Signatures of avoided energy-level crossings in entanglement indicators obtained from quantum tomograms
aa r X i v : . [ qu a n t - ph ] J un Signatures of avoided energy-level crossings inentanglement indicators obtained from quantumtomograms
B. Sharmila , S. Lakshmibala and V. Balakrishnan Department of Physics, Indian Institute of Technology Madras, Chennai 600036,India.E-mail: [email protected]
29 June 2020
Abstract.
Extensive theoretical and experimental investigations on multipartitesystems close to an avoided energy-level crossing reveal interesting features such asthe extremisation of entanglement. Conventionally, the estimation of entanglementdirectly from experimental observation involves either one of two approaches:Uncertainty-relation-based estimation that captures the linear correlation betweenrelevant observables, or rigorous but error-prone quantum state reconstruction ontomograms obtained from homodyne measurements. We investigate the behaviour,close to avoided crossings, of entanglement indicators that can be calculated directlyfrom a numerically-generated tomogram. The systems we study are two genericbipartite continuous-variable systems: a Bose-Einstein condensate trapped in a double-well potential, and a multi-level atom interacting with a radiation field. We alsoconsider a multipartite hybrid quantum system of superconducting qubits interactingwith microwave photons. We carry out a quantitative comparison of the indicators witha standard measure of entanglement, the subsystem von Neumann entropy (SVNE).It is shown that the indicators that capture the nonlinear correlation between relevantsubsystem observables are in excellent agreement with the SVNE.
Keywords : Energy-level crossings, Quantum entanglement, Tomograms, Tomographicentanglement indicator
Submitted to:
J. Phys. B: At. Mol. Phys. omographic indicators at avoided crossings
1. Introduction
The measurement of any observable in a quantum mechanical system yields a histogramof the state of the system in the basis of that observable. In particular, in the context ofquantum optics, measurements of a judiciously chosen quorum of field observables yield aset of histograms (the optical tomogram) from which the density matrix is reconstructed.The latter is needed in standard procedures for estimating the extent of entanglementbetween the subsystems of bipartite or multipartite systems. A standard measure of theentanglement between the two subsystems of a bipartite system is the subsystem vonNeumann entropy ξ svne = − Tr ( ρ log ρ ), where ρ is the density matrix of either one ofthe subsystems [1]. In the case of continuous-variable (CV) quantum systems, an infiniteset of histograms is required, in principle, in order to obtain complete information aboutthe density matrix. In practice, however, only a finite set of histograms (corresponding tomeasurement of a finite set of observables) can be obtained. As state reconstruction fromtomograms [2] typically involves error-prone statistical techniques such as maximumlikelihood estimates, it is preferable to assess the extent of entanglement directly fromtomograms, circumventing detailed state reconstruction.In earlier work [3] we have proposed a tomographic entanglement indicator ξ tei based on mutual information, that is obtained directly from the relevant tomograms.We have tested its efficacy in bipartite CV systems evolving unitarily under nonlinearHamiltonians, by comparing it both with ξ svne and with an entanglement indicator ξ ipr based on inverse participation ratios [4]. (The participation ratio is a measureof delocalization in a given basis.) It has been shown that ξ tei and ξ ipr capturethe gross features of entanglement dynamics. A time-series analysis of the difference | ξ tei − ξ svne | was used to quantify the deviation of ξ tei from ξ svne as a function oftime. Although this is sensitive to the specific choice of initial state and the strength ofthe nonlinearity in the Hamiltonian, it has been shown that ξ tei is a reasonably goodindicator of entanglement in general. In multipartite hybrid quantum (HQ) systemscomprising two-level atoms interacting with radiation fields, too, ξ tei turns out [5] tobe a good estimator of quantum correlations in both the field and the atomic subsystems.