Signatures of bifurcation on quantum correlations: Case of the quantum kicked top
aa r X i v : . [ qu a n t - ph ] J a n Signatures of bifurcation on quantum correlations: Case of quantum kicked top
Udaysinh T. Bhosale ∗ and M. S. Santhanam † Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411 008, India. (Dated: January 31, 2017)Quantum correlations reflect the quantumness of a system and are useful resources for quantuminformation and computational processes. The measures of quantum correlations do not have aclassical analog and yet are influenced by the classical dynamics. In this work, by modelling thequantum kicked top as a multi-qubit system, the effect of classical bifurcations on the measures ofquantum correlations such as quantum discord, geometric discord, Meyer and Wallach Q measure isstudied. The quantum correlation measures change rapidly in the vicinity of a classical bifurcationpoint. If the classical system is largely chaotic, time averages of the correlation measures are in goodagreement with the values obtained by considering the appropriate random matrix ensembles. Thequantum correlations scale with the total spin of the system, representing its semiclassical limit. Inthe vicinity of the trivial fixed points of the kicked top, scaling function decays as a power-law. Inthe chaotic limit, for large total spin, quantum correlations saturate to a constant, which we obtainanalytically, based on random matrix theory, for the Q measure. We also suggest that it can haveexperimental consequences. PACS numbers: 05.45.Mt, 03.65.Ud, 03.67.-a
I. INTRODUCTION
It is well established by more than half a century ofquantum chaos research that many of the properties ofquantum systems can be understood in terms of clas-sical objects such as periodic orbits and their stability[1]. For classically integrable systems, Einstein-Brillouin-Keller quantization method relates the quantum spec-tra and the classical action [2] while for chaotic systemsGutzwiller’s trace formula represents such an approachconnecting the quantum spectra and the classical pe-riodic orbits [3]. The advent of quantum informationand computation has opened up newer scenarios in whichnovel quantum correlations did not have correspondingclassical analogues. Quantum entanglement is one suchphenomena without a classical analogue. The von Neu-mann entropy, a measure of quantum entanglement fora bipartite pure state, captures correlations with purelyquantum origins that are stronger than classical corre-lations. A host of such measures are now widely usedin the quantum information theory to quantify strongerthan classical correlations.Quantum correlations do not have exact classical ana-logues, yet they are surprisingly affected by the classicaldynamics. For instance, in the context of chaotic sys-tems, it is known that upon variation of a parameter, aschaos increases in the system the entanglement also in-creases and saturates to a value predicted based on ran-dom matrix theory [4]. Recently, this was experimentallydemonstrated for an isolated quantum system consistingof three superconducting qubits as a realisation of quan-tum kicked top [5]. It was shown that larger values ofentanglement corresponds to regimes of chaotic dynam- ∗ [email protected] † [email protected] ics [6]. Theoretically, not just the chaotic dynamics butindeed the structure and details of classical phase space,such as the presence of elliptic islands in a sea of chaos,is known to affect the entanglement [7].Quantum entanglement is an important resourcefor quantum information processing and computationaltasks. However, it does not capture all the correlations ina quantum system. It is possible for unentangled states todisplay non-classical behaviour implying that there mightbe residual quantum correlations beyond what is mea-sured by entanglement. In addition, it is now known thatentanglement is not the only ingredient responsible forspeed-up in quantum computing [8–10]. For mixed statequantum computing model, discrete quantum computa-tion with one qubit (DQC1), experiments have shownthat some tasks can be speeded up over their classi-cal counterparts even using non-entangled, i.e., separablestates but having non-zero quantum correlations [11–13].Hence, quantification of all possible quantum correlationsis important. For this purpose, measures like quantumdiscord [14, 15] and geometric discord [16, 17], Leggett-Garg inequality [18] and a host of others are widely used.Quantum discord is independent of entanglement andno simple ordering relations between them is known[19, 20]. Entanglement may be larger than quantum dis-cord even though for separable states entanglement al-ways vanishes but quantum discord may be