Tensor-network approach to compute genuine multisite entanglement in infinite quantum spin chains
Sudipto Singha Roy, Himadri Shekhar Dhar, Aditi Sen De, Ujjwal Sen
TTensor-network approach to compute genuine multisite entanglementin infinite quantum spin chains
Sudipto Singha Roy, , , Himadri Shekhar Dhar, , Aditi Sen(De), and Ujjwal Sen Instituto de F´ısica T´eorica UAM/CSIC, C/ Nicol´as Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain Department of Applied Mathematics, Hanyang University (ERICA),55 Hanyangdaehak-ro, Ansan, Gyeonggi-do, 426-791, Korea Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India Institute for Theoretical Physics, Vienna University of Technology,Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria Physics Department, Blackett Laboratory, Imperial College London, SW7 2AZ London, UK (Dated: June 10, 2019)We devise a method based on the tensor-network formalism to calculate genuine multisite en-tanglement in ground states of infinite spin chains containing spin-1/2 or spin-1 quantum particles.The ground state is obtained by employing an infinite time-evolving block decimation method act-ing upon an initial matrix product state for the infinite spin system. We explicitly show how suchinfinite matrix product states with translational invariance provide a natural framework to derivethe generalized geometric measure, a computable measure of genuine multisite entanglement, in thethermodynamic limit of quantum many-body systems with both spin-1/2 and higher-spin particles.
I. INTRODUCTION
In recent years, entanglement [1] has turned out tobe an important characteristic in the study of low-dimensional strongly-correlated quantum systems, espe-cially from the perspective of critical phenomena in thelow-temperature regime of many quantum many-bodysystems [2–5] and implementation of quantum informa-tion protocols using solid-state, cold gas, and other phys-ical substrates [6–10]. While most of the attention instudying these systems has been bestowed on bipartiteentanglement measures such as entanglement of forma-tion, concurrence, or block entanglement entropy, an im-portant albeit difficult to estimate quantity is the mul-tipartite entanglement in quantum many-body systems(see Ref. [1]). Interestingly, it has often been observedthat there exist some co-operative phenomena where bi-partite entanglement and other known order parametersfail to detect the interesting physics, which are then cap-tured by multipartite entanglement [11–15]. Moreover,the study of multiparty entanglement in quantum sys-tems with higher spins, even for finite-sized systems, re-mains largely unexplored.When expanding the study of multipartite entangle-ment to understand complex quantum phenomena in thethermodynamic limit, for both spin-1/2 and higher-spinquantum particles, the innate difficulty is to character-ize computable entanglement measures (for recent de-velopments, see Refs. [11–21]). In most instances, forquantum many-body systems, the complexity in mea-suring multipartite entanglement scales exponentiallywith increasing dimension of the total Hilbert space,which in turn is associated with the number of quan-tum systems involved in the problem, and can oftenbe unamenable even with approximate methods. In re-cent years, numerical techniques such as density ma-trix renormalization group (DMRG) [22], matrix productstates (MPS) [23], and projected entangled pair states (PEPS) [24] have allowed unprecedented access to phys-ical properties of many-body systems, including estima-tion of global entanglement in low-dimensional spin sys-tems [11–13]. The growth of newer tensor-network meth-ods [25], such as multi-scale entanglement renormaliza-tion ansatz (MERA) [26], along with other significantdevelopments in higher-dimensional [27] and topologi-cal quantum systems [28], provide newer directions toexplore the role of