Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three

TL;DR

This paper explores the existence of four mutually unbiased bases in six dimensions including a complex Hadamard matrix of Schmidt rank three, revealing limitations on entangling power and providing new conditions and examples.

Contribution

It introduces conditions for complex Hadamard matrices of Schmidt rank three in MUBs and constructs examples showing entangling power may be submaximal.

Findings

V and W have no zero entries in the CHM

Maximum entangling power of the controlled unitary is log2 3 ebits

If a CHM of Schmidt rank three is in an MUB, its entangling power may not reach maximum

Abstract

Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation $I_2\otimes V$ and $I_2\otimes W$. We show that $V$ and $W$ have no zero entry, and apply it to exclude examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is $\log_2 3$ ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.