Quantum correlations in B and K meson systems
UUdeM-GPP-TH-14-237
Quantum correlations in B and K meson systems Subhashish Banerjee, ∗ Ashutosh Kumar Alok, † and Richard MacKenzie ‡ Indian Institute of Technology Jodhpur, Jodhpur 342011, India Universit´e de Montr´eal, Pavillon Roger-Gaudry, D´epartement de physique,C.P. 6128, Succursale Centre-ville, Montr´eal, Qc H3C 3J7 Canada (Dated: July 17, 2018)
Abstract
The interplay between the various measures of quantum correlations are well known in stable optical and electronic systems.Here, for the first time, we study such foundational issues in unstable quantum systems. Specifically we study meson-antimesonsystems, which are produced copiously in meson factories. We use the semigroup formalism to compute the time evolutionof several measures of quantum correlations for three meson systems ( K ¯ K , B d ¯ B d and B s ¯ B s ), circumventing difficulties whicharise using other methods due to the instability of these particles. We then compare these measures to one another and findthat the relations between them can be nontrivially different from those of their stable counterparts such as neutrinos. PACS numbers: 03.65.Ta, 3.65.Yz, 14.40.Nd, 14.40.Df
I. INTRODUCTION
Quantum correlations in measurements performed onmultipartite systems provide a fertile testing ground forfoundational aspects of quantum physics. They are alsoof central importance to potential applications such asquantum communication, computation and cryptogra-phy. They have been studied and applied in a varietyof physical contexts such as quantum optics and con-densed matter systems (superconductors, spin systems, etc. ). More recently, attention has also been directed to-wards subatomic physics [1–14], inspired by the technicaladvances in high energy physics experiments, in particu-lar the meson factories, reactor and accelerator neutrinoexperiments.The foundations of quantum mechanics are usuallystudied in optical or electronic systems, where the inter-play between the various measures of quantum correla-tions is well know[15–20]. In these contexts, the detectionefficiency is much lower than that of the correspondingdetectors in high energy physics experiments, such as themeson factories. It is therefore interesting and, as we willsee, fruitful to test the foundations of quantum mechan-ics in unstable massive systems, such as the correlated B ¯ B and K ¯ K systems, for which the interplay betweenthe various measures of quantum correlations have notyet been studied.Of course, testing quantum correlations requires a bi-partite (or multipartite) system. B factories, electron-positron colliders tailor-made to study the productionand decay of B mesons, and φ factories, which performthe same function for K mesons, provide an ideal testingground. In the case of B factories, the collider energy istuned to the Υ resonance, so the first stage of the processis e + e − → Υ. The Υ then decays into b ¯ b ; these form a ∗ [email protected] † [email protected] ‡ [email protected] B q ¯ B q ( q = s, d ) pair through hadronization. All this hap-pens essentially instantaneously. The B mesons then flyapart and decay on a much longer time scale. An impor-tant feature of these systems for the study of correlationsis the oscillations of the bottom and strangeness flavors b ↔ s , giving rise to B ¯ B oscillations, which have beencrucial to the study of CP violation in these systems.Here we study a number of well-established measures ofquantum correlations in the B and K meson systems.Such a study is complicated by the inherently unstablenature of these particles, as a result of which the standardWeisskopf-Wigner approach to time evolution [21, 22] re-sults in ambiguities due to probability loss caused by thedecrease of the trace of the density matrix.The treatment of unstable systems has a long and dis-tinguished history [21–32], and continues to be of inter-est today [33]. The requirement of causality prohibits aphysical state with complex energy [23], while the studyof decay of unstable states is facilitated by the use ofthe density matrix formalism. This view is supported bywork on decaying systems using dynamical semigroups[34], where deep theorems from analysis, such as the Sz.