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arXiv:quant-ph/0411098 · Physical Review A

Class of bound entangled states of N+N qubits revealed by nondecomposable maps

Marco Piani

Published Jan 31, 2006• 32 citations
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Abstract

We introduce a family of linear maps which are positive but not completely positive. We exhibit examples of nondecomposable maps and ${2}^{N}\ensuremath{\bigotimes}{2}^{N}$, $N\ensuremath{\geqslant}2$, bound entangled states---e.g., nondistillable bipartite states of $N+N$ qubits. Such states, as the standard Bell diagonal states, are diagonal in a maximally entangled basis and, apart remaining positive under partial transposition, are not detected by the realignment criterion.

DOI
10.1103/physreva.73.012345

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We introduce a family of linear maps which are positive but not completely positive. We exhibit examples of nondecomposable maps and 2N\ensuremath⨂2N{2}^{N}\ensuremath{\bigotimes}{2}^{N}2N\ensuremath⨂2N, N\ensuremath⩾2N\ensuremath{\geqslant}2N\ensuremath⩾2, bound entangled states---e.g., nondistillable bipartite states of N+NN+NN+N qubits. Such states, as the standard Bell diagonal states, are diagonal in a maximally entangled basis and, apart remaining positive under partial transposition, are not detected by the realignment criterion.
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We exhibit a two-parameter family of bipartite mixed states ${\ensuremath{\rho}}_{\mathrm{bc}},$ in a $d\ensuremath{\bigotimes}d$ Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in $2\ensuremath{\bigotimes}2$ can be distilled by local quantum operations and classical communication (LQ+CC). Evidence for this undistillability is provided by the result that, for certain states in this family, we cannot extract entanglement from any arbitrarily large number of copies of ${\ensuremath{\rho}}_{\mathrm{bc}}$ using a projection on $2\ensuremath{\bigotimes}2.$ These states are canonical NPT states in the sense that any bipartite mixed state in any dimension with NPT can be reduced by LQ+CC operations to a NPT state of the ${\ensuremath{\rho}}_{\mathrm{bc}}$ form. We show that the main question about the distillability of mixed states can be formulated as an open mathematical question about the properties of composed positive linear maps.