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arXiv:1912.06569 · Quantum Reports

Distance between Bound Entangled States from Unextendible Product Bases and Separable States

Marcin Wieśniak, Palash Pandya, Omer Sakarya, Bianka Woloncewicz

Published Jan 13, 2020• 19 citations
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Abstract

We discuss the use of the Gilbert algorithm to tailor entanglement witnesses for unextendible product basis bound entangled states (UPB BE states). The method relies on the fact that an optimal entanglement witness is given by a plane perpendicular to a line between the reference state, entanglement of which is to be witnessed, and its closest separable state (CSS). The Gilbert algorithm finds an approximation of CSS. In this article, we investigate if this approximation can be good enough to yield a valid entanglement witness. We compare witnesses found with Gilbert algorithm and those given by Bandyopadhyay–Ghosh–Roychowdhury (BGR) construction. This comparison allows us to learn about the amount of entanglement and we find a relationship between it and a feature of the construction of UPBBE states, namely the size of their central tile. We show that in most studied cases, witnesses found with the Gilbert algorithm in this work are more optimal than ones obtained by Bandyopadhyay, Ghosh, and Roychowdhury. This result implies the increased tolerance to experimental imperfections in a realization of the state.

DOI
10.3390/quantum2010004

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We discuss the use of the Gilbert algorithm to tailor entanglement witnesses for unextendible product basis bound entangled states (UPB BE states). The method relies on the fact that an optimal entanglement witness is given by a plane perpendicular to a line between the reference state, entanglement of which is to be witnessed, and its closest separable state (CSS). The Gilbert algorithm finds an approximation of CSS. In this article, we investigate if this approximation can be good enough to yield a valid entanglement witness. We compare witnesses found with Gilbert algorithm and those given by Bandyopadhyay–Ghosh–Roychowdhury (BGR) construction. This comparison allows us to learn about the amount of entanglement and we find a relationship between it and a feature of the construction of UPBBE states, namely the size of their central tile. We show that in most studied cases, witnesses found with the Gilbert algorithm in this work are more optimal than ones obtained by Bandyopadhyay, Ghosh, and Roychowdhury. This result implies the increased tolerance to experimental imperfections in a realization of the state.
Gilbert proposed an algorithm for bounding the distance between a given point and a convex set. We apply the Gilbert's algorithm to get an upper bound on the Hilbert-Schmidt distance between a given state and the set of separable states. While Hilbert-Schmidt distance does not form a proper entanglement measure, it can nevertheless be useful for witnessing entanglement. We provide a few methods based on the Gilbert's algorithm that can reliably qualify a given state as strongly entangled or practically separable, while being computationally efficient. The method also outputs successively improved approximations to the closest separable state for the given state. We demonstrate the efficacy of the method with examples.
Abstract Unextendible product basis (UPB) is one of the most fundamental concepts in the field of quantum information theory, and its constructions attract much attention. Recently, Shi et al (2020 Phy. Rev. A 101 062329) constructed UPBs via a tile structure approach. They showed that a tile structure in an m × n grid with certain properties leads to explicit constructions of UPBs in C m ⊗ C n . The size of the UPB is m n − s + 1 if the tile structure contains exactly s tiles. In this paper, we deeply analyze the extent of the tile structure approach, by showing that a desired tile structure in an m × m grid contains at most ⌊ m 2 4 ⌋ + m tiles, and thus the smallest size of a UPB in C m ⊗ C m constructed by this approach is m 2 − ⌊ m 2 4 ⌋ − m + 1 . Such tile structures and UPBs are referred to as extremal tile structures and extremal UPBs. We provide algorithms both for constructing possible candidates of extremal tile structures and checking their validity. For applications, we connect the extremal UPBs to local irreducibility and bound entangled states with large ranks.
It is known that bound entangled states are mixed entangled states from which no pure entanglement can be distilled by local operations and classical communication. This holds true even if large number of identical copies of the state are shared between spatially separated parties. In this work we consider bipartite {\it noisy} bound entangled states with positive partial transpose, that is, such a state can be written as a convex combination of a so-called edge state and a separable state ({\it noise}). Starting from a particular class of unextendible product bases (UPBs) we first show a systematic method to construct low-rank noisy bound entangled states which satisfy the range criterion. Next, we consider different classes of uncompletable product bases (UCPBs) to construct noisy bound entangled states. A subset of these bound entangled states has the property that they satisfy the range criterion. However, UCPBs are important to produce noisy bound entangled states and thus, the problem of constructing UCPBs from different classes of UPBs is also considered in this work. In each case explicit examples are constructed. In fact, present bound entangled states help us to learn about the robustness of edge states. We also discuss about the detection these states.
The existence of genuinely unextendible product bases (GUPBs), incomplete orthogonal sets of fully product states whose orthogonal complements contain no product vector across any bipartition, has remained an open problem. Here we construct GUPBs for any number N≥3N\geq3N≥3 of parties using classical maximum distance separable (MDS) codes. The MDS property imposes a rigidity on the induced product tiling across every bipartition; combined with Fourier mode deletion and a stopper state, this rigidity enforces genuine unextendibility. Consequently, the orthogonal complement of each GUPB is a genuinely entangled subspace whose normalized projector is invariant under partial transposition across every bipartition, yielding an explicit family of multipartite bound entangled states. We further construct GME witnesses that detect these states even though no fully decomposable witness can do so. Moreover, the resulting indistinguishability persists under arbitrary finite tensor powers and measurements separable across any bipartition. These results establish a direct connection between error-correcting codes and multipartite entanglement and provide an algebraic route to certifying genuinely multipartite bound entanglement.
The multiqubit bound entangled states that are uniformly mixed over the subspaces complementary to unextendible product bases (UPBs) are interesting members from the peculiar family of bound entangled states. Their entanglement is delocalized and they look separable with respect to any partition into a single party and all remaining parties. Only one example of quantifying the entanglement of this type of states in the case of three-qubit states has been studied [C. Branciard, H. Zhu, L. Chen, and V. Scarani, Evaluation of two different entanglement measures on a bound entangled state, Phys. Rev. A 82, 012327 (2010)]. We provide a systematic procedure for generating decompositions of such states in terms of minimally entangled states. This procedure enables us to measure entanglement in terms of the concurrence for a wide range of families of these bound entangled (BE) states with each family containing an infinite number of states. In particular, we study one family of three-qubit states in addition to six inequivalent families of four-qubit bound entangled states. These studied families are examples that represent all the known 1446 families of four-qubit UPB-based bound entangled states, which means that our method is applicable to all the 1446 families. In addition, a discussion on the matter of applicability of the proposed strategy to additional families is provided. Specifically, the proposed method looks promising for measuring entanglement of other families of UPB-based BE states in addition to some families of symmetric multipartite positive under partial transposition BE states.