Information diagrams in the study of entanglement in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models
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Spanish Government PGC2018-097831-B-I00
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Quantum Information Processing (2022) 21:223 https://doi.org/10.1007/s11128-022-03524-7 Information diagrams in the study of entanglement in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models Julio Guerrero1,2 ·Alberto Mayorgas3·Manuel Calixto2,3 Received: 30 November 2021 / Accepted: 16 April 2022 / Published online: 24 June 2022 © The Author(s) 2022 Abstract In this paper we pursue the use of information measures (in particular, information diagrams) for the study of entanglement in symmetric multi-quDit systems. We use generalizations to U(D)of spin U(2)coherent states and their adaptation to parity (multicomponent Schrödinger cats), and we analyse oneand two-quDit reduced density matrices. We use these correlation measures to characterize quantum phase transitions occurring in Lipkin–Meshkov–Glick models of D=3-level identical atoms, and we propose the rank of the corresponding reduced density matrix as a discrete order parameter. Keywords Information diagrams ·Entanglement entropies ·Symmetric quDits · Parity adapted coherent states ·Quantum phase transitions ·Many-body systems · Parity adapted states BJulio Guerrero [email protected] Alberto Mayorgas [email protected] Manuel Calixto [email protected] 1Department of Mathematics, University of Jaén, Campus Las Lagunillas s/n, 23071 Jaén, Spain 2Institute Carlos I of Theoretical and Computational Physics (iC1), University of Granada, Fuentenueva s/n, 18071 Granada, Spain 3Department of Applied Mathematics, University of Granada, Fuentenueva s/n, 18071 Granada, Spain 123
223 Page 2 of 21 J. Guerrero et al. 1 Introduction Information diagrams were introduced to discuss the relation between two different information measures, like von Neumann entropy and error probability [1], or von Neumann and linear entropies [2]. In the particular case of linear (L) and von Neumann (S) entropies, pairs (L(ρ), S(ρ)) are usually plotted for any valid probability distribution ρ. Here, ρcan also represent the density matrix of a quantum system (or rather a vector with its eigenvalues), and this is our main interest in this paper. Special attention is paid to the boundaries of the resulting information diagram region, where the associated probability distributions (or density matrices) will be denoted as “extremal”. In Ref. [3], a comparison is made between both entropies in the case of two qubits (see also [4] for the case of the ion-laser interaction). In [5], a detailed study of information diagrams is carried out for arbitrary pairs of entropies. There it is proved that, for certain conditions (satisfied by linear, von Neumann and Rényi entropies), the extremal density matrices are always the same. Counterexamples are given but, in general, the deviation will be very small and we can safely assume that these extremal density matrices have universal character. In this paper we shall use information diagrams to obtain global qualitative information of particle entanglement in symmetric multi-quDit systems described by generalized “Schrödinger cat” (multicomponent DCAT) states (first introduced in [6]as two-component, even and odd, states for an oscillator). These DCAT states turn out to be a ZD−1 2parity adaptation of U(D)-spin coherent (quasi-classical) states, and they have the structure of a quantum superposition of weakly overlapping (macroscopically distinguishable) coherent wave packets with interesting quantum properties. For that purpose we make use of oneand two-quDit reduced density matrices (RDM), obtained by extracting one or two particles/atoms from a composite system of N identical quDits described by a cat state, and tracing out the remaining system. It