How many orthonormal bases are needed to distinguish all pure quantum states?

Claudio Carmeli; Teiko Heinosaari; Jussi Schultz; Alessandro Toigo

doi:10.1140/epjd/e2015-60230-5

EPJ

How many orthonormal bases are needed to distinguish all pure quantum states?

Claudio Carmeli¹, Teiko Heinosaari²^a, Jussi Schultz²^,3 and Alessandro Toigo³^,4

¹ DIME, Università di Genova, Via Magliotto 2, 17100 Savona, Italy
² Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, Finland
³ Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
⁴ I.N.F.N., Sezione di Milano, Via Celoria 16, 20133 Milano, Italy

Received: 13 April 2015
Received in final form: 27 May 2015
Published online: 19 July 2015

Abstract

We collect some recent results that together provide an almost complete answer to the question stated in the title. For the dimension d = 2 the answer is three. For the dimensions d = 3 and d ≥ 5 the answer is four. For the dimension d = 4 the answer is either three or four. Curiously, the exact number in d = 4 seems to be an open problem.