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

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 (\({{{\mathcal {L}}}}\)) and von Neumann (\({{{\mathcal {S}}}}\)) entropies, pairs \(({{{\mathcal {L}}}}(\rho ),{{{\mathcal {S}}}}(\rho ))\) are usually plotted for any valid probability distribution \(\rho \). Here, \(\rho \) can 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 \({\scriptstyle \mathrm {DCAT}}\)) states (first introduced in [6] as two-component, even and odd, states for an oscillator). These \({\scriptstyle \mathrm {DCAT}}\) states turn out to be a \({\mathbb {Z}}_2^{D-1}\) parity 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 one- and 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 one- and 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 one- and two-quDit RDMs can be considered as a discrete order parameter precursor detecting the existence of QPTs.

The paper is organized as follows. Section 2 reviews the notion of information diagram, describing its main properties, particularly with respect to the rank. Section 3 reviews the concept of \({U}(D)\)-spin coherent states and their \({\mathbb {Z}}_2^{D-1}\) parity adapted version, the \({\scriptstyle \mathrm {DCAT}}\). In Sect. 4 we compute one- and two-quDit RDMs for the \({\scriptstyle \mathrm {2CAT}}\) and the \({\scriptstyle \mathrm {3CAT}}\), their Linear and von Neumann entropies, plotting them and constructing the associated information diagrams. In Sect. 5 we use information diagrams to provide qualitative information about QPTs in Lipkin–Meshkov–Glick (LMG) models. Section 6 is devoted to conclusions.

To determine the boundaries in information diagrams [2] we need to show that, for two different measures of entropy (or information) \(E_1\) and \(E_2\), there are maximum and minimum possible values of \(E_1\) (resp. \(E_2\)) for a given value of \(E_2\) (resp. \(E_1\)) [5]. That is, the region \(\Delta \) given by the image of the map \(\rho \mapsto (E_1(\rho ),E_2(\rho ))\) is a bounded set in the plane, where \(\rho \) denotes 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 \(\rho \) with their eigenvalues \((\lambda _1,\lambda _2,\ldots ,\lambda _d)\), 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 matricesrespectively, where \(k= 1,\ldots ,d-1\). 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 \(\rho _k\) the maximally mixed density matrix (or equal probabilities distribution) in dimension k, \(\rho _k=(\frac{1}{k},{\mathop {\ldots }\limits ^{(k)}},\frac{1}{k})=\frac{1}{k}I_k\), where \(I_k\) is the identity matrix in dimension k. Then we have:where \(0_k\) is the null matrix (or vector) in dimension k and \(k=1,\ldots ,d-1\). The relation between \(\epsilon \) and \(\lambda \) is \(\lambda =\frac{1}{d}-\left( 1-\frac{1}{d}\right) \epsilon \) for Eqs. (1, 3) and \(\lambda =\frac{1-\epsilon }{k}\) for Eqs. (2, 4).

In most cases, the pair of entropies \(({{{\mathcal {L}}}},{{{\mathcal {S}}}})\) is considered, where \({{{\mathcal {L}}}}\) and \({{{\mathcal {S}}}}\) denote linear and von Neumann entropies, respectively. We shall consider here normalized linear and von Neumann entropies, i.e.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:andIn Fig. 1a the curves \(\rho \mapsto ({{{\mathcal {L}}}}(\rho ),{{{\mathcal {S}}}}(\rho ))\) are shown for \(\rho \) equal to \(\rho _\mathrm{max}(\epsilon )\) and \(\rho _{\mathrm{min}}^{(k)}(\epsilon )\), delimiting the corresponding region \(\Delta \) (we are setting \(d=5\)).

Note that the density matrices (3) can be seen (for small \(\epsilon \)) as the maximally mixed density matrix \(\rho _d\) perturbed by a rank-1 density matrix, while those of (4) can be seen as maximally mixed density matrix of dimension k, \(\rho _k\), perturbed by a (orthogonal) rank-1 density matrix, for \(k=1,\ldots ,d-1\).

It should be stressed that the range of the parameter \(\epsilon \) in the curves \(\rho _{\mathrm{min}}^{(k)}(\epsilon )\) can be extended to the interval [0, 1]. Let us denote by \({{\bar{\rho }}}_\mathrm{min}^{(k)}(\epsilon )\) the family of density matrices (4) for the range \(\epsilon \in (\frac{1}{1+k},1]\). Their corresponding curves in the information diagram are shown in Fig. 1b.