-Nagy dilation theorem [35], were used to show that a uni-tary evolution of an unstable system along with its decayproducts is not feasible. Thus a decaying system is intrin-sically an open system, even without explicitly invokingan external environment, and as a result it can have sur-prises not seen in its stable counterpart. This work [34]also motivates the development of a feasible theory ofsuch systems using a semigroup approach, which in turnis related to a superselection rule with respect to timereversal symmetry. The Langer-Sz.-Nagy-Foias theorem[35] theorem is used to provide the splitting of a semi-group generator into unitary and non-unitary parts.In this work we make use of the probability-preservingformalism of decaying systems [34, 36] to study variousmeasures of quantum correlations in B and K mesons.We find that the quantum correlations for these unsta-ble systems can be nontrivially different from their sta-ble counterparts. Also, it is not obvious how to perform1 a r X i v : . [ h e p - ph ] F e b xperiments testing nonlocality, as quantified by Bell’sinequality, for B mesons due to the absence of active measurements [7, 8]. Despite this hurdle, the frame-work of open quantum systems [16, 18, 37], adaptedto the probability-preserving formalism of decaying sys-tems, enables us to make quantitative statements aboutBell inequality violations for correlated neutral K and B mesons. The theory of open quantum systems assertsthat any real system interacts with its environment (here,fluctuations of the quantum mechanical vacuum), result-ing in loss of quantum coherence and the transformationfrom pure to mixed states [38]. Thus this work elucidatesfundamental aspects of quantum mechanics of correlatedneutral meson systems, and more generally of unstablequantum systems.The notion of quantum entanglement was implicit inthe seminal 1935 paper of Einstein, Podolsky and Rosen[39]. It was Schr¨odinger who later that year first coinedthe term “entanglement” in a series of papers wherein healso introduced his eponymous cat thought experiment[40]. The subject was further developed with the con-ception of experimental tests of quantum mechanics vs. hidden-variable theories, in particular Bell’s inequality[41] and refinements resulting in the Bell-CHSH (Clauser-Horn-Shimony-Holt) inequalities [42]. Until recently, en-tanglement was considered synonymous with quantumcorrelations in that it was thought that one necessarilyimplied the other. With the advent of quantum discord[43–46], which quantifies the difference between the quan-tum generalizations of two classically equivalent formula-tions of mutual information, it became clear that quan-tum correlations are broader than entanglement. In gen-eral, it is very difficult to obtain an analytical formula forquantum discord because it involves an optimization overlocal measurements, requiring numerical methods [47].To overcome this difficulty, another measure of quantumcorrelation called geometric discord was introduced in[48]. This quantifies the amount of non-classical correla-tion of an arbitrary quantum composite system in termsof its minimal distance from the set of classical states.Along with these measures, another important facet ofquantum correlations is its operational aspect. Telepor-tation fidelity occupies an important place here and wasdeveloped to provide an operational meaning to entan-glement [19, 49].The plan of this work is as follows. In the next sectionwe set up the two-meson system, treating each mesonas a state in a Hilbert space consisting of one-particleand zero-particle sectors. The relevant tool for comput-ing quantum correlations is the density matrix projecteddown to the two-particle sector. In the following sec-tion various correlation measures are computed. We thensummarize our results and highlight the main surprisingresult, which is that the relations between the correla-tion measures we consider can be different from the cor-responding relations for stable systems. II. M ¯ M AS AN OPEN QUANTUM SYSTEM
For the B system, imagine the decay Υ → b ¯ b followedby hadronization into a B ¯ B pair. In the Υ rest frame,the mesons fly off in opposite directions (left and right,say); since the Υ is a spin-1 particle, they are in an anti-symmetric spatial state. The same considerations applyto the K system, with the Υ replaced by a φ meson.The flavor-space wave function of the correlated M ¯ M meson systems ( M = K, B d , B s ) at the initial time t = 0is | ψ (0) (cid:105) = 1 √ (cid:2)(cid:12)(cid:12) M ¯ M (cid:11) − (cid:12)(cid:12) ¯ M M (cid:11)(cid:3) , (1)where the first (second) particle in each ket is the oneflying off in the left (right) direction and | M (cid:105) and (cid:12)(cid:12) ¯ M (cid:11) are