Unextendible entangled bases with fixed Schmidt number

Abstract

The unextendible product basis (UPB) is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number k (UEBk) for any bipartite system

Constructing the unextendible maximally entangled basis from the maximally entangled basis

“…Moreover, it can be used for constructing bound entangled states [6,8,[10][11][12]. Later, unextendible maximally entangled basis (UMEB) [4] and the unextendible entangled basis with Schmidt number k (UEBk) [5] were investigated extensively.…”

“…The existence problem of UPB and UEBk have been resolved completely. The UPB exists in any C d ⊗ C [5,30]. Although considerable progress has been made, the existence of UMEB still remains open.…”

Constructions of unextendible entangled bases

“…UMEBs can be used to construct examples of states for which one-copy entanglement of assistance (EoA) is strictly smaller than the asymptotic, and can be used to find quantum channels that are unital but not convex mixtures of unitary operations [7,8]. Recently, Guo et al [9,10] proposed the concept of special unextendible entangled bases with fixed Schmidt number k (SUEBk), which extends the definitions of both UPB and UMEB. An SUEBk is a set of incomplete orthonormal special entangled states with Schmidt number k in C d ⊗ C d ′ (d ≤ d ′ ), i.e., with the same Schmidt coefficients, whose complementary subspace has no special entangled states with Schmidt number k. An SUEBk is a UPB when k = 1 and is a UMEB when k = d.…”

“…One of the main topics in this field is the existence of different unextendible bases [7,9,[11][12][13][14][15]. Bravyi and Smolin gave a 6-number UMEB in C 3 ⊗ C 3 and a 12-number UMEB in C 4 ⊗ C 4 [7].…”

Unextendible Maximally Entangled Bases and Mutually Unbiased Bases in Multipartite Systems

“…[11,12,13]. In addition, the UMEB problem was generalized to states with given Schmidt numbers [14,15]. Though the bipartite case is well understood, the question of UMEB for multipartite systems is still an open problem.…”