Journal of Algebra and Its Applications

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

Qianqian Yan, Dengming Xu

Published Apr 29, 2022• 0 citations

Abstract

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

DOI: 10.1142/s0219498823501645

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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 Apr 29, 2022• 0 citations

Publisher

Abstract

DOI: 10.1142/s0219498823501645

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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