multipartite entanglement in genericquantum systems.In this work, we employ a tensor-network based ap-proach to estimate the genuine multipartite entangle-ment, which for pure quantum states characterizes thesituation where the many-body system cannot be formedby states that are product across some bipartition(s) ofthe multiparty system. We investigate this behavior inthe thermodynamic limit of infinite chains of both spin-1/2 as well as spin-1 quantum systems. We show thatmatrix product states for infinite one-dimensional quan-tum spin systems, provide a natural framework to esti-mate the generalized geometric measure (GGM) [19] (seealso [16–18]), which is a computable measure of genuinemultipartite entanglement, defined by using the geome-try of the space of multiparty states. To demonstratethe efficacy of our formalism, we first consider a set ofprototypical Hamiltonians of low-dimensional quantumspin systems. For instance, we obtain the ground statesfor spin-1/2 systems such as the transverse Ising and the
XYZ models, using infinite time-evolving block decima-tion ( i TEBD) [29] of an initial state. We show how theGGM in the thermodynamic limit of the system can beestimated from the final infinite matrix product state( i MPS). Subsequently, we extend our study to more com-plex models such as the spin-1 Ising model with trans-verse single-ion anisotropy. Here we observe that thegenuine multipartite entanglement in the thermodynamiclimit can clearly highlight the different quantum phasesof the many-body system and the scaling of entanglement a r X i v : . [ qu a n t - ph ] J un can identify the critical points.The paper is arranged as follows. After the brief intro-duction in Sec. I, we discuss GGM as a measure of gen-uine multiparty entanglement in Section II. We then lookat how expressions for the reduced states can be obtainedfrom the infinite MPS picture in Sec. III. In Section IVwe look at how the ground states of spin chain models,containing spin-1/2 or spin-1 particles, can be derivedusing i TEBD. In Section V, we calculate the GGM forthe ground states of these different models. We concludewith a discussion in Sec. VI.
II. GENERALIZED GEOMETRIC MEASURE
A hierarchy of geometric measures of multiparty en-tanglement [18] of an N -party pure quantum state, | Ψ (cid:105) N ,can be defined in terms of geometric distance between thegiven state and the set of k -separable states, S k , whichis the set of all pure quantum states that are separableacross at least k − k subsystems. Con-sidering fidelity subtracted from unity, which is closelyconnected to the Fubini-Study and the Bures metrics, asour choice of distance measure, one can define the geo-metric measure of multiparty entanglement as [16–19] G k ( | Ψ (cid:105) N ) = 1 − max | χ (cid:105)∈S k |(cid:104) χ | Ψ (cid:105) N | , (1)where 2 ≤ k ≤ N and |(cid:104) χ | Ψ (cid:105) N | is the fidelity. Themaximization ensures that G k measures how entangled(or far away) a state | Ψ (cid:105) N is with respect to the (fromthe) closest k -separable states. In principle, a set of N − { G k } ) can bedefined, by employing the minimum distances from the N − S k . Multipartite entanglement measures, suchas the global entanglement [17], consider the distance of | Ψ (cid:105) N from the set of completely separable or N -separablestates, S k = N . These measures do not detect separabilitythat may occur across lesser number of partitions ( k The maximum Schmidt coefficient across a bipartitionrequired for GGM is the square-root of the maximumeigenvalue of the reduced density matrix of the subsys-tems across the bipartition. Obtaining the reduced den-sity matrices of an infinite-sized system, using the MPSformalism, is the primary motivation of the paper. Letus begin with the preliminary MPS representation of amany-body quantum state, | Ψ (cid:105) N , given by [24, 25] | Ψ (cid:105) N = (cid:88) i i ··· N (cid:88) α ...