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]. The maximum on the right-hand side of (18)isob- The uncertainty about a system S, described by a den- tained from the measurements (on subsystem B)pro- sity operator ρ , is quantified in quantum information viding maximal information about A. An alternative S D theory by the von Neumann entropy [12]: version of QD, A, can be obtained from measure- ments of an observable OA on subsystem A. In general, (ρ ) := − (ρ ρ ). S S Tr S log2 S (11) DA = DB, an asymmetry of QD with several inter- esting physical interpretations as well as information- A direct extension of the mutual information in (9) theoretical implications [16, 17]. is therefore given by the equality If the mutual information is taken as a measure of

I(ρAB) := S(ρA) + S(ρB) − S(ρAB). (12) total correlations, one can verify that QD may be writ- ten as the difference between the mutual informations I(ρ ) AB , named quantum mutual information, can be of the subsystems before and after a complete map of regarded as a measure of the total correlations between von Neumann’s measurements is applied to one of the subsystems A and B when the joint system is in the subsystems: state ρAB [13, 14]. B To extend to the quantum realm, the mutual in- DB(ρAB) = I(ρAB) − max I  (ρAB) , (19) B formation in (6), consider the measurement of an B(ρ ) = (I ⊗ B)ρ (I ⊗ B) observable represented by the following Hermitian where AB j A j AB A j . operator: In this alternative definition, QD can be interpreted as a measure of those correlations that are inevitably = |BB|, OB o j j j (13) destroyed by the measurement. j defined on the state space of subsystem B, HB. 1.3 Nuclear Magnetic Resonance Systems in Quantum If the system is initially in the state ρAB, the value o j Information Science is obtained with probability We expect the relation between nonclassical correla- = I ⊗ Bρ p j Tr A j AB (14) tions and the advantages offered by quantum informa- tion science over classical protocols to play an impor- and the state of subsystem A, immediately after the tant role in the quest for quantum speedup. Nuclear measurement, reads magnetic resonance (NMR) has been one of the lead- 1 ρ A = I ⊗ Bρ I ⊗ B . ing experimental platforms for the implementation of j TrB A j AB A j (15) p j protocols and algorithms in QIS [18] and simulation of Braz J Phys (2013) 43:86–104 89 quantum systems [19]. NMR implementations encode asymmetric with respect to subsystem interchange or, the qubits in nuclear spins that are manipulated by more specifically, if S(ρA) = S(ρB),thenDA(ρAB) = carefully designed sequences of radio-frequency (rf) DB(ρAB) [29]. At first glance, given that the cor- pulses. The information about the system state is ob- relations are associated with the information shared tained directly from the transverse magnetizations, the between subsystems, this asymmetry may seem surpris- natural NMR observables [37]. The NMR density op- ing. The asymmetry is nonetheless compatible with the erator is typically highly mixed. In fact, Braunstein and sharing, because the quantumness limits differently the coauthors showed that no entanglement was generated amount of information available to observers via local in most NMR implementations of QIS protocols [20]. measurements in each subsystem. In another line of work, Vidal proved that a large Only for the class of states in (4) do we have that D (ρcc ) = D (ρcc ) = amount of entanglement must be generated if a pure- A AB B AB 0. A quantity that identifies state quantum computation is to provide exponential and, to some extent, quantifies the quantumness of speedup of information processing [21]. The conjunc- correlations in states that cannot be cast in the form of tion of these results led to doubts contrasting the pos- (4) can be defined by the equality [29] sible classical nature of NMR with the envisioned QIS AB D(ρAB) := I(ρAB) − max I  (ρAB) , (21) implementations. AB Around the same time, however, it was realized that, while necessary, entanglement generation was not where the complete map of local von Neumann mea- surements reads sufficient for gain, even in pure-state quantum compu- tation [22]. In the last decade, it has become progres- AB(ρ ) := A ⊗ B ρ A ⊗ B , AB j k AB j k (22) sively clearer that the quantum speedup in the mixed- j,k state quantum computation scenario depends on more s = I ss = δ s = subtle components [23–26]. In particular, although a with j j s and j k jk j for s A, B. definitive proof is still lacking, QD has been identified This quantum correlation quantifier can be re- as the figure of merit for quantum advantage [27, 28]. garded as a symmetric version of QD. Since the state AB(ρ ) Important as this issue is, to settle it is not the goal AB is classical, we define the classical counter- of this review. We deal only with the theoretical and part of D, experimental quantification or identification of quan- AB C(ρAB) := max I  (ρAB) , (23) tum correlations in NMR systems at room tempera- AB ture. We discuss the signatures of quantumness in the as a measure of the classical correlations in ρ . correlations of highly mixed NMR states and study the AB dynamics of the correlations under decoherence. 2.2 Symmetric Quantum Discord for the Deviation Matrix 2 Quantification and Identification of Quantum We now show how to obtain the classical correlation Correlations (23) and the symmetric quantum discord (21) for sys- tems described by density matrices of the form [33] 2.1 Symmetric Quantum Discord I ρ = AB + ρ , (24) As mentioned in Section 1.2.2, the original definition AB 4 AB of QD is asymmetric with respect to the subsystem we where  1 and ρ is a traceless deviation matrix, choose to measure. As an illustration, let us consider AB Tr(ρ ) = 0.1 the following bipartite state: AB In order to calculate the quantum mutual informa- ρcq = |ψ Aψ A|⊗ρ B, tion, we need to compute the von Neumann entropy. AB pi i i i (20) i To that end, we take advantage of the eigen decompo- sition of ρ to write that { } {|ψ A} ∈ H AB where pi is a probability distribution, i A is ρ B ∈ H (ρ ) = ( / + λ )|λ λ |. an orthonormal basis for subsystem A,and i B is ln AB ln 1 4 j j j (25) the reduced density operator of subsystem B. j D (ρcq ) In this case, A AB vanishes, while the alternative D (ρcq ) definition of QD, B AB , is null if and only if all ρ B [ρ B,ρB]= i commutes, that is, if i j 0 for all i and j. 1Equation (24) contains the typical density operator describing Moreover, numerical calculations indicate that if ρAB is the state of NMR systems. 90 Braz J Phys (2013) 43:86–104

The Taylor expansion of ln(1/4 + λj) then yields 2.3 Symmetric Quantum Discord for Two-Qubit States

