A Note on Mutually Unbiased Unextendible Maximally Entangled Bases in ℂ 2 ⊗ ℂ 3 $\mathbb {C}^{2}\bigotimes \mathbb {C}^{3}$

Abstract

We systematically study the construction of mutually unbiased bases in C 2 C 3 , such that all the bases are unextendible maximally entangled ones. Necessary conditions of constructing a pair of mutually unbiased unextendible maximally entangled bases in C 2 C 3 are derived. Explicit examples are presented. Keywords Mutually unbiased bases · Unextendible maximally entangled bases · Quantum entanglementMutually unbiased bases (MUBs) play important roles in many quantum information processing such as quantum state tomography [1-3], cryptographic protocols [4][5][6][7][8][9], and the mean kings problem [10]. They are also useful in the construction of generalized Bell states. Let B 1 = {|φ i } and B 2 = {|ψ i }, i = 1, 2, · · · , d, be two orthonormal bases of a ddimensional complex vector space C d , φ j |φ i = δ ij , ψ j |ψ i = δ ij . B 1 and B 2 are said to be mutually unbiased if and only ifPhysically if a system is prepared in an eigenstate of basis B 1 and is measured in basis B 2 , then all the measurement outcomes have the same probability.A set of orthonormal bases {B 1 , B 2 , ..., B m } in C d is called a set of MUBs if every pair of bases in the set is mutually unbiased. For given dimension d, the maximum number of MUBs is no more than d + 1. It has been shown that there are d + 1 MUBs when d is a prime power [1,11,12]. However, for general d, e.g. d = 6, it is a formidable problem to determine the maximal numbers of MUBs [13][14][15][16][17][18][19][20][21].

Mutually Unbiased Maximally Entangled Bases for the Bipartite System ℂ d ⊗ ℂ d k $\mathbb {C}^{d}\otimes \mathbb {C}^{d^{k}}$

“…Partial solutions of constructing MUBs have been given for single quantum system or bipartite system with low dimensional cases [19,20]. In this paper, firstly we construct MUBs for C d ⊗ C d for d = 2 to d = 5.…”

Constructing Mutually Unbiased Bases from Unextendible Maximally Entangled Bases

“…Furthermore, a set of states {|φ i ∈ C d ⊗ C d ′ : i = 1, 2, · · · , n, n < dd ′ } is called an n-member unextendible maximally entangled basis (UMEB) if and only if: (i) |φ i , i = 1, 2, · · · , n are maximally entangled; (ii) φ i |φ j = δ ij ,i, j = 1, 2, · · · , n; (iii) if φ i |ψ = 0, ∀ i = 1, 2, · · · , n, then |ψ cannot be maximally entangled. The constructions of MUBs based on UMEBs in C 2 ⊗ C 3 and C 3 ⊗ C 4 were derived in [11][12][13]. The authors proved that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independent of the choice of the set [14].…”

Special Matrices in Constructing Mutually Unbiased Maximally Entangled Bases in C2C4

“…A state ϕ is said to be a d d′ Mutually unbiased bases are recently combined with other bases, such as product basis (PB) [25], unextendible product basis (UPB) [26], unextendible maximally entangled basis (UMEB) [27]- [32] and maximally entangled basis (MEB) [33]- [35]. The MEB is a set of orthonormally maximally entangled states in…”