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arXiv:1501.06400 · Journal of Physics A Mathematical and Theoretical

Entangled bases with fixed Schmidt number

Yu Jing Guo, Shuanping Du, Xiulan Li, Shengjun Wu

Published May 26, 2015• 22 citations
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Abstract

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.

DOI
10.1088/1751-8113/48/24/245301

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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.
Abstract A special entangled basis (SEBk) is a set with orthonormal entangled pure states in whose nonzero Schmidt coefficients are all equal to . When k is equal to the minimum of d and , we get a maximally entangled basis. In this paper, we present how to construct a special entangled basis via some special matrices which are known as weighing matrices. Specifically, using a skew Hadamard matrix of order k + 1, we derive a weighing matrix that is useful for constructing SEBk in whenever . These results are further progress of those studied by Guo et al (2015 J. Phys. A: Math. Theor . 48 245301). We also disprove two conjectures proposed by them.
The unextendible product basis is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number kkk (UEBk) for any bipartite system ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}$ $(2\ensuremath{\le}k<d\ensuremath{\le}{d}^{\ensuremath{'}})$, which can also be regarded as a generalization of the unextendible maximally entangled basis. A general way of constructing such a basis with arbitrary ddd and ${d}^{\ensuremath{'}}$ is proposed. Consequently, it is shown that there are at least $k\ensuremath{-}r$ (here r=dmodkr=d\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}kr=dmodk or $r={d}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k)$ sets of UEBks when ddd or ${d}^{\ensuremath{'}}$ is not a multiple of kkk, while there are at least $2(k\ensuremath{-}1)$ sets of UEBks when both ddd and ${d}^{\ensuremath{'}}$ are multiples of kkk.
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.
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 $D\ensuremath{\leqslant}5$, 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.
Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.