The limited role of mutually unbiased product bases in dimension 6

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].

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“…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].…”

Section: Abstract Mutually Unbiased Bases · Unextendible Maximally Enmentioning

confidence: 99%

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

Nizamidin

Fei

2014

Int J Theor Phys

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“…, B m in C d is said to be a set of mutually unbiased bases if every pair of the bases in the set is mutually unbiased. Recent years, there are many interesting topics combining mutually unbiased bases with other bases, such as product basis (PB) [29], unextendible product basis (UPB) [30] and unextendible maximally entangled basis (UMEB) [31,32]. The UPB is a set of incomplete orthogonal product states in C d ⊗ C d such that whose complementary space has no product states.…”

Section: Introductionmentioning

confidence: 99%

Mutually unbiased maximally entangled bases in $$\mathbb {C}^d\otimes \mathbb {C}^{kd}$$ C d ⊗ C k d

Tao

Nan

Zhang

et al. 2015

Quantum Inf Process

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We study maximally entangled bases in bipartite systems C d ⊗ C kd (k ∈ Z + ), which are mutually unbiased. By systematically constructing maximally entangled bases, we present an approach in constructing mutually unbiased maximally entangled bases. In particular, five maximally entangled bases in C 2 ⊗ C 4 and three maximally entangled bases in C 2 ⊗ C 6 that are mutually unbiased are presented.

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“…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…”

Section: Introductionmentioning

confidence: 99%

Special Matrices in Constructing Mutually Unbiased Maximally Entangled Bases in C2C4

Zhang¹,

Yang²,

Nan³

et al. 2015

OALib

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Some special matrices can really help us to construct more than two mutually unbiased maximally entangled bases in 2 4 C C ⊗ . Through detailed analysis of the necessary and sufficient conditions of two maximally entangled bases to be mutually unbiased, we find these special matrices. Taking one such kind of matrix, we present the steps of constructing five mutually unbiased maximally entangled bases in 2 4 C C ⊗ .

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The limited role of mutually unbiased product bases in dimension 6

Abstract: We show that a complete set of seven mutually unbiased bases in dimension six, if it exists, cannot contain more than one product basis.

Cited by 42 publications

References 26 publications

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

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

Mutually unbiased maximally entangled bases in $$\mathbb {C}^d\otimes \mathbb {C}^{kd}$$ C d ⊗ C k d

Special Matrices in Constructing Mutually Unbiased Maximally Entangled Bases in C2C4

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