nonzero, andthus is less than quantum discord [20–22]. This showsthat discord and in general all quantum correlation mea-sures are more fundamental than entanglement [23]. Itis shown that two-qubit quantum discord in a dissipa-tive dynamics under Markovian environments vanishesonly in the asymptotic limit where entanglement sud-denly disappears [24]. Thus, the quantum algorithmsthat make use of quantum correlations, represented indiscord, might be more robust than those based on entan-glement [24]. This shows that studying quantum correla-tion, in general, in a given system is important from thepoint of view of decoherence which is inevitably presentin almost all experimental setups.In the last decade, many experimental and theoret-ical studies of discord were performed [25]. A recentexperiment realizes quantum advantage with zero en-tanglement but with non-zero quantum discord using asingle photon’s polarization and its path as two qubits[26]. Other experiments have estimated the discord inan anti-ferromagnetic Heisenberg compound [27] and inBell-diagonal states [28]. In the context of chaotic sys-tems, e.g., the quantum kicked top, the dynamics of dis-cord reveals the classical phase space structure [29]. Inthis paper, we show that period doubling bifurcation [30]in the kicked top leaves its signature in the dynamics ofquantum correlation measures such as discord and geo-metric discord, including the multipartite entanglementmeasure Meyer and Wallach Q measure [31].The structure of the paper is as follows: In Sec. II themeasures of quantum correlations used are introduced.In Sec. III the kicked top model is introduced. In Sec. IVresults on the effects of the bifurcation on the time aver-ages of these measures of quantum correlations are given.In Sec. V these results are compared with a suitable ran-dom matrix model. In Sec. VI scaling of these time av-eraged measures is studied as a function of total spin. II. MEASURE OF QUANTUM CORRELATIONSA. Quantum Discord
Quantum discord is a measure of all possible quan-tum correlations including and beyond entanglement ina quantum state. In this approach one removes the classi-cal correlations from the total correlations of the system.For a bipartite quantum system, its two parts labelled A and B , and represented by its density matrix ρ AB , if thevon Neumann entropy is H ( ρ AB ) = − Tr ( ρ AB log ρ AB ),then the total correlations is quantified by the quantummutual information as, I ( B : A ) = H ( B ) + H ( A ) − H ( B, A ) . (1)In classical information theory, the mutual informationbased on Baye’s rule is given by I ( B : A ) = H ( B ) − H ( B | A ) (2)where H ( B ) is the Shannon entropy of B . The condi-tional entropy H ( B | A ) is the average of the Shannonentropies of system B conditioned on the values of A . Itcan be interpreted as the ignorance of B given the infor-mation about A .Quantum measurements on subsystem A are repre-sented by a positive-operator valued measure (POVM)set { Π i } , such that the conditioned state of B given out-come i is ρ B | i = Tr A (Π i ρ AB ) /p i and p i = Tr A,B (Π i ρ AB ) (3) and its entropy is ˜ H { Π i } ( B | A ) = P i p i H ( ρ B | i ). In thiscase, the quantum mutual information is J { Π i } ( B : A ) = H ( B ) − ˜ H { Π i } ( B | A ). Maximizing this over the measure-ment sets { Π i } we get J ( B : A ) = max { Π i } (cid:16) H ( B ) − ˜ H { Π i } ( B | A ) (cid:17) = H ( B ) − ˜ H ( B | A ) (4)where ˜ H ( B | A ) = min { Π i } ˜ H { Π i } ( B | A ). The minimumvalue is achieved using rank 1 POVMs since the condi-tional entropy is concave over the set of convex POVMs[32]. By taking { Π i } as rank-1 POVMs, quantum discordis defined as D ( B : A ) = I ( B : A ) − J ( B : A ), such that D ( B : A ) = H ( A ) − H ( B, A ) + min { Π i } ˜ H { Π i } ( B | A ) . (5)Quantum discord is non-negative for all quantum states[14, 32, 33], and is subadditive [34]. B. Geometric Discord
The calculation of discord involves the maximizationof J ( A : B ) by doing measurements on the subsystem B ,which is a hard problem. A more easily computable formis geometric discord based on a geometric way [16, 17].There are no measurements involved in calculating thismeasure. For the special case of two-qubits a closed formexpression is given [16]. Dynamics of geometric discordis studied under a common dissipating environment [35].For every quantum state there is a set of postmeasure-ment classical states, and the geometric discord is definedas the distance between the quantum state and the near-est classical state, D G ( B | A ) = min χ ∈ Ω k ρ − χ k , (6)where Ω represents the set of classical states, and k X − Y k = Tr[( X − Y ) ] is the Hilbert-Schmidt quadraticnorm. Obviously, D G ( B | A ) is invariant under local uni-tary transformations. Explicit and tight lower bound onthe geometric discord for an arbitrary state of a bipartitequantum system A m × m ⊗ B n × n is available [17, 36, 37].Recently discovered ways to calculate lower bounds ondiscord for such general states do not require tomogra-phy and, hence, are experimentally realisable [36, 37].Following the formalism of Dakic et al. [16] analyticalexpression for the geometric discord for two-qubit statesis obtained. The two-qubit density matrix in the Blochrepresentation is ρ = 14 (cid:16) ⊗ + X i =1 x i σ i ⊗ + X i =1 y i ⊗ σ i + X i,j =1 T ij σ i ⊗ σ j (cid:17) (7)where x i and y i represent the Bloch vectors for the twoqubits, and T ij = Tr[ ρ ( σ i ⊗ σ j )] are the components ofthe correlation matrix. The geometric discord for such astate is D G ( B | A ) = 14 (cid:0) k x k + k T k − η max (cid:1) , (8)where k T k = Tr[ T T T ], and η max is the largest eigenvalueof ~x~x T + T T T , whose explicit form is in [38]. C. Meyer and Wallach Q measure In this work, the effects of bifurcation on multipartiteentanglement is also studied using the Meyer and Wal-lach Q measure [31]. This was used to study the mul-tipartite entanglement in spin Hamiltonians [39–41] andsystem of spin-boson [42]. The geometric multipartite en-tanglement measure Q is shown to be simply related toone-qubit purities [43]. Making its calculation and inter-pretation is straightforward. If ρ i is the reduced densitymatrix of the i th spin obtained by tracing out the rest ofthe spins in a N qubit pure state then Q ( ψ ) = 2 − N N X i =1 Tr( ρ i ) ! . (9)This relation between Q and the single spin reduced den-sity matrix purities has led to a generalization of thismeasure to multiqudit states and for various other bipar-tite splits of the chain [44]. III. KICKED TOP
The quantum kicked top is characterized by an angularmomentum vector J = ( J x , J y , J z ), whose componentsobey the standard angular momentum algebra. Here,the Planck’s constant is set to unity. The dynamics ofthe top is governed by the Hamiltonian [45]: H ( t ) = pJ y + k j J z + ∞ X n = −∞ δ ( t − n ) . (10)The first term represents the free precession of the toparound y − axis with angular frequency p , and the sec-ond term is periodic δ -kicks applied to the top. Eachkick results in a torsion about the z − axis by an angle( k/ j ) J z . The classical limit of Eq. (10) is integrable for k = 0 and becomes increasingly chaotic for k >
0. Theperiod-1 Floquet operator corresponding to Hamiltonianin Eq. (10) is given by U = exp (cid:18) − i k j J z (cid:19) exp ( − ipJ y ) . (11)The dimension of the Hilbert space is 2 j + 1 so that dy-namics can be explored without truncating the Hilbertspace. Kicked top was realized in experiments [46] and FIG. 1. (Color online) Phase-space pictures of the classicalkicked top for p = π/ k = 1, (b) k = 2, (c) k = 3and (d) k = 6. Red solid circles indicates initial position ofthe spin coherent state. the range of parameters used in this work makes it ex-perimentally feasible.The quantum kicked top for given angular momen-tum j can be regarded as a quantum simulation of acollection of N = 2 j qubits (spin-half particles) whoseevolution is restricted to the symmetric subspace un-der the exchange of particles. The state vector is re-stricted to a symmetric subspace spanned by the basisstates {| j, m i ; ( m = − j, − j + 1 , ..., j ) } where j = N/
2. Itis thus a multiqubit system whose collective behavior isgoverned by the Hamiltonian in Eq. 10 and quantum cor-relations between any two qubits can be studied. Kickedtop has served as a useful model to study entanglement[6, 47–51] and its relation to classical dynamics [52].The classical phase space shown in Fig. 1 is a functionof coordinates θ and φ . In order to explore quantum dy-namics in kicked top, we construct spin-coherent states[53–56] pointing along the direction of θ and φ andevolve it under the action of Floquet operator. The quan-tum correlations reported in this paper represent timeaverages obtained from time evolved spin-coherent state.The classical map for the kicked top is [45, 53], X ′ = ( X cos p + z sin p ) cos ( k ( z cos p − X sin p )) − Y sin ( k ( z cos p − X sin p )) (12a) Y ′ = ( X cos p + Z sin p ) sin ( k ( Z cos p − X sin p ))+ Y cos ( k ( Z cos p − X sin p )) (12b) Z ′ = − X sin p + Z cos p. (12c)Since the dynamical variables ( X, Y, Z ) are restrictedto the unit sphere i.e. X + Y + Z = 1, they canbe parameterized in terms of the polar angle θ and theazimuthal angle φ as X = sin θ cos φ , Y = sin θ sin φ and θ φ θ -2 0 20123 -2 0 2 φ (b)(a)(c) (d) FIG. 2. (Color online) Phase-space of the classical kicked topfor p = 1 . k = 1, (b) k = 1 .