α N − Tr( A i α ,α A i α α . . . A Nα N − α N ) × | i , i , i , . . . i N (cid:105) , (3)where i k is the physical index, with the local system di-mension d , and α k being the auxiliary index, each witha bond dimension D . { A i k } are thus D × D matricescorresponding to each k site. For low values of D , theMPS representation of | Ψ (cid:105) N is very efficient as the num-ber of parameters required to express the state scaleswith N as N D d , instead of d N . This can be furtherreduced by considering some potential symmetry in thesystem, such as translational invariance of { A i k } matri-ces. Importantly, in order to obtain the reduced densitymatrices of a quantum many-body system, one should beable to efficiently compute the { A i k } matrices. However,there are only a few cases for which the exact MPS formof the quantum state is known [24, 25]. One such ex-ample is the unnormalized N -qubit Greenberger-Horne-Zeilinger (GHZ) [30] state, | GHZ (cid:105) N = | (cid:105) ⊗ N + | (cid:105) ⊗ N ,which is local unitarily equivalent to the possible en-tangled ground state of the Ising chain at large cou-pling strength [31]. For D = 2 (and d = 2 for qubits),the matrices for the MPS are { A i k } = { A ( k ) , A ( k ) } = { σ + σ x , σ − σ x } , ∀ k , where σ k s are the usual Paulimatrices, and σ ± = ( σ x ± iσ y ). We note that the A i k matrices are translationally invariant. Another ex-ample of TI systems is the ground state of the AKLTHamiltonian [32], where for d = 3 and D = 2, { A i k } = { A ( k ) , A ( k ) , A ( k ) } = { σ z , √ σ + , −√ σ − } , ∀ k . How-ever, in general, the matrices { A i k } can have explicitsite dependence. For example, consider the N -qubit W -state [33], | W (cid:105) N = √ N ( | . . . (cid:105) + | . . . (cid:105) + . . . | . . . (cid:105) ),which is known to be the ground state of the ferro-magnetic XX model with strong transverse field. In-terestingly, although the state is translationally invari-ant, the { A i k } matrices are not, as shown for D =2. Here, { A ( k ) , A ( k ) } = { σ + , I } , for k < N , and { A ( k ) , A ( k ) } = { σ + σ x , σ x } , for k = N , where I is the2 × { A i k } , cal-culation of reduced density matrices of quantum statesbeyond moderate-sized systems may require considerablecomputational effort, especially if the bond dimension D is not small. This is a significant road-block in thecomputation of GGM. However, if the system is TI, i.e. { A i k } = A i , ∀ k , and the A i matrices can be efficientlyestimated, then the reduced density matrices can be ob-tained even for infinite sized systems, thus allowing us tocompute the genuine multipartite entanglement of quan-tum states in the thermodynamic limit. Let us begin withan MPS representation of a TI quantum system with lo-cal dimension d and { A i } , with bond dimension D . TheMPS could be obtained as a ground state of a physicalHamiltonian or a time-evolved quantum state, quenchedfrom some initial product state. To calculate the reduceddensity matrices, we first consider the case for single-site reduced state first from the multi-qubit TI MPS. Fora very small system-size, viz. N = 2, and known A i matrices, the expression for the single-site reduced den-sity matrix, is given by ρ = E tr(( A ⊗ ¯ A ) E ) | (cid:105)(cid:104) | +tr(( A ⊗ ¯ A ) E ) | (cid:105)(cid:104) | + tr(( A ⊗ ¯ A ) E ) | (cid:105)(cid:104) | + tr(( A ⊗ ¯ A ) E ) | (cid:105)(cid:104) | , where E = (cid:80) i A i ⊗ ¯ A i is the transfer ma-trix of the translationally invariant system and ¯ A isthe conjugate transpose of A . Similarly, for N = 3, ρ = E tr(( A ⊗ ¯ A ) E ) | (cid:105)(cid:104) | + tr(( A ⊗ ¯ A ) E ) | (cid:105)(cid:104) | +tr(( A ⊗ ¯ A ) E ) | (cid:105)(cid:104) | + tr(( A ⊗ ¯ A ) E ) | (cid:105)(cid:104) | . For anarbitrary N and local dimension, d = 2 (qubit), the ex-pression for the