Wave-particle duality, a fundamental principle of quantum mechanics, encapsulates the complementary relationship between the wave and particle behaviors of quantum systems. In this paper, we treat quantum coherence and classical distinguishability as complementary resources and uncover a novel duality relation, which is explored through quantum state discrimination under incoherent operations, extending beyond typical interference scenarios. We prove that in an ensemble of mutually orthogonal pure states, the sum of ``co-bits,'' quantifying the coherence preserved under incoherent free operations, and classical bits, representing the distinguishability extracted via quantum state discrimination, is bounded. This coherence-distinguishability duality relation exposes an inherent tradeoff between the simultaneous preservation of a system's quantum coherence (wavelike property) and the extraction of its classical distinguishability (particlelike property). Our findings provide a fresh perspective on wave-particle duality through quantum resource theories, offering complementary insights into manipulating quantum and classical resources, with implications for quantum foundations and quantum technologies.
Wave-particle duality, a fundamental principle of quantum mechanics, encapsulates the complementary relationship between the wave and particle behaviors of quantum systems. In this paper, we treat quantum coherence and classical distinguishability as complementary resources and uncover a novel duality relation, which is explored through quantum state discrimination under incoherent operations, extending beyond typical interference scenarios. We prove that in an ensemble of mutually orthogonal pure states, the sum of ``co-bits,'' quantifying the coherence preserved under incoherent free operations, and classical bits, representing the distinguishability extracted via quantum state discrimination, is bounded. This coherence-distinguishability duality relation exposes an inherent tradeoff between the simultaneous preservation of a system's quantum coherence (wavelike property) and the extraction of its classical distinguishability (particlelike property). Our findings provide a fresh perspective on wave-particle duality through quantum resource theories, offering complementary insights into manipulating quantum and classical resources, with implications for quantum foundations and quantum technologies.
We derive a generalized wave-particle duality relation for arbitrary multipath quantum interference phenomena. Beyond the conventional notion of the wave nature of a quantum system, i.e., the interference fringe visibility, we introduce a quantifier as the normalized quantum coherence, recently defined in the framework of quantum information theory. To witness the particle nature, we quantify the path distinguishability or the which-path information based on unambiguous quantum state discrimination. Then, the Bohr complementarity principle for multipath quantum interference can be stated as a duality relation between the quantum coherence and the path distinguishability. For two-path interference, the quantum coherence is identical to the interference fringe visibility, and the relation reduces to the well-known complementarity relation. The duality relation continues to hold in the case where mixedness is introduced due to possible decoherence effects.
One of the fundamental features of quantum mechanics is the superposition principle, a manifestation of which is embodied in quantum coherence. Coherence of a quantum state is invariably defined with respect to a preferred set of pointer states, and there exist quantum coherence measures with respect to deterministically as well as probabilistically distinguishable sets of quantum state vectors. Here we study the resource theory of quantum coherence with respect to an arbitrary set of quantum state vectors, that may not even be probabilistically distinguishable. Geometrically, a probabilistically indistinguishable set of quantum state vectors forms a linearly dependent set. In quantum optics, the coherent states form an ``overcomplete basis'' of linearly dependent states and are useful in dealing with states that can be prepared in optical systems. Also, the resource theory of magic can be looked upon as a resource theory of quantum coherence with respect to a set of basis vectors that are probabilistically indistinguishable. These motivate us to consider a resource theory of coherence with respect to probabilistically indistinguishable pointers. We find the free states of the resource theory, and analyze the corresponding free operations, obtaining a necessary condition for an arbitrary quantum operation to be free. We identify a class of measures of the quantum coherence and, in particular, establish the monotonicity property of the measures. We find a connection of an arbitrary set of quantum state vectors with positive operator-valued measurements with respect to the resource theory being considered, which paves the way for an alternate definition of the free states. We subsequently examine the wave-particle duality in a double-slit setup in which superposition of probabilistically indistinguishable quantum state vectors is possible. Specifically, we report a complementary relation between quantum coherence and path distinguishability in such a setup.
The concept of wave-particle duality holds significant importance in the field of quantum mechanics, as it elucidates the dual nature encompassing both wavelike and particlelike properties exhibited by microscopic particles. In this paper, we construct generalized measures for the predictability and visibility of n-path interference fringes to quantify the wave and particle properties in quantum high-dimensional systems. By employing the Morozova-Chentsov function, we ascertain that the wave-particle relationship can be delineated by the average coherence. This function exhibits a close correlation with the metric-adjusted skew information, thereby we establish complementary relations between visibility, predictability, and quantum f entropy, which reveal deep connections between wave-particle duality and other physical quantities. Through our methodology, diverse functions can be selected to yield corresponding complementary relationships.
Abstract We propose the notion of quantum coherence for superpositions over states which are not necessarily mutually orthogonal. 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 orthogonal 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 orthogonal 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.
Wave-particle duality, an important and fundamental concept established upon pure quantum systems, is central to the complementarity principle in quantum mechanics. However, due to the environment effects or even the entanglement between the quanton and the which-way detector (WWD), the quanton should be described by a mixed quantum state but not a pure quantum state. Although there are some attempts to clarify the complementarity principle for mixed quantum systems, it is still unclear how the mixedness affects the complementary relation. Here, we give a ternary complementary relation (TCR) among wave, particle and mixedness aspects for arbitrary multi-state systems, which are respectively quantified by the l1 measure for quantum coherence, the which-path predictability, and the linear entropy. In particular, we show how a WWD can transform entropy into predictability and coherence. Through modifying the POVM (positive-operator valued measure) measurement on WWD, our TCR can be simplified as the wave-mixedness and particle-mixedness duality relations. Beyond enclosing the wave-particle duality relation [PRL 116, 160406 (2016)], our TCR relates to the wave-particle-entanglement complementarity relation [PRL 98, 250501 (2008); Opt. Commun. 283, 827 (2010)].
Complementarity, or more specifically, the wave-particle duality, and quantum coherence are two fundamental concepts in quantum mechanics. Recently, motivated by the progress of the quantification of quantum coherence, the complementary relation between coherence and path information is investigated by many authors, and various duality relations between them are established. Such relations not only provide insights into fundamental problems of quantum mechanics, such as the understanding of quantum coherence and wave-particle duality; but also are important in applications of quantum technologies. In this paper, based on the Bures distance and unambiguous quantum state discrimination, systematic analysis of the complementarity between the quantum coherence and path information in two path interferometers is carried out. Similarly as other related works, the wave aspect, or the visibility of the interferometer, is quantified by the l1-norm measure of quantum coherence, and the path information is considered via unambiguous quantum state discrimination. In this way, a novel duality relation in two path interferometers is obtained. Compared with known results, our work considers mixed states as well as pure states; considers the path predictability resulting from the intrinsic path asymmetry of the quantum state, as well as the path distinguishability resulting from the use of a which-path detector. Therefore, our work systematically generalizes known results in two path interferometers by removing all the unnecessary restrictions. Specifically, the most general form of quantum states in two path interferometers is considered and the duality relation between quantum coherence and path information is proved based on the positivity property of density matrices. The cases of path predictability and path distinguishability are considered separately. For path predictability, the proof is straightforward; whereas some advanced mathematical techniques, such as the Schur-Weyl inequality, properties of the fidelity and properties of positive matrices, are required in order to give a rigorous proof of the duality relation between coherence and path distinguishability. Concrete examples are provided to illustrate the abstract method and results. Our work concerns about two path interferometers exclusively and depends heavily that the dimensionality is two, therefore it would be an interesting task to generalize the results in this paper to n-path interferometers.