Correlation Energy and Entanglement Gap in Continuous Models
aa r X i v : . [ qu a n t - ph ] A p r Correlation Energy and Entanglement Gap inContinuous Models
L. Martina , G. Ruggeri, G. Soliani ∗ Dip. Fisica, Universit`a del Salento, I-73100 Lecce, ItalyINFN, Sezione di Lecce, I-73100 Lecce, Italy
Abstract
Our goal is to clarify the relation between entanglement and cor-relation energy in a bipartite system with infinite dimensional Hilbertspace. To this aim we consider the completely solvable Moshinsky’smodel of two linearly coupled harmonic oscillators. Also for small val-ues of the couplings the entanglement of the ground state is nonlinearlyrelated to the correlation energy, involving logarithmic or algebraiccorrections. Then, looking for witness observables of the entangle-ment, we show how to give a physical interpretation of the correla-tion energy. In particular, we have proven that there exists a set ofseparable states, continuously connected with the Hartree-Fock state,which may have a larger overlap with the exact ground state, but alsoa larger energy expectation value. In this sense, the correlation en-ergy provides an entanglement gap, i.e. an energy scale, under whichmeasurements performed on the 1-particle harmonic sub-system candiscriminate the ground state from any other separated state of thesystem. However, in order to verify the generality of the procedure,we have compared the energy distribution cumulants for the 1-particleharmonic sub-system of the Moshinsky’s model with the case of a cou-pling with a damping Ohmic bath at 0 temperature.
PACS 03.65.-Ud, 03.67.-Mn ∗ e-mail: [email protected] Introduction
The concept of entanglement has been recently considered by many authorsin connection with several properties of the quantum systems and as a po-tential resource in quantum computation and information processing both indiscrete and in continuous variable systems [1][2]. Moreover, entanglementhas also been recognized to play an important role in the study of manyparticle systems [3] and experimental and theoretical studies have demon-strated that it can affect thermodynamical properties both of the quantumphase transitions in the condensed matter and in molecular systems [4] [5][6]. However, most of the studies on the subject consider only finite di-mensional Hilbert models, which is not the typical situation occurring inatomic/molecular physics. As pointed out in [2], the theory of the entan-glement for the infinite dimensional setting is full of difficulties, which canbe cured choosing suitable subsets of the density matrices. In particular,the von Neumann entropy is not a continuous function in the Hilbert space,and for any given state of finite entanglement, one can find at least anotherstate closer as we want to the previous one in trace norm, which is infinitelyentangled.However, a new area of research has been opened by [6] [7] [8], where itwas shown that the entanglement, even if it is not a quantum observable,can be used in evaluating the so-called correlation energy: that is the dif-ference between the true eigenvalue energy of a given composite system ofidentical entities, with respect to that one prescribed by the Hartree - Fock(HF) method. In [9] the case of the formation of the Hydrogen molecule wasdiscussed, and a qualitative agreement between the von Neumann entropyof either atom ( as a measure of the entanglement of formation of the wholesystem) and the correlation energy as functions of the inter-nuclear distancewas shown. However, the extension of this idea to multi-atomic moleculesand its effectiveness remains still unclear [8][9]. Actually, the correlation en-ergy is an artifact of the approximation procedure, then it is not a physicalobservable and, by second, it can be modified by the adopted method ofcalculations. Nevertheless, since any other approximating disentangled statehas also a larger energy expectation value with respect to the HF state, one is2ead to look at the correlation energy as an entanglement gap in the sense of[7]. Adding any further correction term at the wave function has to decreasethe energy expectation value to the Hamiltonian eigenvalue and increase theentanglement at the same time up to fill the gap, describing in such a waya peculiar domain in state space, around the exact one. Thus, the first aimis to quantify such a kind of relation, at least on a specific model, finding aquantitative expression of the entanglement in terms of the correlation en-ergy of the ground state for a composite bipartite system. In order to have ananalytically tractable example containing all the desired features, we treatedwith a family of two coupled harmonic oscillators [10] in 3 dimensions. Thecoupling constant of the two parts is in a one-to-one correspondence with thecorrelation energy and and with entanglement estimators.
Viceversa , assum-ing that such a relation is invertible, and then an estimation of the correlationenergy can be expressed in terms of the entanglement, one may ask how directmeasurements on one of component subsystems can provide such quantities.To this aim we have found for the considered model an expression of theconcurrence, in terms of the momenta of the 1-particle subsystem energyprobability distribution, from the knowledge that the composite system isin its ground state. Thus we have an entanglement witness and an a priori estimation of the correlation energy. However, in concrete we have to beable to distinguish the energy probability distribution of the entangled statefrom any other yielded by a mixed state or a pure thermal state. First wegive an upper bound to the environmental temperature, over which all ourprocedure losses validity. Then, we compared the distribution generated bycoupling a single harmonic oscillator with an Ohmic bath at 0-Temperature,by the analysis of all energy distribution cumulants.In Sec. 2 we briefly review the main properties of the model: its exactfundamental state, the HF approximation and the correlation energy. Sec. 3for the fundamental state of the Moshinsky’s model we evaluate the von Neu-mann entropy and the concurrence in terms of the correlation energy. Sincethe concurrence can be expressed in terms of the dispersions of observableconjugated quantities, also the correlation energy takes a well defined phys-ical meaning. A similar relation can be established for the fidelity. In Sec. 4we prove the existence of a continuous manifold of pure separable states, con-3aining the HF state, having an overlap with the exact ground state, whichcan be larger with respect to the former. In Section 5 we describe how the1-particle energy probability distribution for the exact state of the model canbe distinguished from that one for a single harmonic oscillator coupled withan Ohmic bath at 0-Temperature. Some final remarks are addressed in theConclusions.