1 Any two-qubit state can be brought, through local uni- S(ρAB) =− Tr(ρAB ln ρAB) ln 2 tary transformations, to the following form: ⎡ I (2ln2−1) 22 = AB + ρ − ρ 2 +··· 3 Tr AB AB 1 2 ln 2 ln 2 ρ = ⎣I + a σ A ⊗ I + b I ⊗ σ B AB 4 AB j j B j A j 22 j=1 = − (ρ2 ) +··· . ⎤ 2 Tr AB (26) ln 2 3 + c σ A ⊗ σ B⎦ , (32) Since the reduced density matrices have the form j j j j=1 I {σ s} H s where j is the Pauli operator acting on s and ρs = + ρs, (27) 2 a j, b j, c j ∈ R. The measured state, modulo local unitary transformations, can be written in the form with s = A, B, the same procedure can be used to 1 compute AB(ρ ) = I + ασ A ⊗ I + βI ⊗ σ B AB 4 AB 3 B A 3 2  + γσA ⊗ σ B , (33) S(ρ ) = 1 − Tr ρ 2 +··· . (28) 3 3 s ln 2 s with In view of (26)and(28), we can express the quantum α := a zA,β:= b zB,γ:= c zAzB. (34) mutual information in terms of the deviation matrix. j j j j j j j j j j Up to second order in , we have that The parameters 2 s = θ θ φ , I(ρ ) ≈ ρ 2 − ρ 2 − ρ 2 . z1 2sin s cos s cos s AB 2Tr AB Tr A Tr B ln 2 s = θ θ φ , z2 2sin s cos s sin s (29) s = 2 θ − , z3 2cos s 1 ρ The measured state obtained from AB through a with complete set of local projective measurements reads θs ∈[0,π/2] and φs ∈[0, 2π], (35) IAB AB(ρ ) = + η , (30) determine the measurement direction for subsystem AB 4 AB s = A, B. with the measured deviation matrix given by the One might fear that the maximization problem on equality the right-hand side of (23), associated with the compu- tation of the classical correlation, cannot be solved in AB ηAB :=  (ρAB) general. Nevertheless, for Bell-diagonal states, ⎛ ⎞ = A ⊗ B ρ A ⊗ B 3 j k AB j k bd 1 ⎝ A B⎠ , ρ = I + c σ ⊗ σ , (36) j k AB 4 AB j j j j=1 and Tr AB(ρ ) = 0. AB the correlations can be obtained analytically. In this The procedure leading to (29) now yields the fol- case, we have that S(ρbd) = S(ρbd) = 1,and(23)be- lowing expression for the mutual information of the A B comes measured state: C ρbd = 2 − min S AB ρbd , (37) 2 AB γ AB I AB(ρ ) ≈ × 2Tr η2 − Tr η2 AB ln 2 AB A with − η2 , Tr B (31) 4 1 + (−1) jγ 1 + (−1) jγ S AB ρbd =− log . where ηA(B) = TrB(A)(ηAB). The symmetric QD AB 2 = 4 4 and the classical correlation are then computed from j 1 (21)and(23), respectively. (38) Braz J Phys (2013) 43:86–104 91

The minimum of the measured state entropy is ob- All matrices in this article are represented in tained by maximizing |γ |. It results that the standard computational basis {|00, |01, |10, |11}, where |ij:=|i⊗|j and {|0, |1} are the eigenstates |γ |≤κ := max(|c |, |c |, |c |). (39) 1 2 3 of the Pauli matrix σz. The Kraus operators in the operator-sum representation (42) for the phase- An analytical expression for the symmetric QD for damping channel are Bell-diagonal states is then obtained [29]: 1 s = − ps 10 , D(ρbd ) = λbd λbd − C ρbd , K0 1 AB jk log2 4 jk AB (40) 2 01 j,k=0 p 10 Ks = s , (44) with 1 2 0 −1 1 1 where s = A, B and p = p := p is the parametrized C ρbd = 1 + (−1) jκ log 1 + (−1) jκ (41) A B AB 2 2 time. j=0 We have chosen identical environments for the two λbd =[ + (− ) j − (− ) j+k + (− ) j ]/ . subsystems and use the shorthands and jk 1 1 c1 1 c2 1 c3 4 α := ( − )2 , 2.4 Dynamics of Correlations under Decoherence 1 p c1 2 β := (1 − p) c2, We shall use the analytical expressions for the sym- γ := c . metric QD (40) and its classical counterpart (41)in 3 Bell-diagonal states to study the dynamics of these We can see that the time-evolved density matrix (43) correlations under the influence of a local, independent has the Bell-diagonal form (36). Thus, the symmetric Markovian environments that inject phase noise into QD D ρbd (p) and the classical correlation C ρbd (p) the system, as schematically depicted in Fig. 1 [30]. AB AB are given by (40)and(41), respectively, with Under such conditions, the two subsystems A and B, initially prepared in a Bell-diagonal state (36), evolve κ = max(|α|, |β|, |γ |) (45) via the following quantum operation [9]:

and ρbd (p) = E ρbd AB PD AB + λbd =[1 + (−1) jα − (−1) j kβ + (−1) jγ ]/4. (46) † jk = A ⊗ B ρbd A ⊗ B K j Kk AB K j Kk (42) j,k The analytical results define three classes of behavior ⎡ ⎤ + γ α − β for the correlations under decoherence, according to 1 00 = , , ⎢ ⎥ the coefficients c j ( j 1 2 3) in the sum within paren- 1 ⎢ 01− γα+ β 0 ⎥ = ⎣ ⎦ . (43) theses on the right-hand side of (36): 4 0 α + β 1 − γ 0 α − β 001+ γ (i) If |c3|≥|c1|, |c2|, we have that κ =|c3|,which makes the classical correlation independent of the parametrized time p. Since the quantum correlation decays monotonically to zero in the asymptotic-state limit, the classical correlation is equal to the mutual information, i.e.,

C ρbd ( ) = C ρbd ( = ) = I ρbd ( = ) . AB p AB p 0 AB p 1 (47)

Figure 2 shows the evolution of the correlations for an initial state ρbd defined by the parameters Fig. 1 (Color online) Schematic representation of the two sys- AB = 0.06 = 0.3 = 0.33 tems A and B, initially prepared in a state ρAB, evolving under c1 , c2 ,andc3 , which belong the action of local, independent phase channels to this class. 92 Braz J Phys (2013) 43:86–104

1.2 0.15 1.0

0.8 0.10 0.6

Correlations Correlations 0.05 0.4 0.2 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 p p

Fig. 2 (Color online) Total correlation (dotted line, blue online), Fig. 3 (Color online) Total correlation (dotted line, blue online), classical correlation (dashed line, red online), and quantum dis- classical correlation (dashed line, red online), and quantum dis- cord (black solid line) for the Bell-diagonal initial state in (36) cord (black solid line) for a Bell-diagonal initial state evolving = . = . = . with the parameters c1 0 06, c2 0 3,andc3 0 33 evolving under local, independent phase-damping channels. Analogous under local, independent phase-damping channels. Since |c | > 3 to Fig. 2. Here, the initial-state parameters are c1 = 1, c2 = |c |, |c |, the parametrical set belongs to class (i), defined in the 1 2 −0.6,andc3 = 0.6, within class (ii). The behaviors of the partial text. The classical correlation is not affected by the environment, correlations change suddenly at p = psc ≈ 0.22. While constant while the quantum correlation decays monotonically (monotonically decaying) for p < psc, the quantum discord (clas- sical correlation) decays monotonically (remains constant) for p > psc

