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arXiv:quant-ph/0309216 · Physical Review Letters

Noiseless Quantum Circuits for the Peres Separability Criterion

Hilary A. Carteret

Published Jan 31, 2005• 64 citations
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Abstract

In this Letter we give a method for constructing sets of simple circuits that can determine the spectrum of a partially transposed density matrix, without requiring either a tomographically complete positive-operator-valued measurement or the addition of noise to make the spectrum non-negative. These circuits depend only on the dimension of the Hilbert space and are otherwise independent of the state.

DOI
10.1103/physrevlett.94.040502

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In this Letter we give a method for constructing sets of simple circuits that can determine the spectrum of a partially transposed density matrix, without requiring either a tomographically complete positive-operator-valued measurement or the addition of noise to make the spectrum non-negative. These circuits depend only on the dimension of the Hilbert space and are otherwise independent of the state.
We present a method for checking the Peres separability criterion in an arbitrary bipartite quantum state ${\ensuremath{\rho}}_{\mathrm{AB}}$ within local operations and classical communication scenario. The method does not require noise operation which is needed in making the partial transposition map physically implementable. The main task for the two observers, Alice and Bob, is to measure some specific functions of the partial transposed matrix. With these functions, they can determine the eigenvalues of ${\ensuremath{\rho}}_{\mathrm{AB}}^{{T}_{\mathrm{B}}}$, among which the minimum serves as an entanglement witness.
A quantum system consisting of two subsystems is separable if its density matrix can be written as $\ensuremath{\rho}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{A}^{}{w}_{A}{\ensuremath{\rho}}_{A}^{\ensuremath{'}}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}},$ where ${\ensuremath{\rho}}_{A}^{\ensuremath{'}}$ and ${\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}}$ are density matrices for the two subsystems, and the positive weights wA{w}_{A}wA​ satisfy $\ensuremath{\Sigma}{w}_{A}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of \ensuremath{\rho}, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.
The separability from spectrum problem asks for a characterization of the quantum states ρ ∈ Mm ⊗Mn with the property that UρU † is separable for all unitary matrices U. We provide evidence that ρ is separable from spectrum if and only if UρU † has positive partial transpose for all unitary matrices U. In particular, we show that many well-known separability criteria are unable to detect entanglement in any such state, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer–Hall map. We also show that these two properties coincide for the families of isotropic and Werner states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest. 1
It is well known that the positive partial transpose (PPT) criterion for determining separability is an operational separability criterion. However, in a high-dimensional (>6) situation, this criterion is not sufficient. How can we judge whether an entangled state in any high-dimensional quantum system is PPT or not? Here we propose a linear algebra method for checking the positivity of the partial transpose of a state and present a set of entangled mixed states with non-positive partial transpose.
We consider low-rank density operators $\ensuremath{\varrho}$ supported on a $M\ifmmode\times\else\texttimes\fi{}N$ Hilbert space for arbitrary M and N (M< N),(M<~N),(M< N), and with a positive partial transpose (PPT) ${\ensuremath{\varrho}}^{{T}_{A}}>~0.$ For rank $r(\ensuremath{\varrho})<~N$ we prove that having a PPT is necessary and sufficient for $\ensuremath{\varrho}$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank-3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\ensuremath{\varrho}$ and ${\ensuremath{\varrho}}^{{T}_{A}}$ satisfies $r(\ensuremath{\varrho})+r({\ensuremath{\varrho}}^{{T}_{A}})<~2MN\ensuremath{-}M\ensuremath{-}N+2.$ This separability condition has the form of a constructive check, thus also providing a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.