Supplementary material from "Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three. 6 November 2019 26 February 2020"

Published on 2020-03-23T05:26:43Z (GMT) by
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<sub>2</sub>⊗V and I<sub>2</sub>⊗W. We show that V and W have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log<sub>2</sub>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.

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Hu, Mengyao; Chen, Lin; Sun, Yize (2020): Supplementary material from "Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three. 6 November 2019 26 February 2020". The Royal Society. Collection. https://doi.org/10.6084/m9.figshare.c.4886775.v2