Random qubit-states and how best to measure them

Abstract

We consider the problem of measuring a single qubit, known to have been prepared in either a randomly selected pure state or a randomly selected real pure state. We seek the measurements that provide either the best estimate of the state prepared or maximise the accessible information. Surprisingly, any sensible measurement turns out to be optimal. We discuss the application of these ideas to multiple qubits and higher-dimensional systems.

Keywords:

• quantum measurement
• quantum information

Acknowledgements

I thank Daniel Oi, Sarah Croke and Adrian Kent for helpful comments and suggestions. I also thank an anonymous referee who kindly posed the questions addressed in Section 6. I gratefully acknowledge the support of the Royal Society and the Wolfson Foundation.

Notes

Notes

1. It is possible, however, to obtain estimates of each of a number of properties, but this gives, at best, only some indication of the state 2.

2. It is interesting to note that we can derive this distribution using a Bayesian analysis 31. The maximum entropy method naturally gives this isotropic distribution if we define the set of states as the isotropic limit of a suitable discrete set 32.