Trade-off between predictability and quantum coherence for multi-path interferometry and its operational interpretation

Abstract

The complementarity principle is a cornerstone of quantum mechanics. In this work, we investigate a complementarity relation between the predictability and quantum coherence, respectively, representing the particle-like and wave-like behaviour in wave--particle duality, in a multi-path setup. We introduce a basis-dependent predictability defined by the Bures distance between the state dephased in the chosen basis and the maximally mixed state. The predictability depends only on the observed basis statistics and admits a closed form in terms of the Bhattacharyya overlap. We derive a trade-off relation between the predictability and a coherence measure defined based on the nonclassicality of the Kirkwood--Dirac quasiprobability. For pure states, the trade-off relation is an exact equality. Remarkably, this wave--particle duality relation endows the coherence measure with an operational interpretation as the classically irreducible part of measurement randomness, yielding a tight worst-case bound on the guessing probability in source-independent QRNG.