Entangled bases with fixed Schmidt number

Abstract

An entangled basis with fixed Schmidt number k (EBk) is a set of orthonormal basis states with the same Schmidt number k in a product Hilbert spaceIt is a generalization of both the product basis and the maximally entangled basis. We show here that, for any k ≤ min{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 case and find out that the multipartite EBk can be constructed similarly.

Superactivation of monogamy relations for nonadditive quantum correlation measures

“…Moreover, there are special classes of states, which are always monogamous for any quantum correlation measures. For example, the generalized n-partite GHZ-class state admitting the multipartite Schmidt decomposition [44,45], |ψ A1A2···An = i=1 λ i |i 1 ⊗ |i 2 ⊗ · · · ⊗ |i n , i λ 2 i = 1, λ i > 0, for which we always have E(|ψ A1|A2···An ) > 0, while E(ρ A1Ai ) = 0 for all i = 2, · · · , n, for any quantum entanglement measures E.…”

Constructing the unextendible maximally entangled basis from the maximally entangled basis

“…III and IV. The Lemma 1 below is borrowed from [29], which reveals that SV1B exists in M d×d ′ for any d and…”

Constructions of unextendible entangled bases

“…The Schmidt decomposition of a pure state |ψ ∈ Let M d×d ′ be the space of all d × d ′ complex matrices equipped with an inner product defined by (A, B) = Tr(A † B). There is a one-to-one relation between the space C d ⊗ C d ′ and the space M d×d ′ [10,18]:…”

“…Due to the one-to-one relation, {|ψ i } is an SEBk if and only if {A i } is an SV1Bk; and {|ψ i } is an SUEBk if and only if {A i } is a USV1Bk.Lemma 1[18] If k|dd ′ , then there is an SEBk in C d ⊗ C d ′ , and consequently there is an SV1Bk in M d×d ′ .…”