In order to evaluate how good the HF mean field method is in computingquantum states, Moshinsky proposed a simple, but non trivial, model of twocoupled spin- harmonic oscillators in 3 dimensions in [10]. In dimensionlessunities, the Hamiltonian of the model readsˆ H = 12 (cid:16) ˆ ~p + ˆ ~p + ˆ ~r + ˆ ~r (cid:17) − K (cid:16) ˆ ~r − ˆ ~r (cid:17) , (1)where ˆ ~r i and ˆ ~p i denote the position and the momentum operators of the i-thparticle, respectively. The constant K parametrizes the interaction strengthof a supplementary quadratic potential between the two oscillators (noticethe difference of sign with respect to [10]). The model describes a systemof two identical particles in the same harmonic potential, interacting by asmooth effective repulsive coupling, which is truncated at the second orderin a Taylor expansion for small interparticle distances. Thus, we will dwellupon 0 ≤ K < /
2, where the upper bound will correspond to a breaking ofthe model, since no bound states can exist. This signals that the model isfar to be realistic and it is intended only as a toy model shaped to our aims.The model energy spectrum is E n,m = 32 (cid:0) χ (cid:1) + n + mχ , m, n ∈ N ∪ { } (2)where χ = (1 − K ) , (3)plays the role of effective coupling constant, parametrizing a sort of doublewell potential, with an increasingly higher (or wider) barrier for χ →
0. The4ormalized position wave function of the fundamental level is given byΨ (cid:16) ~R, ~r (cid:17) = (cid:16) χπ (cid:17) / e − R / e − χ r / , (4)where the mean and relative positions ~R = ~r + ~r √ , ~r = ~r − ~r √ χ = 0, i.e. for the limitingvalue of the coupling K → /
2. Moreover, crossings occur for higher eigen-values at isolated points of K , but we are not interested to them. Finally,since the function (4) is symmetric in the interchange ~r ↔ ~r , the total spinmust be necessarily into the singlet state. Thus, the spinorial aspect of theproblem is not relevant at this stage, and it can be ignored.Applying the standard HF mean field approximation for the ground stateof the Hamiltonian (1), one is led to the wave functionΨ HF (cid:16) ~R, ~r (cid:17) = π − / (1 − K ) / e − (1 − K ) / ( R + r )/ , (6)corresponding to the approximated eigenvalue E HF = 3(1 − K ) / . (7)Defining the correlation energy ( positive, by Ritz’s theorem ) as E corr = E HF − E , = 3 √ − K − (cid:16) √ − K (cid:17) . (8)Moreover, the explicit expression of the overlap (or the squared fidelity)between the exact and the HF wave function is |h Ψ HF | Ψ i| = 64(1 − K ) / (1 − K ) / (cid:0)(cid:0) √ − K (cid:1) (cid:0) √ − K (cid:1) − K (cid:1) ≤ , (9)Thus, one can figure out that adding to the HF state further corrections,surely the estimation of the energy eigenvalue improves and the fidelity in-creases, but the simplest factorized expression in (6) will be lost. Differently5o what happens in the approximated state, the two oscillators in the correctfundamental state are entangled. From analytic point of view this happensbecause of the different coefficients of ~R and ~r in the wave function (4).From a different point of view, one can see the expressions (4) and (6) astwo distinct continuous curves in the Hilbert space, parametrized by K (or χ ). They have only one common point at K = 0. The main property of thelatter curve is to contain only factorized states. Since we are dealing with pure states, the entanglement estimator is theentanglement entropy, given in term of the von Neumann entropy S vN [ ˆ ρ r ] = − Tr [ ˆ ρ r log ˆ ρ r ] = − X i µ i log ( µ i ) , (10)of the reduced to 1-particle density matrix ˆ ρ r = Tr [ ˆ ρ ] = Tr [ ˆ ρ ], denotingby µ i the corresponding eigenvalues.On the other hand, the von Neumann Entropy S vN satisfies the additiverelation S vN [ ˆ ρ ⊗ ˆ σ ] = S vN [ ˆ ρ ] + S vN [ˆ σ ] , for any factorized density operatorˆ ρ ⊗ ˆ σ . But this is precisely the structure of the reduced density matrix, whichfactorizes into positional and spinorial contribution, where the latter takesthe form ˆ σ = for the singlet spin state. Thus, it contributes to an addi-tive constant term (equal to 1), which measures only the equal uncertaintyin attributing one of the two possible quantum states to each spin. Followingthe ideas in [11] for fermions, anti-symmetrizing the product of 1-particleorthogonal states into a spin stationary state contains all information aboutentanglement by definition. In conclusion, here we will compare only the con-tributions to the entanglement coming from the space configurations factorof the 2-particle fundamental state.In the position representation the exact 2-particles density matrix ˆ ρ forthe fundamental state (4) is given by the integral kernel ρ ( ~r , ~r , ~r ′ , ~r ′ ) = (cid:16) χπ (cid:17) e − ( R + R ′ ) / e − χ ( r + r ′ ) / , (11)6here the supplementary variables ~R ′ and ~r ′ are in analogy with (5). A simi-lar expression holds for the | Ψ HF i state (6), where the density matrix is givenby the product of gaussian normal distributions with the same variance. Theconsequences of such different structure can be seen also by the comparisonof the 1-particle space distribution densities, which are given by ρ ( ~r ) = (cid:18) χ π ( χ + 1) (cid:19) / e − χ χ r , ρ HF1 ( ~r ) = (1 − K ) / π / e −√ − K r . (12)Thus, because of