Abstract Complementarity relations for wave-particle duality are saturated only for pure, single-quanton, quantum states. For a completely incoherent state, it is known that wave and particle quantifiers can reach zero, and hence no information about the wave and particle aspects of the system can be obtained. This means that the information is being shared with other systems, and quantum correlations can be seen as responsible for the loss of purity of the quanton, provided that the quanton is part of a multipartite pure quantum system. In this paper, by exploring the purity of bi- and tri-partite pure quantum states, we show that it is possible to obtain complete complementarity relations. This procedure allows us to create a general framework for obtaining complete complementarity relations for a subsystem that belongs to an arbitrary multi-partite quantum system in a pure state. By some simple examples, we show that if the predictability measure is changed then the correlation measure must also be changed in order to obtain complete complementarity relations for pure cases.
Abstract Complementarity relations for wave-particle duality are saturated only for pure, single-quanton, quantum states. For a completely incoherent state, it is known that wave and particle quantifiers can reach zero, and hence no information about the wave and particle aspects of the system can be obtained. This means that the information is being shared with other systems, and quantum correlations can be seen as responsible for the loss of purity of the quanton, provided that the quanton is part of a multipartite pure quantum system. In this paper, by exploring the purity of bi- and tri-partite pure quantum states, we show that it is possible to obtain complete complementarity relations. This procedure allows us to create a general framework for obtaining complete complementarity relations for a subsystem that belongs to an arbitrary multi-partite quantum system in a pure state. By some simple examples, we show that if the predictability measure is changed then the correlation measure must also be changed in order to obtain complete complementarity relations for pure cases.
Abstract Bohr’s complementarity and Schrödinger’s entanglement are two prominent physical characters of quantum systems. In this article, we formally connect them. It is known that complementarity relations for wave-particle duality are saturated only for pure, single-quanton, quantum states. For mixed states, the wave-particle quantifiers never saturate a complementarity relation and can even reach zero for a maximally mixed state. To fully characterize a quanton, it is not enough to consider its wave-particle aspect; we have also to regard its quantum correlations with other systems. Here we prove that for any complete complementarity relation involving predictability and visibility measures that satisfy the criteria established in the literature, the corresponding quantum correlations are entanglement monotones. Therefore, we formally connect entanglement monotones with complementarity relations without appealing to a particular measure.
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)].
Abstract Wave-particle duality is an essential character of quantum systems. In the last few years, much progress has been made towards formally quantifying these quantum features. The properties of the quantum density matrix were shown to lead to duality inequalities for single-quanton mixed states and to triality identities, also known as complete complementarity relations (CCRs), for two-quanton pure states. In this article, we establish connections between CCRs and Einstein-Podolsky-Rosen (ir)realism, as formalized by Bilobran and Angelo (see Bilobran A. L. O. and Angelo R. M., EPL , 112 (2015) 40005). Besides, we show that, for tripartite pure states , CCRs can be applied to obtain the system-environment entanglement of formation, , through the local information of a system A and its classical correlation with an auxiliary system B . Moreover, we discuss CCRs in the context of the emergence of the pointer basis in the decoherence process identified via the constancy of system-apparatus classical correlations. We also obtain CCRs for purified mixed bipartite quantum-classical states. In addition, we derive CCRs for mixed-bipartite two-qudit quantum states and discuss its interpretations.
We derive two complementarity relations that constrain the individual and bipartite properties that may simultaneously exist in a multi-qubit system. The first expression, valid for an arbitrary pure state of n qubits, demonstrates that the degree to which single particle properties are possessed by an individual member of the system is limited by the bipartite entanglement that exists between that qubit and the remainder of the system. This result implies that the phenomenon of entanglement sharing is one specific consequence of complementarity. The second expression, which holds for an arbitrary state of two qubits, pure or mixed, quantifies a tradeoff between the amounts of entanglement, separable uncertainty, and single particle properties that are encoded in the quantum state. The separable uncertainty is a natural measure of our ignorance about the properties possessed by individual subsystems, and may be used to completely characterize the relationship between entanglement and mixedness in two-qubit systems. The two-qubit complementarity relation yields a useful geometric picture in which the root mean square values of local subsystem properties act like coordinates in the space of density matrices, and suggests possible insights into the problem of interpreting quantum mechanics.
Complementarity relations in composite bipartite quantum systems of arbitrary dimensions are proposed. Genuine bipartite quantum properties whose information content is quantified by the generalized concurrence mutually exclude the single-partite properties of the subsystems. The single-partite properties are determined by generalized predictabilities and visibilities, i.e. by the traditional concepts of wave-particle duality and define two complementary realities. These properties combined together are complementary to the generalized I-concurrence which quantifies genuine quantum correlations in n ⊗ m-dimensional bipartite systems. An important feature of the proposed quantitative complementarity relation is its upper bound which is given by a dimensional-dependent factor that exceeds unity in case of qudits with dimensionality d > 2. In d > 2-dimensional systems the information content can exceed one bit of information.
In this work, we will deal with pure, bipartite quantum systems of two q-dits (arbitrary dimensions). In these systems, we will study the complementarity between the local entities Predictability and Visibility (that quantifie the properties of particle and wave of each subsystem) and the Concurrence, (that quantifies the entanglement between the q-dits). In a previous work [Quantitative complementarity relations in bipartite systems. Optics Communications, Vol 283, (2010); and arXiv.org, e-print quant-ph/0302075 (2003)] concerning systems of two q-bits (2x2), Jakob and Bergou have found that this complementarity can be quantified by a beautiful equality to one. But when the system is two q-dits there is a disagreement in the literature concerning the normalization of the local information (Preditability and Visibility) of the subsystems [Quantitative wave-particle duality in multibeam interferometers. Phys. Rev. A, Vol 64, 042113 (2001)] e [Complementarity and entanglement in bipartite qudit systems. Phys. Rev. A, Vol 76, 052107 (2007)]. In this last paper, Jakob and Bergou have chosen the non-normalization and have found an expression that quantifies the complementarity between local and non-local information, but not a so simple one, like it has been found in the two q-bits case. We have tried to see what happens if we choose normalized versions of the Preditability, Visibility and Concurrence and we have found an alternative relation that quantifies the complementarity, in terms of an equality to one