Constructions of mutually unbiased entangled bases

TL;DR

This paper constructs mutually unbiased entangled bases in bipartite quantum systems, introduces recursive methods for bases with fixed Schmidt number, and explores their existence in various dimensions, advancing quantum information theory.

Contribution

It provides the first example of mutually unbiased maximally entangled bases in systems where dimensions are not divisible, and develops recursive constructions for bases with fixed Schmidt number.

Findings

Constructed two MUMEBs in 2 3 dimensions.

Showed MUMEBs cannot be extended to four in 6.

Established existence of multiple MUMEBs in various dimensions.

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 $\mathbb{C}^{d}\otimes \mathbb{C}^{d'}$ 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 $d\nmid d'$.