the repulsive interaction, the exact average distance betweenthe particles is larger than in the approximated estimation, being their ratio (cid:18) √ − K ( √ − K ) √ − K (cid:19) / , with a divergence for K → ( χ → ρ r = R ρ r ( ~r, ~r ′ ) · d ~r ′ has the kernel ρ r ( ~r, ~r ′ ) = (cid:18) χ π ( χ + 1) (cid:19) / exp χ − ~r · ~r ′ − (cid:16) χ + ( χ + 1) (cid:17) ( ~r + ~r ′ )8 ( χ + 1) . (13)That can be rewritten in the usual gaussian form [4] [15] ρ (∆ p, ∆ q ) r ( ~r, ~r ′ ) = (cid:18) π ∆ q (cid:19) / exp " − ( ~r + ~r ′ ) q + ∆ p ( ~r − ~r ′ ) ! , (14)where the squared mean values for the 1-particle position and momentum∆ q = χ + 14 χ , ∆ p = 14 (cid:0) χ + 1 (cid:1) , (15)have been introduced.The system becomes disentangled when the 1-particle state is the pureminimal packet, i.e. when ∆ p ∆ q = . But this occurs only for K = 0 ( χ =1). In this sense ∆ p and ∆ q contains information about the entanglementof the system, as remarked in [4]. However, differently from [4], by the simplealgebraic relation χ = ∆ p ∆ q the squared mean values contain also information7bout the form and the strength of the interaction. Thus, one would arisethe question if the analysis of (14) not only provides information about anentangled harmonic oscillator, but also the main properties of the coupling:is it coupled to a small system or to a thermodynamic bath?From expression (13) the kernel of the eigenvalue equation for ˆ ρ r is sym-metric and of Hilbert-Schmidt type, since the coefficient of ~r (and ~r ′ ) isnegative. So the spectrum is real and discrete, as for the tensor product ofthree independent oscillators. Accordingly, the eigenfunctions of ˆ ρ r can befactorized in the product of three functions, each of them depending only onone real variable and the eigenvalues as a product u l,m,n ( ~r ) = w l ( x ) w m ( y ) w n ( z ) , µ l,m,n = ν l ν m ν n , (16)so that the problem is reduced to solve the 1-dimensional integral spectralproblem Z ρ r ( x, x ′ ) w l ( x ′ ) dx ′ = ν l w l ( x ) , (17)which has the non degenerate spectrum and eigen-solutions of the form ν l = Cc l , w l ( x ) = H l ( √ χx ) e − χ x , (18)where C = χ (1+ χ ) , c = (cid:16) − χ χ (cid:17) and H l (cid:0) √ χx (cid:1) denote the Hermite polyno-mials. Of course, these positive eigenvalues sum up to 1, because they arerelated to a matrix density operator. On the other hand, accordingly to (16)the eigenvalues of the one-particle density matrix ˆ ρ r are given by µ k = C c l + m + n = C c k , k ǫ N , (19)with degeneration order deg [ µ k ] = k ( k + 3) / ρ ( ~r ) = P lmn C c l + m + n | w l ( x ) w m ( y ) w n ( z ) | is represented as amixed state in the basis of the ”natural orbitals”, using the terminology byL¨odwin [12], which describe a 3D single harmonic oscillator of frequency χ .The weights µ k describe a system at the equilibrium temperature k B T ∗ =3 χ/ (2 ln χ − χ ), which is a decreasing function of χ . Since experiments are al-ways performed at a finite temperature, performing particle position/momentummeasurements on such a system we need to work at T < T ∗ ( χ ), in order tohighlight the quantum behavior discussed here.8or comparison, the spectrum of the reduced density matrix in the theHF approximation is given by { , } ,with the corresponding eigenfunctions (for each of the three space variables) n H l (cid:16) (1 − K ) / x (cid:17) exp h − (1 − K ) / x io .Hence, if we are allowed to look at the eigenvalues of the density matrixoperator ˆ ρ r as the probabilities to find the 1-particle subsystem in one ofthe states of a K -parametrized family of harmonic oscillators, for small K it can be found very likely in the fundamental one. But this probabilitydecreases rapidly to 0 for K → , while the higher states become significantlymore accessible. Notice that at K = the system is meaningless, since alleigenvalues of ˆ ρ r become 0 except one. However, for 0 ≤ K < one cananalytically sum up T r (ˆ ρ r ) = 1 pointlike, taking into account the degeneracy.On the other hand, the lack of coherence can be estimated also by computingthe T r [ ˆ ρ r ], which is 1 only for pure states. In the present case one hasexplicitly Tr (cid:2) ˆ ρ (cid:3) = 8 χ (1 + χ ) , (20)which is a monotonically decreasing from 1 to 0 function on K .Complementary to this quantity there is the so-called linear entropy [13],analogous to the concurrence [14] C [ ˆ ρ r ] = 1 − Tr[ ˆ ρ r ] , (21)which takes values in the range [0 , ∞ -dimensional Hilbertspace.The entropy of entanglement (10) can be explicitly written as S vN [ ˆ ρ r ] = 3ln 4 χ (cid:2) ( χ + 1) ln ( χ + 1) − χ ln 4 χ − ( χ − ln | χ − | (cid:3) . (22)For K → ( χ → S vN [ ˆ ρ r ] is logarithmic divergent, accordingto the expansion S vN [ ˆ ρ r ] ≈ − χ ]ln [2] + O (1) . (23)9his is a well known result for harmonic chains [15], indicating the degeneracyof the ground state in the considered limit.On the other hand, the behavior of S vN [ ˆ ρ r ] near K → S vN [ ˆ ρ r ] = − K (1 + 2 K )8 ln(2) ln( K ) + O (cid:0) K (cid:1) , (24)approaching 0, because of the oscillators decoupling. However, this approx-imation becomes inaccurate very rapidly. From the above expression, for K →