flavor eigenstates. As can be seen from (1), the initialstate of the neutral meson system is a singlet (maximallyentangled) state. The usual analysis of such systems isdone using a trace-decreasing density matrix descriptionof the state (see for example [31]). However, such an ap-proach may not be very useful for calculating quantumcorrelations, the subject of interest in this work. Thisis because the usual methods for computing quantumcorrelations require a trace-preserving, completely posi-tive description of the system. The semigroup formalismenables the calculation of a trace-preserving density ma-trix. We also incorporate the effects of decoherence in ourcalculations, providing a uniform formalism for studyingcorrelations in neutral meson systems.The Hilbert space of a system of two correlated neutralmesons, as in (1), is H = ( H L ⊕ H ) ⊗ ( H R ⊕ H ) , (2)where H L,R are the Hilbert spaces of the left-moving andright-moving decay products, each of which can be eithera meson or an anti-meson, and H is that of the zero-particle (vacuum) state. Thus the total Hilbert spacecan be seen to be the tensor sum of a two-particle space,two one-particle spaces, and one zero-particle state. Theinitial density matrix of the full system is ρ H (0) = | ψ (0) (cid:105) (cid:104) ψ (0) | . (3)The system, initially in the two-particle subspace,evolves in time into the full Hilbert space, eventually (af-ter the decay of both particles) finding itself in the vac-uum state. As can be appreciated from basic notions ofquantum correlations such as entanglement, one needs tohave two parties to correlate. For this we need to projectfrom the full Hilbert space (2) down to the two-particlesector H L ⊗H R , resulting in the following density matrixfor the correlated neutral meson system: ρ ( t ) = 14 a − − a − a + − a + − a + a + − a − a − , (4)2here we have used the basis {| M M (cid:105) , (cid:12)(cid:12) M ¯ M (cid:11) , (cid:12)(cid:12) ¯ M M (cid:11) , (cid:12)(cid:12) ¯ M ¯ M (cid:11) } and a ± = 1 ± e − λt .This expression is trace-preserving and is obtained bywriting ρ H ( t ) in the operator-sum representation andthen performing the partial trace. We have neglectedthe effects of CP violation; however, its inclusion wouldnot affect our results significantly.The approach used here can also be effectively appliedto study observables of central importance in particlephysics [50]. For example, several important observablesthat are used to characterize meson decay processes canbe developed using the above density matrix. Any physi-cal observable of the neutral B -meson system is describedby a suitable hermitian operator O . Its evolution in timecan be obtained by taking its trace with the density ma-trix ρ ( t ), Tr [ O f ρ ( t )], and from this standard results per-tinent to the meson systems can be derived. Hence, bothquantum correlations as well as standard studies in par-ticle physics can be carried out in a unified manner withthe formalism used in this work. III. INTERPLAY OF QUANTUM CORRELA-TIONS IN NEUTRAL MESON SYSTEMS
Using the density matrix (4), we study the interplayof quantum correlations in our systems. We begin withBell’s inequality which was one of the first tools used toanalyze and detect entanglement. Its physical content isthat a system that can be described by a local realistictheory will satisfy this inequality. However, quantummechanics seems to take delight in violating it [51]! Itis worth testing for such a violation in EPR-correlated B and K meson systems. Given a pair of qubits in thestate ρ , we define the correlation matrix T by T mn = T r [ ρ ( σ m ⊗ σ n )]; let u i ( i = 1 , ,
3) be the eigenvalues ofthe matrix T † T . Then the Bell-CHSH inequality can bewritten M ( ρ ) <
1, where M ( ρ ) = max( u i + u j ) ( i (cid:54) = j )[19]. For the state (4), M ( ρ ) is given by M ( ρ ) = (1 + e − λt ) . (5)Since Bell’s inequality is not able to detect all possibleentangled states, there is a need for some kind of measurewhich will quantify the amount of entanglement presentin the system. A well-known measure of entanglement isconcurrence, which for a two-qubit system is equivalentto the entanglement of formation. For a mixed state ρ oftwo qubits, the concurrence is [52] C = max( λ − λ − λ − λ , , (6)where λ i are the square root of the eigenvalues, indecreasing order, of the matrix ρ ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) ρ where ρ is computed in the computational basis {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} . For the state (4), concurrence hasthe simple analytical form C = e − λt . (7) A similar result was also obtained in [4]. The entangle-ment of formation can then be expressed as a monotonicfunction of concurrence C as E F = − √ − C ( 1 + √ − C − − √ − C ( 1 − √ − C . (8)As noted above, entanglement and quantum correla-tions need not be identical. Quantum discord attempts toreveal the quantum advantage over classical correlations.For the case of two qubits, geometric discord was shown[48] to be D G ( ρ ) = [ (cid:107) (cid:126)x (cid:107) + (cid:107) T (cid:107) − λ max ( (cid:126)x(cid:126)x † + T T † )]where T is the correlation matrix defined above, (cid:126)x is thevector whose components are x m = Tr( ρ ( σ m ⊗ I )), and λ max ( M ) is the maximum eigenvalue of the matrix M .Here it can be seen that D G ( ρ ) = M ( ρ ) /
3, in consistencewith the findings in [53].Teleportation provides an operational meaning to en-tanglement. The classical fidelity of teleportation in theabsence of entanglement is 2 /
3. Thus, whenever F max > /
3, teleportation is possible. Interestingly, this does notrule out the possibility of entangled states that, whilethey do not violate Bell’s inequality, can nonetheless beuseful for teleportation. F max is easily computed in termsof the eigenvalues { u i } of T † T mentioned earlier: it is F max = (cid:0) N ( ρ ) (cid:1) where N ( ρ ) = √ u + √ u + √ u .The calculation of F max for the state (4) gives F max = 112 (cid:104) e − λt + √ (cid:113) α − (cid:112) β + √ (cid:113) α + (cid:112) β (cid:105) , (9)where α = 1 + cosh(4 λt ) − sinh(4 λt ) , (10) β = 3 − α + cosh(8 λt ) − sinh(8 λt ) . (11)Thus we see that for λ = 0, E F = F max . A useful in-equality involving M ( ρ ) and F max is [19]: F max ≥ (cid:18) M ( ρ ) (cid:19) ≥
23 if M ( ρ ) > . (12)We will see that this can be violated for unstable systems.To take into account the effect of decay in the systemsunder study, the various correlations are modified by theprobability of survival of the pair of particles up to thattime, which can be shown to be e − t , where Γ is themeson decay width. For the K meson, Γ = (Γ S + Γ L )(where Γ S and Γ L are the decay widths of short andlong neutral kaon states, respectively); its value is 5 . × s − [54]. The decay widths for B d and B s mesons are6 . × s − and 6 . × s − , respectively [55].The parameter λ models the effect of decoherence. Inthe case of the K meson system, its value has been ob-tained by the KLOE collaboration by studying the in-terference between the initially entangled kaons and thedecay product in the channel φ → K S K L → π + π − π + π − [56]. The value of λ is restricted to 1 . × s − at3 σ . In the case of B meson systems, the decoherenceparameter is determined by using time-integrated dilep-ton events. The value of λ , for B d mesons, is determinedby the measurement of ratio of the total same-sign toopposites-sign semileptonic rates, R d , and is restrictedto 2 . × s − at 3 σ [57]. However there has beenno experimental update for R d since [58] two decades.For B s mesons, to the best of our knowledge, there is noexperimental information about λ so we will take it tobe zero in what follows. The results for K and B mesonsystems are shown in Fig. 1.The nonclassicality of quantum correlations, in theneutral mesons, can be characterized in terms of nonlo-cality (which is the strongest condition), entanglement,teleportation fidelity or weaker nonclassicality measureslike quantum discord. The fall in the pattern of the av-erage value of these correlations, as shown in Fig. 1, arein accord with the fact that here we are dealing with un-stable particles, which decay with time. From the leftpanel of Fig. 1, one can see that until about 50% of theaverage life time of K S meson in the presence of deco-herence and about 60% in its absence, Bell’s inequalityis violated. This means that, in the conventional sense,until this time, the time evolution cannot be simulatedby any local realistic theory. However, we find that evenif Bell’s inequality is violated ( M ( ρ ) > F max could be below the classical value of2/3. For example, from the left panel of Fig. 1, it is seenthat, in the absence of decoherence, F max drops below2/3 as M ( ρ ) drops below 1.3, in violation of the inequal-ity (12) [19] according to which the cutoff is M ( ρ ) = 1.This violation is slightly reduced, but nonetheless stilloccurs, even in the presence of decoherence, starting at M ( ρ ) (cid:39) .