single-site density matrix is given by ρ = (cid:88) i,j =0 tr(( A i ⊗ ¯ A j ) E N − ) E N | i (cid:105)(cid:104) j | . (4)At this stage, our aim is to generalize Eq. (4) for verylarge, and eventually, infinite systems. To this end, wefirst consider the spectral decomposition of the trans-fer matrix, in the MPS formalism for infinite system,known as i MPS, E N = (cid:80) i λ Ni | L i (cid:105)(cid:104) R i | , where | L i (cid:105) and | R i (cid:105) are the left and right eigenvectors, respectively. For N → ∞ , E has 1 as a non-degenerate eigenvalue andall other eigenvalues have modulus smaller than 1, i.e. E N = | L (cid:105)(cid:104) R | + (cid:80) D j =2 λ Nk | L k (cid:105)(cid:104) R k | . Hence, as N → ∞ , E N → | L (cid:105)(cid:104) R | . Thus, the elements of ρ , as expressedin Eq. (4), are given by ρ ij = (cid:104) L | A i ⊗ ¯ A j | R (cid:105)(cid:104) L | R (cid:105) . (5)Similarly, one can obtain the form of all m -consecutivesite l, l + 1 , l + 2 . . . ( m ≥ 2) reduced density matrices,using the relation ρ mij = (cid:104) L | A i A i . . . A i m ⊗ ¯ A j ¯ A j . . . ¯ A j m | R (cid:105)(cid:104) L | R (cid:105) , (6)where i = i i . . . i m and j = j j . . . j m . For non-consecutive sites, l, l + r , l + r + r , . . . the expressionof the m -site reduced density matrix is given by ρ mij = (cid:104) L | ˜ A E r − ˜ A E r − .. ˜ A m | R (cid:105)(cid:104) L | R (cid:105) , where ˜ A k = ( A i k ⊗ ¯ A j k ).This has remarkable significance as the number of pa-rameters required to represent the m -site density ma-trices is reduced from d m to D d . The reduced densitymatrix can thus be used to estimate the genuine multisiteentanglement in systems described using infinite MPS. IV. GROUND STATE MPS USING i TEBD We briefly describe the algorithm to simulate theground state of an infinite, one-dimensional quantummany-body Hamiltonian, H , using the infinite MPS for-malism. We start with an arbitrary MPS, | Ψ (cid:105) N , as ex-pressed in Eq. (3), and then eventually build the groundstate i MPS using infinite time-evolving block decima-tion method. To this end, starting from | Ψ (cid:105) N , we per-form an imaginary time-evolution: | Ψ (cid:105) N → e − τ H | Ψ (cid:105) N .The ground state configuration | Ψ (cid:105) N is then obtainedwhen τ becomes very large i.e. | Ψ (cid:105) N ∼ | Ψ (cid:105) N + (cid:80) d N i =1 e − τ ( E i − E ) | Ψ i (cid:105) N τ →∞ −−−−→ | Ψ (cid:105) N . In order to performthe i TEBD, we first use second order Suzuki-Trotter (ST)decomposition [35] on the exponential unitary operationand express each term in the TI matrix product operator(MPO) form [36, 37]. This essentially helps to change theoptimization problem of the energy for the total system,to the optimization associated with each decomposed TIMPO. After one such ST iteration, we obtain an MPS, | Ψ t (cid:105) , which, in general, has a bigger bond dimension thanthe initial MPS. Therefore, one needs to truncate this tothe allowed bond dimension D . We then normalize theimaginary time evolved state and choose that as a seedfor the next time iteration. After each such ST step, en-ergy per site ( E /N = N (cid:104) Ψ t |H| Ψ t (cid:105) ) is calculated andthe expressions of the { A i } matrices for the i MPS of theground state of the given Hamiltonian are then obtainedby minimizing the energy. In general, energy per sitescales with the size of the system. However, throughsome intermediary steps, one can show that for N → ∞ ,it converges, to E ∞ (say). Hence, the final { A i } matricesare obtained when the energy per site converges.To apply the above i MPS formalism we begin with aone-dimensional quantum system consisting of spin-1/2particles. Such a quantum many-body Hamiltonian canbe written, with a certain degree of genericity, as H = (cid:88) (cid:104) ij (cid:105) (cid:0) J x S xi S xj + J y S yi S yj + ∆ S zi S zj (cid:1) + (cid:88) i