Constructing the unextendible maximally entangled basis from the maximally entangled basis

Abstract

A new way of constructing unextendible maximally entangled basis (UMEB) from maximally entangled basis (MEB) is proposed. Consequently, it is shown that if there is an N -member UMEB in In addition, a very easy way of constructing UMEB in′ is presented, which promotes and covers all the previous related work.

Constructions of unextendible entangled bases

“…In Section 5, we show that there is a (d(d ′ − 1) + m)-number UMEB in C d ⊗ C d ′ if there exists an unextendible partial Hadamard matrix H m×d . Our constructions generalize and improve most results in [8,9,[11][12][13][14][15][16], and provide new SUEBks and UMEBs.…”

“…In fact, Case (3) provides a 72-number SUEB4, a 60-number SUEB4 and a 48-number SUEB4 in C 7 ⊗ C 12 , while [9] only gives a 48-number SUEB4 in C 7 ⊗ C 12 . Also, Our constructions cover all of the results in [8,12,13,16]. See Table 1.…”

“…Let L denote a subspace of M d×d ′ , L ⊥ denote the complementary space of L, and ⊕ denote the direct sum. Inspired by Lemmas 2 and 3 of [13], we get the following two lemmas.…”

“…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].…”

“…For d < d ′ , Li et al showed that UMEBs exist in any C d ⊗ C d ′ and gave explicit constructions [12]. There are several recursive constructions for UMEBs, such as a construction of a (pqd 2 − p(d 2 − N ))-number UMEB in C pd ⊗ C qd (p ≤ q) from an N -number UMEB in C d ⊗ C d , and a construction of a (pqd 2 − d(pq − N ))-number UMEB in C pd ⊗ C qd (p ≤ q) from an N -number UMEB in C p ⊗ C q [13][14][15]. It was shown that SUEBks exist in any bipartite system when 2 ≤ k < d ≤ d ′ [9].…”