From Classical Definiteness to Geometric Predictability: Complementarity, Coherence, and Thermodynamic Triality

Wave–particle complementarity is one of the central principles of quantum mechanics, traditionally quantified through the Englert–Greenberger–Yasin relation between which-way information and interference visibility. In higher-dimensional and resource-theoretic settings, however, visibility is no longer unique, and it becomes natural to reformulate complementarity in terms of basis-dependent predictability, coherence, and mixedness.

In this talk, I present two related works along this line. First, I discuss an exact complementarity relation between classical definiteness and quantumness, where definiteness is defined operationally through the resilience of a quantum state under nonselective dichotomic yes/no measurements, while the complementary quantum contribution is quantified using a Kirkwood–Dirac-based notion of coherence/interference motivated by recent KD-based coherence measures. Second, I introduce a geometric predictability defined by the Bures distance between the dephased state and the maximally mixed state. This predictability depends only on the observed measurement statistics and admits a closed form in terms of the Bhattacharyya overlap. For pure states, it satisfies an exact complementarity relation with nonclassical Kirkwood–Dirac coherence; for mixed states, this motivates a convex-roof extension whose operational meaning is the classically irreducible part of measurement randomness, with implications for guessing probability and min-entropy. Finally, motivated by the decomposition of entropy production into population and coherence contributions in quantum thermodynamics, and by standard wave–particle–mixedness triality relations, I show how the usual predictability–coherence duality can be promoted into a triality relation involving predictability, coherence, and mixedness.

Altogether, the talk connects wave–particle duality, coherence resource theories, operational guessing tasks, and thermodynamic balance relations within a unified framework.