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arXiv:1004.1633 · Physical Review A

Construction of equally entangled bases in arbitrary dimensions via quadratic Gauss sums and graph states

Vlad Gheorghiu, Shiang Yong Looi

Published Jun 25, 2010• 6 citations
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Abstract

Recently, Karimipour and Memarzadeh [Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space, each of dimension $D$, with the following properties: (i) The family continuously interpolates between a product basis and a maximally entangled basis as some parameter $t$ is varied, and (ii) for a fixed $t$, all basis states have the same amount of entanglement. The authors derived a necessary condition and provided explicit solutions for $D\ensuremath{\leqslant}5$, but the existence of a solution for arbitrary dimensions remained an open problem. We prove that such families exist in arbitrary dimensions by providing two simple solutions, one that employs the properties of quadratic Gauss sums and the other that uses graph states. The latter can be generalized to more than two parties---we can construct equally entangled families that vary continuously between a product basis and a graph-state basis.

DOI
10.1103/physreva.81.062341

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Recently, Karimipour and Memarzadeh [Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space, each of dimension DDD, with the following properties: (i) The family continuously interpolates between a product basis and a maximally entangled basis as some parameter ttt is varied, and (ii) for a fixed ttt, all basis states have the same amount of entanglement. The authors derived a necessary condition and provided explicit solutions for D\ensuremath⩽5D\ensuremath{\leqslant}5D\ensuremath⩽5, but the existence of a solution for arbitrary dimensions remained an open problem. We prove that such families exist in arbitrary dimensions by providing two simple solutions, one that employs the properties of quadratic Gauss sums and the other that uses graph states. The latter can be generalized to more than two parties---we can construct equally entangled families that vary continuously between a product basis and a graph-state basis.
Recently [Karimipour and Memarzadeh, PhysRevA 73, 012329 (2006)] posed the problem of finding a continuous family of orthonormal bases in a bipartite space of two identical systems with the following properties: i) in each basis, all states have to be equally entangled, and ii) the family continuously interpolate between the product basis and the maximally entangled basis. The authors provided a necessary condition and simple examples for relatively small dimensions, but questioned the existence of a general solution for arbitrary dimensions. Employing the properties of quadratic Gauss sums, we prove that such a family of bases exists for all dimensions and provide an explicit simple parametrization. We illustrate the behaviour of our solution with particular examples.
For the space of two identical systems of arbitrary dimensions, we introduce a continuous family of bases with the following properties: (i) the bases are orthonormal, (ii) in each basis, all the states have the same values of entanglement, and (iii) they continuously interpolate between the product basis and the maximally entangled basis. The states thus constructed may find applications in many areas related to the quantum information science including quantum cryptography, optimal Bell tests, and the investigation of the enhancement of channel capacity due to entanglement.
We study an unextendible maximally entangled basis in arbitrary bipartite spaces. A systematic way of constructing a set of d2{d}^{2}d2 orthonormal maximally entangled states in ${\mathbb{C}}^{d}⨂{\mathbb{C}}^{{d}^{\ensuremath{'}}}(\frac{{d}^{\ensuremath{'}}}{2}<d<{d}^{\ensuremath{'}})$ is provided. The complementary space of the set of these d2{d}^{2}d2 orthonormal maximally entangled states contains no maximally entangled states that are orthogonal to all of them. Furthermore, we investigate mutually unbiased bases in which all the bases are unextendible maximally entangled ones. We present two unextendible maximally entangled bases in C2⨂C3{\mathbb{C}}^{2}⨂{\mathbb{C}}^{3}C2⨂C3 which are mutually unbiased.
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.
An entangled basis with a fixed Schmidt number k (EBk) is a set of orthonormal basis states with the same Schmidt number k in a product Hilbert space . It is a generalization of both the product basis and the maximally entangled basis. We show here that for any , EBk exists in for any d and d '. Consequently, general methods of constructing SEBk (EBk with the same Schmidt coefficients) and EBk (but not SEBk) are proposed. Moreover, we extend the concept of EBk to multipartite cases and find out that the multipartite EBk can be constructed similarly.
One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon manifests itself most strongly when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independent of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe the basic properties of mutually unbiased bases composed of product states only. The constructions are illustrated with explicit examples in low dimensions. We believe that the properties of entanglement in mutually unbiased bases may be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.