Minimal Determination of a Pure State through Adaptive Tomography

TL;DR

This paper proves Peres' conjecture for low-dimensional cases, showing two unbiased measurements can determine a pure state with finite ambiguities, and proposes an adaptive minimal measurement scheme for pure state tomography.

Contribution

It proves Peres' conjecture for dimensions 3 and 4, and introduces an adaptive three-measurement protocol for pure state tomography that minimizes measurement settings.

Findings

Two unbiased measurements determine a pure state up to 6 and 16 candidates in 3 and 4 dimensions.

An adaptive 3-measurement scheme can uniquely determine a pure state.

The protocol simplifies previous methods by reducing the number of measurement bases.

Abstract

Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. In the fixed measurement scheme four orthonormal bases are required even though there are far less parameters in a pure state. Peres conjectured that two unbiased bases suffice to determine a pure state up to some finite ambiguities. Here we shall at first prove Peres conjecture in the case of $d=3,4$, namely, two unbiased measurements determine a pure state up to to 6 and 16 candidates for a qutrit and ququad, respectively. And then, taking Peres' conjecture for established, we propose an adaptive 3-measurement scheme involving the minimal number of measurements, based on the observation that the ambiguities can be removed by an adaptive two-outcome projective measurement. With the help of this observation, we simplify a recent five-basis protocol $[Phys. Rev. Lett. 115, 090401 (2015)]$ to a three-basis one at the cost of the extra dichotomic measurement.