EPJ D Colloquium - How many orthonormal bases are needed to distinguish all pure quantum states?

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Published on 22 July 2015

How many different measurement settings are needed in order to uniquely determine a pure quantum state, and how should such measurements be chosen? This problem goes back to a famous remark by Wolfgang Pauli in 1933, in which he raised the question whether or not the position and the momentum distributions are enough to define the wave function uniquely modulo a global phase. The original Pauli problem has a negative answer, but it has evolved into many interesting variants and has been studied from several fruitful perspectives.

In this review article the authors concentrate on a specific form of the Pauli problem, which is concerned with the minimal number of orthonormal bases in a finite dimensional Hilbert space that is needed in order to distinguish all pure quantum states. This problem is clearly connected to the fundamental structures of quantum theory. It has been raised several times in the past but the answer has been elusive until recently. They review some recent results that together provide an almost complete answer to the question. For the dimension d = 2 the minimal number of orthonormal bases is three. For the dimensions d = 3 and d ≥ 5 the minimal number is four. For the dimension d = 4 the minimal number is either three or four. Curiously, the exact number in d = 4 seems to be an open problem. They also demonstrate with spin-1 measurements the fact that even if four bases can distinguish all pure quantum states, these bases must be chosen appropriately and it may happen that some “natural" choices are not suitable for this task.