9, (c) k = 2 . k = 6. Red solid circles indicates initial position of the spincoherent state. Z = cos θ . We evolve the map in Eq. (12) and determinethe values of ( θ, φ ) using the inverse relations (not shownhere). For p = π/ p = π/ p = 1 . IV. EFFECT OF BIFURCATION
Firstly, we consider the case of p = π/
2. If kickstrength is k = 1, then the phase space is largely domi-nated by invariant tori as seen in Fig. 1(a). In particular,the trivial fixed points of the map at ( θ, φ ) = ( π/ , ± π/ k = 2. As k increases further, the new fixed points born at k = 2move away (see Fig. 1(c)). For k = 6, the phase space islargely chaotic with no islands visible in 1(d)). In kickedtop, the period doubling bifurcation is the route for reg-ular to chaotic transition.Second case that is studied here is p = 1 .
7. As seen inFig. 2(a-d), the phase space displays similar features asin the case of p = π/ θ, φ ) = ( π/ , − π/
2) now loses stability at numericallydetermined k = 1 .
76 while ( θ, φ ) = ( π/ , π/
2) loses at k = 2 .
2. The dark circle, marking the point ( θ , φ ) =( π/ , − π/
2) in Figs. 1 and 2, is the initial position of thespin-coherent state wavepacket.To study the effect of bifurcation on the quantum cor-relation and multipartite entanglement measures, multi-qubit representation of the system is used. For particularvalue of j the system can be decomposed into N = 2 j qubits. The reduced density matrix of two qubits is cal-culated by tracing out all other N − D D G k Q k FIG. 3. (Color online) Average discord, geometric discord and Q measure as a function of k for p = π/
2. Left (right) columnis for j = 50 ( j = 120). For comparison purposes, j = 10case is shown in every graph as square (green) symbols. Thevertical line marks the position of bifurcation at k = 2. reduced density matrix to compute the various measuresof correlation. As all the qubits are identical, the correla-tions measures do not depend on the actual choice of twoqubits. Similarly, while calculating Q measure one needsto compute reduced density matrix of only one qubit.The spin-coherent state at time t = 0 denoted as | ψ (0) i is placed at the fixed point ( θ, φ ) = ( π/ , − π/
2) (red solidcircle in Figs. 1 and 2) undergoing a period doubling bi-furcation. The state | ψ (0) i is evolved by the Floquetoperator ˆ U as | ψ ( n ) i = U n | ψ (0) i . We apply the nu-merical iteration scheme given in refs. [49, 59] for timeevolving the initial state. At every time step, discord D ,geometric discord D G and, Meyer and Wallach Q mea-sure is calculated for given value of k . The results shownin Figs. 3 and 4 represent time averaged values of D , D G and Q for every k .For both cases of p = π/ p = 1 . j , namely, j = 50 and j = 120. For comparison, the case of j = 10qubits is also shown in Fig. 3. Broadly, in all the cases,the quantum correlation measures D , D G and Q respondto the classical bifurcation in a similar manner; by dis-playing a jump in the mean value from about 0 to a non-zero value. This can be understood as follows. When theelliptic islands are large, as is the case when 0 < k < p = π/ < k < .
76 for p = 1 . θ, φ ) = ( π/ , − π/
2) is largely confined to the same el-liptic islands. As the bifurcation point is approached, thelocal instability in the vicinity of the fixed point evolvespart of the coherent state into the chaotic layers of phasespace. This leads to an increase in the values of corre-lation measures. Note that increasing chaos leads to anincrease in entanglement too. When j is increased, thewidth of coherent state σ ∝ / √ j becomes narrower and D D G k Q k FIG. 4. (Color online) Average discord, geometric discord and Q measure as a function of k for p = 1 .
