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arXiv:0911.4090 · Physical Review A

Unextendible maximally entangled bases

Sergey B. Bravyi, John A. Smolin

Published Oct 4, 2011• 52 citations
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Abstract

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 ${d}^{2}$ vectors which have no additional maximally entangled vectors orthogonal to all of them. We prove that UMEBs do not exist for $d=2$ and give explicit constructions for a six-member UMEB with $d=3$ and a 12-member UMEB with $d=4$.

DOI
10.1103/physreva.84.042306

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We introduce the notion of the unextendible maximally entangled basis (UMEB), a set of orthonormal maximally entangled states in Cd\ensuremath⨂Cd{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}Cd\ensuremath⨂Cd 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.
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Constructions of Unextendible Maximally Entangled Bases in Cd⊗Cd′{{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }Cd⊗Cd′
Abstract We study unextendible maximally entangled bases (UMEBs) in Cd⊗Cd′{{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }Cd⊗Cd′ ℂ d ⊗ ℂ d ′ (d < d′). An operational method to construct UMEBs containing d(d′ − 1) maximally entangled vectors is established, and two UMEBs in C5⊗C6{{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{6}C5⊗C6 ℂ 5 ⊗ ℂ 6 and C5⊗C12{{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{12}C5⊗C12 ℂ 5 ⊗ ℂ 12 are given as examples. Furthermore, a systematic way of constructing UMEBs containing d(d′ − r) maximally entangled vectors in Cd⊗Cd′{{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }Cd⊗Cd′ ℂ d ⊗ ℂ d ′ is presented for r = 1, 2, …, d − 1. Correspondingly, two UMEBs in C3⊗C10{{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}C3⊗C10 ℂ 3 ⊗ ℂ 10 are obtained.
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We investigate the unextendible maximally entangled bases (UMEBs) in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ and present a 30-number UMEB construction in ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$. For the higher-dimensional case, we show that for a given NNN-number UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, there is an $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}$-number, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}={(qd)}^{2}\ensuremath{-}({d}^{2}\ensuremath{-}N)$, UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ for any $q\ensuremath{\in}\mathbb{N}$. As an example, for ${\mathbb{C}}^{12n}\ensuremath{\bigotimes}{\mathbb{C}}^{12n}$ systems, we show that there are at least two sets of UMEBs which are not equivalent.
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.
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.