2. This is consistent with the degradation ofcorrelations with decoherence.The violation can be understood mathematically sincethe individual measures of correlation are modulated bya factor e − t . Since (12) contains constant terms whichare not modulated, it is affected by these modulations.Thus we see that the study of quantum correlationsin unstable systems is nontrivially different from theirstable counterparts. However, there are some similaritiesas well. In particular, the average geometric discord isalways bounded from above by the average entanglementof formation. This is consistent with the fact that discordis a weaker measure of quantum correlations comparedto entanglement [53]. From the middle and right panelof Fig. 1, we see that the above conclusions hold also for B d and B s meson systems, respectively. Here we wouldlike to point out that the nontrivial differences betweenthe meson systems and their stable counterparts is onlydue to the decaying nature of the system and not due tooscillations. This is borne out by the fact that a study ofquantum correlations in a stable, oscillating subatomicsystem such as a neutrino, shows no such deviation [14].From Fig. 1, we see that the spread in the variouscorrelations, corresponding to 3 σ upper bound on thedecoherence parameter λ , is more prominent for the B d , compared to the K meson system. This is because ofthe choice of time scales used in the plots. In the caseof K mesons, the time scale is 10 − s, which is roughlythe average lifetime of the K S mesons; whereas in thecase of B d and B s mesons, the time scale is picosecond,which is roughly the average life time of these mesons.As can be seen from (4), the coherences in the system,of which the correlations would be a function, dependupon the evolution time t and the decoherence parameter λ , as a function of λt . Thus from the values of λ ’s forthese systems (see below (4)), it can be easily inferredthat the effect of decoherence is more prominent, in thechosen time scales, for the B d meson as compared to the K meson system. This is consistent with Fig. 1. IV. CONCLUSIONS
To conclude, in this work we have studied a number ofaspects of quantum correlations in correlated neutral me-son systems, viz. neutral K , B d and B s mesons. This isnontrivial given the fact that the mesons decay with time.This was accomplished by using the semigroup formal-ism, well known in the study of open quantum systems.We have also studied the impact of decoherence on vari-ous correlation measures in these systems. We found thatthe quantum correlations here can be nontrivially differ-ent from their stable counterparts. This was made ex-plicit by the interplay between Bell’s inequality and tele-portation fidelity. On average, Bell’s inequality in thesecorrelated-meson systems is violated for about half of themeson lifetime. One particularly surprising result is thatteleportation fidelity does not exceed the classical thresh-old of 2/3 for all Bell’s inequality violations. This behav-ior, not seen in stable systems, is interesting since one ofthe cornerstones in the field of quantum information isthe interplay between Bell’s inequality and teleportationfidelity. This surprising behavior is due to the decayingnature of the parent system and not due to flavor os-cillations. There are some similarities as well. This isparticularly seen by the fact that entanglement providesan upper bound to discord at all times. This is the cruxthat provides insight into the differences/similarities inquantum correlations in these meson systems in compar-ison to their stable counterparts. Thus this work providesimportant