7. Left (right) column isfor j = 50 ( j = 120). The solid horizontal line represents thelong time average of an initial state from the bifurcation pointevolved using the operator U CUE . The dashed line representthe standard deviation from the average value. Vertical linemarks the position of bifurcation approximately at k = 1 . closely mimics the classical evolution [53]. Thus, as j in-creases, we expect the quantum correlations to sharplyrespond to classical bifurcation at k = 2. Indeed, as seenin Fig. 3, the quantum correlations changes sharply at k = 2 for = 120 in comparison with the case of j = 10.To understand the details of Fig. 3 consider two valuesof j , e.g., j = j and j = j such that j > j . The slowdecay of σ as j → ∞ implies that the response of quan-tum correlations to classical bifurcation becomes percep-tible only when | j − j | >>
1. Thus, relative changesare easily seen when quantum correlations for j = 120is compared with j = 10 case rather than with that of j = 50. The approach to semiclassics, ~ → V. CORRELATION MEASURES ANDRANDOM MATRIX THEORY
Next, we show that the saturated values for D , D G and Q after bifurcation has taken place at k = k b , can be ob-tained from random matrix considerations. The kickedtop is time-reversal invariant and as a consequence itsFloquet operator in the globally chaotic case has the sta-tistical properties of a random matrix chosen from thecircular orthogonal ensemble (COE) [60]. For kicked top,the statistical properties of eigenvectors of its Floquetoperator are in good agreement with COE of randommatrix theory [60]. Apart from time-reversal symme-try, the kicked top additionally has the parity symmetry, b R y = exp( − iπj y ) that commutes with the Floquet oper-ator for all values of p . As b R y = I , the eigenvalues of b R y D D G t Q t FIG. 5. (Color online) Time variation of the correlation mea-sures using kicked-top Floquet operator for j = 50 (left) andfor j = 120 (right) for the globally chaotic case ( k = 10 and p = 1 . are +1 and −
1. Thus, in the basis of the parity operator,the Floquet operator has a block-diagonal structure con-sisting of two blocks associated with the positive-(+1) ornegative-parity ( −
1) eigenvalues. Thus, due to the paritysymmetry, the kicked top is statistically equivalent to ablock-diagonal random matrix (block diagonal in the ba-sis in which the parity operator is diagonal) whose blocks(corresponding to the eigenvalues ±
1) are sampled fromthe COE [4]. If p = π/ p = 1 . b R y , this matrix is thenwritten in the | j, m i basis. Finally, this matrix is usedto evolve the coherent state and compared with the evo-lution done using the Floquet operator in the globallychaotic case ( k = 10). The results are presented in Fig. 5and summarised in Table I.Fig. 5 shows the evolution of 2-qubit discord, geometricdiscord and Meyer-Wallach Q measure for j = 50 and j = 120 when acted by kicked-top Floquet operator with k = 10. At this kick strength the classical phase spaceof kicked top is largely chaotic with no visible regularregions. As Fig. 5 and Table I reveal, the dynamics ofvarious correlations measures under the action of COEmatrix is similar to that of kicked-top Floquet operator inits chaotic regime with k = 10. While this is not entirelyunexpected, the values of the three measures listed inTable I closely agree with those obtained after bifurcationtakes place at k = k b , but at values of kick strengthsmuch less than 10 considered in Fig. 5. Time averageslisted in Table I are plotted in Fig. 4 along with the j l n D j -6 l n D G j -3-2.5-2 l n Q FIG. 6. (Color online) The variation of time-averaged quantum correlations (circles) as a function of j . The lines are the powerlaw fits given in Eq. (13). standard deviation of the individual measures. It can beseen that the agreement between these values and that ofFloquet operator begins to emerge at around k = 4 whichis much less than k = 10. The position of the coherentstate in this case is ( θ, φ ) = ( π/ , − π/ j → ∞ .Hence, these deviations can be attributed to finite j ef-fect. Similar systematic deviations from RMT were ob-served in the study of the log-negativity in kicked rotorsystem [61]. In this case too, the deviations decreasedas the corresponding Hilbert space dimensions were in-creased. j = 50 j = 50 j = 120 j = 120Measure Floquet COE Floquet COEDiscord 0 .
205 0 .
209 0 .
217 0 . .
045 0 .
047 0 .
049 0 . Q measure 0 .
986 0 .
991 0 .
994 0 . k = 10) and the COE matrix. The COEvalues are represented in Fig. 4 as horizontal lines. VI. SCALING WITH PLANCK VOLUME
Kicked top is a finite dimensional quantum system andthe volume of its Planck cell is V = 4 π/ (2 j + 1). Forlarge j , V ∝ /j . It is natural to ask how the measuresof quantum correlation scale with this volume when kickstrength corresponds to k = k b where k b is a bifurca-tion point. In Fig. 6, we show the variation in the timeaverage of D , D G and Q as a function of j for k = k b .Here, k b = 2 and p = π/
2. The coherent state is placed atthe corresponding trivial fixed point ( θ, φ ) = ( π/ , − π/ j >>
1, the correlation measures scale with j approximately in apower-law of the form j − µ , µ is the scaling exponent. Thepower law fits through linear regression for the numeri-cally computed correlations measures shown in Fig. 6 areconsistent with D ∝ j − µ , D G ∝ j − µ and Q ∝ j − µ , (13)where µ = 0 . ± . µ = 0 .