is well known (see [3] and references therein) that the entropy of these RDMs provides information about the entanglement of the system. We shall plot the information diagrams associated with these RDMs and extract qualitative information about oneand two-quDit entanglement, and also about the rank of the corresponding RDM, which also provides information on the entanglement of the original system [7]. We shall apply these results to the characterization of quantum phase transitions (QPT) occurring in Lipkin–Meshkov–Glick models of 3-level identical atoms, complementing the results of [8]. In particular, we have seen that the rank of the oneand two-quDit RDMs can be considered as a discrete order parameter precursor detecting the existence of QPTs. The paper is organized as follows. Section 2reviews the notion of information diagram, describing its main properties, particularly with respect to the rank. Section 3reviews the concept of U(D)-spin coherent states and their ZD−1 2parity adapted version, the DCAT. In Sect. 4we compute oneand two-quDit RDMs for the 2CAT and the 3CAT, their Linear and von Neumann entropies, plotting them and constructing the associated information diagrams. In Sect. 5we use information diagrams to provide qualitative information about QPTs in Lipkin–Meshkov–Glick (LMG) models. Section 6is devoted to conclusions. 123
Information diagrams in the study of entanglement... Page 3 of 21 223 2 Information diagrams To determine the boundaries in information diagrams [2] we need to show that, for two different measures of entropy (or information) E1and E2, there are maximum and minimum possible values of E1(resp. E2) for a given value of E2(resp. E1) [5]. That is, the region given by the image of the map ρ→ (E1(ρ), E2(ρ)) is a bounded set in the plane, where ρdenotes all possible probability distributions (or density matrices) for a given dimension d. Since usual measures of entropy for density matrices are based on the trace, they are invariant under changes of basis. Hence, the only relevant information of a density matrix is contained in its eigenvalues; thus, in this paper we shall identify density matrices ρwith their eigenvalues (λ1,λ 2,...,λ d), the order being irrelevant. Therefore, for our purposes, we can identify probability distributions and density matrices using a vector notation in terms of eigenvalues, referring to both of them as density matrices for short. Notwithstanding, we shall continue to treat density matrices as matrices in some situations. In [5] it was proved that, under rather general assumptions on the convexity/concavity of the entropy measures, the maximum and minimum values are always attained in two standard forms of density matrices ρmax(λ) =(λ, ¯ λ, (d−1) ... ,¯ λ) , ¯ λ=1−λ d−1≤λ, λ∈1 d,1,(1) ρ(k) min(λ) =(λ, (k) ...,λ,¯ λ, 0,...,0), ¯ λ=1−kλ<λ, λ∈1 k+1,1 k,(2) respectively, where k=1,...,d−1. Let us write the previous equations as (convex) sums of density matrices, that in turn can be seen as lower dimensional density matrices. For that purpose denote by ρkthe maximally mixed density matrix (or equal probabilities distribution) in dimension k,ρk=(1 k,(k) ..., 1 k)=1 kIk, where Ikis the identity matrix in dimension k. Then we have: ρmax() =(1−)ρd+ρ 1⊕0d−1,∈[0,1)(3) ρ(k) min() =(1−)ρk⊕0d−k+0k⊕ρ1⊕0d−1−k,∈0,1 1+k(4) where 0kis the null matrix (or vector) in dimension kand k=1,...,d−1. The relation between and λis λ=1 d−1−1 dfor Eqs. (1,3) and λ=1− kfor Eqs. (2,4). In most cases, the pair of entropies (L,S)is considered, where Land Sdenote linear and von Neumann entropies, respectively. We shall consider here normalized linear and von Neumann entropies, i.e. L(ρ) =d d−11−Tr(ρ2),S(ρ) =−Tr(ρ logdρ), (5) 123