0, the asymptotic behavior of the entropy is controlled by a logarith-mic term, differently from the correlation energy (8), which has a pure powerexpansion. Then, we cannot expect a great similarity between the two func-tions, also at very small values of K . This results breaks the conjecturedexistence of a simple relationship between the two quantities. On the otherhand, let us observe that both functions S vN [ ˆ ρ r ] and E corr are monotonicallyincreasing in K . Thus, the entanglement is an increasing function of thecorrelation energy (see Fig. 1). In order to have analytic expressions, wesolve algebraically the coupling constant in terms of E corr as χ ( E corr ) = (cid:20) − (cid:18)q E corr ( E corr + 3) (2 E corr + 3) − E corr ( E corr + 3) (cid:19)(cid:21) / (25)and replacing into S vN [ ˆ ρ r ], we obtain a one-to-one correspondence ˜ S vN ( E corr ).In particular, one can look for asymptotic expressions of the entanglementfor small values of E corr , corresponding to small couplings. Indeed, includinglogarithmic corrections at the lowest order near E corr ≈
0, one obtains˜ S vN ( E corr ) ≈ (1 + ln(6) − ln ( E corr ))2 ln(2) E corr + O (cid:0) E / (cid:1) , (26)for the Moshinsky’s oscillators. One verifies that similar expressions canbe obtained studying other systems (for instance the 2-points Ising model),but up to now does not exist a general procedure to compute directly thecoefficients appearing in the above developments. Moving to the upper limit K → / χ → E corr → E corr = (cid:0) − √ (cid:1) , the entropy10 .1 0.2 0.3 0.4 0.5 0.6 E corr vN @ Ρ` r D Figure 1: The entanglement as a function of the correlation energy for theMoshinsky’s model.diverges logarithmic as˜ S vN ( E corr ) ≈ − E corr − E corr )ln(4) + O (1) , (27)which is the specific behavior for degenerate ground state, as remarked for(23).However, the singular behavior near 0 of the entanglement as a functionof the correlation energy does not seem related to the specific way of its esti-mation. In fact, by using Eq. (25) into (20), as a function of the correlationenergy the concurrence for the Moshinsky’s model takes the form C ( E corr ) = 1 − √ f / ( f / +3 ) , (28) f = 9 + 12 E corr ( E corr + 3) − √ q E corr ( E corr + 3) (2 E corr + 3) , a graph of which is shown in Fig. 2. This function is regular in the origin,but it is not in its second derivative. Again a singularity is signaling afaster increase of the entanglement for small values of the correlation energy.But, the expression (28) is algebraic and it can be manipulated more easily.Specifically, for small values of the concurrence one gets the correlation energyas an half-integer power series of C E corr ≈ C + r C / + 2 C O ( C / ) , (29)11 .1 0.2 0.3 0.4 0.5 0.6 E corr H E corr L Figure 2: The entanglement expressed in terms of the concurrence as a func-tion of the correlation energy for the Moshinsky’s model.while for 0 ≪ C ≤ E corr ≈ E corr −
38 (1 − C ) / + 364 (cid:16) √ − (cid:17) (1 − C ) / + O ((1 − C )) . (30)These expressions give direct relations between the correlation energyand the entanglement, which may suggest an experimental measure of theentanglement and of the correlation energy. In fact, let us suppose to performtwo independent series of measurements of position and linear momentum onone particle of the system. Their results are distributed with squared meanvalues ∆ q and ∆ p , respectively. On the other hand, by resorting to therelations (15) in terms of the coupling constant χ and to the expression (20),one gets Tr (cid:2) ˆ ρ r (cid:3) = p ∆ p ∆ q (∆ p + ∆ q ) ! . (31)Thus the entanglement is related to the ratio between the uncertainty andthe energy mean value of the observed subsystem. Furthermore, by thedefinition (21) and in the range of validity of expansion (29) (or (30) ), onemay obtain a relation among observable quantities and the mathematicalartifact E corr . On the other hand, relation (31) has to be used carefullysince, if applied to a generic gaussian separate pure state, it does not givea measure of entanglement, of course. The point to be remarked is that its12alue depends by the special relation of the mean squared deviations on thecoupling constant, not only on the preparation of the state.Finally, the fidelity of the fundamental state of the Moshinsky’s modelwith the corresponding HF state or, equivalently, the overlap (9) can beexpressed as a function of the entanglement. In some sense, we are comparingtwo different ways to measure the “distance” between the two curves of states,even if neither quantities actually have the properties of a distance. However,also in this case a monotonic function can be obtained for any pair of statescorresponding to the same coupling constant K , or correlation energy E corr .This function is regular, even if at the extrema a singularity in its higherderivatives appears. Here we would like to elucidate the special role played by the HF statein the set of all separable states, which may be closer, in the sense of thetrace - norm, to the exact solution. To this aim and since we are lookingto a neighborhood of the ground state in the Hilbert space, let us restrictourselves to the pure separable states, which are symmetric with respect tothe change 1 ↔ ~r ) ˜Φ ( ~r ) , (32)where for convenience we assume that ˜Φ is normalized to 1. Of course, moregeneral choices are possible, compatibly with the assumed identity of theparticles. In the class of states (32) there exists the 1-parameter curve givenby the gaussian functions˜Ψ a = (cid:16) aπ (cid:17) / exp h − a (cid:0) R + r (cid:1)i , (33)certainly containing the HF wave function (6). Its overlap with the groundstate (4) is such that |h ˜Ψ a | Ψ i| = 64 a χ ( a + 1) ( χ + a ) ≥ |h Ψ HF | Ψ i| (34)13see eq. (9)) for a ∈ h a low ( χ ) , q ( χ −