(ii) If |c1|≥|c2|, |c3| or |c2|≥|c1|, |c3|,and|c3| = 0 the dynamics of the correlations under decoher- ence exhibits a sudden change at the parame- separable from entangled states. It is natural to ask trized time whether the same approach can be used to identify classical states. It turns out that the set formed by states | | c3 with null QD is not convex. As a consequence, linear psc = 1 − . (48) max(|c1|, |c2|) witnesses can no longer be defined. To verify this assertion, consider a linear Hermitian < κ = ( − )2 (| |, For p psc, we have that 1 p max c1 operator W whose mean value is nonnegative for clas- | |) C l c2 , so that the classical correlation decays sical states, i.e., monotonically. By contrast, for p ≥ psc, we have κ =| | C   = ( ρcc ) ≥ , that c3 and is constant: Wl ρcc Tr Wl AB 0 (50) AB C ρbd (p > p ) = I ρbd (p = 1) , (49)   < AB sc AB while Wl ρAB 0 for quantum-correlated states. Since D = the separable states in (3) satisfy the inequality while the decay rate of changes abruptly at p p . sc Wlρsep = p jWlρ A⊗ρ B ≥ 0, (51) AB j j Figure 3 shows an example. The initial state j ρbd = =− . AB is now defined by c1 1, c2 0 6,and c3 = 0.6. In this example, not only does the clas- sical correlation remain constant in a certain 0.10 time interval, but also the quantum correlation in unaffected by decoherence in another inter- 0.08 D = C = val. The regimes in which constant and 0.06 constant were named classical and quantum de- 0.04

coherence regimes, respectively, in [31]. Correlations = C D (iii) Finally, if c3 0, both correlations and decay 0.02 monotonically. Figure 4 shows an example of this

kind of behavior, for a Bell-diagonal initial state 0.0 0.2 0.4 0.6 0.8 1.0 p with c1 = c2 = 0.25 and c3 = 0.0. Fig. 4 (Color online) Total correlation (dotted line, blue online), 2.5 Witness of Classicality classical correlation (dashed line, red online), and quantum dis- cord (black continuous line) for a Bell-diagonal initial state evolv- The set composed by separable states of the form (3) ing under local independent phase-damping channels. Analogous to Fig. 2. The initial-state parameters are now c1 = c2 = 0.25 and is convex. There is hence a linear Hermitian operator, c3 = 0.0, within class (iii). All correlations decay monotonously known as the entanglement witness, that distinguishes with time Braz J Phys (2013) 43:86–104 93

the linear operator Wij cannot identify separable dis- the natural observables accessed in NMR experiments. cordant states. We therefore note that To overcome this difficulty, one introduces a nonlin- A ear classicality witness providing a sufficient condition Oiρ = σ ⊗ IB (53) AB 1 ξi for the absence of quantumness in the correlations of two-qubit states of the form (32)[32]. To this end, with consider the following set of Hermitian operators † ξ = U → R (θ )ρ R (θ ) U → , (54) = σ A ⊗ σ B, i A B ni i AB ni i A B O j j j = · σ ⊗ I + I ⊗ w · σ , O4 z A B A B where = , , , w ∈ R3 | |=| w|= where j 1 2 3, and the z ( z 1) B U → =|00|⊗I +|11|⊗σ (55) should be randomly chosen. Consider next the follow- A B B 1 ing relation among the averages of these operators: is the controlled-NOT gate with qubit A as control 3 4 and R (θ ) = RA(θ ) ⊗ RB(θ ),whereRs (θ ) is a local ni i ni i ni i ni i ρ = | ρ  ρ |, = θ θ = θ = W AB Oi AB O j AB (52) rotation of qubit s A, B by an angle i ( 1 0, 2 = = + i 1 j 1 1 θ3 = π/2) in the direction nˆi (nˆ 2 = yˆ, nˆ 3 =ˆz). where |x| is the absolute value of x. = We can see that WρAB 0 if and only if the mean values of at least three of the above-defined four ob- 3 Quantum Discord in NMR Systems servables are zero. Since 3.1 Physical System   = = , , , Oi ρAB ci for i 1 2 3   = · + w · , In order to experimentally follow the dynamics of O4 ρAB z a b correlations under decoherence and to implement the with a = (a1, a2, a3) and b = (b 1, b 2, b 3), the only way classicality witness discussed in Section 2.5, we have to = ρ to insure that WρAB 0 is to impose that AB be of prepare, manipulate, and measure the state of a two- the type qubit system in the laboratory. In the NMR picture, two-qubit systems can be achieved using samples pre- 1 χ = I + c σ A ⊗ σ B for i = 1 or i = 2 or i = 3, senting either two J-coupled nuclear spins-1/2,whereJ i 4 AB i i i ⎡ ⎤ denotes the scalar spin–spin coupling, or one quadrupo- 3 / 1 lar spin-3 2 system, called quadrupolar spin system, for χ = ⎣I + σ A ⊗ I + I ⊗ σ B ⎦ . 4 AB a j B b j A brevity, in a local electric field gradient. Many examples 4 j j j=1 of two dipolar J-coupled spins-1/2 are available: 1H and 13C nuclei in chloroform (CHCl )or1Hand31P nu- All theses states can be cast in the form of (4) 3 clei in phosphoric acid (H PO )[18], for instance. The and hence possess no quantum correlations. It follows 3 4 two-qubit quadrupolar systems used so far in quantum that Wρ = 0 is a sufficient condition for ρ to be AB AB information processing (QIP) usually comprise spin- classically correlated or to have no correlations at all. 3/2 nuclei in single crystals [38] or in lyotropic liquid For Bell-diagonal states, Wρ = 0 is also a necessary AB crystals [37, 39]. Examples are 23Na and 7Li nuclei in condition for the absence of quantum discord. To show ρbd lyotropic liquid crystals based on sodium dodecyl sul- this, we note that the classically correlated state AB (I + σ A ⊗ σ B)/ = = fate (SDS) [37] and lithium tetrafluoroborate (LiBF4) must have the form AB ci i i 4 with i 1, i 23 [39], as well as Na in sodium nitrate (NaNO3)single 2,ori = 3, which implies that Wρ = 0. AB crystals [38]. For the experimental implementation of the classi- All of these are equally good representations of two- cality witness (52) in the NMR context2 that will be qubit systems. The nuclear spin interactions driving presented below, it proves useful to rewrite the witness their quantum evolution being nonetheless distinct, Wρ in terms of the qubits magnetizations, which are AB different state preparation, manipulation, and readout techniques are required. Besides, the characteristics of each spin interaction lead to unique features in the 2A modified version of this classicality witness was implemented decoherence behavior. For these reasons, each system in the optical context [36]. deserves separate discussion. 94 Braz J Phys (2013) 43:86–104

