How well can you know the edge of a quantum pyramid?

Abstract

We consider a symmetric quantum communication scenario in which the signal states are edges of a quantum pyramid of arbitrary dimension and arbitrary shape, and all edge states are transmitted with the same probability. The receiver could employ different decoding strategies: he could minimize the error probability, or discriminate without ambiguity, or extract the accessible information. We state the optimal measurement scheme for each strategy. For large parameter ranges, the standard square-root measurement does not extract the information optimally.

^†We dedicate this work to János Bergou–friend, colleague, and grandmaster of unambiguous discrimination–on the occasion of his 60th birthday

Keywords:

• quantum state discrimination
• minimum-error measurement
• unambiguous discrimination
• accessible information

Acknowledgements

We are very grateful for the valuable discussions with Dagomir Kaszlikowski, Ajay Gopinathan, Frederick Willeboordse, Shiang Yong Looi, and Sergei Kulik. J.Ř. wishes to thank B.-G. Englert and his group for the kind hospitality received during his visits to Singapore. This work was supported by Grant MSM6198959213 from the Czech Ministry of Education, and by NUS Grant WBS: R-144-000-109-112. Centre for Quantum Technologies is a Research Centre of Excellence funded by the Ministry of Education and National Research Foundation of Singapore.

Notes

^†We dedicate this work to János Bergou–friend, colleague, and grandmaster of unambiguous discrimination–on the occasion of his 60th birthday

Notes

1. We regard the identity operator on the right-hand side of Equation (11) as the projector on the joint range of the signal states ρ _j and ignore the orthogonal complement of the Hilbert space of kets, if there is one.

2. We note that, if the underlying Hilbert space is infinite dimensional, U must be isometric but not necessarily unitary.