Constructions of mutually unbiased entangled bases

Abstract

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$s$ in ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ when $d\nmid d^{\prime }$, namely $d^{\prime }$ 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 ${\mathbb{C}}^{pd}\otimes {\mathbb{C}}^{q{d}^{\prime }}$ can be constructed from $t_1$ MUSEB$k_1$s in ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ and $t_2$ MUSEB$k_2$s in ${\mathbb{C}}^p\otimes {\mathbb{C}}^q$ for any $d,d^{\prime },p,q$. Further, we show that three MUMEB$s$ exist in ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ for any $d,d^{\prime }$ with $d\mid d^{\prime }$, and two MUMEB$s$ exist in ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ for infinitely many $d,d^{\prime }$ with $d\nmid d^{\prime }$.

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