223 Page 4 of 21 J. Guerrero et al. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Linear entropy von Neumann entropy (a) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Linear entropy von Neumann entropy (b) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Linear entropy von Neumann entropy (c) Fig. 1 aInformation diagram for linear and von Neumann entropies in dimension d=5, where the region is bounded by the curves associated with the density matrices given by Eqs. (3) (above) and (4) (below). All curves except ρmax are traced from left to right when increases. bCurves associated with density matrices ¯ρ(k) min() for k=1,...,d−1, which are traced from right to left. Note that in the case k=1 the associated curve is the same as in a, but traced backwards. Also, for k=d−1 the associated curve coincides with that of ρmax.cPlot of the asymptotic curves (8,9) for density matrices near a pure state (bottom-left, red and pink, respectively) and the asymptotic curve (10) near the maximally mixed state (upper-right, green) (Color figure online) in such a way that both entropies range from 0 (pure states) to 1 (maximally mixed states). The values of both entropies for each family of curves are: L(ρmax()) =1−2, S(ρmax()) =−(d−1)1− dlogd1− d−1+(d−1) d ×logd1+(d−1) d,(6) and L(ρ(k) min()) =d d−11−2−(1−)2 k, S(ρ(k) min()) =−(1−)logd(1−) −logd() +(1−)logd(k). (7) In Fig. 1a the curves ρ→ (L(ρ), S(ρ)) are shown for ρequal to ρmax() and ρ(k) min(), delimiting the corresponding region (we are setting d=5). Note that the density matrices (3) can be seen (for small ) as the maximally mixed density matrix ρdperturbed by a rank-1 density matrix, while those of (4) can be seen as maximally mixed density matrix of dimension k,ρk, perturbed by a (orthogonal) rank-1 density matrix, for k=1,...,d−1. It should be stressed that the range of the parameter in the curves ρ(k) min() can be extended to the interval [0,1]. Let us denote by ¯ρ(k) min() the family of density matrices (4) for the range ∈(1 1+k,1]. Their corresponding curves in the information diagram are shown in Fig. 1b. 123
Information diagrams in the study of entanglement... Page 5 of 21 223 Fig. 2 Coloured plot of a sample of 1000 density matrices of dimension d=5 randomly generated following aχ2distribution for the eigenvalues in an information diagram where the different colours represent the rank of the density matrix (warmer colours represent higher ranks) (Color figure online) 2.1 Information diagrams and rank of density matrices As it can be seen in Fig. 1b, there are only d−3 distinct ¯ρ(k) min curves, for k=2,...,d− 2. These curves divide the region into d−2 subregions, k,k=2,...,d−1, bounded by the curves ρ(k) min,¯ρ(k) min and ¯ρ(k−1) min . Each subregion kcontains density matrices of rank greater than k. Density matrices of rank 1 (pure states) lie on the origin, while density matrices of rank 2 lie on the curve ρ(1) min =¯ρ(1) min. See Fig. 2for a plot of a sample of 20000 density matrices of dimension d=5 randomly generated following a χ2distribution for the eigenvalues where the colour of the corresponding point in the information diagram is associated with its rank (warmer colours correspond to higher rank). From the expression of the extremal density matrices (1,2), or their alternative expressions (3,4), and the expression of the inner curves ¯ρ(k) min(), it is clear that, for a given value of the linear entropy and a fixed rank k+1, the extreme values of the von Neumann entropy are reached for kidentical eigenvalues. If the remaining eigenvalue is larger than the rest (i.e. we are in ¯ρ(k) min), then there is a maximum, and if it is smaller than the rest (in ρ(k) min), then it is a minimum of von Neumann entropy. 2.2 Asymptotic curves It is interesting to obtain approximate expressions for the function S(L)in some regions of the information diagram. Near a pure state (bottom left of the information diagram), we have the following asymptotic expressions for the curves ρmax and ρ(1) min: S(L)=d−1 2dlog(d)(1+log(2d))L−Llog(L),(8) S(L)=d−1 2dlog(d)(1+log(2d)−log(d−1))L−Llog(L),(9) 123