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 N identical (indistinguishable) quDits, namely, D-level identical atoms. Denoting by \(a^\dag _i\) (resp. \(a_i\)) 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)which generate the unitary symmetry \({U}(D)\). The operator \(S_{ii}\) represents the number of quDits in the level i, whereas are raising and lowering (tunnelling) operators. The fully symmetric representation space of \({U}(D)\) is embedded into Fock space, with Bose–Einstein–Fock basis ( denotes the Fock vacuum)when fixing \(n_1+\dots +n_D=N\) (the linear Casimir \(C_1=S_{11}+\dots +S_{DD}\)) to the total number N of quDits. Collective \({U}(D)\)-spin operator (11) matrix elements are given byThe expansion of a general symmetric N-particle state \(\psi \) in the Fock basis will be written aswhere \(\sum '\) 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 form¹and are labelled by complex points \({\mathbf {z}}=(z_2,\dots ,z_D)\in {\mathbb {C}}^{D-1}\). These DSCSs can be seen as Bose–Einstein condensates (BECs) of D modes, generalizing the spin \({U}(2)\) (binomial) coherent states of two modes introduced by [9, 10] long ago. If we order levels \(i=1,\dots ,D\) from lower to higher energies, the state \(|{\mathbf {z}}=0\rangle \) would be the ground state, whereas general \(|{\mathbf {z}}\rangle \) could be seen as coherent excitations. Coherent states are sometimes called “quasi-classical” states, and we shall see in Sect. 5 that \(|{\mathbf {z}}\rangle \) turns 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 \(N\rightarrow \infty \).

Expanding the multinomial (15), we identify the coefficients of the expansion (14) of the DSCS in the Fock basis aswhere we have written \(|{\mathbf {z}}|\equiv ({{\mathbf {z}}}\cdot {\mathbf {z}})^{1/2}=(1+\sum _{i=2}^D |z_i|^2)^{1/2}\) for the “length” of \({\mathbf {z}}\). Note that DSCS are not orthogonal (in general) sinceis not zero, in general. However, contrary to the standard (canonical, harmonic oscillator) CSs, they can be orthogonal when \({{\mathbf {z}}}'\cdot {\mathbf {z}}=0\).

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:where we added the superscript (N) to the N-particle coherent state (15) and \(|{\mathbf {z}}\rangle _i\) 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 \(N\rightarrow \infty \). In [8], the even parity multi-quDit cat state \({\scriptstyle \mathrm {DCAT}}\) have been constructed for general D, and here we shall reproduce the construction to fix notation.

The parity operators are defined asNote that \(\Pi _i^{-1}=\Pi _i\) and \(\Pi _1\dots \Pi _D=(-1)^N\), a constraint that says that the parity group for symmetric quDits is not \({\mathbb {Z}}_2\times {\mathop {\dots }\limits ^{D}}\times {\mathbb {Z}}_2\) but \({\mathbb {Z}}_2\times {\mathop {\dots }\limits ^{D-1}}\times {\mathbb {Z}}_2={\mathbb {Z}}_2^{D-1}\) instead. Therefore, we can discard in our discussion one of the parity operators, and we select \(\Pi _1\) (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 \(n_j\) in each level \(j=1,\dots ,D\), 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 \(|{\mathbf {z}}\rangle \) is thenThe projector onto the even parity subspace is given by:where the binary string labels the elements of the parity group \({\mathbb {Z}}_2^{D-1}\). We shall also denote the symbol for the string \((0,\ldots ,0)\).

Let us define the even parity generalized Schrödinger cat statewhere and we are using as a shorthand for . The \({\scriptstyle \mathrm {DCAT}}\) is just the projection of a DSCS onto the even parity subspace. The normalization factor is given bywhere . We shall also use the alternative notation for for convenience.

As an illustration, let us provide the particular expressions of \(|{\scriptstyle \mathrm {DCAT}}\rangle \) for \(D=2\) and \(D=3\). Denoting by \(|{\mathbf {z}}\rangle =|z_2\rangle =|\alpha \rangle \) the coherent state (15) for \(D=2\), the corresponding even parity \({\scriptstyle \mathrm {2CAT}}\) state is given bywith normalization factorwith \(L_\pm =1\pm |\alpha |^2\). Note that the overlap \(\langle \alpha |-\alpha \rangle =(L_-/L_+)^N{\mathop {\longrightarrow }\limits ^{N\rightarrow \infty }} 0\) for \(\alpha \ne 0\), which means that \(|\alpha \rangle \) and \(|-\alpha \rangle \) are macroscopically distinguishable wave packets for any \(\alpha \ne 0\) (they are orthogonal for \(|\alpha |=1\)).