3.1.1 Two-Dipolar J-Coupled Nuclear Spin-1/2 In NMR experiments, the nuclear spins are placed

System—Liquid Sample in a strong static magnetic field B0, whose direction defines the positive z-axis. The spins are manipulated Figure 5 schematically depicts a 13C-enriched chloro- by time-dependent rf fields applied to both spins, which form molecule (CHCl3). As already mentioned, the two interact with each other and their local environments. 1 −13 ωH qubits are constituted by the nuclear H Cspinpair In a doubly rotating frame with rf frequencies rf and ωC in the molecule. To generate adequate signal strength, rf, the Hamiltonian of the two coupled spins reads many replicas of the two-qubit unit are needed, i.e., an H =− ω − ωH H − ω − ωC C + π H C ensemble of spin pairs is required, each spin interacting H rf Iz C rf Iz 2 JIz Iz only with its counterpart in the pair. This is achieved H H H H H 13 + ω I cos ϕ + I sin ϕ by diluting the C-enriched CHCl3 molecules in a 1 x y deuterated solvent. Since the deuteration and the low + ωC C ϕC + C ϕC + H ( ), natural abundance of 13C make the solvent molecules 1 Ix cos Iy sin Env t (56) magnetically inert, from the NMR viewpoint, the sys- H C tem is well described as an ensemble of isolated spin where Iu Iv is the spin angular momentum operator 1 13 pairs, i.e., numerous replicas of the two-qubit unit. in the u,v = x, y, z direction for H( C) and the angle ϕH ϕC ωH ωC The results discussed in this section were obtained and frequency 1 1 define the phase and from a sample prepared by dissolving 100 mg of 99 % power, respectively, of the rf field driving the 1H(13C) 13 C-labeled CHCl3 in 0.2 ml of 99.8 % acetone-d6 nuclei. and placing this in a 5-mm Wildmad LabGlass tube. The first two terms on the right-hand side describe 1 13 Both samples were provided by the Cambridge Isotope the free precession of Hand C nuclei around B0, Laboratories, Inc. NMR experiments were performed with Larmor frequencies at 25 ◦Cusinga500-MHz Varian Spectrometer and a ω ω 5-mm double-resonance probe tuned to the 1Hand13C H ≈ C ≈ . π 500 MHz and π 125 MHz (57) nuclei. 2 2 The third term describes the scalar spin–spin cou- pling (also referred to as J coupling in the NMR lit- erature), converted to the frequency unit:

J ≈ 215.1 Hz. (58)

The fourth and fifth terms are the external rf fields applied to manipulate the 1Hand13C nuclear spins, re- μ μ H ( ) S I spectively. Finally, Env t represents time-dependent fields resulting from the random fluctuating in the in- teractions between the spins and their environment. This term, which includes interactions with the chlorine nuclei and higher order spin–spin couplings, leads to spin relaxation and decoherence. The density operator for any quantum system in contact with a thermal bath can be written in the form

13 1 exp(−βH) C H ρ = , (59) Tr exp(−βH)

0 0 where β = 1/kBT. 400 300 200 100 0 -100 -200 -300 -400 400 300 200 100 0 -100 -200 -300 -400 Frequency Frequency For two-qubit NMR systems at room temperature, the thermal energy dwarves the magnetic energy: Fig. 5 (Color online) Top panel: Schematic representation of chloroform molecule (HCCl ). The halo around the central (left- 3 γ B0 − hand) sphere depicts perturbation and response of the carbon ε = ≈ 10 5, (60) (hydrogen) nucleus at its respective resonance frequency. Bot- 4kBT tom: Normalized equilibrium spectra of carbon (left) and hydro- gen (right) nuclei where γ is the gyromagnetic ratio. Braz J Phys (2013) 43:86–104 95

We can therefore write the density operator in the so-called high-temperature expansion Iab Sodium ρ ≈ + ερ, (61) 4 where ρ is the deviation matrix, the part of the system density matrix that can be manipulated (via rf pulses) Sulfur and accessed in NMR experiments. Two unitary transformations are common in NMR experiments: rf pulses and free evolutions of the spin systems under the coupling and static fields. The rf Chlorine pulses are characterized by intrinsic parameters: ampli- tude (power), duration, frequency, and phase. Proper setting of these parameters induces nuclear spin rota- Oxygen tions about any axis by any rotation angle. For example, a nonselective 1Hspinrotation(ahardpulse,inNMR jargon) of π/2 is the result of a 7.4-μs rf pulse, from Carbon which we acquire the 1H equilibrium spectrum (i.e., the NMR spectrum resulting from rf pulses applied to the thermal equilibrium state) shown by the bottom-right panel in Fig. 5. Another simple example is the preces- Hydrogen sion of the nuclear spins around B0 in free evolution.

3.1.2 Nuclear Quadrupolar System—Liquid Crystal Sample 23Na We have already mentioned that 23Na (I = 3/2)ina SDS molecule of a lyotropic liquid crystal represents a two-qubit quadrupolar spin system well. As depicted 0 23 15 10 5 0 -5 -10 -15 in Fig. 6,the Na nucleus in this molecule is far from Frequency (kHz) other NMR-active highly abundant nuclei, which are mostly 1H. The magnetic dipolar interaction between 23 Fig. 6 (Color online) Top left: Labels of the atomic components Na and the other nuclei in the SDS molecule can of the molecules. Bottom left: Normalized equilibrium spectrum therefore be neglected. The liquid crystal is moreover of the sodium nucleus. Right: Cartoon representing the sodium prepared with heavy water (D2O), to weaken the dipo- dodecyl sulfate molecule (SDS). The elliptical halo around the lar interaction with the 1H nuclear spins of water, since top atom represents the perturbation and response of sodium nucleus at its resonance frequencies deuterium has a small gyromagnetic ratio. Therefore, to a good approximation, in the SDS D2O-based liq- uid crystal, the interaction between the 23Na quadru- pole moment and the electric field gradient produced strength of the quadrupolar coupling, facilitating the by the electrical charges in its vicinity (usually called manipulation of the spin system by rf pulses. Under quadrupolar coupling) is the only internal spin interac- these conditions, in a rotating frame with frequency ωrf, tion affecting the evolution of the quantum system. the 23Na nuclear spin Hamiltonian can be described by The local electric field gradient is defined by the the expression charge configuration in the vicinity of the 23Na nu- cleus. It follows that, to replicate the 23Na environ- ω H =−(ω −ω ) I + Q 3I2 −I2 +ω I cos ϕ+I sin ϕ ment throughout the sample, all SDS molecules must L rf z 6 z 1 x y have the same orientation. Fortunately, the strong sta- + HQ (t), (62) tic magnetic field in NMR experiments induces the Env alignment of the SDS molecules in the lyotropic liquid crystal, naturally leading to the same local electric field where ωQ is the strength of the quadrupolar coupling 23 gradient at each Na site. Furthermore, the anisotropic expressed in the frequency unit and ωL is the Larmor internal motions of the SDS molecules reduces the frequency (|ωL|  |ωQ|). The spin angular momentum 96 Braz J Phys (2013) 43:86–104