(In the atom sector, ξ tei is extracted directly from the corresponding qubit tomograms).Several indicators of correlations between different parts of a classical system havebeen used extensively in various applications such as automated image processing. Thesecorrelators are obtained from classical tomograms. Their definitions, however, are notintrinsically classical in nature, and it is worth examining their applicability in quantumcontexts. Since correlations are inherently present in entangled states of quantumsystems, a natural question that arises is whether the performance of entanglementquantifiers obtained from these correlators is comparable to that of standard indicatorssuch as ξ svne .We examine quantum systems where the spacings between energy levels changesignificantly with changes in the parameters, with two or more levels moving close to eachother for specific values of the parameters and then moving away as these values change. omographic indicators at avoided crossings ξ svne ) is generically at an extremum at an avoided crossing. Typically, theenergy spectrum and the spacing between the energy levels depend on the strengths ofthe nonlinearity and the coupling between subsystems. With changes in the values ofthese parameters, the spacing between adjacent levels can decrease, and even tend tozero, resulting in an energy-level crossing. According to the von Neumann-Wigner no-crossing theorem, energy levels within a multiplet generically avoid crossing, providedonly one of the parameters is varied in the Hamiltonian governing the system. In thispaper, we investigate how effectively some of these entanglement indicators mimic thebehaviour of ξ svne close to avoided crossings.Energy-level crossings display other interesting features. Since they affect the levelspacings and their probability distribution [9], they are also important from the pointof view of non-integrability and quantum chaos (see, for instance, [10]). In addition,avoided crossings point to phase transitions which trigger a change in the quantumcorrelations in the system [6, 7, 11, 12]. This aspect has been investigated extensivelyboth theoretically and in experiments [13–16].We examine two experimentally relevant bipartite CV systems: a Bose-Einsteincondensate (BEC) in a double-well trap [17], and a multilevel atom interacting witha radiation field [18]. We also investigate a multipartite HQ system [19, 20] that iseffectively described by the Tavis-Cummings model [21]. The rest of this paper isorganized as follows: In the next section we introduce the entanglement indicators tobe employed. In section 3, we investigate how these indicators behave close to avoidedcrossings in the two bipartite CV models mentioned above. In section 4, we extend ouranalysis to the multipartite HQ model. Concluding remarks are made in section 5.
2. Entanglement indicators from tomograms
We first consider generic CV systems. A typical example of a bipartite CV system istwo coupled oscillators (equivalently, a single-mode radiation field interacting with amultilevel atom modelled as an oscillator). The tomogram is obtained from the quorumof observables that contain complete information about the state. These observablesare represented by the rotated quadrature operators X θ a = ( a e − iθ a + a † e iθ a ) / √ , X θ b = ( be − iθ b + b † e iθ b ) / √ . (1)Here 0 θ a , θ b < π , and ( a, a † ) [respectively, ( b, b † )] are the oscillator annihilationand creation operators corresponding to the two subsystems A and B. The bipartitetomogram is given by w ( X θ a , θ a ; X θ b , θ b ) = h X θ a , θ a ; X θ b , θ b | ρ ab | X θ a , θ a ; X θ b , θ b i , (2)where ρ ab denotes the bipartite density matrix. Here X θ i | X θ i , θ i i = X θ i | X θ i , θ i i ( i =A,B), and the product basis state | X θ a , θ a i ⊗ | X θ b , θ b i is written as omographic indicators at avoided crossings | X θ a , θ a ; X θ b , θ b i . The normalization condition is given by Z ∞−∞ d X θ a Z ∞−∞ d X θ b w ( X θ a , θ a ; X θ b , θ b ) = 1 (3)for each θ a and θ b . The reduced tomogram for subsystem A is w a ( X θ a , θ a ) = Z ∞−∞ d X θ b w ( X θ a , θ a ; X θ b , θ b )= h X θ a , θ a | ρ a | X θ a , θ a i , (4)where ρ a = Tr B ( ρ ab ) is the corresponding reduced density matrix. A similar definitionholds for subsystem B. In order to estimate the degree of correlation between thesubsystems, we use the following tomographic entropies. The bipartite tomographicentropy is given by S ( θ a , θ b ) = − Z ∞−∞ d X θ a Z ∞−∞ d X θ b w ( X θ a , θ a ; X θ b , θ b ) × log w ( X θ a , θ a ; X θ b , θ b ) . (5)The subsystem tomographic entropy is S ( θ i ) = − Z ∞−∞ d X θ i w i ( X θ i , θ i ) log [ w i ( X θ i , θ i )] ( i = A,B ) . (6)Some of the correlators that we examine in this paper are obtained from a sectionof the tomogram corresponding to specific values of θ a and θ b . The efficacy ofsuch a correlator as a measure of entanglement is therefore sensitive to the choiceof the tomographic section. We now define these correlators, and the correspondingentanglement indicators.The mutual information ε tei ( θ a , θ b ) which we get from the