1) + 1 = √ − K i . Of course, theupper bound is exactly the value involved in the HF wavefuction (see eq.(6)). The lower bound a low ( χ ) is an algebraic positive monotonic increasingfunction, going from 0 to χ →
0, like a low ( χ ) ≈ γ χ + O [ χ ] ( γ = const ), to 1 for χ → K → a max = χ , forwhich one has |h ˜Ψ a max | Ψ i| = χ ( χ +1) , equal to 1 only at χ = 1 ( K = 0).It should be noticed that a max is exactly the same exponent appearing inthe eigenfunctions of the 1-particle reduced density matrix operator (see eq.(18)), accordingly with the notion of ”natural orbital”. In conclusion, the HFstate is not the closest (in the sense of the trace-norm) pure separable stateto the exact fundamental state and one may wonder if other states arbitrarilyclose to it may be found. Of course, by deforming the pure gaussian form(33) with maximal overlapping, in the base of the Hermite polynomials onecan construct symmetric factorized wavefunctions of the form˜Ψ c = Y i =1 N ( i ) n i X j =0 c ( i ) j H j (cid:16) √ χx ( i )1 (cid:17) ! n i X j =0 c ( i ) j H j (cid:16) √ χx ( i )2 (cid:17)! ˜Ψ a max , (35)for arbitrary complex constants n c ( i ) j o and for suitable normalization con-stants N ( i ) . Thus, it is not difficult to find |h ˜Ψ c | Ψ i| = |h ˜Ψ a max | Ψ i| Y i =1 | P n i j =0 j j ! c ( i ) j (cid:16) χ − χ +1 (cid:17) j | (cid:16)P n i j =0 j j ! | c ( i ) j | (cid:17) . (36)The three factors in the r.h.s. of the above expression are ≤
1, then theoverlapping of the generalized wave-packets (35) with the exact state cannotexceed the maximal one. In conclusion, we have proved that there existsa dense set of pure and separated states, containing the HF state, havingoverlap |h Ψ HF | Ψ i| ≤ |h ˜Ψ | Ψ i| <
1, except for χ = 1 ( K = 0), whenΨ HF ≡ Ψ . Then, the exact state cannot be approached arbitrarily close bya separated state, except when it is itself separate. This result is complemen-tary to the statement that entanglement entropy of a continuous model is a adiscontinuous function, diverging at infinity in any neighborhood of any pure14tate [15]. The maximal overlapping is provided by taking the suitable tensorproduct of the natural orbitals [12]. Finally, because of the convexity of theset of all separable mixed states, i. e. of the form ρ = P n p n | ˜Ψ c n ih ˜Ψ c n | , with p n ≥ P n p n = 1, one can extend the previous statement to the entirespace of states.On the other hand, the HF state has been selected in the class of sepa-rable states by the minimal energy requirement. But in the domain of thepure separated states of form (33) the relation h b H i ˜Ψ a − E HF = ( a −√ − K ) a holds. Then, the expectation value of the Moshinsky’s Hamiltonian b H getsits absolute minimum indeed at the HF state. Moreover, this can be seenalso considering general factorized states as in (35). Now, because of theconvexity of the set of separable states, the minimum in the spectrum of abounded observable from below is always achieved by a pure separable state.Thus, we conclude that the above introduced correlation energy is not simplya mathematical artifact, but it looks analogous to the concept of entangle-ment gap introduced in Ref. [7]. Since this is a global result, not dependingon a particular computation procedure, we claim that the HF state for theMoshinsky’s model provides the minimum separable energy as introducedin [7] E sep = min ρ separable T r h b Hρ separable i = E HF . Moreover, the observable Z EW = b H − E HF is an entanglement witness, the spectrum of which is nonnegative on all separable states and there exists the ground state (entangled)of the Moshinsky’s model for which its expectation value is − E corr <
0. Ac-tually, for χ >
K < /
2) isolated eigenvalues of Z EW may exist in the gap[ − E corr , Z EW we obtain a negative value, we can still affirm thatthe the system is in an entangled state, even if not necessarily in the groundstate. In the previous Sections we have shown that the entanglement gap is themain energetic scale that dictate if a composed system is, or not, entangled.For the Moshinsky’s model we have shown that this gap is given by the15orrelation energy, derived from the HF calculations. However, this relieson the full knowledge of the density matrix, while for pure states all neededinformation is encoded into the reduced matrix of a selected subsystem: inour case one of the harmonic oscillators. Thus the question if one can estimatethe entanglement by energy measurements on the single harmonic oscillatorsarises, conditionally to the knowledge that the whole system is not in aseparated ground state. These will be subjected to statistical fluctuations,which in principle contains the required information, i.e. the entanglementof original ground state of the composite system. This Section is devoted tohow extract this result and how to distinguish the energy distribution