Wave-particle duality employing quantum coherence in superposition with non-orthogonal pointers

Abstract

We propose the notion of quantum coherence for superpositions over states which are not necessarily mutually distinguishable. This anticipatedly leads to a resource theory of non-orthogonal coherence. We characterize free states and free operations in this theory, and connect the latter with free operations in the resource theory of quantum coherence for distinguishable bases. We show that the concept of non-orthogonal coherence naturally furnishes us with a wave-particle duality in quantum double-slit experiments where the channels beyond the slits are leaky between them. Furthermore, we demonstrate existence of a unique maximally coherent qubit state corresponding to any given purity. In addition, and in contradistinction with the case of distinguishable bases, there appears a non-trivial minimally coherent qubit state for a given purity. We also study the behavior of quantum coherence for some typical configurations of non-orthogonal bases which have no analogs for orthogonal bases. We further investigate the problem of determining the energy cost of creating non-orthogonal coherence, and find that it scales linearly with the non-orthogonal coherence created.

Resource theory of quantum coherence with probabilistically non-distinguishable pointers and corresponding wave-particle duality

“…These states, representing different paths, form a natural basis for this set-up. Generally considered to be mutually orthogonal, these states can also be mutually nonorthogonal and linearly independent in realistic leaky set-ups [36]. Quantum coherence with respect to mutually orthogonal bases as also the more general case of linearly independent bases, have already been studied in detail [7-9, 35, 36].…”

“…Let us now consider the set-up used in [36]. The superposition of the quantum system was considered there in a linearly independent basis.…”

“…Let the coherence in this case be denoted by C li , the probability of unambiguous discrimination of the detector states by P li , and the path distinguishability by D li . It is to be noted that in [36], a similar phase damping of the detector state was employed as considered here. The probability of the discarded POVM outcome was not multiplied to the probability of unambiguous discrimination of the detector states, P li , and instead P li itself was considered as path distinguishability.…”

“…We remember that a set of mutually orthogonal states is of course linearly independent. The resource theory of quantum coherence in superpositions over such linearly independent sets of states has been explored recently in [35,36], with a double-slit set-up where such a superposition can naturally appear being provided in [36]. There also exist a different formalism of coherence, based on measurements [7,[37][38][39].…”

“…The corresponding duality relations for superpositions over orthogonal and linearly independent bases have been studied earlier in Refs. [36,[46][47][48][49][50]. In a double-slit set-up, the wave nature of the quantum system is quantified by the quantum coherence of the corresponding state, measured in the basis that is preferred by the apparatus considered.…”