Determining quantum coherence with minimal resources

TL;DR

This paper identifies minimal measurement strategies for efficiently validating quantum coherence in unknown states, using mutually unbiased bases, and provides explicit constructions for optimal measurements across different dimensions.

Contribution

It introduces the optimal measurement scheme for quantum coherence validation using mutually unbiased bases and provides explicit constructions for any dimension.

Findings

Optimal measurement strategy involves measuring $d$ mutually unbiased bases.

The same measurement setup can verify if coherence exceeds a threshold.

Explicit constructions of optimal measurements are provided for all dimensions.

Abstract

We characterize minimal measurement setups for validating the quantum coherence of an unknown quantum state. We show that for a $d$-level system, the optimal strategy consists of measuring $d$ orthonormal bases such that each measured basis is mutually unbiased with respect to the reference basis, and together with the reference basis they form an informationally complete set of measurements. We show that, in general, any strategy capable of validating quantum coherence allows one to evaluate also the exact value of coherence. We then give an explicit construction of the optimal measurements for arbitrary dimensions. Finally, we show that the same measurement setup is also optimal for the modified task of verifying if the coherence is above or below a given threshold value.