Journal of Algebra and Its Applications

Mutually unbiased special entangled bases with Schmidt number 2d in ℂ2d+1 ⊗ ℂ4d

Qianqian Yan, Dengming Xu

Published Jan 30, 2023• 0 citations

Abstract

In this paper, we construct mutually unbiased special entangled bases with Schmidt number 2d in [Formula: see text]. Precisely, we first provide a necessary and sufficient condition for two special entangled bases with Schmidt number 2d are mutually unbiased, and then use the condition to construct two mutually unbiased special entangled bases with Schmidt number 2 in [Formula: see text] and two mutually unbiased special entangled bases with Schmidt number 4 in [Formula: see text].

DOI: 10.1142/s0219498823500548

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Journal of Algebra and Its Applications

Mutually unbiased special entangled bases with Schmidt number 2d in ℂ2d+1 ⊗ ℂ4d

Qianqian Yan, Dengming Xu

Published Jan 30, 2023• 0 citations

Publisher

Abstract

DOI: 10.1142/s0219498823500548

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Mutually unbiased special entangled bases with Schmidt number 2 in ${\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}$ C 3 ⊗ C 4 kMutually unbiased special entangled bases with Schmidt number 2 in $${\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}$$ C 3 ⊗ C 4 k

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We construct two mutually unbiased bases by maximally entangled states (MUMEB

s

) in

\mathbb{C}^{2}\otimes \mathbb{C}^{3}

. This is the first example of MUMEB

s

\mathbb{C}^{d}\otimes \mathbb{C}^{d'}

when

d\nmid d'

, namely

d'

is not divisible by

d

. We show that they cannot be extended to four MUBs in

\mathbb{C}^6

. We propose a recursive construction of mutually unbiased bases formed by special entangled states with a fixed Schmidt number

k

(MUSEB

k

s). It shows that

\min \{t_{1},t_{2}\}

MUSEB

k_{1}k_{2}

s in

\mathbb{C}^{pd}\otimes \mathbb{C}^{qd'}

can be constructed from

t_{1}

MUSEB

k_{1}

s in

\mathbb{C}^{d}\otimes \mathbb{C}^{d'}

and

t_{2}

MUSEB

k_{2}

s in

\mathbb{C}^{p}\otimes \mathbb{C}^{q}

for any

d,d',p,q

. Further, we show that three MUMEB

s

exist in

\mathbb{C}^{d}\otimes \mathbb{C}^{d'}

for any

d,d'

with

d\mid d'

, and two MUMEB

s

exist in

\mathbb{C}^{d}\otimes \mathbb{C}^{d'}

for infinitely many

d,d'

with

d\nmid d'

Mutually unbiased special entangled bases with Schmidt number 2 in

{\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}

C 3 ⊗ C 4 k