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Discussão sobre complementaridade em sistemas quânticos de dois q-dits

Gabriel Augusto Pires de Pinho

Published Jan 24, 2014• 0 citations
OpenAlex

Abstract

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

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•Jan 1, 2004•17 citations
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
The notions of predictability and visibility are essential in the mathematical formulation of wave particle duality. The work of Jakob and Bergou [Phys. Rev. A 76, 052107] generalises these notions for higher-dimensional quantum systems, which were initially defined for qubits, and subsequently proves a complementarity relation between predictability and visibility. By defining the single-party information content of a quantum system as the addition of predictability and visibility, and assuming that entanglement in a bipartite system in the form of concurrence mutually excludes the single-party information, the authors have proposed a complementarity relation between the concurrence and the single-party information content. We show that the information content of a quantum system defined by Jakob and Bergou is nothing but the Hilbert-Schmidt distance between the state of the quantum system of our consideration and the maximally mixed state. Motivated by the fact that the trace distance is a good measure of distance as compared to the Hilbert-Schmidt distance from the information theoretic point of view, we, in this work, define the information content of a quantum system as the trace distance between the quantum state and the maximally mixed state. We then employ the quantum Pinsker's inequality and the reverse Pinsker's inequality to derive a new complementarity and a reverse complementarity relation between the single-party information content and the entanglement present in a bipartite quantum system in a pure state. As a consequence of our findings, we show that for a bipartite system in a pure state, its entanglement and the predictabilities and visibilities associated with the subsystems cannot be arbitrarily small as well as arbitrarily large.
We consider complementarity in a bipartite quantum system of arbitrary dimensions. Single-partite and bipartite properties turn out as mutually exclusive quantities. The single-partite properties can be related to a generalized predictability and visibility which compose two complementary realities for themselves. These properties combined become mutually exclusive to the genuine quantum mechanical bipartite correlations of the system which can be quantified with the generalized III concurrence that defines a proper entanglement measure. Consequently, the complementary relation quantifies entanglement in the bipartite system. The concept of complementarity determines entanglement as a property which mutually excludes any single-partite reality. As an application, we provide a proper definition of distinguishability in an nnn-port interferometer.
We introduce a complete set of complementary quantities in bipartite, two-dimensional systems. Complementarity then relates the quantitative entanglement measure concurrence which is a bipartite property to the single-particle quantum properties predictability and visibility, for the most general quantum state of two qubits. Consequently, from an interferometric point of view, the usual wave-particle duality relation must be extended to a ``triality'' relation containing, in addition, the quantitative entanglement measure concurrence, which has no classical counterpart and manifests a genuine quantum aspect of bipartite systems. A generalized duality relation, that also governs possible violations of the Bell's inequality, arises between single- and bipartite properties.
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.
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.
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.