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arXiv:1909.10043

Special unextendible entangled bases with continuous integer cardinality

Yan-Ling Wang

Published Sep 22, 2019• 2 citations
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Abstract

Special unextendible entangled basis of "type $k$" (SUEBk), a set of incomplete orthonormal special entangled states of "type $k$" whose complementary space has no special entangled state of "type $k$". This concept can be seem as a generalization of the unextendible product basis (UPB) introduced by Bennett et al. in [ Phys. Rev. Lett. \textbf{82}, 5385(1999) ] and the unextendible maximally entangled basis (UMEB) introduced by Bravyi and Smolin in [Phys. Rev. A \textbf{84}, 042306(2011)]. We present an efficient method to construct sets of SUEBk. The main strategy here is to decompose the whole space into two subspaces such that the rank of one subspace can be easily upper bounded by $k$ while the other one can be generated by two kinds of the special entangled states of type $k$. This method is very effective for those $k=p^m\geq 3$ where $p$ is a prime number. For these cases, we can otain sets of SUEBk with continuous integer cardinality when the local dimensions are large. Moreover, one can find that our method here can be easily extended when there are more than two kinds of the special entangled states of type $k$ at hand.

DOI
10.48550/arxiv.1909.10043

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Special unextendible entangled basis of "type kkk" (SUEBk), a set of incomplete orthonormal special entangled states of "type kkk" whose complementary space has no special entangled state of "type kkk". This concept can be seem as a generalization of the unextendible product basis (UPB) introduced by Bennett et al. in [ Phys. Rev. Lett. \textbf{82}, 5385(1999) ] and the unextendible maximally entangled basis (UMEB) introduced by Bravyi and Smolin in [Phys. Rev. A \textbf{84}, 042306(2011)]. We present an efficient method to construct sets of SUEBk. The main strategy here is to decompose the whole space into two subspaces such that the rank of one subspace can be easily upper bounded by kkk while the other one can be generated by two kinds of the special entangled states of type kkk. This method is very effective for those k=pm≥3k=p^m\geq 3k=pm≥3 where ppp is a prime number. For these cases, we can otain sets of SUEBk with continuous integer cardinality when the local dimensions are large. Moreover, one can find that our method here can be easily extended when there are more than two kinds of the special entangled states of type kkk at hand.
Unextendible product basis (UPB), a set of incomplete orthonormal product states whose complementary space has no product state, is very useful for constructing bound entangled states. Naturally, instead of considering the set of product states, Bravyi and Smolin considered the set of maximally entangled states. They introduced the concept of unextendible maximally entangled basis (UMEB), a set of incomplete orthonormal maximally entangled states whose complementary space contains no maximally entangled state [Phys. Rev. A 84, 042,306 (2011)]. An entangled state whose nonzero Schmidt coefficients are all equal to 1/k is called a special entangled state of “type k”. In this paper, we introduce a concept named special unextendible entangled basis of “type k” which generalizes both UPB and UMEB. A special unextendible entangled basis of “type k” (SUEBk) is a set of incomplete orthonormal special entangled states of “type k” whose complementary space has no special entangled state of “type k”. We present an efficient method to construct sets of SUEBk. The main strategy here is to decompose the whole space into two subspaces such that the rank of each element in one subspace can be easily upper bounded by k while the other one can be generated by two kinds of the special entangled states of “type k”. This method is very effective when k = pm ≥ 3 where p is a prime number. For these cases, we can obtain sets of SUEBk with continuous integer cardinality when the local dimensions are large.
Unextendible product bases (UPBs) provide a versatile tool with various applications across different areas of quantum information theory. Their comprehensive characterization is thus of great importance and has been a subject of vital interest for over two decades now. An open question asks about the existence of UPBs, which are genuinely unextendible, i.e., they are not extendible even with biproduct vectors. In other words, the problem is to verify whether there exist genuinely entangled subspaces (GESs), subspaces composed solely of genuinely multiparty entangled states, complementary to UPBs. We solve this problem in the negative for many sizes of UPBs in different multipartite scenarios. In particular, in the all-important case of equal local dimensions, we show that there are always forbidden cardinalities for such UPBs, including the minimal ones corresponding to GESs of the maximal dimensions.
It is known that some two-qutrit entangled states of rank 4 with positive partial transpose can be built from the unextendible product bases (UPB) [C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett. 82, 5385 (1999)]. We show that this fact is indeed universal, namely, all such states can be constructed from UPB as conjectured recently by Leinaas, Myrheim, and Sollid. We also classify the five-dimensional subspaces of two qutrits which contain only finitely many product states (up to scalar multiple), and in particular those spanned by an UPB.
It is known how to construct, in a bipartite quantum system, a unique low-rank entangled mixed state with positive partial transpose (a PPT state) from an unextendible product basis (UPB), defined as an unextendible set of orthogonal product vectors. We point out that a state constructed in this way belongs to a continuous family of entangled PPT states of the same rank, all related by nonsingular unitary or nonunitary product transformations. The characteristic property of a state $\ensuremath{\rho}$ in such a family is that its kernel $\text{Ker} \ensuremath{\rho}$ has a generalized UPB, a basis of product vectors, not necessarily orthogonal, with no product vector in $\mathrm{Im} \ensuremath{\rho}$, the orthogonal complement of $\text{Ker} \ensuremath{\rho}$. The generalized UPB in $\text{Ker} \ensuremath{\rho}$ has the special property that it can be transformed to orthogonal form by a product transformation. In the case of a system of dimension $3\ifmmode\times\else\texttimes\fi{}3$, we give a complete parametrization of orthogonal UPBs. This is then a parametrization of families of rank 4 entangled (and extremal) PPT states, and we present strong numerical evidence that it is a complete classification of such states. We speculate that the lowest rank entangled and extremal PPT states also in higher dimensions are related to generalized, nonorthogonal UPBs in similar ways.
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.
We introduce the notion of the unextendible maximally entangled basis (UMEB), a set of orthonormal maximally entangled states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ consisting of fewer than d2{d}^{2}d2 vectors which have no additional maximally entangled vectors orthogonal to all of them. We prove that UMEBs do not exist for d=2d=2d=2 and give explicit constructions for a six-member UMEB with d=3d=3d=3 and a 12-member UMEB with d=4d=4d=4.