944 and µ = 0 . µ and µ are of the order of 10 − and hence negligible. Iden-tical power-law scaling is obtained for the other trivialfixed point at ( θ, φ ) = ( π/ , π/
2) with exponents µ i ap-proximately same as given in Eq. (13). The quantumcorrelations tend to zero as V → j → ∞ ) indicatingthat for any finite j quantum correlations, however smallit might be, would continue to exist. As the wavepacketbecomes more ’classical’ and the underlying dynamics isregular, we expect the quantum correlations to decreasewith j . This is another indication that the regular regionsin the vicinity of the fixed point undergoing bifurcationsaffect the quantum correlations deep in the semiclassicalregime.The appearance of power-law scaling can be under-stood for the case k = 2 when the regular region is largeand the chaotic layer is a tiny fraction of the entire phasespace. The presence of chaotic layer has a strong influ-ence on quantum correlations. Note that for j >> σ = j − / becomessmall and its evolution is mostly confined to the large el-liptic islands in Fig. 1(a,b). As a result, it can be arguedthat the strength of the overlap of coherent state with thechaotic layer is indicative of quantum correlations. Since σ = j − / , for j >>
1, this overlap is small. The slowpower-law decay of σ might possibly be the reason forsimilar decay of quantum correlations as well, as shown inEq. (13). Since quantum correlations are affected by thelocal phase space features, a complete quantitative ex-planation for power-law scaling might require a detailedsemiclassical analysis.Next, we consider the case of a coherent state placedinitially at a bifurcation point leading to a period-2 cy-cle. The origin of this bifurcation point is as follows. The -2 0 2 φ θ FIG. 7. (Color online) Phase-space picture of the classicalkicked top for k = √ π . Red solid circles indicates initialposition of the spin coherent state. trivial fixed points at ( θ, φ ) = ( π/ , ± π/
2) are easily vis-ible in Figs. 1(a-b) and 2(a-b). If p = π/
2, these fixedpoints bifurcate at k = 2 through a period doubling bi-furcation and become unstable. In the process, the point( θ, φ ) = ( π/ , π/
2) gives rise to two new period-1 stablefixed points while the point ( θ, φ ) = ( π/ , − π/
2) givesrise to a period-2 cycle. For k > k and they are sta-ble for k ≤ √ π . For k > √ π , the two fixed pointsbifurcate into two new period-2 cycles while the period-2cycle gives rise to a new period-4 cycle. Their positionsfor k = √ π are shown in Fig. 7. Our interest lies in thefixed point located at ( θ, φ ) = ( π/ , j for the initialcoherent state placed at this fixed point. It can be seenthat after initial fluctuations the correlations start to de-crease for larger values of j . It should be noted that thearea of elliptic islands are continually shrinking as k → ∞ consistent with the predominance of chaotic regions inthe phase space. The width of the spin-coherent state | ψ (0) i is equal to 1 / √ j . For small values of j , the widthof | ψ (0) i is much larger than that of the regular ellipticisland as shown in Fig. 7. Hence, there is a pronouncedoverlap of the state | ψ (0) i with the chaotic sea. Hence weexpect that for small j the quantum correlations will bereasonably close to their random matrix averages. Thisis indeed seen in Figs. 8 for 1 ≤ j ≤
50 as the width of | ψ (0) i are at least twice the size of elliptic island. For j >>
1, the width of | ψ (0) i has become much smallerthan that of elliptic island. Thus, under these condi-tions we expect smooth decay with increasing j , similarto what is seen in Fig. 6(a-c). Fig. 8 do show smoothdecay for j & j and the area of theregular region surrounding the fixed point undergoing bi-furcation strongly affects the quantum correlations. Now, we consider kick strength k = 10 and place thespin-coherent state | ψ (0) i at an arbitrary position in thechaotic sea, namely, ( θ, φ ) = (1 . , . j . Based on the results from figs. 4 and 5 wecan expect that at every value of j the time averaged D , D G and Q agree with those found using apropriate COEensemble.For a coherent state the quantum correlation measuresare zero. However, after time evolution, the correlationvalues will depend on the corresponding measures for Flo-quet eigenstates. Thus, it is important to study the typ-ical values of these measures for these eigenstates. Thiscan be analytically obtained for the average Q measure.An exact analytical formula for the average Q measure isderived (see Appendix A for the detailed derivation) fora typical COE ensemble modelling the Floquet operatorin the globally chaotic case. It is given by h Q i E = 1 − j ( j + 1)3(2 j + 3)(2 j + 1) . (14)For large j , h Q i E ≈ − / (3 j ) implying that the mea-sure tends to one for large j . The numerically computedcorrelations for the eigenvectors of COE ensemble andfor the eigenvectors of the Floquet operator under con-ditions of globally classical chaos are compared with theanalytical result in Eq. (14) in Fig. 10.For generating sufficient statistics for the eigenvectorsof Floquet operator, we