Predictability as a quantum resource

Just recently, complementarity relations (CRs) have been derived from the basic rules of Quantum Mechanics. The complete CRs are equalities involving quantum coherence, $C$, quantum entanglement, and predictability, $P$. While the first two are already quantified in the resource theory framework, such a characterization lacks for the last. In this article, we start showing that, for a system prepared in a state $ρ$, $P$ of $ρ$, with reference to an observable $X$, is equal to $C$, with reference to observables mutually unbiased (MU) to $X$, of the state $Φ_{X}(ρ)$, which is obtained from a non-revealing von Neumann measurement (NRvNM) of $X$. We also show that $P^X(ρ)>C^{Y}(Φ_{X}(ρ))$ for observables not MU. Afterwards, we provide quantum circuits for implementing NRvNMs and use these circuits to experimentally test these (in)equalities using the IBM's quantum computers. Furthermore, we give a resource theory for predictability, identifying its free quantum states and free quantum operations and discussing some predictability monotones. Besides, after applying one of these predictability monotones to study bipartite systems, we discuss the relation among the resource theories of quantum coherence, predictability, and purity.

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