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Redalyc.Quantum Discord in Nuclear Magnetic Resonance Systems At Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Maziero, J.; Auccaise, R.; Céleri, L. C.; Soares-Pinto, D. O.; deAzevedo, E. R.; Bonagamba, T. J.; Sarthour, R. S.; Oliveira, I. S.; Serra, R. M. Quantum Discord in Nuclear Magnetic Resonance Systems at Room Temperature Brazilian Journal of Physics, vol. 43, núm. 1-2, abril, 2013, pp. 86-104 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46425766006 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Braz J Phys (2013) 43:86–104 DOI 10.1007/s13538-013-0118-1 GENERAL AND APPLIED PHYSICS Quantum Discord in Nuclear Magnetic Resonance Systems at Room Temperature J. Maziero · R. Auccaise · L. C. Céleri · D. O. Soares-Pinto · E. R. deAzevedo · T. J. Bonagamba · R. S. Sarthour · I. S. Oliveira · R. M. Serra Received: 10 December 2012 / Published online: 26 February 2013 © Sociedade Brasileira de Física 2013 Abstract We review the theoretical and the experi- Previous theoretical analysis of quantum discord deco- mental researches aimed at quantifying or identifying herence had predicted the time dependence of the dis- quantum correlations in liquid-state nuclear magnetic cord to change suddenly under the influence of phase resonance (NMR) systems at room temperature. We noise. The experiment attests to the robustness of the first overview, at the formal level, a method to deter- effect, sufficient to confirm the theoretical prediction mine the quantum discord and its classical counterpart even under the additional influence of a thermal envi- in systems described by a deviation matrix. Next, we de- ronment. Finally, we discuss an observable witness for scribe an experimental implementation of that method. the quantumness of correlations in two-qubit systems and its first NMR implementation. Should the nature, not the amount, of the correlation be under scrutiny, the witness offers the most attractive alternative. J. Maziero · · Universidade Federal do Pampa, Campus Bagé, Keywords Quantum information Quantum discord 96413-170, Bagé, Rio Grande do Sul, Brazil Nonclassical correlations · Nuclear magnetic resonance e-mail: [email protected] R. Auccaise Empresa Brasileira de Pesquisa Agropecuária, 1 Introduction Rua Jardim Botânico 1024, 22460-000 Rio de Janeiro, Rio de Janeiro, Brazil 1.1 Bell’s Inequalities, Entanglement, and the L. C. Céleri No-Local-Broadcasting Theorem Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia, Goiás, Brazil The study of quantum correlations is among the most D. O. Soares-Pinto · E. R. deAzevedo · T. J. Bonagamba active research branches of quantum information sci- Instituto de Física de São Carlos, ence (QIS). The seminal works of Einstein et al. [1]and Universidade de São Paulo, P.O. Box 369, 13560-970 São Carlos, São Paulo, Brazil Schrödinger [2] first drew attention to the subject, in 1935, and placed the nonlocal nature of the correlations R. S. Sarthour · I. S. Oliveira at the focus of initial research. The name entanglement Centro Brasileiro de Pesquisas Físicas, was coined by Schrödinger [2]. In 1964, rigorous rela- Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil tions among the average values of composite-system observables were presented by Bell [3]. The Bell in- R. M. Serra (B) equalities, derived under the assumptions of realism Centro de Ciências Naturais e Humanas, and locality, are violated by quantum systems in certain Universidade Federal do ABC, R. Santa Adélia 166, 09210-170 Santo André, São Paulo, Brazil entangled states, which are said to possess nonlocal e-mail: [email protected] quantum correlations [4–6]. Braz J Phys (2013) 43:86–104 87 { } {|ψ A} In 1989, Werner reported a result that unveiled new where p jk is a probability distribution and the j perspectives in the study of nonclassical correlations {|φ B} ( k ) defines an orthonormal basis for subsystem in multipartite systems [7]. By defining entanglement A (B). operationally as those correlations that cannot be gen- ρcc Comparison of (4)with(3) shows that AB forms a erated via local quantum operations assisted by classi- non convex set that is a subset of the separable states. cal communication, Werner identified entangled states It follows that quantum correlations, i.e., correlations satisfying all Bell inequalities. Another surprise came that cannot be locally broadcast, can be found even in with the work of Popescu, who showed that with certain separable states. Nonclassical correlations of this kind nonseparable local states, quantum states allow