tomogram of a quantumsystem can carry signatures of entanglement. This quantity is expressed in terms of thetomographic entropies defined above as ε tei ( θ a , θ b ) = S ( θ a ) + S ( θ b ) − S ( θ a , θ b ) . (7)Indicators based on the inverse participation ratio (IPR) are also found to be goodcandidates for estimating the extent of entanglement [4, 22]. The IPR corresponding toa bipartite system in the basis of the rotated quadrature operators is defined as η ab ( θ a , θ b ) = Z ∞−∞ d X θ a Z ∞−∞ d X θ b [ w ( X θ a , θ a ; X θ b , θ b )] . (8)The IPR for each subsystem is given by η i ( θ i ) = Z ∞−∞ d X θ i [ w i ( X θ i , θ i )] ( i = A,B ) . (9)The entanglement indicator in this case is given by ε ipr ( θ a , θ b ) = 1 + η ab ( θ a , θ b ) − η a ( θ a ) − η b ( θ b ) . (10) omographic indicators at avoided crossings X and Y , given by PCC ( X, Y ) = Cov(
X, Y ) σ x σ y . (11)Here σ x , σ y are the standard deviations of X and Y respectively, and Cov( X, Y ) is theircovariance. Of direct relevance to us is
PCC ( X θ a , X θ b ) calculated for fixed values of θ a and θ b . Since the quantifier of entanglement between two subsystems must be non-negative, a simple definition of the entanglement indicator in this case would be ε pcc ( θ a , θ b ) = | PCC ( X θ a , X θ b ) | . (12)This indicator captures the effect of linear correlations. Our motivation for assessingthis indicator arises from the fact that, in recent experiments on generating and testingthe extent of entanglement in CV systems, the variances of suitably chosen conjugateobservables and the corresponding standard quantum limit alone are used [24]. Wereiterate that these merely capture the extent of linear correlations between two states.The second indicator (to be denoted by ε bd ) that we introduce and use is arrivedat as follows. In probability theory, the mutual information [25] between two continuousrandom variables X and Y can be expressed in terms of the Kullback-Leibler divergence D kl [26] between their joint probability density p XY ( x, y ) and the product of thecorresponding marginal densities p X ( x ) = R p XY ( x, y ) dy and p Y ( y ) = R p XY ( x, y ) dx ,as [27] D kl [ p XY : p X p Y ] = Z dx Z dy p XY ( x, y ) log p XY ( x, y ) p X ( x ) p Y ( y ) , (13)The quantity ε tei ( θ a , θ b ) defined in Eq. (7) is precisely the mutual information in thecase of optical tomograms (which are continuous probability distributions): ε tei ( θ a , θ b ) = D kl [ w ( X θ a , θ a ; X θ b , θ b ) : w a ( X θ a , θ ) w b ( X θ b , θ b )] . (14)A simpler alternative for our purposes is provided by the Bhattacharyya distance D b [28]between p XY and p X p Y , defined as D b [ p XY : p X p Y ] = − log n Z dx Z dy [ p xy ( x, y ) p X ( x ) p Y ( y )] / o . (15)Using Jensen’s inequality, it is easily shown that D b D kl . D b thus gives us anapproximate estimate (that is an underestimate) of the mutual information. Based onthis quantity, we have an entanglement indicator that is the analogue of Eq. (14),namely, ε bd ( θ a , θ b ) = D b [ w ( X θ a , θ a ; X θ b , θ b ) : w a ( X θ a , θ ) w b ( X θ b , θ b )] . (16)The dependence on θ a and θ b of each of the foregoing entanglement indicators ε is removed by averaging over a representative set of values of those variables. Wedenote the corresponding averaged value by ξ . In the context of bipartite CV models,we have shown in earlier work [3, 4] that averaging ε tei ( θ a , θ b ) over 25 different values omographic indicators at avoided crossings θ a , θ b ) selected at equal intervals in the range [0 , π ) yields a reliable entanglementindicator ξ tei . A similar averaging of each of the quantities ε ipr , ε pcc and ε bd yields ξ ipr , ξ pcc and ξ bd , respectively.Next, we turn to hybrid systems of field-atom interactions. For a two-level atomwith ground state | g i and excited state | e i , the quorum of observables is [29] σ x = ( | e i h g | + | g i h e | ) , σ y = i ( | g i h e | − | e i h g | ) ,σ z = ( | e i h e | − | g i h g | ) , (17)where σ i is a Pauli matrix. Let σ z | m i = m | m i . Then U ( ϑ, ϕ ) | m i = | ϑ, ϕ, m i , where U ( ϑ, ϕ ) is a general SU(2) transformation parametrized by ( ϑ, ϕ ). Denoting ( ϑ, ϕ ) bythe unit vector n , the qubit tomogram is given by w ( n , m ) = h n , m | ρ s | n , m i (18)where ρ s is the qubit density matrix. Corresponding to each value of n there existsa complete basis set. The atomic tomograms are obtained from these, and thecorresponding entanglement properties are quantified using appropriate adaptations ofthe indicators described above. The extension of the foregoing to the multipartite caseis straightforward [30], and the tomograms obtained can be examined on similar lines.