hS zi , (7)where J x , J y are the coupling constants along x - and y - directions respectively, ∆ is the “anisotropy” alongthe z - direction, h is the strength of the transverse field, S k = σ k are the Pauli spin matrices, and (cid:104) ij (cid:105) denotes thenearest-neighbor sites. Two important models that canbe derived from H are the transverse Ising (in the limit J y = ∆ = 0), the anisotropic XYZ model ( J x ( y ) = J ± γ ,and h = 0) [38–40]. Note that in the limit γ = 0, the XYZ model reduces to the anisotropic XXZ model, which hasgained some attention in studies on strongly-correlatedsystems [41]. We note that in recent years, cooperativephenomena in quantum spin chains have been widely ex-plored in the context of quantum information theory, es-pecially in terms of entanglement [2–5] and other quan-tum correlations [42].We next look at the i MPS representation for more com-plex quantum spin systems. For instance, we considera quantum many-body chain with higher-spin particles,viz. the spin-1 Ising model with a transverse field akin toparameters arising from single-ion anisotropy generatedby crystal fields [43]. These systems can also be con-sidered to be a derivative of the Blume-Emery-Griffithsmodel [44], where the quadratic terms have been ne-glected. Such models have lately been used to studyphase transitions in multicomponent fluids and semicon-ductor systems [45]. The Hamiltonian of the spin-1 modelis thus given by¯ H = J z (cid:88) (cid:104) ij (cid:105) S zi S zj + K ( S xi ) , (8)where S i ’s are generalizations of the Pauli matrices fora spin-1 system, J z denotes the coupling along the z -direction and K denotes the strength of the single-ionanisotropy parameter due to the crystal field in the trans-verse direction. The model undergoes a quantum phasetransition at J z K = 2 [43].In implementing the i MPS form and the i TEBD algo-rithm for obtaining the ground state of these Hamiltoni-ans, we fix the bond dimension at D = 10 and choose theinitial Trotter step to be τ = 10 − , which is then grad-ually changed to 10 − to improve accuracy. The conver-gence of the ground state energy is determined with anaccuracy 10 − . Once the ground state i MPS is obtained,one can access the Schmidt coefficients across all possi-ble bipartitions of the quantum state by contracting thetensors efficiently, as shown in Eq. (6). The behavior ofgenuine multisite entanglement in ground state phases of N = 8 N = 10 N = 12 i MPS G h/J x FIG. 1. (Color online.) Variation of GGM ( G ) with fieldstrength ( h ) for the transverse Ising model, for differentone-dimensional lattice sizes, viz. N = 8 (green-diamonds),10 (blue-squares), 12 (black-triangles) and infinite N (red-circles). Both axes represent dimensionless quantities. the Hamiltonian, in the thermodynamic limit, can thenbe estimated from the generalized geometric measure. V. GENUINE MULTISITE ENTANGLEMENTIN THE THERMODYNAMIC LIMIT For transverse Ising model, we consider a region awayfrom critical point ( h/J x = 1), viz. 1 . ≤ h/J x ≤ 2. Thevariation of GGM ( G ) with respect to the transverse fieldstrength h/J x , is depicted in Fig. 1. The thermodynamiclimit of the genuine multisite entanglement, in the infinitespin lattice, is compared with the corresponding valuesobtained for finite-sized lattices ( N = 8, 10, and 12) us-ing exact diagonalization. In order to compute the valueof GGM ( G ) using exact diagonalization method, in allthe cases ( N = 8 , , (cid:80) i =1 (cid:0) Ni (cid:1) for N = 12). We notethat for the transverse field Ising model, maximum valueof Schmidt coefficient always comes from the single-sitereduced density matrices. We use this fact to computethe value of GGM ( G ) in the thermodynamic limit us-ing i MPS. Therefore, in our case, Eq. (4) will serve thepurpose. For this model, in the region parametrized by0 ≤ h/J x ≤ . 