of theentangled composite system, from the effects of couplings to a more genericenvironments, like an Ohmic bath, even if the latter is at 0 temperature.First step concerns the calculation of the energy distribution for one singleharmonic oscillator included into the Moshinsky’s model. To this aim itis useful to have the above expressions in the simple harmonic oscillatorHamiltonian eigenvector basis B = (cid:8) ϕ il,m,n (cid:9) ( i = 1 , h ϕ l,m,n ϕ l ′ ,m ′ ,n ′ | Ψ i = (cid:18) χ / π (cid:19) I l,l ′ I m,m ′ I n,n ′ √ l + l ′ m + m ′ n + n ′ l ! l ′ ! m ! m ′ ! n ! n ′ ! , (37)where the matrix { I m,m ′ } has a sort of chessboard structure, given by therelation I m,m ′ = 2 πǫ ( m + m ′ ) ( − m ′ ( m + m ′ − ζ ) / ζ m + m ′ , (38)where the expression ζ = − χ χ and the scaled step function ǫ ( m ) = (cid:26) / m even0 m oddhave been introduced.On the other hand, in the basis B the amplitudes of ˜Ψ a are given by h ϕ l,m,n ϕ l ′ ,m ′ ,n ′ | ˜Ψ a i = a / π A l A l ′ A m A m ′ A n A n ′ √ l + l ′ m + m ′ n + n ′ l ! l ′ ! m ! m ′ ! n ! n ′ ! , (39)where A l = ǫ ( l ) 2 l +1+ α − l − (cid:0) − α (cid:1) l/ Γ (cid:18) l + 12 (cid:19) (40)16ith α = √ a for brevity. The expression (40) establishes that the nonvanishing terms occur only for even principal quantum numbers l, l ′ , . . . , butthey are not correlated among them.In the representation (37)-(38) the elements of the full matrix densityoperator for the exact ground state are given by taking the tensor productin three dimensions of the 1-dimensional factors ρ l,l ′ ,m,m ′ = 4 (1 + ζ ) χ ζ l + l ′ + m + m ′ (41)( − l ′ + m ′ ε ( l + l ′ ) ε ( m + m ′ ) ( l + l ′ − m + m ′ − l + l ′ + m + m ′ l ! l ′ ! m ! m ′ !] / . The corresponding reduced density matrix ρ r,l,l ′ = P m ρ l,m,l ′ ,m can be com-puted from the above expression, or using the continuous basis representa-tion, contracting with respect the suitable states of the uncoupled harmonicoscillator. In particular, we are interested in the evaluation of the diagonalelements ρ r,l,l , which represent the probabilities to find the 1-particle sub-system into the energy eigenstates of the uncoupled harmonic oscillator. Itresults that these quantities are related by the following recursion relation ρ r,l +1 ,l +1 = l !( l + 1)! χ ( ζ + 1) ζ l +1 ∂ ζ (cid:18) ρ r,l,l χ ( ζ + 1) ζ l (cid:19) (42)with the expressions for the fundamental state ρ r, , = 2 χ ( ζ + 1) p − ζ . (43)Thus, the general structure of the considered distribution is ρ r,l,l = 2 ζ l χ ( ζ + 1)(4 − ζ ) l +12 Q l ( ζ ) , (44)where Q l ( ζ ) is a polynomial of degree l in the ”scaled” coupling constant ζ . This distribution of probability has its own peculiarities, which make itdifferent from a generic factorized state or from a pure equilibrium ther-modynamical distribution. Then, for comparison one computes the energy17robability distribution for a factorized gaussian state by using the formulae(39) - (40). For each eigen-state label one obtains the expression ρ r,l,l = 16 a / ε ( l ) ( l − l ! ( a + 1) ( a + 3) / (cid:18) − a a (cid:19) l . (45)The first observation is that this distribution is different from 0 only for even l : this is a common character of all factorized gaussian states, included theHF approximated wave-packet, so it could be used to make an experimentalcomparison with the entangled state.On the other hand, one can compute such a kind of quantity by the directuse of the 1-particle reduced matrix (14) [4]. In fact, by using the generatingmatrix for the Hermite polynomial, the diagonal elements of the reducedmatrix in the basis of the pure harmonic oscillator are given by ρ (∆ p, ∆ q ) r,l,l = 2 [(2∆ p −
1) (2∆ q − l/ [(2∆ p + 1) (2∆ q + 1)] ( l +1) / P l p ∆ q − p (4∆ p −
1) (4∆ q − ! , (46)where P l denotes the l -th Legendre polynomial. The parameters ∆ p and ∆ q are independent quantities, limited only by the minimal uncertainty condi-tion ∆ p ∆ q ≥ . Of course, substituting the expressions of ∆ p and of ∆ q given in (15), one recovers the formula (44): there the emphasis is on the de-pendency by the coupling strength. In Fig. (3) we give a set of contour plotsof the probabilities to find the single harmonic subsystem ( of frequency ω = 1) in one of the first six eigenvalues as functions of the uncertainties(∆ p, ∆ q ), accordingly to expression (46). The bold dashed curve is given bythe equations (15) of the uncertainties in the Moshinsky’s model. Of course,the efficacy of the above procedure to measure the entanglement has to beevaluated by comparison with other situations. For instance, one may ask ifis it possible to distinguish the above distribution of energy eigenvalues froma sufficiently general mixed one. Specifically we consider that one obtainedcoupling one of the harmonic oscillator to an Ohmic bath [16]. To this aim,we propose two different methods for this comparison.The first way is based on the position and momentum measurements, fromwhich we can reproduce the graph of Fig. 3, for fixed values of the coupling18 D q D p Ρ r ,0,0 D q D p Ρ r ,1,1 D q D p Ρ r ,2,2 D q D p Ρ r ,3,3 D q D p Ρ r ,4,4 D q D p Ρ r ,5,5 Figure 3: The probability distributions (46) to find into one of the firstsix modes of the simple harmonic oscillator a 1-particle subsystem, de-scribed by a reduced density matrix (14) in the (∆ q, ∆ p ) plane. In thearea between two contours the probability varies by 1 /
