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AIP conference proceedings

Effect of Bound Entanglement on the Convertibility of Pure States

Satoshi Ishizaka

Published Jan 1, 2004• 0 citations
Publisher

Abstract

I show that bound entanglement strongly influences the quantum entanglement processing of pure states: If N distant parties share appropriate bound entangled states with positive partial transpose, all N‐partite pure entangled states become inter‐convertible by stochastic local operations and classical communication (SLOCC) at the single copy level. This implies that the Schmidt rank of a bipartite pure entangled state can be increased, and that two incomparable tripartite entanglement of the GHZ and W type can be inter‐converted by the assistance of bound entanglement. Further, I propose the simplest experimental scheme for the demonstration of the corresponding bound‐entanglement‐assisted SLOCC. This scheme does not need quantum gates and is feasible for the current experimental technology of linear optics.

DOI
10.1063/1.1834430

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I show that bound entanglement strongly influences the quantum entanglement processing of pure states: If N distant parties share appropriate bound entangled states with positive partial transpose, all N‐partite pure entangled states become inter‐convertible by stochastic local operations and classical communication (SLOCC) at the single copy level. This implies that the Schmidt rank of a bipartite pure entangled state can be increased, and that two incomparable tripartite entanglement of the GHZ and W type can be inter‐converted by the assistance of bound entanglement. Further, I propose the simplest experimental scheme for the demonstration of the corresponding bound‐entanglement‐assisted SLOCC. This scheme does not need quantum gates and is feasible for the current experimental technology of linear optics.
I show that two distant parties can transform pure entangled states to arbitrary pure states by stochastic local operations and classical communication (SLOCC) at the single copy level, if they share bound entangled states. This is the effect of bound entanglement since this entanglement processing is impossible by SLOCC alone. A similar effect of bound entanglement exists in three qubits where two incomparable entangled states of GHZ (Greenberger, Horne, and Zeilinger) and W can be interconverted. In general, multipartite settings composed by N distant parties, all N-partite pure entangled states are interconvertible by SLOCC with the assistance of bound entangled states with positive partial transpose.
I show that two distant parties can transform a single copy of their pure entangled state to arbitrary pure state by stochastic local operations and classical communication (SLOCC), if they share a single copy of bound entangled states. It has been known that this entanglement processing is impossible by SLOCC alone. This is the first example of the effect of bound entangled states on the entanglement processing for bipartite pure states. Further, I obtain the optimal attainable probability for several classes of states.
We consider the transformation of multipartite states in the single-copy setting under positive-partial-transpose-preserving operations (PPT operations) and obtain both qualitative and quantitative results. First, for some pure-state transformations that are impossible under local operations and classical communication (LOCC), we demonstrate that they become possible with a surprisingly large success probability under PPT operations. Furthermore, we clarify the convertibility of arbitrary multipartite pure states under PPT operations and show that a drastic simplification in the classification of pure-state entanglement occurs when the set of operations is switched from LOCC to PPT operations. Indeed, the infinitely many types of LOCC-incomparable entanglement are reduced to only one type under the action of PPT operations. This is a clear manifestation of the increased power afforded by the use of PPT-bound entanglement. In addition, we further enlarge the set of operations to clarify the effect of another type of bound entanglement, multipartite unlockable bound entanglement, and show that a further simplification occurs. As compared to pure states a more complicated situation emerges in mixed-state settings. While single-copy distillation becomes possible under PPT operations for some mixed states it remains impossible for other mixed states.
It is shown that the order property of pure bipartite states under stochastic local operations and classical communications (SLOCC) changes radically when dimensionality shifts from finite to infinite. In contrast to finite-dimensional systems where there is no pure incomparable state, the existence of infinitely many mutually SLOCC incomparable states is shown for infinite-dimensional systems even under the bounded energy and finite information exchange condition. These results show that the effect of the infinite dimensionality of Hilbert space, the ``infinite workspace'' property, remains even in physically relevant infinite-dimensional systems.
Abstract Quantum state merging is one of the most important protocols in quantum information theory. In this task two parties aim to merge their parts of a pure tripartite state by making use of additional singlets while preserving correlations with a third party. We study a variation of this scenario where the shared state is not necessarily pure, and the merging parties have free access to local operations, classical communication, and positive partial transpose (PPT) entangled states. We provide general conditions for a state to admit perfect merging, and present a family of fully separable states which cannot be perfectly merged if the merging parties have no access to additional singlets. We also show that free PPT entangled states do not give any advantage for merging of pure states, and the conditional entropy plays the same role as in standard quantum state merging quantifying the rate of additional singlets needed to perfectly merge the state.
The tensor rank (also known as generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investigate the tensor rank of symmetric states such as the tripartite state |W3>=1/√3(|100> + |010> + |001>) and its N-partite generalization |W(N)>. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of |W3> have a rank of either 15 or 16, (ii) two copies of |W(N)> have a rank of 3N - 2, and (iii) n copies of |W(N)> have a rank of O(N). A remarkable consequence of these results is that certain multipartite transformations, impossible even probabilistically, can become possible when performed in multiple-copy bunches or when assisted by some catalyzing state. This effect is impossible for bipartite pure states.