 ${d}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$.

arXiv:quant-ph/0409095•Physical Review A•Sep 16, 2005•44 citations
For finite-dimensional bipartite quantum systems, we find the size of the largest balls, in spectral lpl_plp​ norms for 1lepleinfty1 \\le p \\le \\infty1lepleinfty, of separable (unentangled) matrices around the identity matrix. We discuss corollaries and applications to density matrices and bulk quantum information processing such as high-temperature nuclear magnetic resonance.
For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral lp{l}_{p}lp​ norms for $1<~p<~\ensuremath{\infty},$ of separable (unentangled) matrices around the identity matrix. This implies a simple and intuitively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\mathrm{tr}{\ensuremath{\rho}}^{2}$ 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 nuclear magnetic resonance quantum computation implementations or other experimental situations.
We show that for an m-partite quantum system, there is a ball of radius ${2}^{\ensuremath{-}(m/2\ensuremath{-}1)}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices. This can be used to derive an \ensuremath{\epsilon} below which mixtures of \ensuremath{\epsilon} of any density matrix with $1\ensuremath{-}\ensuremath{\epsilon}$ of the maximally mixed state will be separable. The \ensuremath{\epsilon} thus obtained is exponentially better (in the number of systems) than existing results. This gives a number of qubits below which nuclear magnetic resonance with standard pseudopure-state preparation techniques can access only unentangled states; with parameters realistic for current experiments, this is 23 qubits (compared to 13 qubits via earlier results). A ball of radius 1 is obtained for multipartite states separable over the reals.
Given a bipartite quantum system represented by a Hilbert space ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2},$ we give an elementary argument to show that if either $\mathrm{dim}{\mathcal{H}}_{1}=\ensuremath{\infty}$ or $\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ then the set of nonseparable density operators on ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}$ is trace-norm dense in the set of all density operators (and the separable density operators nowhere dense). This result complements recent detailed investigations of separability, which show that when $\mathrm{dim}{\mathcal{H}}_{i}<\ensuremath{\infty}$ for i=1,2,i=1,2,i=1,2, there is a separable neighborhood (perhaps very small for large dimensions) of the maximally mixed state.
We show that for an mmm-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∕21∕21∕2. For normalized mmm-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 mmm-partite systems of fixed local dimension d0{d}_{0}d0​, although approaching our earlier exponent as ${d}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$.