use a range of k values such thatthe corresponding classical section does not have any sig-nificant regular islands and is highly chaotic. The ana-lytical result in Eq. (14) agrees with that for the eigen-vectors of the COE ensemble. In order to derive similarexpressions for the average discord and geometric discordfor the eigenvectors of COE, analytical expression for thedistribution of the matrix elements of the two-qubit den-sity matrix for these class of states is required. Such anexpression is not known yet, to the best of our knowledge.Thus, the derivation of the average discord and geometricdiscord as a function of j remains an open question.It is instructive to compare these results with otherwell-studied ensembles such as the Gaussian ensembles.In this case, the states are distributed uniformly, alsoknown as Haar measure, on the unit sphere. Consider atripartite random pure state. The entanglement betweenany of its two subsystems shows a transition from beingentangled to separable state as the size of the third sub-system is increased [61, 62]. Another example is that ofdefinite particle states. This shows algebraic to exponen-tial decay of entanglement when the number of particlesexceed the size of two subsystems [63]. For both thesecases, discord and geometric discord between two qubitsin a tripartite system goes to zero as the size of the thirdsubsystem is increased. It is known that average Q mea-sure for Haar distributed states of N qubits, for large N ,goes as 1 − / N [62]. In terms of j (= N/
2) it equals j D j D G j Q FIG. 8. (Color online) The variation of time-averaged quantum correlations (circles connected with lines) as a function of j for k = √ π and initial position of the spin coherent state as shown in Fig. 7. j D j D G j Q FIG. 9. (Color online)The variation of time-averaged quantum correlations (circles) as a function of j for the globally chaoticcase ( k = 10).. j 〈 Q 〉 E Analytical resultEigenvectors of COE ensembleEigenvectors of Floquet operator
FIG. 10. (Color online) Average Q measure for the eigenvec-tors of COE ensemble and that of Floquet operator in theglobally chaotic case for the parameter range 10 ≤ k ≤ p = 1 . − / j implying that the measure tend to 1 for large j .But, the rate at which it approaches 1 is much faster thanthat for eigenvectors of COE ensemble corresponding tothe kicked top in globally chaotic case. In contrast tothe standard Gaussian or circular ensemble, the randommatrix ensemble appropriate for the kicked top is COEwith additional particle exchange symmetry. Hence, this ensemble displays different properties from the standardcircular or Gaussian ensembles as far as the quantumcorrelations are concerned.Interestingly, it is found numerically in the globallychaotic case that D G = 0 . D − .
018 holds good. Thisis seen in Figs. 3, 4, 5 and 9. Such simple relation relating Q measure and discord or geometric discord could not bediscerned. It is known that for two-qubit states, discordand geometric discord are related to each other by D G ≥ D / VII. SUMMARY AND CONCLUSIONS
In this paper, we have investigated the effect of classi-cal bifurcations on the measures of quantum correlationssuch as quantum discord, geometric discord and Meyerand Wallach Q measure using kicked top as a model ofquantum chaotic system. In a related work [29], signa-ture of classical chaos in the kicked top was found in thedynamics of quantum discord and this work explores thisrelation in the more general context of quantum correla-tions including multipartite entanglement. The suitabil-ity of kicked top is due to the fact that it can be repre-sented as a collection of qubits. Most importantly, thissystem has been realised in experiments [46]. A promi-nent feature in its phase space is the period-1 fixed pointwhose bifurcation is associated with the quantum discordclimbing from nearly 0 to a value that is in agreementwith the numerically determined random matrix equiv-alent. The transition in the quantum discord reflectsthe qualitative change in the classical phase space; frombeing dominated by elliptic island to a largely chaoticsea with a few small elliptic islands. The other mea-sures we have reported here, namely the geometric dis-cord and Meyer and Wallach Q measure, both displaysimilar trends as the quantum discord. Other measuresof quantum correlations can be expected to display quali-tatively similar results. We have also presented numericalresults for the random matrix averages of these quantumcorrelation measures.In general, as a function of chaos parameter, quantumdiscord can be expected to increase under the influenceof a period-doubling bifurcation. However, after the bi-furcation has taken place, the saturation to the randommatrix average will depend on the qualitative nature ofdynamics in the larger neighbourhood around the fixedpoint. It must also be pointed out that these results havebeen obtained through time evolution of a spin-coherentstate placed initially on an elliptic island undergoing bi-furcation. For reasonably large elliptic islands, equiva-lent results could have been obtained by considering theFloquet states of the kicked top as well.We have also investigated the fate of quantum corre-lations in the semiclassical limit as Planck volume tendsto zero. In the context of kicked top, this limit trans-lates as j → ∞ . In the case of bifurcation associatedwith larger islands, as in k ≤