3. Avoided energy-level crossings in bipartite CV models
The effective Hamiltonian for the system and its diagonalisation are as follows [17].Setting ~ = 1, H bec = ω N tot + ω ( a † a − b † b ) + U N − λ ( a † b + ab † ) . (19)Here, ( a, a † ) and ( b, b † ) are the respective boson annihilation and creation operators ofthe atoms in wells A and B (the two subsystems), and N tot = ( a † a + b † b ). U is thestrength of nonlinear interactions between atoms within each well, and also betweenthe two wells. U >
0, ensuring that the energy spectrum is bounded from below. λ is the linear interaction strength, while ω is the strength of the population imbalancebetween the two wells. The Hamiltonian is diagonalised by the unitary transformation V = e κ ( a † b − b † a ) / where κ = tan − ( λ/ω ), to yield V † H bec V = e H bec = ω N tot + λ ( a † a − b † b ) + U N , (20)with λ = ( λ + ω ) / . e H bec and N tot commute with each other. Their commoneigenstates are the product states | k i ⊗ | N − k i ≡ | k, N − k i . Here N = 0 , , , . . . isthe eigenvalue of N tot , and | k i is a boson number state, with k running from 0 to N fora given N . The eigenstates and eigenvalues of H bec are given by | ψ N,k i = V | k, N − k i (21)and E ( N, k ) = ω N + λ (2 k − N ) + U N . (22) omographic indicators at avoided crossings E ( N , k ) ω ξ SV N E ω Figure 1. (a) E ( N, k ) vs. ω for N = 4 and k = 0 , , ξ svne vs. ω for N = 4 , k = 0 , ,
2. The curves correspond to k = 0 (red solid), 1 (bluedashed), 2 (green dotted) and 4 (orange dot-dashed). λ = 0 . For numerical analysis we set ω = 1 , U = 1.In figure 1(a), E ( N = 4 , k ) is plotted against ω for k = 0 , ,
4, with λ = 0 . E ( N, N − k ) is the reflection of E ( N, k ) about the value ω N + U N . Avoided energy-level crossings are seen at ω = 0. In order to set the reference level for the extent ofentanglement between the two wells, we compute ξ svne = − Tr ( ρ a log ρ a ), where ρ a isthe reduced density matrix of the subsystem A. ( ξ svne is also equal to − Tr ( ρ b log ρ b ),since | ψ N,k i is a bipartite pure state.) Plots of ξ svne corresponding to the state | ψ ,k i for k = 0 , , | ψ , i and | ψ , i have the same ξ svne ,(as do the states | ψ , i and | ψ , i ), owing to the k ↔ N − k symmetry. It is evident thatthere is a significant extent of entanglement close to the avoided crossing, and ω = 0is marked by a local maximum or minimum in ξ svne .Figure 2 depicts θ a = 0 , θ b = π sections of the tomograms corresponding to thestates | ψ ,k i for k = 0 , , ω = 0 , . ,
1. It is clear that, for a given value of ω ,the qualitative features of the tomograms are altered considerably as k is varied. Thepatterns in the tomograms also reveal nonlinear correlations between the quadraturevariables X θ a and X θ b (top panel). For instance, the tomogram slice on the top rightshows a probability distribution that is essentially unimodal and symmetric about theorigin with the annular structures diminished in magnitude. It is clear that this caseis less correlated than the tomogram in the top left corner. This conforms to theobserved trend in the extent of entanglement (compare ξ svne corresponding to k = 0and k = 2 at ω = 0 in figure 1 (b)). Again, in the bottom panel of the figure, thesub-structures in the patterns increase with increasing k , signifying a higher degree ofnonlinear correlation. This is in consonance with the trend in the entanglement at ω = 1 (figure 1 (b)). We therefore expect ε tei and its averaged version ξ tei to bemuch better entanglement indicators than ε pcc and ξ pcc . We also mention here thatthe current experimental techniques of testing CV entanglement based on the variancesand covariances of suitably chosen observables [24] are not as effective as calculatingnonlinear correlators, for the same reason.Our detailed investigations reveal that ξ tei and ξ ipr follow the trends in ξ svne reasonably well for generic eigenstates of H bec . This is illustrated in figure 3, which omographic indicators at avoided crossings -4-2 0 2 4 -4 -2 0 2 4 X θ B X θ A X θ B X θ A X θ B X θ A X θ B X θ A X θ B X θ A X θ B X θ A X θ B X θ A X θ B X θ A X θ B X θ A Figure 2. θ a = 0 , θ b = π slice of the tomogram for N = 4 in the BEC model. Leftto right, k = 0 , ω = 0 , . shows plots of these indicators as functions of ω . Apart from examining the suitabilityof ε pcc as an entanglement indicator, we have also checked for the extent of linearcorrelation between any two indicators based on the corresponding PCC, as follows. Wehave obtained 100 values each of ξ tei and ξ svne for different values of ω in the range( − ,
1) in steps of 0 .