operator is represented by its u-component Iu (u = the field gradients eliminate undesired coherences. The x, y, z) and square modulus I2. The first term on the gradient pulses generate magnetic field distributions right-hand side of (62) describes the Zeeman interac- that randomize the phases of the individual quantum tion between the static magnetic field and nuclear spin, coherences and average out their contribution to the while the second term accounts for the static first-order final state. quadrupolar interaction. As in (56), the next-to-last and As an example, consider the preparation of the last terms correspond to an externally applied rf field quantum state |11 in the computational basis, which is and the time-dependent coupling of the 23Na nuclear accomplished by the following pulse sequence, applied spin with its environment, respectively. The latter is to the thermal equilibrium state: mainly due to the motion-induced fluctuations in the local electric field gradients, with smaller contributions π H,C π H,C π H,C 23 → 1 → → 1 → → from the weak dipolar interactions between the Na 2 −x 4J 2 y 4J 2 −x nucleus and other nearby NMR-active spins [40, 42]. , , The following experimental results were obtained { } → π H C → 1 → π H C → { } , Gz 4 −y 2J 6 x Gz from a liquid crystal sample prepared with 20.9 %of SDS (95 % of purity), 3.7 % of decanol, and 75.4 % (θ)H,C deuterium oxide, following the procedure described in where d denotes simultaneous rotations of the [43]. The 23Na NMR experiments were performed in a 1Hand13C nuclear spins by the angle θ around the 400-MHz Varian Inova spectrometer with a 7 mm solid- d-direction, while (1/nJ) indicates the duration of a ◦ state NMR probe head at 26 C. The bottom-left panel free evolution and {Gz} represents a magnetic field in Fig. 6 shows the 23Na equilibrium spectrum of our gradient in the z-direction. In practice, each rotation SDS liquid-crystal sample. The quadrupole coupling is implemented by a hard rf pulses with appropriate frequency is obtained directly from the separation be- phase (subindex) and amplitude (rotation angle be- tween the central and the satellite lines [44, 45], which tween parentheses), applied to 1Hand13C (superindex) yields nuclear spins. Figure 7 compares a numerical simulation of the ω Q pulse sequence (rf pulses, free evolutions, and mag- νQ = = 15 kHz. (63) 2π netic field gradients) to prepare the state |11 with the experimental procedure. The real and the imaginary Having overviewed the general features of the two parts of the computed (measured) deviation matrix are main two-qubit NMR systems, we now to turn to their displayed in the bar plots on the left (right) panels. applications in QIP and start out with descriptions of The experimental deviation matrix was obtained by the three central QIP steps, state preparation, ma- the quantum state tomography procedure described in nipulation, and readout, in Sections 3.2, 3.3,and3.4, Section 3.3. respectively State preparation by spatial averaging can equally well be achieved by replacing the hard rf pulses and free 3.2 State Preparation evolutions by numerically optimized low-power pulses (soft pulses, in NMR terminology) followed by mag- Most of the NMR QIP implementations rely on netic gradient pulses. Given the excellent experimental effectively pure states, usually referred to as pseudo- control over the NMR variables, the system evolution pure states. Among other methods [46–48], the main is accurately described by NMR Hamiltonians defined procedures for pseudo-pure state preparation are the by the parameters of the rf pulses (the power ω1, dura- so-called spatial [47] and temporal averagings [46], tion t, and phase ϕ). One can therefore let numerical which we discuss in this section. optimization routines seek the set of pulses that will Spatial averaging starts with a system in thermal drive the system from thermal equilibrium to a state equilibrium. A set of rf and magnetic field gradient such that the diagonal elements of its deviation matrix pulses are applied to manipulate the spin populations coincide with those of the target-state deviation matrix. and coherences (i.e., the diagonal and off-diagonal Subsequently, the field-gradient pulse averages out the elements of the density matrix in the computational off-diagonal deviation matrix elements and yields the basis, respectively) and ultimately construct the desired desired state. Two numerical procedures, named gra- quantum state. dient ascent pulse engineering (GRAPE) [49, 50]and While the rf pulses impose specific spin rotations that strongly modulated pulses (SMP) [51], are commonly change the density matrix populations and coherences, used for pulse optimization. Braz J Phys (2013) 43:86–104 97

Fig. 7 (Color online) Bar Numerical procedure Experimental procedure representation of the prepared deviation matrix. Real Real The states were generated by 1 1 hard pulses, free evolutions, 0.5 0.5 and magnetic field gradients. The top (bottom)partofthe 0 0 figure corresponds to the real -0.5 -0.5 (imaginary) part of the deviation matrix. The left -1 -1 (right) panels correspond to 00 00 the simulated (experimental) 10 10 results for the prepared initial 01 11 01 11 01 01 quantum state 11 10 11 10 00 00 Imaginary Imaginary 1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

00 00 10 10 01 11 01 11 01 01 11 10 11 10 00 00

An example of a SMP pulse sequence implementing rf pulses, to wash out the off-diagonal coherences in the a specific initial quantum state is presented in Table 1. deviation matrix. The first column indicates the order of application for The application of the pulse sequence in Table 1 to a certain set of parameters. The second (fifth) and the thermal equilibrium state, followed by the gradient third (sixth) columns show the power and phase of the pulse, was numerically simulated. The resulting real rf pulses applied to the 1H(13C) nuclei. The fourth and imaginary parts of the deviation matrix are shown column shows the duration of each rf pulse. As already in the left part of Fig. 8. For comparison, the right- mentioned, a magnetic field gradient is applied after the hand panels show the deviation matrix measured by quantum state tomography in an experiment using the parameters in the table. The main advantage of applying SMP or GRAPE Table 1 Power, duration, and phase of the rf pulses imple- pulses followed by magnetic field gradient pulses for menting the SMP technique for the preparation of a predefined state preparation via spatial averaging is versatility. The quantum state two techniques yield pulses optimized to produce any SMP Parameters desired spin population and are hence more general pulse Hydrogen Time Carbon than the application of hard pulses, which calls for a step ωH ϕH (ms) ωC ϕC 1 1 specific pulse sequence for each desired state. 1 1,092.6 5.94 0.518 1,962.9 1.88 The preparation of a quantum state via temporal av- 2 1,679.8 5.73 1.546 1,276.7 5.26 eraging requires a specific number of time-uncorrelated 3 112.0 1.64 0.226 619.3 2.55 4 1,998.0 2.63 1.130 868.9 3.91 experiments. Each experiment is designed so that the 5 211.3 4.84 1.996 835.8 5.46 desired state results from the sum of the outcomes of all the experiments. As in spatial averaging, two As explained in the text, the SMP technique is one of the methods that generates quantum states suitable for the NMR study of approaches have been developed: one based on hard quantum discord. Of particular interest is the generation of diag- pulses [46] and the other based on soft pulses [38, 51]. onal initial quantum states, as illustrated by the example in Fig. Time averaging has advantages. In particular, it can 11. The tabulated parameters define an rf pulse sequence that generate nondiagonal states, which can be prepared yields a Bell-diagonal state satisfying the inequality |c |, |c | ≥ 1 2 directly from thermal equilibrium since no magnetic |c3|. Similar procedures were followed to obtain other states of interest gradient pulses are applied. 98 Braz J Phys (2013) 43:86–104

Fig. 8 (Color online) Bar Numerical procedure Experimental procedure representation of the prepared deviation matrix, Real Real obtained from states 1 1 produced by the SMP 0.5 0.5 technique with the parameters in Table 1. 0 0