223 Page 6 of 21 J. Guerrero et al. respectively. Near the maximally mixed density matrix (upper right of the information diagram), both ρmax and ρ(d−1) min collapse into the same curve, with asymptotic expression: S(L)=1−d−1 2log(d)(1−L). (10) See Fig. 1c for a plot of these asymptotic curves in an information diagram with d=5. Once we have explained what the information diagrams are, and their main features, we shall use them in the study of oneand two-quDit entanglement of generalized Schrödinger cat states, which arise as a parity adaptation of U(D)-spin (symmetric multi-quDit) coherent states. 3U(D)-spin coherent states and their adaptation to parity in symmetric multi-quDit systems In this section we introduce the main ingredients and notation required to define parity adapted U(D)-spin coherent states in symmetric multi-quDit systems. These kinds of states were introduced long ago in [6] as nonclassical (even and odd) states of light. We shall particularize to D=2 and D=3 for practical cases. See [8] for a more detailed study of the general case. We consider a system of Nidentical (indistinguishable) quDits, namely, D-level identical atoms. Denoting by a† i(resp. ai) the creation (resp. annihilation) operator of an atom in the i-th level (namely, i=1,2 for ground and excited—or spin up and down—in the case D=2, or i=1,2,3 for a 3-level atom in the case of D=3), the collective U(D)-spin operators can be expressed (in the fully symmetric representation) as bilinear products of creation and annihilation operators as (Schwinger representation) Sij =a† iaj,i,j=1,...,D,(11) which generate the unitary symmetry U(D). The operator Sii represents the number of quDits in the level i, whereas Sij,i= jare raising and lowering (tunnelling) operators. The fully symmetric representation space of U(D)is embedded into Fock space, with Bose–Einstein–Fock basis (| 0denotes the Fock vacuum) |n=|n1,...,nD=(a† 1)n1...(a† D)nD √n1!...nD!| 0,(12) when fixing n1+···+nD=N(the linear Casimir C1=S11 +···+SDD)tothe total number Nof quDits. Collective U(D)-spin operator (11) matrix elements are given by 123
Information diagrams in the study of entanglement... Page 7 of 21 223 m|Sii|n=niδm,n, m|Sij|n=(ni+1)njδmi,ni+1δmj,nj−1 k=i,j δmk,nk,∀i= j.(13) The expansion of a general symmetric N-particle state ψin the Fock basis will be written as |ψ= n cn|n= n1+···+nD=N cn1,...,nD|n1,...,nD,(14) where is a shorthand for the restricted sum. Among all symmetric multi-quDit states, we shall pay special attention to U(D)-spin coherent states (DSCSs for short), which adopt the multinomial form1 |z=|z2,...,zD= 1 √N!a† 1+z2a† 2+···+zDa† D 1+|z2|2+···+|zD|2N | 0,(15) and are labelled by complex points z=(z2,...,zD)∈CD−1. These DSCSs can be seen as Bose–Einstein condensates (BECs) of Dmodes, generalizing the spin U(2) (binomial) coherent states of two modes introduced by [9,10] long ago. If we order levels i=1,...,Dfrom lower to higher energies, the state |z=0would be the ground state, whereas general |zcould be seen as coherent excitations. Coherent states are sometimes called “quasi-classical” states, and we shall see in Sect. 5that |zturns out to be a good variational state that reproduces the energy and wave function of the ground state of multilevel LMG atom models in the thermodynamic (classical) limit N→∞. Expanding the multinomial (15), we identify the coefficients cnof the expansion (14)oftheDSCS|zin the Fock basis as cn(z)=N! D i=1ni!D i=2zni i |z|N,(16) where we have written |z|≡(z·z)1/2=(1+D i=2|zi|2)1/2for the “length” of z. Note that DSCS are not orthogonal (in general) since z|z= (z·z)N (z·z)N/2(z·z)N/2,z·z≡1+¯z 2z2+···+¯zDzD,(17) is not zero, in general. However, contrary to the standard (canonical, harmonic oscillator) CSs, they can be orthogonal when z·z=0. 1In Eq. (15) and the following ones we have put z1=1, where z1is the parameter multiplying a† 1,see [8]. Consequently, it has been removed from the expression of |z. 123