02. Treating the two sets of values as two sets of random numbers,we obtain the PCC between them, as defined in Eq. (11). The PCC between ξ tei and ξ svne (respectively, ξ ipr and ξ svne ) estimates the extent of linear correlationbetween the two indicators, and is found to be 0 .
97 (resp., 0 .
99) in the case shownin figure 3 corresponding to | ψ , i . (In general, the PCC ranges from 1 for completecorrelation, to − ξ svne and various indicators, for the eigenstates | ψ ,k i where k = 0 , , , ,
4. From figure 4(a), we see that ξ ipr , ξ tei and ξ bd are verygood entanglement indicators. We have also found that all these indicators improve withincreasing N . The performance of the ε -indicators depends, of course, on the specificchoice of the tomographic section. For instance, ε tei and ε bd perform marginally betterfor the slice θ a = 0 , θ b = 0 than for the slice θ a = 0 , θ b = π . It is also evident that ξ pcc does not fare as well as the other indicators. This is to be expected, since ξ pcc only captures linear correlations, as already emphasised.We have verified that the sensitivity of all the indicators decreases with an increasein λ , the strength of the coupling between the two subsystems (as in Eq. 19). ξ ipr , omographic indicators at avoided crossings ξ SV N E ω ξ T E I ω ξ I P R ω Figure 3. ξ svne (red solid line), ξ tei (blue dashed line) and ξ ipr (green dotted line)vs. ω , for the state | ψ , i in the BEC model. ξ TEI ξ IPR ξ BD ξ PCC k -1-0.5 0 0.5 1 0 1 2 3 4 ε TEI ε IPR ε BD ε PCC k -1-0.5 0 0.5 1 0 1 2 3 4 ε TEI ε IPR ε BD ε PCC k -1-0.5 0 0.5 1 Figure 4.
Correlation of ξ svne with ξ -indicators (left), with ε -indicators for the slice θ a = 0 , θ b = π (centre), and with ε -indicators for the slice θ a = 0 , θ b = 0 (right),for the eigenstates | ψ ,k i , k however, remains closer to ξ svne than the other indicators. This fact is consistent withinferences drawn from our earlier work [4] about the relation between the Hammingdistance [31] and the efficacy of ξ ipr . We recall that the Hamming distance between twobipartite qudits | u i⊗| u i and | v i⊗| v i attains its maximum value of 2 when h u | v i = 0and h u | v i = 0. A straightforward extension to CV systems implies that the Hammingdistance between | k , N − k i and | k , N − k i is 2 (so that these states are Hamming-uncorrelated), if k = k . Participation ratios are valid measures of entanglementfor superpositions of Hamming-uncorrelated states in spin systems [22]. We havedemonstrated in our earlier work that ξ ipr effectively mimics standard measures ofentanglement in CV systems as well. In the present instance, the eigenstates | ψ N,k i aresuperpositions of the states {| j, N − j i} which are Hamming-uncorrelated for differentvalues of j . This is the reason for the usefulness of ξ ipr as an entanglement indicatoreven for larger values of λ .We now proceed to examine quantitatively the efficacy of the entanglementindicators as functions of λ . For numerical computation we have set ω = 0 . E (4 , k ) ( k = 0 , ,
4) as functionsof λ . These plots are exactly the same as those in figure 1(a), with ω replaced by λ on the horizontal axis, since E ( N, k ) only depends on the parameters ω and λ in thesymmetric combination λ = ( λ + ω ) / . The avoided crossing of energy levels nowoccurs at λ = 0. But this symmetry between ω and λ does not extend to the unitary omographic indicators at avoided crossings ξ SV N E λ Figure 5. ξ svne vs. λ for N = 4 , k = 0 , ,
2, for the BEC model. The curvescorrespond to k = 0 (red solid), 1 (blue dashed) and 2 (green dotted). ω = 0 . transformation V , and hence to the eigenstates of H bec . (Recall that V involves theparameter κ = tan − ( λ/ω ).) When λ = 0 there is no linear interaction between thetwo modes. V then reduces to the identity operator, and H bec is diagonal in the basis {| k, N − k i} . We therefore expect the entanglement to vanish at the avoided crossing.This is borne out in figure 5 in which ξ svne for the state | ψ (4 , k ) i is plotted for differentvalues of k . As before, it suffices to depict the cases k = 0 , k ↔ N − k symmetry. We observe that, in the case k = 0, while there is a minimum in ξ svne at λ = 0, there is a maximum in this quantity at ω = 0 (figure 1(b)).We have also calculated the PCC between various indicators and ξ svne for the setof states | ψ ,k i , k