The panels are arranged -0.5 -0.5 as in Fig. 7 -1 -1

00 00 10 10 11 11 01 01 01 01 11 10 11 10 00 00

Imaginary Imaginary 1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

00 00 10 10 11 11 01 01 01 01 11 10 11 10 00 00

Figure 9 shows an example of a state prepared by ments to the reading positions in a controlled fashion. temporal averaging. This state was implemented using After the rotations, the spin magnetizations become 23Na quadrupolar nuclei (spin 3/2) in a lyotropic liquid dependent on these “nondirectly detectable” elements; crystal sample as described in Section 3.1.2.ASMP proper data processing is then sufficient to determine pulse was also applied to prepare the state, but in this the latter. The art of finding the minimum number of case, the parameters were optimized on the basis of the rotations that allows mapping of the full system density Hamiltonian (62). Different experiments are labeled by matrix, or the deviation matrix, in the context of NMR, the capital letters A, B, C, and D, each with the five sets has led to a variety of QST procedures. ω ϕ of rf pulse parameters ( 1, t and )inTable2.Each Among other sequences in the NMR QIP literature, pulse was applied separately to the thermal equilib- we find combined spin rotations [53, 54], global spin- rium state, resulting in transformed deviation matrices system rotations [52], and transition-selective excita- whose sum is the final ρ. The simulated final ρ is tions [38, 41]. Generally speaking, each spin system is shown by the bar plots in the left panels of Fig. 9.The best served by a specific QST method. For example, ρ experimental is shown by plots in the right panels. while sets of single-spin rotations are more suitable for spin-1/2 systems, transition-selective pulses or global 3.3 Quantum State Tomography spin rotations are more appropriate for quadrupolar nuclei. Quantum state tomography (QST) uses the observables As an illustration, we will briefly discuss the QST of a quantum system to measure each element of the technique introduced by Long et al. [53]. Widely used corresponding density matrix. In NMR, the observables for QST of two-coupled spin-1/2 systems, this method are the spin magnetizations, which depend on combina- has more recently been adapted to systems with more tions of the density matrix elements corresponding to spins [54, 55]. The original version is depicted in Fig. 10. single quantum transitions (m =±1). Therefore, only Two sequences of nine spectra running along the left those elements located at certain reading positions in and the right borders of the figure represent the QST the density matrix are readily accessible via experiment. procedure, a pair of horizontally aligned spectra cor- To determine the other elements, special sequences responding to each of the nine steps in the procedure. of rotations must be applied to bring the desired ele- Before each step, the system has to be prepared in Braz J Phys (2013) 43:86–104 99

Fig. 9 (Color online) Bar Numerical procedure Experimental procedure representation of the prepared deviation matrix. Real Real The panels are arranged as in 1 1 Fig. 7. The quantum states 0.5 were produced by hard 0.5 pulses, free evolutions, and 0 0 magnetic field gradients, with the parameters in Table 2 -0.5 -0.5 -1 -1

00 00 10 10 01 11 11 01 01 01 11 10 11 10 00 00

Imaginary Imaginary 1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

00 00 10 10 11 11 01 01 01 01 11 10 11 10 00 00

the same state, the deviation matrix of which must be To be more specific, let us consider the following characterized. deviation matrix: In the first step, the NMR signal, i.e., the free in- ⎡ ⎤ duction decay (FID), of both nuclei is acquired, in the a1 a2 + ia11 a3 + ia12 a4 + ia13 absence of rf pulses. The line intensities in the Fourier- ⎢ ⎥ ⎢ a2 − ia11 a5 a6 + ia14 a7 + ia15 ⎥ transformed FID are recorded, which are represented ρ = ⎣ ⎦ , (64) a3 − ia12 a6 − ia14 a8 a9 + ia16 by the top spectra (labeled II) in Fig. 10. In each of the a4 − ia13 a7 − ia15 a9 − ia16 a10 subsequent steps, rf pulses with the phases specified in Table 3 are applied to each or to both nuclei, and the same reading procedure is carried out. containing 16 unknowns. Each plot along the sequence on the left-hand (right- The rotation imposed on the spins by the rf pulses in hand) side of Fig. 10 represents the carbon (hydrogen) step n of the tomographic process can be represented ( , ) spectra resulting from each tomography step, carried by the operator Un X Y . For specified pulse duration and phase, the analytical form of U (X, Y) is known. out after a Bell-diagonal state is prepared in the CHCl3 n two-qubit spin system. The spectra are labeled II, XX, Hence, after the nth rotation, the transformed devia- IX, IY, XI, YI, XY, YX, YY, to indicate no pulse (I), a tion matrix is determined by the expression π/2 pulse in the positive x-direction (X)oraπ/2 pulse in the positive y-direction (Y). In the case under study, ρ = ( , ) ρ † ( , ) . n Un X Y Un X Y (65) each spectrum has two spectral lines, each of which has a real and an imaginary component. Altogether, 72 line intensities are therefore recorded. After the nth transformation, the real (Mx)and Each intensity is associated with one or more ele- imaginary (My) parts of the NMR magnetization for { , } ments of the deviation matrix. Therefore, to obtain the each nuclear spin A B are therefore given by the full deviation matrix, we have to relate the line inten- equalities sities, i.e., the NMR readouts, to the deviation matrix elements. The rf pulse phases on Table 3 are designed A = ρ ⊗ I ; Mx,y;n Tr n Ix,y (66a) to give access to all elements, with some redundancy to B = ρ I ⊗ . minimize the error. Mx,y;n Tr n Ix,y (66b) 100 Braz J Phys (2013) 43:86–104

Table 2 Power, duration, and phase of the rf pulses used to prepare quantum states by the SMP technique and the time- I I I I averaging procedure X X X X Group SMP Parameters of sodium nuclei set pulse ω1 ϕ Time (μs) step Real A 1 37,485.4 6.208 15.11 I X 1 I X 2 11,624.8 4.243 36.53 3 31,867.5 3.293 4.15 0.5 4 38,906.5 6.027 47.99 0

5 38,681.0 1.927 31.62 I Y -0.5 I Y

-1 B 1 12,063.7 3.919 7.95 2 15,399.8 3.581 33.74 00 X I 10 X I 3 38,721.2 5.845 30.47 11 01 01 11 10 4 24,148.3 5.117 3.38 00 5 38,916.9 3.762 48.08 Imaginary Y I Y I C 1 39,208.2 1.788 38.82 1 2 7,589.9 4.984 27.38 0.5