10 of the maximavalues { . , . , . , . , . , . } , respec-tively for each plot, decreasing from black to lightest gray. The black thickdashed curve represents the values of (∆ q, ∆ p ) given by (15), while the blackthick continuous curve corresponds to the oscillator coupled to an Ohmicbath at T = 0 o accordingly to (50). The dashed gray boundary curve isdetermined by the hyperbola ∆ q ∆ p = , which represents the gaussian fac-torized states. When the coupling constants of both models vanishes, thecorresponding wave-packets are minimal.19onstant. In this case, the curves corresponding to the parametric represen-tations of (∆ q, ∆ p ) characterize the two models. Thus, we can distinguishthe Ohmic model from the Moshinsky’s model by knowing the position andmomentum uncertainty behaviour in the (∆ q, ∆ p ) plane. Because of suchmeasurements, this method produces a lack of information about the ener-getics of the system. Then, the second method, suggested by [4], is based onthe analysis of the cumulants of the simple harmonic oscillator Hamiltonian H HO , namely hh H nHO ii = ( − n d n dξ n ln Z ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 , (47)where the partition function Z ( ξ ) is evaluated by tracing the harmonic prop-agator in the imaginary time ξ , with respect the generic gaussian densitymatrix (14). Following the standard calculations [4] [17], one knows that Z ( ξ ) = h e − ξH HO i ρ (∆ p, ∆ q ) r = (48) (cid:20) (∆ p + ∆ q ) sinh ξ + 2∆ q ∆ p (cosh ξ −
1) + 1 + cosh ξ (cid:21) − / . Then, a list of the cumulants can be algorithmically computed as polynomialsof even dergee on the uncertainties (∆ q, ∆ p ), as for instance hh H nHO ii = 32 (cid:0) ( n − (cid:0) ∆ q n + ∆ p n (cid:1) + O (cid:0) ∆ q n − + ∆ p n − (cid:1)(cid:1) , (49)For the Moshinsky’s model one substitutes the uncertainties given in (15)in (49), while for the Ohmic bath (in the underdamped limit) one uses themean squared values [4]∆ q = 12 p − β − π arctan β p − β ! , ∆ p = (1 − β )∆ q + 2 βπ ln ω C ω , (50)where β is the coupling to the dissipative environment, in units of the oscil-lator frequency, and ω C is a cutoff frequency.In order to have a unique parameter, which measures the interactionstrength between the singled out oscillator with the remaining of the com-posite system (the environment) in both considered cases, let us assume the20 .0 0.2 0.4 0.6 0.8 1.0 Χ YY H HO2 ]] Figure 4: The second energy cumulants hh H HO ii as a function of the couplingconstant χ with ω C = 10 ω . Dashed line represents the Ohmic bath, whilesolid line corresponds to system (1).relation β = (1 − χ ) χ ) . (51)This relation is suggested by the derivation of the classical Langevin equationfrom a pure Hamiltonian system of coupled oscillators [16]. Thus, we canrewrite the above cumulants in terms of the coupling constant χ for bothsystems and compare them, as it has been shown in Fig. 4 for those oforder 2. There, we can see that the two functions are similar for χ = 1,i.e. when they describe free harmonic oscillator in both cases. The situationdramatically changes when χ decreases, i.e. for stronger interactions. Infact, for χ →
0, the second energy cumulant associated with the Ohmicbath remains finite, while for the Moshinsky’s model it is divergent. Similarconsiderations can be made for the higher order cumulants. Thus, we haveprovided a method for distinguishing the two classes of states.A different approach concerns the analysis the logarithm of the cumulants,at a fixed value of the coupling parameter χ , as function of their order n . Infact, it results that ln hh H nHO ii is approximatively a linear function of n . But,from expression (49), the relevant physical information is contained in theirslope and in the corresponding differences between the two models. Then,21 .2 0.4 0.6 0.8 1.0 Χ R H Χ L Figure 5: The relative slope difference R ( χ ) as a function of the couplingconstant χ .we introduce the function R ( χ ) = 1 − (cid:0) ln hh H nHO − Oh ii (cid:1) χ (cid:0) ln hh H nHO − Mosh ii (cid:1) χ , (52)which gives the relative difference of the ln hh H nHO − J ii ( J = Oh, M osh ) slopesfor the two models at different χ values (see Fig. 5). By inspection, we candeduce that for stronger or weaker interactions, i.e. for χ ≈ χ ≈ R ( χ ) < . ρ ]as Tr (cid:2) ˆ ρ (cid:3) = p hh H HO ii − hh H HO ii − hh H HO ii ! (53)and then of the concurrence only in terms of measured energy distributionproperties. Moreover, from the arguments of Section 3 we get also an a priori Oh C Mosh Χ Figure 6: The concurrence for the Moshinsky’s model and the Ohmic modelaccordingly with the expression (53) as a function of the coupling constant χ .estimation of the correlation energy. Now, using the parametrization of thecoupling constant (51), one can compare the resulting concurrencies for theconsidered models (see Fig. 6). Qualitatively one can establish the intensityof the entanglement generated in those different models. In the present article we have clarify the relation between the entanglementand correlation energy in a bipartite system with infinite dimensional Hilbertspace. We have considered the completely solvable Moshinsky’s model of