4, using 100 values of each of the ξ -indicators calculated foreach λ in the range [ − ,
1] with a step size of 0 .
02. The results are very similar to thosealready found (see figure 4) using ω as the variable parameter instead of λ . We turn next to the case of a multi-level atom (modelled by an anharmonic oscillator)that is linearly coupled with strength g to a radiation field of frequency ω f . The effectiveHamiltonian (setting ~ = 1 ) is given by [18] H af = ω f a † a + ω a b † b + γb † b + g ( a † b + ab † ) . (23) ω a and γ ( > a, a † ) and ( b, b † ) are the annihilation andcreation operators for the field mode and the oscillator mode, respectively. As before, N tot = a † a + b † b and [ H af , N tot ] = 0. As in the BEC model of the preceding section,the eigenvalues E af ( N, k ) and the common eigenstates | φ N,k i of these two operators arelabelled by N = 0 , , . . . (the eigenvalue of N tot ) and, within each ( N + 1)-dimensionalsubspace for a given N , by the index k that runs from 0 to N .We find | φ N,k i and E af ( N, k ) numerically. Figures 6(a) and (b) show plots of E af ( N, k ) and ξ svne versus g for N = 4 and k = 0 , , ω f = 1 . , ω a = 1.Avoided crossings occur at g = 0, with a corresponding minimum in ξ svne that dropsdown to zero for each of the three states | φ , i , | φ , i and | φ , i . These states aretherefore unentangled at g = 0, i.e., in the absence of interaction between the twomodes of the bipartite system, as one might expect. omographic indicators at avoided crossings E A F ( N , k ) g 0 0.5 1 1.5 -1 -0.5 0 0.5 1(b) ξ SV N E g Figure 6. (a) E af ( N, k ) and (b) ξ svne vs. g for N = 4, k = 0 , , k = 0 (red solid), 1 (blue dashed) and 2(green dotted). ω f = 1 . , ω a = 1 , γ = 1. E A F ( N , k ) g 0 0.5 1 1.5 -1 -0.25 0.5 1.25(b) ξ SV N E g Figure 7. (a) E af ( N, k ) and (b) ξ svne vs. g for N = 4, k = 0 , , ω f = ω a = 1. The curves correspond to k = 0(red solid), 1 (blue dashed) and 2 (green dotted). γ = 1. In order to examine what happens when there is a crossing of energy levels, weintroduce a degeneracy by setting ω f = ω a . Figures 7 (a) and (b) are plots of E af ( N, k )and ξ svne versus g for N = 4 and k = 0 , ,
2, with γ , ω a and ω f set equal to 1. Both alevel crossing and an avoided crossing are seen to occur at g = 0, signalled by a minimumin ξ svne for each of the three states concerned. The crossing of E af (4 ,
0) and E af (4 , | p, − p i denote the product state | p i ⊗ | − p i , where | p i is aphoton number state of the field mode and | − p i is an oscillator state of the atommode. When γ = ω a = ω f = 1 and g = 0, the Hamiltonian reduces to a † a + ( b † b ) . Theenergy levels E af (4 ,
0) and E af (4 ,
1) become degenerate at the value 4. The degeneracyoccurs because the operator | , i h , | + | , i h , | commutes with H af when ω a = ω f and g = 0. Mixing of the states | , i and | , i occurs, and the corresponding energyeigenstates are given by the symmetric linear combination | φ , i = ( | , i + | , i ) / √ | φ , i = ( | , i − | , i ) / √
2. As the symmetriesof the two states are different, the level crossing does not violate the von Neumann-Wigner no-crossing theorem. At the crossing, each of the states | φ , i and | φ , i remainsa manifestly entangled state that is, in fact, a Bell state. This is why the corresponding ξ svne does not vanish at that point, but merely dips to a local minimum with value 1,characteristic of a Bell state. It is interesting to note that the degeneracy that occurs omographic indicators at avoided crossings ξ TEI ξ IPR ξ BD ξ PCC k -1-0.5 0 0.5 1 0 1 2 3 4 ε TEI ε IPR ε BD ε PCC k -1-0.5 0 0.5 1 0 1 2 3 4 ε TEI ε IPR ε BD ε PCC k -1-0.5 0 0.5 1 Figure 8.