2, the measures of quan-tum correlations decreases as a function of j and tendsto 0 through a slow, approximately power-law decay. Inthe case of bifurcation associated with smaller islandsand creation of higher order periodic cycles the averagedecay of quantum correlations is evident but marked bystrong fluctuations. The quantum correlation measuresreported here have been obtained as that for the time av-erage of an evolving spin-coherent state placed initially ata chosen position in phase space. However, we note thatif the spin-coherent state is placed instead in the chaoticsea initially, then a different behaviour is obtained. As afunction of j , in this case, the quantum correlation mea-sures increases and saturates to a constant value thatcan be understood based on eigenvectors of appropriaterandom matrix ensemble. Evaluation of exact analyti-cal expression for average Q measure for the eigenvectorsof the corresponding circular unitary ensemble is carriedout and agrees very well that for the eigenvectors of theFloquet operator in the globally chaotic case.All the results presented in this work emphasise thespecial role played by the bifurcations and the associatedregular phase space regions in modifying general expecta-tions for the quantum correlations based on random ma-trix equivalents. These results are important from theexperimental point of view as the kicked top was firstimplemented in a system of laser cooled Caesium atoms[46]. Recently this model was implemented using super-conducting qubits [5]. Here the time-averaged von Neu- mann entropy has shown very close resembalance, despitepresence of the decoherence, with the corresponding clas-sical phase-space structure for given parameter values [5].Hence, the detailed effects of bifurcations presented hereshould be amenable to experiments as well. The scalingof quantum correlations with the total spin should alsobe observable with less than about ten superconductingqubits. ACKNOWLEDGMENTS
We are very grateful to acknowledge many discussionswith Vaibhav Madhok, T. S. Mahesh, Jayendra Bandy-opadhyay and Arul Lakshminarayan. UTB acknowledgesthe funding from National Post Doctoral Fellowship(NPDF) of DST-SERB, India file No. PDF/2015/00050.
Appendix A: Exact evaluation of h Q i E In this Appendix an exact evaluation of the ensem-ble average of the Meyer and Wallach Q measure iscalculated. The states in the ensemble have ideticalqubits and remains unchanged under qubit exchange.As explained in Section III one needs to use symmet-ric subspace spanned by the basis states {| j, m i ; ( m = − j, − j + 1 , ..., j ) } . Any pure state | φ i in this basis isgiven as | φ i = j X m = − j a m | j, m i where j X m = − j | a m | = 1 . (A1)In this case the Q measure is given as follows: Q = 1 − j + 1) (cid:0) h S z i + h S + ih S − i (cid:1) (A2)where S z and S ± are collective spin operatorssuch that S z | j, m i = m | j, m i and S ± | j, m i = p ( j ∓ m )( j ± m + 1) | j, m ± i [58]. The ensemble av-erage is carried over the states such that they have thestatistical properties of the eigenvectors of the COE en-semble. For the state | φ i one obtains the following ex-pression for the expectation: h S z i = j X m = − j m | a m | . (A3)This gives h S z i = j X m,n = − j mn | a m | | a n | = X m = n m | a m | + X m = n mn | a m | | a n | . (A4)0Now, an exact RMT ensemble-average is carried out[45, 65]. Firstly, one obtains hh S z i i E = X m = n m h| a m | i E + X m = n mn h| a m | | a n | i E . (A5)It should be noted that the first expectation is for agiven state | φ i and second expectation with subscript E denotes the ensemble average over all | φ i having statis-tical properties of COE eigenvectors. Using the RMTensemble averages [45, 65] h| a m | i E = 3(2 j + 1)(2 j + 3) , h| a m | | a n | i E = 1(2 j + 1)(2 j + 3) (A6)one obtains hh S z i i E = 3(2 j + 1)(2 j + 3) X m = n m +1(2 j + 1)(2 j + 3) X m = n mn. (A7)The first summation in the above equation is calcu-lated as follows: j X m = − j m = 2 j X m =1 m = j ( j + 1)(2 j + 1)3 . (A8)The second summation is now calculated. Consider theequality: j X m = − j m j X m = − j n = 0 . (A9)This gives X m,n mn = X m = n m + X m = n mn = 0 (A10)Thus, X m = n mn = − X m = n m = − j ( j + 1)(2 j + 1)3 . (A11)The ensemble average in Eq. (A5) is given as follows: hh S z i i E = 2 j ( j + 1)(2 j + 1)3(2 j + 1)(2 j + 3) . (A12) Considering the average of operators S ± for the state | φ i h S ± i = X a m a ∗ m ± p ( j ∓ m )( j ± m + 1) . (A13)This gives h S + ih S − i = X m,n a m a ∗ m +1 a n a ∗ n − p ( j − m )( j + m + 1)( j + n )( j − n + 1) . (A14)It can be seen that the ensemble average will havenonzero contribution only when m = n −
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