insight into foundational issues in the contextof quantum mechanics of unstable subatomic particles,and presumably other unstable systems as well.On the experimental front, our results can impactprobing the nonlocality of B mesons, in particular theBell’s inequality, which, at present, leads to difficultiesdue to the lack of active measurements. Also, the mea-surement of decoherence parameter in the B s mesons canprovide insight into the fundamental nature of these sys-tems. This work could hopefully motivate future experi-ments (or reanalysis of past experimental results such asthe wealth of results from decommissioned B factories)to probe these issues.4 .5 1.0 1.5 2.0 2.5 3.0 t C (cid:143)(cid:143)(cid:143) t C (cid:143)(cid:143)(cid:143) t C (cid:143)(cid:143)(cid:143) FIG. 1. (color online) Average correlation measures as a function of time t . The left, middle and right panels correspond to thecorrelations of a K ¯ K , B d ¯ B d and B s ¯ B s pair created at t = 0, respectively. The four correlation measures are (top to bottom): M ( ρ ) (Bell’s inequality; blue band), F max (teleportation fidelity; red band), E F (entanglement of formation; grey band) and D G (geometric discord; green band). For K ¯ K pairs, left panel, time is in units of 10 − seconds whereas for the B d ¯ B d and B s ¯ B s pairs, time is in units of 10 − seconds (in all cases, the approximate lifetime of the particles). In the left and middlepanels, the bands represent the effect of decoherence corresponding to a 3 σ upper bound on the decoherence parameter λ . Theright panel has no such bands because there is currently no experimental evidence for decoherence in the case of B s mesons;for this case, F max = E F . ACKNOWLEDGMENTS
SB and AKA thank R. Srikanth, A. Pathak and S.Uma Sankar for useful comments on the manuscript. Thework of RM is supported by the Natural Science and En-gineering Research Council of Canada and by the Fondsde recherche du Qu´ebec – Nature et technologies. [1] A. Bramon and M. Nowakowski, Phys. Rev. Lett. , 1(1999) [hep-ph/9811406].[2] A. Bramon and G. Garbarino, Phys. Rev. Lett. ,040403 (2002) [quant-ph/0108047].[3] A. Bramon and G. Garbarino, Phys. Rev. Lett. ,160401 (2002) [quant-ph/0205112].[4] R. A. Bertlmann, K. Durstberger and B. C. Hiesmayr,Phys. Rev. A , 012111 (2003) [quant-ph/0209017].[5] A. Bramon, G. Garbarino and B. C. Hiesmayr, Phys.Rev. A , 062111 (2004) [quant-ph/0402212].[6] B. C. Hiesmayr, Eur. Phys. J. C , 73 (2007).[7] R. A. Bertlmann, A. Bramon, G. Garbarino andB. C. Hiesmayr, Phys. Lett. A , 355 (2004) [quant-ph/0409051].[8] A. Go et al. [Belle Collaboration], Phys. Rev. Lett. ,131802 (2007) [quant-ph/0702267 [QUANT-PH]].[9] M. Blasone, F. Dell’Anno, S. De Siena and F. Illuminati,Europhys. Lett. , 0002 (2009) [arXiv:0707.4476 [hep-ph]].[10] M. Blasone, F. Dell’Anno, S. De Siena, M. Di Mauroand F. Illuminati, Phys. Rev. D , 096002 (2008)[arXiv:0711.2268 [quant-ph]].[11] G. Amelino-Camelia, F. Archilli, D. Babusci, D. Badoni,G. Bencivenni, J. Bernabeu, R. A. Bertlmann andD. R. Boito et al. , Eur. Phys. J. C , 619 (2010)[arXiv:1003.3868 [hep-ex]].[12] B. C. Hiesmayr, A. Di Domenico, C. Curceanu,A. Gabriel, M. Huber, J. A. Larsson and P. Moskal, Eur.Phys. J. C , 1856 (2012) [arXiv:1111.4797 [quant-ph]]. [13] M. Blasone, F. Dell’Anno, S. De Siena and F. Illumi-nati, Europhys. Lett. , 30002 (2014) [arXiv:1401.7793[quant-ph]].[14] A. K. Alok, S. Banerjee and S. U. Sankar,arXiv:1411.5536 [quant-ph].[15] W. H. Louisell, Quantum Statistical Properties of Radia-tion (Wiley, New York, 1973).[16] H-P. Breuer and F. Petruccione,
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