Erwin Schrödinger Center for Quantum Science & Technology (ESQ)

Mutually unbiased bases – the notorious case of dimension 6

A long-standing problem in quantum information theory is the existence of mutually unbiased bases (MUB) in arbitrary finite dimension. The existence of different measurement bases for the same degree of freedom, that mutually reveal no information about one another, is at the heart of quantum mechanics. It is well known that in a Hilbert space of dimension d there exist at most d+1 such bases, this being the exact quantity for prime-power dimensions. In other dimensions, the simplest case being d = 6, the exact total of mutually unbiased bases is unknown. In this project, we propose two new methods, based on semidefinite programming, to tackle this much studied problem. The best feature of our methods is that they do not attack the problem head-on, as it has been attempted before. Instead, they solve alternative problems that have been proven to be closely connected to the MUB problem - the problem of joint measurability and the problem of separability of quantum states.

The ESQ discovery project provided me with cutting edge equipment to tackle a high-risk problem in which I have been interested for a long time but lacked the right tools to approach. It allowed me to put my ideas to work and to aim higher for follow-up projects that will continue exploring these resources.