Correlation of ξ svne with ξ -indicators (left), with ε -indicators for the slice θ a = 0 , θ b = π (centre), and with ε -indicators for the slice θ a = 0 , θ b = 0 (right), forthe eigenstates | φ ,k i , k ω f = ω a = γ = 1. when ω f = ω a ensures entanglement even in the absence of any interaction between thetwo modes.The level E af (4 , g = 0.The corresponding eigenstate | φ , i becomes the unentangled product state | , i at theavoided crossing, and ξ svne drops to zero in this case, as expected.In figure 8, we plot the correlation between various indicators and ξ svne . Forthis purpose, 80 values of each of the ξ -indicators were calculated with g varied in therange [ − , .
4] in steps of 0 .
03. Treating these as sets of random numbers, we obtainthe PCC between the various indicators and ξ svne , as described in the foregoing. Theperformance of the entanglement indicators in this case is similar to that found in theBEC system. Increasing γ marginally decreases the efficacy of all the indicators.
4. Avoided crossings in multipartite HQ systems
As our third and final example, we consider hybrid quantum systems comprising severalqubits interacting with an external field. These systems are described by the class ofTavis-Cummings models [21] in a variety of diverse physical situations which includeinherent field nonlinearities and inter-qubit interactions. The model we consider belowis generic, applicable to a system of several two-level atoms with nearest-neighbourcouplings interacting with an external radiation field in the presence of a Kerr-likenonlinearity, or to a chain of M superconducting qubits interacting with a microwavefield of frequency Ω f . In the latter case, the model Hamiltonian (setting ~ = 1) is givenby [19, 20] H tc = Ω f a † a + χa † a + M X p =1 Ω p σ pz + Λ( a † σ − p + aσ + p )+ M − X p =1 Λ s ( σ − p σ +( p +1) + σ − ( p +1) σ + p ) . (24)Here, χ is the strength of the field nonlinearity, Λ is the coupling strength between thefield and each of the M qubits, σ ± p are the ladder operators of the p th qubit, and Λ s is omographic indicators at avoided crossings ξ TEI ξ IPR ξ BD ξ PCC k -1-0.5 0 0.5 1 0 8 16 24 ε TEI ε IPR ε BD ε PCC k -1-0.5 0 0.5 1 Figure 9.
Correlation of ξ svne with ξ -indicators (left) and with ε -indicators for theslice corresponding to θ = π for the field and the σ x basis for each qubit (right). Thefigures are for the eigenstates | ψ , ,k i , k − the strength of the interaction between nearest-neighbour qubits. Ω p = (∆ p + ǫ ) / isthe energy difference between the two levels of the p th qubit, where ∆ p is the inherentexcitation gap and ǫ is the detuning of the external magnetic flux from the flux quantum h/ (2 e ). In our numerical computations we have used the experimentally relevant [20]parameter values Ω f / (2 π ) = 7 .
78 GHz and ǫ/ (2 π ) = 4 .
62 GHz. The level separations∆ p of the individual qubits have been drawn from a Gaussian distribution with a meangiven by h ∆ i / (2 π ) = 5 . . h ∆ i .We have considered three cases, namely, (i) Λ s = χ = 0 (ii) Λ s / (2 π ) = 1 MHz , χ =0 (iii) Λ s / (2 π ) = χ/ (2 π ) = 1 MHz. In each case, Λ / (2 π ) is varied from − . . .