tele- can be identified and quantified by the measure we portation with more fidelity than the classical alterna- discuss next. tives [8]. The most general quantum operation describing changes of quantum states has the following form, in 1.2 Quantum Discord the operator-sum representation [9]: Our review is focused on quantum discord (QD). This E(ρ) = ρ †, quantity stands out among the measures of quantum K j K j (1) j correlations because it was proposed first and has been most widely studied. Detailed discussions of other where K is the Kraus operator acting on the system j quantum correlation measures and their properties state space H, which satisfies the inequality have been presented in two recent reviews [16, 17]. † ≤ I, Before turning to QD, we find it instructive to briefly K j K j (2) j review certain aspects of classical information theory, to set the notation and define relevant quantities. where I is the identity operator in H. The equality on the right-hand side of (2)isa sufficient condition for the map E : H → H to be trace 1.2.1 Concepts from Classical Information Theory preserving, that is, to insure that Tr [E(ρ)] = 1. The most general bipartite state that can be created In classical information theory, the uncertainty of a via local quantum operations (EA and EB) and classical random variable A is quantified by its Shannon en- communication has the form tropy [11] ρsep = ρ A ⊗ ρ B, AB p j j j (3) ( ) := − , H A pa log2 pa (5) j a { } ≥ where p j is a probability distribution (i.e., p j 0 and A(B) where pa := Pr(A = a) stands for the probability that p = 1)andρ ∈ H ( ) is the reduced density j j j A B A takes the value a. The binary logarithm expresses operator of subsystem A (B). States that can be written information in bits. intheformof(3) are said to be separable. All other Two random variables are said to be correlated if states are said to be entangled (or nonseparable). they share information. Knowledge about one of them, The above-discussed nonlocal and nonseparable as- say B, yields information about the other, A.The pects of the correlations in a composite system are difference in the uncertainties of A before and after we distinctive features of the quantum realm. It is tempting know B, to believe that they thoroughly describe the quantum nature of the multipartite state, but a characterization J(A:B) := H(A) − H(A|B), (6) attentive only to entanglement would be incomplete. To construct a more encompassing description, we have a quantity named mutual information, is a measure to examine the possibility of locally redistributing the of the correlation between the two random variables correlations. In 2008, Piani and coauthors proved the (A and B). so-called no-local-broadcasting theorem [10], showing In (6), the conditional entropy reads that local operations can redistribute the correlations in a bipartite state ρ if and only if the latter can be AB ( | ) := − , cast in the following form: H A B pa,b log2 pa|b (7) a,b ρcc = |ψ Aψ A|⊗|φ Bφ B|, AB p jk j j k k (4) j,k where pa,b := Pr(A = a, B = b). 88 Braz J Phys (2013) 43:86–104 B Given the definition for the conditional probability In (14)and(15), j is a complete set of von Neu- for A to take the value a when B is equal to b, namely mann’s measurements on subsystem B satisfying the pa|b := pa,b /pb , we have that B = I BB = δ B relations j j B and j k jk j . The average quantum conditional entropy of subsys- H(A|B) = H(A,B) − H(B), (8) tem A, given the measurement of the observable OB ( , ) =− with H A B a,b pa,b log2 pa,b . on subsystem B, is determined by the expression Hence, the following equivalent expression for the (ρ ) = ρ A . mutual information between A and B can be written: S A|B p jS j (16) j I(A:B) = H(A) + H(B) − H(A,B). (9) A quantum extension of J(A:B) can therefore be In classical information theory, the relation defined as follows: ( ) = ( ) J A:B I A:B (10) J (ρAB) := S(ρA) − S(ρA|B). (17) is always valid; the two expressions for the mutual The two expressions for the quantum mutual infor- information are equivalent. The same is not true, in mation, (12)and(17), are in general inequivalent. The general, of the extensions of J(A:B) and I(A:B) to difference quantum states. This nonequivalence is at the heart D (ρ ) := I(ρ ) − J (ρ ), of the definition of quantum discord, which therefore B AB AB max AB (18) OB measures the quantum aspects of correlation. a measure of quantum correlation in bipartite systems, 1.2.2 Original Def inition of Quantum Discord was named quantum discord [15].
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