Likewise, denoting by \(|{\mathbf {z}}\rangle =|z_2,z_3\rangle =|\alpha ,\beta \rangle \) the coherent state (15) for \(D=3\), the corresponding even parity \({\scriptstyle \mathrm {3CAT}}\)s state is explicitly given bywherewith \(L_{\sigma _1\sigma _2}= 1+\sigma _1|\alpha |^2+\sigma _2|\beta |^2\), for \(\sigma _1,\sigma _2=\pm \). We shall use (26) and (27) in Sect. 5, when discussing a LMG model of atoms with \(D=3\) levels. The \({\scriptstyle \mathrm {3CAT}}\) 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].

One of the most important applications of entropy measures is to quantify the entanglement of the state of a system. For that purpose, we define several types of bipartition of the whole system, computing the corresponding RDMs and entanglement measures for symmetric multi-quDit states \(\psi \) in terms of linear \({\mathcal {L}}\) and von Neumann \({\mathcal {S}}\) entropies. We shall focus on one- and two-quDit entanglement, computing the one- and two-particle RDMs (\(\rho _1\) and \(\rho _2\)) for a single and a pair of particles extracted at random from a symmetric N-quDit state \(\psi \). The procedure is straightforwardly extended to \(\rho _M\) for an arbitrary number \(M\le N/2\) of quDits. However, as we shall see, it is not necessary to go beyond two particles since the two-particle RDMs provides enough information for small values of D. Actually, in the particular case of \(D=2\), the one-particle RDM contains all necessary information about the entanglement of the system.

In [8] we gave the general expression of the one-quDit RDM of any normalized symmetric N-quDit state \(\psi \) like (14), expressed in terms of expectation values of \({U}(D)\)-spin operators \(S_{ij}\) aswhere \(E_{ij}\) represent \(D^2\), \(D\times D\)-matrices with entries \((E_{{ij}})_{lk}=\delta _{{il}}\delta _{{jk}}\) (1 in row i, column j, and 0 elsewhere). Likewise, the two-particle RDM of a symmetric state \(\psi \) of \(N>2\) quDits is written as [8]The matrices \(E_{ij}\) are the generalization to arbitrary D of standard Pauli matrices for qubits (\(D=2\)), namely \(E_{12}=\sigma _+, E_{21}=\sigma _-, E_{11}-E_{22}=\sigma _3\) and \(E_{11}+E_{22}=\sigma _0\) (the \(2\times 2\) identity matrix). Actually, the one- and two-qubit RDMs for \(D=2\) were already considered time ago by Wang and Mølmer in [15]. Here we shall consider both cases, \(D=2\) (qubits) and \(D=3\) (qutrits), in order to discuss the similitudes and differences.

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 Hamiltonianwritten in terms of collective \({U}(3)\)-spin operators \(S_{ij}\). 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 \(\epsilon /N\). For simplicity, we consider equal interactions, with coupling constant \(\lambda \), for atoms in different levels, and vanishing interactions for atoms in the same level (i.e. we discard interactions of the form \(S_{ij}S_{ji}\)). Therefore, H is invariant under parity transformations \(\Pi _j\) in (19), since the interaction term scatters pairs of particles conserving the parity of the population \(n_j\) in each level \(j=1,\dots ,D\). 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 H an intensive quantity, since we are interested in the thermodynamic limit \(N\rightarrow \infty \). 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 H is 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 formwhere we have used the parametrization \({\mathbf {z}}=(\alpha ,\beta )\), as in Eq. (26), for \({U}(3)\)-spin coherent states \(|{\mathbf {z}}\rangle \). Note that this energy surface is invariant under \(\alpha \rightarrow -\alpha \) and \(\beta \rightarrow -\beta \), which is a consequence of the inherent parity symmetry of the Hamiltonian (46) and the transformation (20) of \(|{\mathbf {z}}\rangle \) under parity.

The minimum energyis attained at the stationary (real) phase-space values \(\alpha _0^\pm =\pm \alpha _0\) and \(\beta _0^\pm =\pm \beta _0\) withIn Figs. 3 and 5 we plot (in magenta colour) the stationary-point curve \((\alpha _0(\lambda ),\beta _0(\lambda ))\) on top of one- and two-qutrit entanglement entropies, noting that \((\alpha _0(\lambda ),\beta _0(\lambda ))\rightarrow (1,1)\) for \(\lambda \rightarrow \infty \) (high interactions). Inserting (49) into (47) gives the ground state energy density at the thermodynamic limitHere we clearly distinguish three different phases: I, II and III, and two second-order QPTs at \(\lambda ^{(0)}_{\mathrm {I}\leftrightarrow \mathrm {II}}=\epsilon /2\) and \(\lambda ^{(0)}_{\mathrm {II}\leftrightarrow \mathrm {III}}=3\epsilon /2\), respectively, where \(\frac{\partial ^2E_0(\epsilon ,\lambda )}{\partial \lambda ^2}\) are discontinuous. In the stationary (magenta) curve \((\alpha _0(\lambda ),\beta _0(\lambda ))\) shown in Figs. 3a, b, 4, 5a, b and 6, the phase I corresponds to the origin \((\alpha _0,\beta _0)=(0,0)\) (square point), phase II corresponds to the horizontal part \(\beta _0=0\) up to the star point, and phase III corresponds to \(\beta _0\ne 0\).