8, energy gap closes and as discussed ear-lier, it is not possible to compute the multiparty entan-glement using the measure GGM for non-unique groundstates. The figure shows a distinct scaling of G at fieldstrengths closer to the critical point, h/J x = 1. In thisregion, difference between the GGM ( G ) values, obtainedusing exact diagonalization method ( N = 12) and i MPS,turns out to be at most ≈ − . Away from it, G quicklybecomes scale invariant, and approaches its thermody-namic limit even for low N . Here, difference betweenthe GGM ( G ) values computed for N = 12 and i MPS,becomes (cid:46) − . N = 8 N = 10 N = 12 i MPS G ∆ /J FIG. 2. (Color online.) Variation of GGM ( G ) with ∆ /J forthe XYZ model with γ = 0 . 5, for different one-dimensionallattice sizes. Both axes represent dimensionless quantities. Inthe inset we plot the same quantities for γ = 0 case ( XXZ model). Let us now consider the XYZ Hamiltonian in absenceof magnetic field, i.e., J x , J y , ∆ (cid:54) = 0, h = 0. The behav-ior of G with ∆ /J , for the anisotropic XYZ Hamiltonianwith γ = 0 . 5, is depicted in Fig. 2. Unlike the Ising case,from the exact diagonalization results for this model, wenote that the maximum value of Schmidt coefficient al-ways comes from the consecutive two-site reduced densitymatrices. We again use this result to compute the valueof GGM in the thermodynamic limit using i MPS. There-fore, in this case, we use Eq. (6) for computation of themaximum Schmidt coefficients. Like as the Ising case.here also degeneracy hinders us to find a unique groundstate for the region − ≤ ∆ /J ≤ 0. Therefore, for thismodel, we consider following region between two criticalpoints, for both finite and infinite lattices, parametrizedby 0 . ≤ ∆ /J ≤ . 0. Figure 2 shows that in contrast tothe transverse Ising model, no scale invariance is achievedfor G even away from the critical points, and it is notpossible to achieve the thermodynamic limit by exactlydiagonalizing a spin model with small system size. Inthis case, difference between the GGM values computedfor N = 12 and i MPS, at small values of ∆ /J becomes (cid:46) − , which further increases to (cid:46) − as ∆ /J tendsto 1. For the XXZ model ( γ = 0) (see the inset of Fig. 2),where the critical points are known to exist in the vicin-ity of ∆ /J = ± 1, a similar absence of scale invariance isobserved. Here difference between the GGM values com-puted for N = 12 and i MPS never decreases below 10 − .Thus, i MPS plays a significant role in computing genuinemultipartite entanglement in these systems.We now look at the genuine multipartite entanglementproperties of the more complex higher-spin model, viz.the spin-1 Ising model with single-ion anisotropy as ex-pressed in Eq. (8). We note that in contrast to the spin-1/2 models, behavior of multipartite entanglement in thisspin-1 model is unexplored even for finite spin systems.Here, we look at the behavior of GGM in the thermo- N = 4 N = 6 N = 8 i MPS G K / J x z FIG. 3. (Color online.) Variation of GGM ( G ) with KJ z for the spin-1 model described in Eq. (8), for different one-dimensional lattice sizes, viz. N = 4 (green-diamonds),6 (blue-squares), 8 (black-triangles), and infinite N (red-circles). Both axes represent dimensionless quantities. In theinset, we show the GGM in the region close to the transitionpoint ( KJ z ≈ dynamic limit of the system using the i MPS formalism.The behavior of genuine multiparty entanglement is plot-ted in Fig. 3. We again note that like the transverseIsing model, the maximum Schmidt coefficient in thiscase also comes from the single-site reduced density ma-trices. Moreover, as in the previous cases, we observethat GGM starts decreasing monotonously with the in-crease of the strength of the single-ion anisotropy or thecrystal field. However, for the spin-1 model, the scalingpattern of GGM in the thermodynamic limit shows sev-eral interesting