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arXiv:1508.03753 · New Journal of Physics

Quantum state merging with bound entanglement

Alexander Streltsov

Published Jan 29, 2020• 2 citations
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Abstract

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.

DOI
10.1088/1367-2630/ab70d7

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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.
Understanding the resource consumption in distributed scenarios is one of the main goals of quantum information theory. A prominent example for such a scenario is the task of quantum state merging, where two parties aim to merge their tripartite quantum state parts. In standard quantum state merging, entanglement is considered to be an expensive resource, while local quantum operations can be performed at no additional cost. However, recent developments show that some local operations could be more expensive than others: it is reasonable to distinguish between local incoherent operations and local operations which can create coherence. This idea leads us to the task of incoherent quantum state merging, where one of the parties has free access to local incoherent operations only. In this case the resources of the process are quantified by pairs of entanglement and coherence. Here, we develop tools for studying this process and apply them to several relevant scenarios. While quantum state merging can lead to a gain of entanglement, our results imply that no merging procedure can gain entanglement and coherence at the same time. We also provide a general lower bound on the entanglement-coherence sum and show that the bound is tight for all pure states. Our results also lead to an incoherent version of Schumacher compression: in this case the compression rate is equal to the von Neumann entropy of the diagonal elements of the corresponding quantum state.
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.
Two important subsets of tripartite pure states, namely {\\em triseparable} states (i.e. states such that the bipartite density matrices obtained by tracing out any party's subsystem are separable) and {\\em triorthogonal} states (i.e. states with a three-party Schmidt-like decomposition) are introduced. It is proved that any triseparable state is triorthogonal and vice-versa. The relevance of this result to quantifying tripartite pure-state entanglement is discussed. It is also proved that any purification of a bipartite bound entangled state with positive partial transpose is {\\em tri-inseparable}. This is a new necessary condition for such bound entanglement.
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.
We introduce the concentrated information of tripartite quantum states. For three parties Alice, Bob, and Charlie, it is defined as the maximal mutual information achievable between Alice and Charlie via local operations and classical communication performed by Charlie and Bob. We derive upper and lower bounds to the concentrated information, and obtain a closed expression for it on several classes of states including arbitrary pure tripartite states in the asymptotic setting. We show that distillable entanglement, entanglement of assistance, and quantum discord can all be expressed in terms of the concentrated information, thus revealing its role as a unifying informational primitive. We finally investigate quantum state merging of mixed states with and without additional entanglement. The gap between classical and quantum concentrated information is proven to be an operational figure of merit for mixed state merging in the absence of additional entanglement. Contrary to the pure state merging, our analysis shows that classical communication in both directions can provide an advantage for merging of mixed 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.