3 19,817.7 0.889 9.27 0 4 38,915.2 1.682 25.45 X Y X Y -0.5 5 38,755.6 1.200 21.98 -1

D 1 13,372.2 4.333 15.24 00 2 2,192.1 1.506 37.89 Y X 10 Y X 11 01 01 3 39,094.7 4.433 29.35 11 10 4 11,495.7 3.294 10.49 00 5 18,867.5 0.213 36.92 Y Y Y Y The pulse sequences generate the diagonal initial quantum state, as illustrated by the example in Fig. 11. The quantum states | | ≥ resulting from the listed parameters satisfy the inequality c1 300200 100 0 -100 -200 -300 300200 100 0 -100 -200 -300 ν (Ηz) ν (Ηz) |c2|, |c3| = 0 Fig. 10 (Color online) Outer panels: NMR spectra of a two- spin system encoded in the chloroform molecule. The spectra are Equation (66) linearly relate the magnetizations generated by the QST pulse sequence in Table 3.Thespectra to the elements of ρ. In other words, for each n, in the sequence running along the left-hand (right-hand) bor- (66a)and(66b) yield two equations of a linear system der corresponds to the carbon (hydrogen) nuclear spin. Central panels: Bar plots representing the real (top) and imaginary (bot- in which the elements of ρ are the independent vari- tom) parts of the experimental deviation matrix resulting from A,B ables, the magnetizations Mx,y;n are dependent vari- the QST ables, and the elements of the operators Un(X, Y) are coefficients. The magnetizations are directly related to the line intensities and hence experimentally accessible, and the Table 3 Ordered pulse sequence applied simultaneously to two nuclear species Un(X, Y) elements are analytically determined by the rf Pulse order Carbon Hydrogen pulse parameters. Equation (66) is therefore equivalent to the linear system 1st II 2nd XX = , 3rd IX b k Xkja j (67) 4th IY 5th XI where b k (k = 1,...,73)isthekth recorded line inten- 6th YI sity, the a j ( j = 1,...,16) are linearly related by (64) 7th XY to the ρ elements before the transformation, and the 8th YX 9th YY elements Xkj are obtained from the pulse parameters in each QST step. ≡ π π/ X 2 +x represents an rf pulse of 2 in the positive x- ≡ π π/ We solve (67)fora j to reconstruct the deviation direction. Y 2 +y represents an rf pulse of 2 in the positive matrix ρ. The bar plots in Fig. 10 show an example. y-direction. I indicates the absence of rf pulses Braz J Phys (2013) 43:86–104 101

n( ) The QST methods for quadrupolar nuclei are where K j t is the Kraus operator. analogous. Sequences of readouts are obtained after For the generalized amplitude-damping channel, the specifically designed rf pulses are applied, the line in- Kraus operators take the form tensities are recorded, and a linear system is solved to √ √ 10 √ 0 p yield the deviation matrix elements. For details, which Ka = γ √ , Ka = γ , will not be presented here because they are somewhat 0 0 1 − p 1 00 √ more intricate see [52]. − a = − γ 1 p 0 , a = −γ √00 , K2 1 K3 1 3.4 Measurements of Quantum Correlations in NMR 01 p 0 Systems (70)

We now discuss the experimental efforts to quantify where, in the NMR context, and identify quantum correlations in NMR systems at room temperature. First, we verify that the peculiar 1 − ε γ ≈ (71) dynamics of quantum discord under decoherence, the- 2 oretically predicted for phase-noise channels [30], can occur even under the additional influence of a thermal and source [34]. Next, we review the first experimental im- = 1 − exp (− / ) . plementation of a classicality witness [32], which iden- p t T1 (72) tifies the nature of the correlations in NMR systems Here, T is the longitudinal relaxation time of the even without full QST [35]. 1 qubit under consideration. 3.4.1 Experimental Dynamics of Correlations In our case, each qubit has a distinct Larmor fre- quency and distinct relaxation times. The measured Here, we discuss an experimental verification of the spin–lattice relaxation times are sudden change in the quantum discord under decoher- 1 13 ence theoretically discussed in Section 2.4. While that T1 H = 2.5 sandT1 C = 7 s. (73) effect can in principle be observed in the quadrupo- lar system described in Section 3.1.2 [33], in practice, The Kraus operators for the phase-damping channel serious difficulties arise. In the quadrupolar system, are given by (44)withp = 1 − exp (−t/T2), where the the phase-noise environment is global [56], and the measured transverse relaxation times associated with transversal relaxation time, very short. the two qubits are The transverse relaxation times in the system dis- 1 13 cussed in Section 3.1.1 are much longer. For this reason, T2 H = 1.8 sandT2 C = 0.29 s. (74) to experimentally demonstrate the peculiar behavior of correlations under decoherence, the two-qubit system Since no refocusing pulse was used, the effective 1 13 encoded in the Hand C nuclear spins of a CHCl3 transverse relaxation times are molecule was used. As already mentioned, relaxation ∗ 1 = . ∗ 13 = . . process causing phase decoherence and energy dissipa- T2 H 0 31 sandT2 C 0 12 s (75) tion in this system is mainly due to internal molecular or atomic motions. As a result, the electromagnetic field To attest to the sudden change in the decay rate and in which the qubits are immersed fluctuates randomly. robustness of correlations under phase-noise environ- To model the resulting decoherence, we combine a ments, a suitable initial state was prepared by mapping ρ phase-damping channel with a generalized amplitude- the NMR deviation matrix onto a Bell-diagonal | |, | | > | | damping channel, i.e., describe the quantum state by state with c1 c2 c3 on the right-hand side of (36). the expression Figure 11 depicts the pulse sequence applied to monitor the correlation dynamics. With help of the deviation ρ ( ) = E ◦ E (ρ ), AB t p a AB (68) matrices obtained from quantum state tomography, the where the quantum operations E (n = p, a) are writ- correlations were numerically computed at each time n = / = , ,..., ten in the operator-sum representation as discussed in step tm m 4J (m 0 1 250). Section 1, i.e., The results in Fig. 12 clearly show the sudden change in the correlation decay rates, visible even under the † E (ρ) = n( )ρ n( ) . strong influence of the thermal noise of the fluctuating n K j t K j t (69) j fields in the nuclear spin environments. As pointed out 102 Braz J Phys (2013) 43:86–104

Pseudo Tomography E P R Pulse

SMP X Y FID

δ 1 m 2J 4J 12 3 4 5

Pseudo Tomography E P R Pulse

X FID SMP δ 1 m 2J 4J 12 3 4 5 Fig. 13 Pulse sequence implementing the classicality witness.