Note that the ground state is fourfold degenerated in the thermodynamic limit since the four \({U}(3)\)-spin coherent states \(|{\mathbf {z}}_0^{\pm \pm }\rangle =|\pm \alpha _0,\pm \beta _0\rangle \) have the same energy density \(E_0\). 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 \({\scriptstyle \mathrm {3CAT}}\) (26) as an ansatz for the ground state, minimizing \(\langle {\scriptstyle \mathrm {3CAT}}|H|{\scriptstyle \mathrm {3CAT}}\rangle \), or 2) we restore the parity symmetry of the coherent state \(|\alpha _0,\beta _0\rangle \) for finite N by 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 \(N\rightarrow \infty \). 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 \({\scriptstyle \mathrm {3CAT}}\) (26), evaluated at \(\alpha =\alpha _0\) and \(\beta =\beta _0\), as a variational approximation \(|{\scriptstyle \mathrm {3CAT}}_0\rangle \) to the numerical (exact) ground state \(|\psi _0\rangle \) for 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 \(\lambda \)) of the 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) \((\alpha _0(\lambda ),\beta _0(\lambda ))\) for \(N\rightarrow \infty \). We can conclude that they do not lie in the inferior part of the region \(\Delta \), as the variational one, but as N grows 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. 7 the rank of the RDMs as a function of \(\lambda \) for both variational (\(N\rightarrow \infty \)) and numerical (\(N=50\)) solutions for the ground state of Hamiltonian (46). The QPT critical points \(\lambda ^{(0)}_{\mathrm {I}\leftrightarrow \mathrm {II}}=\epsilon /2\) and \(\lambda ^{(0)}_{\mathrm {II}\leftrightarrow \mathrm {III}}=3\epsilon /2\), are clearly marked in the case of the variational curve, with a jump from rank 1 to rank 2 at \(\lambda =\epsilon /2\) and another jump from rank 2 to rank 4 (3 in the case of 1 qutrit RDMs) at \(\lambda =3\epsilon /2\). 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 \(\lambda \simeq 0.5\), whereas the second jump takes place at slightly larges values of \(\lambda =1.5\) (in \(\epsilon =1\) 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.

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 (\({\scriptstyle \mathrm {DCAT}}\)s) using one- and two-quDit reduced density matrices, and we have applied it to the study of atom entanglement in the ground state (both variational, in the \(N\rightarrow \infty \), and numerical, with finite N) of the 3-level atom LMG model.

We have shown how the allowed region \(\Delta \) 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 \({\scriptstyle \mathrm {3CAT}}\) state is smaller that the maximum one corresponding to a maximally mixed RDM or order \(3^2\). We have already seen that this maximally entangled \({\scriptstyle \mathrm {3CAT}}\) is attained for the values \((\alpha ,\beta )=(1,1)\) (or \(\alpha =1\) for \(D=2\)), 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 \((\alpha _0(\lambda ),\beta _0(\lambda ))\) practically all the time lies in the inferior part of the information diagram subregion filled by all \({\scriptstyle \mathrm {3CAT}}\) states. We conjecture that this is due to the variational character of these states (minimum of the energy surface (47)) and the universality character of the extremal states lying at the boundary of the region \(\Delta \).

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 \(\alpha \) and \(\beta \) of \({\scriptstyle \mathrm {3CAT}}\) states (see Sect. 4), indicating that the one- and two-quDit RDMs have in general lower ranks than the maximal rank allowed by the corresponding dimension. Focusing on the variational analytic curve \((\alpha _0(\lambda ),\beta _0(\lambda ))\), and in the numerical solution for the ground state for finite N, Fig. 7 shows that the rank of one- and two-qutrit RDMs has jumps precisely at the points where QPTs occurs (or near these values in the numerical finite N case). 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 \(2^{D-1}\) different 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 \(2^{D-1}\) parity invariant states) when we extract 1, 2, or in general M quDits, and find the basis realizing the Schmidt decomposition in the larger factor.