025 MHz. It is easily shown that the total number operator N tot = a † a + M X p =1 σ + p σ − p , (25)commutes with H tc . For each value of Λ we have numerically solved for the complete set {| ψ M,N,k i} of common eigenstates of N tot and H tc , where N = 0 , , . . . is the eigenvalueof N tot and k = 0 , , . . . , M −
1. Considering the total system as a bipartite compositionof the field subsystem and a subsystem comprising all the qubits, we have computedthe entanglement indicators. Figure 9 shows the correlation between the indicators and ξ svne in Case (i). The associated Pearson correlation coefficients are 0 .
97 for ε tei , 0 . ε ipr , 0 .
97 for ε bd , correct to two decimal places. (The accuracy of the ε -indicatorsdepends, of course, on the basis chosen.) On averaging, we obtain the corresponding ξ -indicators with a PCC equal to 0 .
99, showing that these indicators track ξ svne veryclosely. We have carried out a similar exercise in Cases (ii) and (iii). The results andthe inferences drawn from them are broadly similar to those found in Case (i).Finally, with Λ / (2 π ) set equal to 1 . p by varying the standard deviation of ∆ p from 0to 0 . h ∆ i in steps of 2 × − h ∆ i . Calculating the entanglement indicators for eachdisorder strength in Ω p , we have found the correlations between the ξ -indicators and ξ svne in Cases (i), (ii), and (iii). ξ tei and ξ bd turn out to be significantly closer to omographic indicators at avoided crossings ξ svne , and hence more accurate indicators of entanglement, than the other indicators.
5. Concluding remarks
We have considered generic bipartite continuous-variable systems and hybrid quantumsystems in the presence of nonlinearities, and tested quantitatively the efficacy ofvarious indicators in estimating entanglement directly from quantum state tomogramsclose to avoided energy-level crossings. We find that the nonlinear correlationbetween the respective quadratures of the two subsystems reflects very reliably theextent of entanglement in bipartite CV systems governed by number-conservingHamiltonians. We have shown that if the eigenstates of the Hamiltonian are Hamming-uncorrelated, the inverse-participation-ratio-based quantifier ξ ipr is an excellentindicator of entanglement near avoided crossings. In fact, even ε ipr (the correspondingindicator for a single section of the tomogram) suffices to estimate entanglement reliably.The tomographic entanglement indicator ξ tei and the Bhattacharyya-distance-basedindicator ξ bd are also good indicators at avoided crossings, in contrast to the linearcorrelator ξ pcc which is based on the Pearson correlation coefficient. Entanglementindicators seem to perform better with increasing h N tot i . The conclusions drawn areboth significant and readily applicable in identifying optimal entanglement indicatorsthat are easily obtained from tomograms, without employing state-reconstructionprocedures. References [1] Nielsen M A and Chuang I L 2010
Quantum Computation and Quantum Information (CambridgeUniversity Press, Cambridge)[2] Paris M and Rehacek J 2004
Quantum State Estimation (Springer, Berlin)[3] Sharmila B, Saumitran K, Lakshmibala S and Balakrishnan V 2017
J. Phys. B: At. Mol. Opt. Quantum Inf. Process. Quantum Inf. Process. Phys. Rev. Lett. Phys. Rev. A Phys. Rev. A Phys. Rev. Lett. Quantum Signatures of Chaos (Springer, Berlin)[11] Cejnar P and Jolie J 2000
Phys. Rev. E J. Phys. A: Math. Theor. New J. Phys. Phys. Rev. A Phys.Rev. Lett. Phys. Rev. Lett. Ann. Phys. (N.Y.)
Phys. Rev. A omographic indicators at avoided crossings [19] Macha P, Oelsner G, Reiner J M, Marthaler M, Andr´e S, Sch¨on G, H¨ubner U, Meyer H G, Il’ichevE and Ustinov A V 2014 Nat. Commun. Photonics Phys. Rev.
J. Phys. A: Math. Theor. Stat Nanophotonics Elements of Information Theory (Wiley-Interscience, Hoboken)[26] Kullback S and Leibler R A 1951
Ann. Math. Stat. Pattern Recognition and Machine Learning (Springer, New York)[28] Kailath T 1967
IEEE T. Commun. Techn. Phys. Rev. A Phys. Scripta Phys. Rev. Lett.87