223 Page 8 of 21 J. Guerrero et al. In [8] we have shown that DSCSs are separable and exhibit no quDit entanglement (although they do exhibit interlevel entanglement). In fact they can be written as a tensor product of 1-quDit coherent states: |z(N)=|z1⊗|z2⊗···⊗|zN,(18) where we added the superscript (N)to the N-particle coherent state (15) and |zi denotes the one-particle coherent state for the i-th quDit. Note that this state is explicitly symmetric under the interchange of quDits and therefore there is no need to symmetrize it. The situation changes when we deal with parity adapted DSCSs, sometimes called “Schrödinger cat states”, since they are a quantum superposition of weakly overlapping (macroscopically distinguishable) quasi-classical coherent wave packets. These kind of cat states arise in several physical situations and display interesting nonclassical properties. The case of even parity cat states is particularly important since they turn out to be good variational states [10], reproducing the energy of the ground state of quantum critical models in the thermodynamic limit N→∞.In[8], the even parity multi-quDit cat state DCAT have been constructed for general D, and here we shall reproduce the construction to fix notation. The parity operators are defined as j=exp(iπSjj), j=1,...,D.(19) Note that −1 i=iand 1...D=(−1)N, a constraint that says that the parity group for symmetric quDits is not Z2×D ...×Z2but Z2×D−1 ... ×Z2=ZD−1 2instead. Therefore, we can discard in our discussion one of the parity operators, and we select 1(since we will use level 1 as reference level in Sect. 5). Parity operators are conserved when the Hamiltonian scatters pairs of particles conserving the parity of the population njin each level j=1,...,D, like in the D-level LMG model considered in Sect. 5. Using the multinomial expansion (15), it is easy to see that the effect of parity operators on symmetric DSCSs |zis then i|z=i|z2,...,zi,...,zD=|z2,...,−zi,...,zD,i=2,...,D.(20) The projector onto the even parity subspace is given by: even =21−D b∈{0,1}D−1 b2 2b3 3... bD D,(21) where the binary string b=(b2,...,bD)∈{0,1}D−1labels the elements of the parity group ZD−1 2. We shall also denote the symbol 0for the string (0,...,0). Let us define the even parity generalized Schrödinger cat state |DCAT= 1 N(DCAT)even|z= 21−D N(DCAT) b|zb,(22) 123
Information diagrams in the study of entanglement... Page 9 of 21 223 where |zb≡|(−1)b2z2,...,(−1)bDzDand we are using bas a shorthand for b∈{0,1}D−1.TheDCAT is just the projection of a DSCS onto the even parity subspace. The normalization factor is given by N(DCAT)2=21−DbLN b LN 0 (23) where Lb=1+(−1)b2|z2|2+···+(−1)bD|zD|2. We shall also use the alternative notation Lσ≡Lbfor σ=(−1)b=((−1)b2,...,(−1)bD)for convenience. As an illustration, let us provide the particular expressions of |DCATfor D=2 and D=3. Denoting by |z=|z2=|αthe coherent state (15)forD=2, the corresponding even parity 2CAT state is given by |2CAT= 1 2N(2CAT)|α+|−α,(24) with normalization factor N(2CAT)2=1 21+1−|α|2 1+|α|2N=1 2 LN ++LN − LN + ,(25) with L±=1±|α|2. Note that the overlap α|−α=(L−/L+)NN→∞ −→ 0forα= 0, which means that |αand |−αare macroscopically distinguishable wave packets for any α= 0 (they are orthogonal for |α|=1). Likewise, denoting by |z=|z2,z3=|α, βthe coherent state (15)forD=3, the corresponding even parity 3CATs state is explicitly given by |3CAT= 1 4N(3CAT)|α, β+|−α, β+|α, −β+|−α, −β,(26) where N(3CAT)2=1 41+(1−|α|2+|β|2)N+(1+|α|2−|β|2)N+(1−|α|2−|β|2)N (1+|α|2+|β|2)N =1 4 LN ++ +LN −+ +LN +− +LN −− LN ++ ,(27) with Lσ1σ2=1+σ1|α|2+σ2|β|2,forσ1,σ 2=±. We shall use (26) and (27)in Sect. 5, when discussing a LMG model of atoms with D=3levels.The3CAT state has also been used in U(3)vibron models of molecules [11,12] and Dicke models of 3-level atoms interacting with a polychromatic radiation field [13,14]. 123