features. For instance, before K < 2, inmost of the regions, GGM increases with system size. Onthe other hand, for K > 2, the trend is reversed, i.e., thevalue of GGM decreases with the increase of N . How-ever, the variation of GGM with the anisotropy parame-ter clearly detects the critical points in the system. Weobserve that near the value K ≈ 2, GGM becomes almostscale invariant, which is a known value at which quan-tum phase transition occurs in the system. Therefore,our study shows that the scaling of GGM can identifythe vital characteristics of the critical phenomena in thespin-1 model. VI. DISCUSSION In this work, we have shown how the tensor-networkapproach provides a natural structure to study genuinemultiparty entanglement, quantified by generalized ge-ometric measure, in many-body quantum systems. Inparticular, the method involved matrix product states toefficiently obtain the reduced density matrices of infinitequantum spin lattices, which upon making use of sym-metries such as translational invariance of the matrices,allowed us to accurately estimate the generalized geomet-ric measure of systems consisting of both spin-1/2 andhigher-spins. The method thus provided us a viable the-oretical framework to look at interesting cooperative andcritical phenomena by investigating multiparticle phys-ical quantities in the thermodynamic limit of quantummany-body systems.Importantly, this approach to compute generalized ge-ometric measure using tensor networks is in principle alsoapplicable for higher-dimensional lattices, provided therelevant tensors under the i MPS formalism are accessi-ble using available numerical techniques. Finally, we alsonote that the formalism presented in the work may pro-vide useful directions in investigating genuine multipar-tite entanglement properties in several quantum systems,including condensed matter, photonic, and other topolog-ical systems, where tensor-network methods have turnedout to be successful in studying physical properties. ACKNOWLEDGMENTS The research of SSR was supported in part by the IN-FOSYS scholarship for senior students. HSD acknowl-edges funding by the Austrian Science Fund (FWF),project no. M 2022-N27, under the Lise Meitner pro-gramme of the FWF. Appendix A: Proof of GGM as a measure of genuinemultiparty entanglement Here, we present a very concise proof for the GGM tobe a measure of genuine multiparty entanglement, start-ing from the concept of k -separability and the definitionof the geometric measures of multiparty entanglementin Eq. (1). An important point to note is that G is the minimum distance from the set of all k -separablequantum states, S k ∀ k . However, in principle, as mea-surements over general entangled bases yield higher orequal values as compared to those over product bases,the maximum fidelity in Eq. (1), can always be consid-ered from the set S k with lowest k , as they contain moreclustered partitions. Hence, for G , the set S of bi-separable states contains a closest separable state. Let { λ i A : B } di =1 and {| φ i (cid:105) A , | ˜ φ i (cid:105) B } di =1 be the set of real, non-negative Schmidt coefficients and corresponding orthog-onal vectors, respectively, across the bipartition A : B ,where d = max { d A , d B } . A bi-separable state, in gen-eral, can be written as, | χ (cid:105) = | η (cid:105) A | ˜ η (cid:105) B . The fidelity isthen given by |(cid:104) χ | Ψ (cid:105) N | = | (cid:88) i λ i A : B (cid:104) η | φ i (cid:105) A (cid:104) ˜ η | φ i (cid:105) B | = | (cid:88) i λ i A : B f i A g i B | . (A1)A value of fidelity, possibly non-maximal, corresponds to | η (cid:105) A = | φ k (cid:105) A and | ˜ η (cid:105) B = | ˜ φ k (cid:105) B , such that f k A = g k B = 1,where k gives λ A : B = λ k A : B = max { λ i A : B } . Thus we have, |(cid:104) χ | Ψ (cid:105) N | ≥ λ A : B . 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