Constructions of unextendible entangled bases

Abstract

We provide several constructions of special unextendible entangled bases with fixed Schmidt number k (SUEBk) inWe generalize the space decomposition method in Guo [Phys. Rev. A 94, 052302 (2016)], by proposing a systematic way of constructing new SUEBks inIn addition, we give a construction of a (pqdd ′ − p(dd ′ − N ))-number SUEBpk in C pd ⊗ C qd ′ from an N -number SUEBk in C d ⊗ C d ′ for p ≤ q by using permutation matrices. We also connect a (d(d ′ − 1) + m)-number UMEB in C d ⊗ C d ′ with an unextendible partial Hadamard matrix H m×d with m < d, which extends the result in [Quantum Inf. Process. 16(3), 84 (2017)].

New results on unextendible product bases

“…UPB 也展示了无纠缠的非局域性 [5] 和无量子违反的 Bell 不等式 [6] . 进一步地, 不可 扩展最大纠缠基 [7] 和固定 Schmidt 数的不可扩展纠缠基 [8,9] ⊗ m i=1 C ki 中的规模最小的 UPB 所包含的态的数目. 文献 [1] 在提出 UPB 的定义的 同时也提出了关于此数目的一个平凡下界.…”

“…令 H 2 包含以下 5 条边: v (1) ∼ v (7) , v (3) ∼ v (7) , v (2) ∼ v (8) , v (4) ∼ v (8) , (8) 到 v (7) 的一条交错 的路径. 不失一般性, 设定此路径为 v (8) ∼ v (9) (7) , 则 H 3 也被唯一决定, 即所绘图的补图,…”

“…v (1) = a ⊗ c ⊗ (0, 1, 0), v (2) = a ⊗ d ⊗ (1, 0, 2), v (3) = a ⊥ ⊗ c ⊥ ⊗ (0, 1, 2), v (4) = a ⊥ ⊗ c ⊗ (0, 1, 1), v (5) = a ⊥ ⊗ c ⊗ (0, 1, −1), v (6) = b ⊗ c ⊥ ⊗ (2, 2, −1), v (7) = b ⊥ ⊗ d ⊥ ⊗ (1, 0, 0). v (1) = a ⊗ c ⊗ (1, 0, 0, 0), v (2) = a ⊗ c ⊥ ⊗ (1, 1, 1, 0), v (3) = a ⊗ d ⊗ (0, 1, −1, 1), v (4) = a ⊥ ⊗ c ⊥ ⊗ (30, 37, 16, 21), v (5) = a ⊥ ⊗ d ⊗ (−10, 11, −29, 17), v (6) = a ⊥ ⊗ c ⊗ (1, 2, 1, 1), v (7) = b ⊗ c ⊗ (0, 2, −1, −3), v (8) = b ⊗ c ⊥ ⊗ (−3, 1, 2, 1), v (9) = b ⊥ ⊗ d ⊥ ⊗ (0, −1, 1, 1). v (1) = a ⊗ c ⊗ (3, −2, 2, 3, 3), v (2) = a ⊗ d ⊗ (11, −21, −18, −26, 13), v (3) = a ⊥ ⊗ e ⊗ (1, 0, 0, 0, 0), v (4) = a ⊥ ⊗ e ⊥ ⊗ (0, 1, 0, 0, 0), v (5) = a ⊥ ⊗ c ⊗ (0, 0, 1, 2, 1), v (6) = b ⊗ e ⊗ (0, 1, 1, −1, 1), v (7) = b ⊥ ⊗ d ⊥ ⊗ (0, 0, 3, −1, −1), v (8) = b ⊗ e ⊥ ⊗ (1, 0, −3, 2, −1), v (9) = b ⊥ ⊗ c ⊥ ⊗ (0, 0, 13, 7, 32).…”

“…v (1) = a ⊗ c ⊗ (1, 0, 0, 0), v (2) = a ⊗ c ⊥ ⊗ (1, 1, 1, 0), v (3) = a ⊗ d ⊗ (0, 1, −1, 1), v (4) = a ⊥ ⊗ c ⊥ ⊗ (30, 37, 16, 21), v (5) = a ⊥ ⊗ d ⊗ (−10, 11, −29, 17), v (6) = a ⊥ ⊗ c ⊗ (1, 2, 1, 1), v (7) = b ⊗ c ⊗ (0, 2, −1, −3), v (8) = b ⊗ c ⊥ ⊗ (−3, 1, 2, 1), v (9) = b ⊥ ⊗ d ⊥ ⊗ (0, −1, 1, 1). v (1) = a ⊗ c ⊗ (3, −2, 2, 3, 3), v (2) = a ⊗ d ⊗ (11, −21, −18, −26, 13), v (3) = a ⊥ ⊗ e ⊗ (1, 0, 0, 0, 0), v (4) = a ⊥ ⊗ e ⊥ ⊗ (0, 1, 0, 0, 0), v (5) = a ⊥ ⊗ c ⊗ (0, 0, 1, 2, 1), v (6) = b ⊗ e ⊗ (0, 1, 1, −1, 1), v (7) = b ⊥ ⊗ d ⊥ ⊗ (0, 0, 3, −1, −1), v (8) = b ⊗ e ⊥ ⊗ (1, 0, −3, 2, −1), v (9) = b ⊥ ⊗ c ⊥ ⊗ (0, 0, 13, 7, 32). v (1) = a ⊗ c ⊗ (1, 0, 0, 0, 0, 0), v (2) = a ⊗ c ⊥ ⊗ (1, 1, 1, 0, 0, 0), v (3) = a ⊗ e ⊗ (0, 1, −1, 1, 2, 1), v (4) = a ⊥ ⊗ d ⊗ (−10, −1, −1, 3, 2, 1), v (5) = a ⊥ ⊗ d ⊗ (1, −26, 3, −9, −6, 26), v (6) = a ⊥ ⊗ e ⊗ (1, 1, −2, 1, 2, 2), v (7) = b ⊗ d ⊥ ⊗ (0, 1, −1, −1, 0, −1), v (8) = b ⊗ e ⊥ ⊗ (0, −1, 1, −1, 2, −1), v (9) = b ⊥ ⊗ c ⊥ ⊗ (−6, 9, −3, −21, 3, 3), v (10) = b ⊥ ⊗ c ⊗ (0, 1, 1, 1, −1, 1), v (11) = b ⊥ ⊗ d ⊥ ⊗ (0, −3, 3, −1, 2, 3).…”

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

“…We remind the reader in this regard that a partial Hadamard matrix is an r × n (binary) matrix H with r ≤ n such that HH t = nI r . The recent implementation of these types of matrices in cryptography [19], experimental design [20], and quantum information [21] has awakened the interest in describing different ways of constructing them [22][23][24][25]. In addition, Latin rectangles may be implemented in Internet of Things (IoT) studies [26], coding theory [27,28], and modern 5G wireless networks [29].…”