223 Page 16 of 21 J. Guerrero et al. (a) (b) (c) (d) Fig. 5 Contour plots of alinear L∞ 2and bvon Neumann S∞ 2entanglement entropies of the two-qutrit RDM ρ2(3CAT)of a U(3)Schrödinger cat (26)forN→∞, as a function of the phase-space coordinates α, β (they just depend on moduli). The meaning of the magenta curve is the same as in Fig. 3. The asymptotic behaviour of cL∞ 2and dS∞ 2for large values of |α|and |β|displays isentropic curves θ=constant, according to the expression of ρ∞ 2(3CAT)in Eq. (44) (Color figure online) 5 Information diagrams and quantum phase transitions in Lipkin–Meshkov–Glick models of 3-level identical atoms Now we apply the previous results to the study of QPTs of D-level Lipkin–Meshkov– Glick atom models. The standard case of D=2 level atoms has already been studied in the literature (see e.g. [17]). We shall restrict ourselves to D=3 level atoms for practical calculations, although the procedure can be easily extended to general D.In particular, we propose the following LMG-type Hamiltonian H= N(S33 −S11)−λ N(N−1) 3 i=j=1 S2 ij,(46) written in terms of collective U(3)-spin operators Sij. Hamiltonians of this kind have already been proposed in the literature [18–22] [see also [23] for the role of mixed symmetry sectors in QPTs of multi-quDit LMG systems]. We place levels symmetrically about i=2, with intensive energy splitting per particle /N. For simplicity, we consider equal interactions, with coupling constant λ, for atoms in different levels, 123
Information diagrams in the study of entanglement... Page 17 of 21 223 and vanishing interactions for atoms in the same level (i.e. we discard interactions of the form SijSji). Therefore, His invariant under parity transformations jin (19), since the interaction term scatters pairs of particles conserving the parity of the population njin each level j=1,...,D. Energy levels have good parity, the ground state being an even state. We divide the two-body interaction in (46) by the number of atom pairs N(N−1)to make Han intensive quantity, since we are interested in the thermodynamic limit N→∞. We shall see that parity symmetry is spontaneously broken in this limit. As already pointed long ago by Gilmore and coworkers [10,24], coherent states constitute in general a powerful tool for rigorously studying the ground state and critical properties of some physical systems in the thermodynamic limit. The energy surface associated with a Hamiltonian density His defined in general as the coherent state expectation value of the Hamiltonian density in the thermodynamic limit. In our case, the energy surface acquires the following form E(α,β)(, λ) =lim N→∞z|H|z =β¯ β−1 α¯α+β¯ β+1−λα2¯ β2+1+β2+1¯α2+¯ β2+β2 α¯α+β¯ β+12,(47) where we have used the parametrization z=(α, β),asinEq.(26), for U(3)-spin coherent states |z. Note that this energy surface is invariant under α→−αand β→−β, which is a consequence of the inherent parity symmetry of the Hamiltonian (46) and the transformation (20)of|zunder parity. The minimum energy E0(, λ) =minα,β∈CE(α,β)(, λ) (48) is attained at the stationary (real) phase-space values α± 0=±α0and β± 0=±β0with α0(, λ) =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0,0≤λ≤ 2, 2λ− 2λ+, 2≤λ≤3 2, 2λ 2λ+3,λ≥3 2, β0(, λ) =0,0≤λ≤3 2, 2λ−3 2λ+3,λ≥3 2.(49) In Figs. 3and 5we plot (in magenta colour) the stationary-point curve (α0(λ), β0(λ)) on top of oneand two-qutrit entanglement entropies, noting that (α0(λ), β0(λ)) → (1,1)for λ→∞(high interactions). Inserting (49)into(47) gives the ground state energy density at the thermodynamic limit E0(, λ) =⎧ ⎪ ⎨ ⎪ ⎩ −, 0≤λ≤ 2,(I) −(2λ+)2 8λ, 2≤λ≤3 2,(II) −4λ2+32 6λ,λ≥3 2.(III) (50) 123