1 t The bottom (top) sequence corresponds to the pulses applied to the 1H(13C) nucleus. In the first step, the SMP technique Fig. 11 (Color online) Pulse sequence implementing the dynam- and a z-gradient pulse implement a diagonal state. In the second ics of quantum and classical correlations under decoherence. step, a pseudo-EPR gate is implemented. In the third step, the The bottom (top) sequence corresponds to the representation system interacts with the environment, which leads to decoher- of pulses applied to the 1H(13C) nuclei. First, the SMP tech- ence of the quantum states. Finally, tomography reads out the nique and a z-gradient pulse implement a diagonal state. Next, quantum state at the uniformly distributed time intervals m/4J a pseudo-EPR gate is performed. In the third step, the nuclear (m = 0, 1,...,11) system and environment interact, resulting in decoherence of the quantum states. Finally, the quantum state is read out at equally spaced times tm = m/4J (m = 0, 1,...,250)withQST system (67), which give rise to the oscillations in the experimental curves in Fig. 12. in Section 2.4, the classical correlation should remain 3.4.2 Experimental Implementation of a Classicality constant under phase damping. The weak decay in Witness Fig. 12 is due to the thermal environment. Overall, the theoretical predictions agree well with the experiment. Sections 3.3 and 3.4.1 dealt with QST measurements of The figure defines two decoherence regimes [31]. In quantum correlations. In certain situations, however, the first regime, the decoherence affects more strongly it suffices to identify the nature of correlations. For the classical correlation; the quantum correlations this purpose, correlations witnesses are convenient. As decay, insignificantly. After the sudden change, the explained in Section 2.5, the space of classical states classical correlation becomes more robust against de- not being convex, linear witnesses are inadequate to coherence, while the quantum discord is progressively identify the quantumness of correlations in separable reduced by the noise. The experimental deviation ma- states, and one must look for nonlinear witnesses. Here, trices have small coherences, which introduce small os- we discuss an experimental implementation of the non- cillations in the recorded spectral line intensities. Small linear classicality witness introduced in Section 2.5 [32]. perturbations are consequently introduced in the linear The experiment used the same NMR apparatus that evidenced the sudden change of quantum discord under decoherence [35]. The pulse sequence implementing the classicality 12 witness protocol is sketched in Fig. 13. Three states 10 were prepared: a thermal equilibrium state ρT,a 8 quantum-correlated state ρQC, and a classically correlated 6 4 Table 4 Witness, quantum discord, and classical correlation mea- Correlations 2 sured in three different initial states: the quantum correlated 0 state ρQC, the classically correlated state ρCC, and the thermal 0.05 0.10 0.15 0.20 0.25 equilibrium state ρT Time (s) ρQC ρCC ρT Fig. 12 (Color online) Sudden change in behavior and robust- Witness 3.13 0.04 0.05 ness of classical and quantum correlations under decoherence. Quantum discord 4.02 0.00 0.00 The f illed triangles (green online) depict the measured quan- Classical correlation 2.09 7.15 0.00 tum mutual information. The f illed circles (blue online)and diamonds (orange online) represent the classical and quantum The witness was directly measured with the sequence of pulses correlations, respectively. The solid lines are the theoretical pre- in Fig. 13. The classical correlation and the symmetric quantum dictions. The initial state is analogous to the state defined by (36), discord were computed by means of QST followed by numer- with |c1|, |c2| > |c3|. The correlations are displayed in units of ical extremization. The correlations are displayed in units of (ε2/ ln 2)bit (ε2/ ln 2)bit Braz J Phys (2013) 43:86–104 103

Witness 6 Quantum Discord physics, specially quantum mechanics and thermody- Classical Correlation 5 namics, but also in technology. Basic studies of quan- tum information therefore have great importance. 4 Quantum correlations are currently associated with 3 the gain offered by quantum mechanics in comparison with classical protocols. Our understanding of the dis- 2 tinction between the quantum and the classical aspects 1 of correlated system has dramatically changed since the

0 seminal EPR paper in 1935 [1]. Initially, quantum cor- 0 0.1 0.2 0.3 0.4 0.5 0.6 relations were identified with nonlocal aspects of quan- Time (s) tum systems. After that, entanglement was raised to the Fig. 14 (Color online) Measured witness and computed corre- status of quantum correlations. In the last decade, it lations for ρQC as a function of free-evolution time. Each open was realized that even separable, nonentangled states circle (red online) was obtained from a witness protocol carried can entail quantum correlations and hence carry an ad- = δ out after the state had freely evolved for a time tn n t since ini- vantage in quantum information protocols that classical tial preparation, where δt = 55.7 ms, and n is an integer (0 ≤ n ≤ 11). The downward triangles (gray online) indicate the amount systems cannot match. The most popular quantifier of of classical correlation, and the upward triangles (blue online) this kind of correlation is called quantum discord [15]. represent the symmetric quantum discord. The correlations are We have discussed the theoretical and the experi- (ε2/ ) in units of ln 2 bit mental aspects of quantum discord in NMR at room temperature. After dwelling on the dynamics of quan- tum discord under decoherence, we presented a wit- ρ state CC. The measured values of the witness for the ness for quantum correlation. The experimental inves- three different states are listed in Table 4. It is assumed tigations of both the QD and the witness were then ρ = . that the classicality cutoff limit is W T 0 05, i.e., the surveyed, with a brief review of control techniques equilibrium-state witness value. For the classical state, in NMR as well as of quantum state interrogation, a ρ = . the witness value W CC 0 04 lies below the classical- procedure known as quantum state tomography. ity cutoff. By contrast, the quantum-correlated state = . yielded WρQC 3 13, which is far above the classicality Acknowledgements The authors acknowledge the financial bound 0.05. As this example shows, the classicality wit- support from UFABC, CNPq, CAPES, FAPESP, FAPERJ, and ness reliably sorts out quantum correlated states from Brazilian National Institute of Science and Technology for Quan- tum Information (INCT-IQ). The warm hospitality of Instituto classically correlated ones. For comparison, Table 4 de Física de São Carlos—Universidade de São Paulo (IFSC-USP) also shows the quantum discord computed from the and Centro Brasileiro de Pesquisas Físicas (CBPF), where the experimentally reconstructed deviation matrices, the experiments were performed, is also acknowledge. result of QST followed by numerical extremization. As one would expect, while vanishing in the last two columns, which contain the witnesses within the classi- References cal bound, the QD is nonzero in the first column. Finally, we discuss the decoherent dynamics of the 1. A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935) 2. E. Schrödinger, Proc. Camb. Philos. Soc. 31, 555 (1935) witness. Following initial preparation in the state ρQC, 3. J.S. Bell, Physica 1, 195 (1964) we let the system evolve freely in the NRM environ- 4. J.F. Clauser, A. Shimony, Rep. Prog. Phys. 41, 1882 (1978) = δ δ = . ment during a time interval tn n t,where t 55 7 ms 5. A. Aspect, http://arxiv.org/abs/quant-ph/0402001 (2004) and n is an integer, and then implement the witness 6. N.D. Mermin, Rev. Mod. Phys. 65, 803 (1993) protocol. The experimental results for n = 0, 1,...,11 7. R.F. Werner, Phys. Rev. A 40, 4277 (1989) 8. S. Popescu, Phys. Rev. Lett. 72, 797 (1994) are shown in Fig. 14, along with the correlations. The 9. M.A. Nielsen, I.L. Chuang, Quantum Computation and identification of classical states on the basis of the Quantum Information (Cambridge University Press, Cam- witness values agrees fairly well with the identification bridge, 2000) provided by the correlation quantifiers. 10. M. Piani, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 100, 090502 (2008) 11. C.E. Shannon, Bell Syst Tech J 27, 379 (1948) 12. B. Schumacher, Phys. Rev. A 51, 2738 (1995) 4 Concluding Remarks 13. B. Groisman, S. Popescu, A. Winter, Phys. Rev. A 72, 032317 (2005) 14. B. Schumacher, M.D. Westmoreland, Phys. Rev. A 74, Quantum information processing has the potential 042305 (2006) to promote advances not only in the foundations of 15. H. Ollivier, W.H. Zurek, Phys. Rev. Lett. 88, 017901 (2001) 104 Braz J Phys (2013) 43:86–104

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