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Physical Review A

Concurrence of multiqubit bound entangled states constructed from unextendible product bases

Mohamed Nawareg

Published Mar 25, 2020• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.101.032342

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While we possess a basic understanding of bipartite entangled systems, multipartite entanglement remains mostly unexplored. Specifically, it is quite unclear how several concepts developed for the bipartite case can be generalized in a meaningful way to study multipartite scenarios. At the heart of this research on multiparty quantum states lies the problem of testing the entanglement present in a given state. It has been proved, that in the most general scenario, the question of determining the entanglement or separability of a state is NP-hard. From this perspective, it is desirable to have a construction of states for which some prior knowledge about entanglement properties already exist. One particular approach to this problem relies on the construction of Completely Entangled Subspaces (CESes), i.e., subspaces void of product states. It follows that states in such subspaces are necessarily entangled and form a convex set. 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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.
While we possess a basic understanding of bipartite entangled systems, multipartite entanglement remains mostly unexplored. Specifically, it is quite unclear how several concepts developed for the bipartite case can be generalized in a meaningful way to study multipartite scenarios. At the heart of this research on multiparty quantum states lies the problem of testing the entanglement present in a given state. It has been proved, that in the most general scenario, the question of determining the entanglement or separability of a state is NP-hard. From this perspective, it is desirable to have a construction of states for which some prior knowledge about entanglement properties already exist. One particular approach to this problem relies on the construction of Completely Entangled Subspaces (CESes), i.e., subspaces void of product states. It follows that states in such subspaces are necessarily entangled and form a convex set. This means we possess prior knowledge of entanglement of all states in that subspace. In our study of multiparty systems, we have constructed a specific subclass of the CESes such that all the states residing in the newly constructed subspace is not only entangled but also Genuinely Entangled. A given state is Genuinely Entangled if it is not just entangled for any pair among the multiparty but is entangled with all the parties present and at the same time. The notion of a CES is intimately connected with the idea of Unextendible Product Bases (UPBs). UPBs are sets of twisted orthogonal product states/vectors spanning a proper subset of a given Hilbert space. They have the property that no other product vector exists in the ortho-complement of their span. While UPBs have been relatively well studied for bipartite systems, no known multipartite orthogonal UPBs give rise to Genuinely Entangled Subspace (GES). In our study, we provide schemes for the construction of GES for a tripartite scenario. The construction proposed is linked to a Symmetric Block structure such that it can be extended to any arbitrary dimension.One of the primary motivations for construction of UPBs is to create entangled states which remain Positive under Partial Transposition (PPT). These classes of Entangled States are known to be un-distillable and hence called Bound Entangled, i.e., states in which the resource of Entanglement is not free to be used. These Bound Entangled states are essential to our understanding the basics of Entanglement Theory such as distillability, quantification of entanglement, and the relationship between nonlocality and entanglement (apart from their applications in other quantum information tasks). The construction itself guarantees the PPTness of the state residing in the complementary subspace i.e. formed by the equal mixture of the basis spanning the complementary space. In our study, we propose a symmetric and an asymmetric construction of the Biseperable and Genuinely Entangled states. Via the symmetric construction, we have derived two subspaces, an eight-dimensional CES and a five-dimensional GES within the CES mentioned above. The state formed by the equal mixtures of the CES is a Genuinely Bound Entangled PPT state i.e., it entangled as well as PPT in each cut. To the best of author’s knowledge, the proposed state is the only know Genuinely Entangled PPT state derived from a UPB construction. Additionally, the state lying in the ortho-complement of UPB (i.e., states formed by an equal mixture of the basis of CES) is known to be a PPT Bound Entangled state. However, unlike in case of UPBs, the state in the ortho-complement of Unextendible Biseparable Product basis (i.e. states formed by an equal mixture of the basis of GES) is found to be NPT states across all bipartitions. This is striking and prompts us to believe that the state in the ortho-complement of CES is the only examples of bound entangled state and distinct from other states in the space. We further expect that the state we obtain must exhibit some distinctive features. We verify that this state is one copy undistillable. However, the n-copy undistillablity needs to be probed further, and no conclusive statement regarding the same can be made with our current analysis. The existence of NPT-bound entangled state would prove that the distillable entanglement is nonadditive and not convex. The constructed subspace of Genuinely Entangled States in-turn reveals some peculiar features. As mentioned above, unlike the Bennett et al., 1999 case, the state in the ortho-complement is an NPT state across all bipartition. Furthermore, it was numerically verified by taking one hundred thousand random probability distributions that the entire five-dimensional genuine subspace obtained happens to be NPT. This leads us to conjecture that the subspace obtained is a Genuinly Entangled NPT subspace. To the best of the author’s knowledge, no known genuine NPT subspace exists for the multipartite scenario. The exists only one other NPT subspace for Biseparable States and was suggested by Johnston, 2013. Here, the construction is not related to UPBs. We further argue that if our conjecture on the NPTness of the entire subspace is correct, any four-dimensional subspace of the constructed tripartite NPT GES is also distillable. This would then be the only known distillable subspace in a multipartite scenario. We believe that our study brings us closer to a systematic analysis of multipartite entanglement and provides us with tools to probe further into the current mess of an entangled (read communicating) world of multiple actors.
In quantum information theory, it is a fundamental problem to construct multipartite unextendible product bases (UPBs). We show that there exist two families UPBs in Hilbert space C2⊗C2⊗C2⊗C2⊗C2⊗C4\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^4C2⊗C2⊗C2⊗C2⊗C2⊗C4 by merging two different systems of an existing 777-qubit UPB of size 111111. Moreover, a new family of 777-qubit positive-partial-transpose (PPT) entangled states of rank 27−112^7-1127−11 is constructed. We analytically derive a geometric measure of entanglement of a special PPT entangled states. Also an upper bound are given by two methods.
The 4-qubit unextendible product bases (UPBs) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. Based on this study we show that there is only one UPB of size 6 and six UPBs of size 9 in H=C2⊗C2⊗C4, three UPBs of size 9 in K=C4⊗C4, and no UPB of size 7 in H and K. Furthermore we construct a new family of 4-qubit positive-partial-transpose (PPT) entangled states of rank seven, and show that they are also PPT entangled states in H and K, respectively. We analytically derive a geometric measure of a special PPT entangled states.
Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a construction of bound entangled states or Bell inequalities without quantum violation. They are closely related to another important notion, completely entangled subspaces (CESs), which are those that do not contain any fully separable pure state. Among CESs one finds a class of subspaces in which all vectors are not only entangled but genuinely entangled. Here we explore the connection between UPBs and such genuinely entangled subspaces (GESs) and provide classes of nonorthogonal UPBs that lead to GESs for any number of parties and local dimensions. We then show how these subspaces can be immediately utilized for a simple general construction of genuinely entangled states in any such multipartite scenario.
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.