223 Page 18 of 21 J. Guerrero et al. (a) (b) Fig. 6 Information diagram for the family of a1-qutrit RDMs and b2-qutrit RDMs for 3CAT in the limit N→∞, for all values of |α|and |β|, where the curves of numerical RDMs, as a function of λfor different values of N, has been added, as well as the analytical stationary curve for N→∞(in magenta). Observe that as Ngrows, the numerical (green) curves approach the (magenta) analytical one (Color figure online) Here we clearly distinguish three different phases: I, II and III, and two secondorder QPTs at λ(0) I↔II =/2 and λ(0) II↔III =3/2, respectively, where ∂2E0(,λ) ∂λ2are discontinuous. In the stationary (magenta) curve (α0(λ), β0(λ)) shown in Figs. 3a, b, 4,5a, b and 6, the phase I corresponds to the origin (α0,β 0)=(0,0)(square point), phase II corresponds to the horizontal part β0=0 up to the star point, and phase III corresponds to β0= 0. Note that the ground state is fourfold degenerated in the thermodynamic limit since the four U(3)-spin coherent states |z±± 0=|±α0,±β0have the same energy density E0. These four coherent states are related by parity transformations and, therefore, parity symmetry is spontaneously broken in the thermodynamic limit. In order to have good variational states for finite N, to compare with numerical calculations, we have two possibilities: 1) either we use the 3CAT (26) as an ansatz for the ground state, minimizing 3CAT|H|3CAT, or 2) we restore the parity symmetry of the coherent state |α0,β 0for finite Nby projecting on the even parity sector. Although the first possibility offers a more accurate variational approximation to the ground state, it entails a more tedious numerical minimization than the one already obtained in (48) for N→∞. Therefore, we shall use the second possibility which, despite being less accurate, it is straightforward and good enough for our purposes. That is, we shall use the 3CAT (26), evaluated at α=α0and β=β0, as a variational approximation |3CAT0 to the numerical (exact) ground state |ψ0for finite N. Let us apply the tools developed in previous sections to this model and draw the main conclusions. Firstly, in Fig. 6a, b, we have added to the information diagrams for 1 and 2 qutrits RDMs already shown in Fig. 4, the curves (as a function of λ)ofthe numerically computed ground states of the 3-level LMG model for different values on N(in green colours), together with the already shown analytical variational curve (in magenta) (α0(λ), β0(λ)) for N→∞. We can conclude that they do not lie in the inferior part of the region , as the variational one, but as Ngrows the numerical curves approach the analytical one. Secondly, suggested by the results about the rank of 1 and 2 quDits RDMs of Sect. 4, we plot in Fig. 7the rank of the RDMs as a function of λfor both variational (N→∞) 123
Information diagrams in the study of entanglement... Page 19 of 21 223 Fig. 7 Plot of rank of 1-quDit and 2-quDit RDMs along the stationary curve both for analytical (variational, N→∞) and numerical (N=50) solution of the 3-level LMG model as a function of λ(in =1 units) and numerical (N=50) solutions for the ground state of Hamiltonian (46). The QPT critical points λ(0) I↔II =/2 and λ(0) II↔III =3/2, are clearly marked in the case of the variational curve, with a jump from rank 1 to rank 2 at λ=/2 and another jump from rank 2 to rank 4 (3 in the case of 1 qutrit RDMs) at λ=3/2. In the case of the numerical curve, where a small threshold has been applied to the eigenvalues to suppress spurious oscillations, the first jump continues to be at λ≃0.5, whereas the second jump takes place at slightly larges values of λ=1.5(in=1 units). This behaviour is the same as with other precursors of QPTs, like susceptibility of fidelity in the 3-level LMG model [23]. From this, it is clear that the rank of the RDMs is a good precursor of a QPT, with the advantage of being a discrete parameter. 