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arXiv:quant-ph/0002089 · Physical Review A

Operational criterion and constructive checks for the separability of low-rank density matrices

Paweł Horodecki, Maciej Lewenstein, Guifré Vidal, J. I. Cirac

Published Aug 17, 2000• 150 citations
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Abstract

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 $(M<~N),$ 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.

DOI
10.1103/physreva.62.032310

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We consider low-rank density operators \ensuremathϱ\ensuremath{\varrho}\ensuremathϱ supported on a M\ifmmode×\else\texttimes\fiNM\ifmmode\times\else\texttimes\fi{}NM\ifmmode×\else\texttimes\fiN Hilbert space for arbitrary M and N (M< N),(M<~N),(M< N), and with a positive partial transpose (PPT) \ensuremathϱTA> 0.{\ensuremath{\varrho}}^{{T}_{A}}>~0.\ensuremathϱTA​> 0. For rank r(\ensuremathϱ)< Nr(\ensuremath{\varrho})<~Nr(\ensuremathϱ)< N we prove that having a PPT is necessary and sufficient for \ensuremathϱ\ensuremath{\varrho}\ensuremathϱ 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 \ensuremathϱ\ensuremath{\varrho}\ensuremathϱ and \ensuremathϱTA{\ensuremath{\varrho}}^{{T}_{A}}\ensuremathϱTA​ satisfies r(\ensuremathϱ)+r(\ensuremathϱTA)< 2MN\ensuremath−M\ensuremath−N+2.r(\ensuremath{\varrho})+r({\ensuremath{\varrho}}^{{T}_{A}})<~2MN\ensuremath{-}M\ensuremath{-}N+2.r(\ensuremathϱ)+r(\ensuremathϱTA​)< 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.
Deciding whether a mixed quantum state is separable or entangled is a difficult problem in general. Separable states are positive under partial transposition [they are positive-partial-transpose (PPT) states], but this simple test does not exclude all entangled states. In order to understand the entangled PPT states, having so-called bound entanglement, we want to study the extremal PPT states. An extremal PPT state is either a pure product state, then it is separable, or it is entangled, in which case the state and its partial transpose must still be rather low-rank density matrices, although it is known that the rank must be at least 4. In a previous paper we presented a complete classification of the rank-4 entangled PPT states in dimension $3\ifmmode\times\else\texttimes\fi{}3$, generalizing the construction by Bennett et al. from unextendible sets of orthogonal product vectors. In the present paper we continue our investigations of the low-rank entangled PPT states, mostly in dimension $3\ifmmode\times\else\texttimes\fi{}3$, using a combination of analytical and numerical methods. We use perturbation theory in order to construct rank-5 entangled PPT states close to the known rank-4 states, and in order to compute dimensions and study the geometry of surfaces of low-rank PPT states. We exploit the close connection between low-rank PPT states and product vectors. In particular, we show how to reconstruct a low-rank PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.
We analyze the separability properties of density operators supported on ${C}^{2}\ensuremath{\bigotimes}{C}^{N}$ whose partial transposes are positive operators. We show that if the rank of $\ensuremath{\rho}$ equals N then it is separable, and that bound entangled states have ranks larger than N. We also give a separability criterion for a generic density operator such that the sum of its rank and the one of its partial transpose does not exceed 3N.3N.3N. If it exceeds this number, we show that one can subtract product vectors until it decreases to 3N,3N,3N, while keeping the positivity of $\ensuremath{\rho}$ and its partial transpose. This automatically gives us a sufficient criterion for separability for general density operators. We also prove that all density operators that remain invariant after partial transposition with respect to the first system are separable.
We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.
One of the most important problems in quantum information is the separability problem, which asks whether a given quantum state is separable. We investigate multipartite states of rank at most 4 which are PPT (i.e., all their partial transposes are positive semidefinite). We show that any PPT state of rank 2 or 3 is separable and has length at most 4. For separable states of rank 4, we show that they have length at most 6. It is six only for some qubit–qutrit or multiqubit states. It turns out that any PPT entangled state of rank 4 is necessarily supported on a 3⊗3 or a 2⊗2⊗2 subsystem. We obtain a very simple criterion for the separability problem of the PPT states of rank at most 4: such a state is entangled if and only if its range contains no product vectors. This criterion can be easily applied since a four-dimensional subspace in the 3⊗3 or 2⊗2⊗2 system contains a product vector if and only if its Plücker coordinates satisfy a homogeneous polynomial equation (the Chow form of the corresponding Segre variety). We have computed an explicit determinantal expression for the Chow form in the former case, while such an expression was already known in the latter case.
We investigate optimal separable approximations (decompositions) of states rho of bipartite quantum systems A and B of arbitrary dimensions MxN following the lines of Ref. [M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 (1998)]. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two qubit systems (M=N=2) the best separable approximation has a form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in the best separable approximation is not maximally entangled. We demonstrate that the weight of the entangled state in the best separable approximation in arbitrary dimensions provides a good entanglement measure. We prove in general for arbitrary M and N that the best separable approximation corresponds to a mixture of a separable and an entangled state which are both unique. We develop also a theory of optimal separable approximations for states with positive partial transpose (PPT states). Such approximations allow to decompose any density operator with positive partial transpose as a sum of a separable state and an entangled PPT state. We discuss procedures of constructing such decompositions.