twolinearly coupled harmonic oscillators, which may constitute a simple case be-fore studying more complicated systems, like double well potentials. Thesystem has a coupling constant, which can be varied in a finite range. Thus,it continuously parametrizes two special curves in the space states: one con-taining the exact ground states, the other the separable HF states. Of course,for vanishing coupling the two curves emerge from the same state, but theirseparation can be described in terms of norm, entanglement and energy cor-relation. The peculiarity of the second curve is to lie always in a set of 0entropy entanglement states, while along the first one it increases monoton-ically (in K ), with a logarithmic divergence when the ground state becomes23egenerated. On the other hand, a similar description is given in terms of thecorrelation energy, which in principle is defined only for pairs of correspond-ing states (at the same coupling constant) in the two curves. We have provedthat entanglement and correlation energy are one-to-one along these curves,at least for the considered model. However, they are not simply proportional,but at small couplings they have a quite different rate of increasing. Thisphenomenon occurs not only if one uses the entropy of entanglement for purestates, but also if one introduces the concurrence. However, in the consideredmodel certain algebraic approximated expressions of the correlation energyin terms of the concurrence are given, so that an artifact of the calculationmethods can take a physical interpretation. However, at the moment wehave not a general method to compute directly the coefficients of such typeof expansions. These could be very useful in order to have an alternative apriori estimation of the errors made in numerical computations of the correctexpectation values of the energy. Such a type of relation may be useful inthe studies of bipartite systems with many inner degrees of freedom, like thedimers of complex molecules (see [9] for instance). In this respect the ex-plored concept of entanglement gap and its identification we made with thecorrelation energy may play an important role: it represents the energy rangewe have to be able to measure, in order to establish if a composed system is,or not, entangled. However, the theory of the entanglement gap for systemswith infinite dimensional Hilbert space does not seem completely developedas for the finite dimensional case and further investigations are needed. Inthe final section we have shown that, conditionally to the knowledge that thewhole system is not in a separated ground state, one can estimate the en-tanglement by energy measurements on the single harmonic oscillators. Thiscan be done by two sequence of position and momentum measurements, aswell by energy measurements. The distribution of the energy measurementsis sufficiently characterized in terms of its cumulants. This analysis enable usto compare among different systems at 0 temperature and distinguish theirability to generate entanglement, for instance by using the parameter R in-troduced in (52). Finally, via the formula (53) we propose a new estimatorof the entanglement, based on the first two momenta of energy distributionof the considered subsystem. We plan to check the how good is the present24pproach in considering bipartite multi-particle systems and non linearly cou-pled systems. In particular, we would like to consider integrable systems, likein [18], in which a complete analytic control of the calculations is at the hand.Another direction of research is to consider a different entropy entanglementparameter, like the quantum version of the Tsallis entropy [19]. Acknowledgments
The authors acknowledge the Italian Ministry of Scientific Researches (MIUR)for partial support and the INFN for partial support under the project Inizia-tiva Specifica LE41. We are grateful to S. Pascazio for helpful discussions.
References [1] M.A. Nielsen and I.L. Chuang
Quantum Computation and QuantumInformation (Cambridge University Press, Cambridge, 2000)[2] J. Eisert and M.B. Plenio,
Int. J. Quant. Inf. (2003) 479[3] Y. Chen, P. Zanardi, Z.D. Wang and F.C. Zhang, New J. Phys. (2006)97[4] A.N. Jordan and M. B¨uttiker, Phys. Rev. Lett. (2004) 247901[5] D.M. Collins, Z. Naturforsch. A (1993) 68[6] Z. Huang and S. Kais, Chem. Phys. Lett. (2005) 1[7] M.R. Dowling, A.C. Doherty and S.D. Bartlett,
Phys. Rev. A (2004)062113[8] A. Mohajeri and M. Alipour, Int. J. Quant. Inf. (2009) 801[9] T. Maiolo, F. Della Sala, L. Martina and G. Soliani, Theor. Math. Phys. (2007) 1146[10] M. Moshinsky,
Am. J. Phys. (1968) 522511] G.C. Ghirardi and L. Marinatto, Phys. Rev. A (2004 ) 012109[12] P. - O. L¨owdin, Phys. Rev. (1955) 1474[13] F. Buscemi, P. Bordone and A. Bertoni, Phys. Rev. A (2006) 052312[14] S. Hill and W.K. Wootters, Phys. Rev. Lett. (1997) 5022; W.K.Wootters, Phys. Rev. Lett. (1998) 2245[15] K. Audenaert, J. Eisert, M.B. Plenio and R.F. Werner, Phys. Rev. A (2002) 042327[16] U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore,2008)[17] J. Zinn–Justin,
Quantum Field Theory and Critical Phenomena (Claren-don Press, Oxford, 1993)[18] J.S. Dehesa, A. Martinez-Finkelshtein, V.N. Sorokin,
J. Math. Phys. (2003) 36[19] Xinhua Hu and Zhongxing Ye, J. Math. Phys.47