6 Conclusions In this paper we have used an information-theoretic tool like the information diagrams to extract qualitative information about the quDit entanglement (and rank) of parity adapted U(D)-spin coherent states (DCATs) using oneand two-quDit reduced density matrices, and we have applied it to the study of atom entanglement in the ground state (both variational, in the N→∞, and numerical, with finite N) of the 3-level atom LMG model. We have shown how the allowed region of information diagrams is completely filled in the case of one-qutrit RDMs, while only the lower part of it is partially filled in the case of two-qutrits RDMs. This indicates that the maximum pairwise (2-qutrit) entanglement attained in a 3CAT state is smaller that the maximum one corresponding to a maximally mixed RDM or order 32. We have already seen that this maximally entangled 3CAT is attained for the values (α, β) =(1,1)(or α=1forD=2), and these are precisely the values obtained for the variational analytical approximation to the ground state of a 3-level LMG model in the high coupling regime. In addition, we have shown that the variational curve (α0(λ), β0(λ)) practically all the time lies in the inferior part of the information diagram subregion filled by all 3CAT states. We conjecture that this is due to the variational character of these states 123
223 Page 20 of 21 J. Guerrero et al. (minimum of the energy surface (47)) and the universality character of the extremal states lying at the boundary of the region . Information diagrams also provide qualitative information about the rank of the RDMs. This has motivated us to study with detail their rank for different values of the parameters αand βof 3CAT states (see Sect. 4), indicating that the oneand two-quDit RDMs have in general lower ranks than the maximal rank allowed by the corresponding dimension. Focusing on the variational analytic curve (α0(λ), β0(λ)), and in the numerical solution for the ground state for finite N,Fig.7shows that the rank of oneand two-qutrit RDMs has jumps precisely at the points where QPTs occurs (or near these values in the numerical finite Ncase). Therefore the rank can be used as a discrete precursor of a QPT in the LMG model, but this conclusion can be probably extended to other critical models. All these results motivate us to further study the application of information diagrams and rank of RDMs to other parity adapted U(D)-spin coherent states, but with different parity character. Here we have restricted ourselves to the even case, but remember that there are 2D−1different parity adapted U(D)-spin coherent states, the even one just being a particular case. For example, odd parity cat states (for D=2) are known to be well suited as variational states to approximate excited states in, for example, the Dicke model of superradiance [25]. Since the rank of a RDM is equal to the Schmidt number, by the Schmidt decomposition theorem (see, for instance, [26]), it would be interesting to study with detail the Schmidt decomposition of parity adapted U(D)-spin coherent states (not only of the even one, but for all 2D−1parity invariant states) when we extract 1, 2, or in general MquDits, and find the basis realizing the Schmidt decomposition in the larger factor. Acknowledgements We thank the support of the Spanish MICINN through the Project PGC2018-097831B-I00 and Junta de Andalucía through the Projects SOMM17/6105/UGR, UHU-1262561, FQM-381 and FEDER/UJA-1381026. AM thanks the Spanish MIU for the FPU19/06376 predoctoral fellowship. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Feder, M., Merhav, N.: Relations between entropy and error probability. IEEE Trans. Inf. Theory 40, 259 (1994). https://doi.org/10.1109/18.272494 2. Harremoes, P., Topsoe, F.: Inequalities between entropy and index of coincidence derived from information diagrams. IEEE Trans. Inf. Theory 47, 2944 (2001). https://doi.org/10.1109/18.959272 3. Wei, T.-C., Nemoto, K., Goldbart, P.M., Kwiat, P.G., Munro, W.J., Verstraete, F.: Maximal entanglement versus entropy for mixed quantum states. Phys. Rev. A 67, 022110 (2003). https://doi.org/10.1103/ PhysRevA.67.022110 123
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