Multipartite entanglement serves as a faithful detector for quantum phase transitions

We investigate quantum phase transitions (QPTs) in various spin chain systems using the multipartite entanglement measure τSEF based on the monogamy inequality of squared entanglement of formation (EOF). Our results demonstrate that τSEF is more effective and reliable than bipartite entanglement or bipartite correlation measures such as EOF, von Neumann entropy, and quantum discord in characterizing QPTs. τSEF not only detects critical points that may go unnoticed by other detectors but also avoids the issue of singularity at non-critical points encountered by other measures. Furthermore, by applying τSEF , we have obtained the phase diagram for the XY spin chain with three and four interactions and discovered a new quantum phase.


INTRODUCTION
Entanglement is a pure quantum phenomenon and an essential feature that distinguishes quantum mechanics from classical mechanics [1].Its nonlocality has always puzzled scientists [2][3][4][5], and essentially all significant experiments to date have supported the quantummechanical predictions [6][7][8][9][10][11]. On the other hand, entanglement can serve as a carrier of quantum information and is considered a crucial resource in quantum information processing.Therefore, the study of entanglement has always held a significant position in the field of quantum information [12][13][14][15].
Quantum phase transitions (QPTs) [16], which are purely driven by quantum fluctuations, occupy a significant position in quantum many-body systems.Researchers even believe that QPTs play significant roles in understanding many puzzles in physics, such as heavy-fermion metals and high-temperature superconductors [17].
Many-body systems often exhibit multipartite entanglement or correlation (it is possible for more than two parties to be entangled in different inequivalent ways, and multipartite entanglement can inevitably reflect a greater amount of diverse and sophisticated quantum information [37]).Therefore, it should obviously have an advantage in characterizing quantum phase transitions, as reflected by changes in the degree of entanglement.However, most of the QPT detectors currently being studied fall into the category of bipartite correlation.Although there have been some studies on the relationship between multipartite entanglement and quantum phase transitions [38][39][40][41][42], most of them focus on identifying specific quantum phase transitions and there is still lack of research on the universality of the ability in detecting QPTs of multipartite entanglement.Therefore, in this paper, we focus on the effectiveness of multipartite entanglement in determining quantum phase transitions for different systems.The corresponding quantum phase transitions, which vary depending on the system being studied, include the first-order type and BKT-type QPTs observed in the XXZ model, as well as the topologicaltype QPT seen in the SSH model, among others.We also explore the characteristics and advantages of using multipartite entanglement compared to bipartite entanglement or bipartite correlation detectors for determining quantum phase transitions.

II. METHODS
Entanglement monogamy is one of the most important properties in multipartite systems, which implies that quantum entanglement cannot be freely shared in many-body systems [43].Based on quantitative entanglement monogamy inequalities [44][45][46][47], one can construct effective measures or indicators for multipartite entanglement.The entanglement of formation possesses operational physical meaning and is related to quantum data compression [48].It has been proven that the squared entanglement of formation obeys the monogamy inequality and its residual entanglement has been demonstrated to be a trustworthy measure for genuine multipartite entanglement in quantum many-body systems [47,49].
In an N -qubit multipartite system, genuine multipartite entanglement based on the residual entanglement of formation can be expressed as [47] where E f ρ A1|A2•••AN quantifies the entanglement between one arbitrary selected qubit A 1 and the rest qubits of the system, and can be written as is the von Neumann entropy derived from the singlequbit reduced density matrix ρ A1 and the minimum runs over all the pure state decompositions [42,50] (if the state considered is itself a pure state, there is no need to perform the decomposition as we do in this paper); E f (ρ A1A k ) is the entanglement of formation (EOF) between the two qubits A 1 and A k , and can be defined as [26,51] [48], where λ n (n = 1, 2, 3, and 4) are the square roots of the eigenvalues of ρ A1A k ρA1A k in descending order, and A1A k is the complex conjugation of ρ A1A k , and σ y refers to the y component of the Pauli operator.
In addition to EOF, for a comprehensive comparison, we also consider other bipartite correlation detectors that are currently more effective in determining quantum phase transitions: quantum discord and quantum entanglement based on von Neumann entropy.Here, we consider the two-site entanglement entropy [19] where ρ r represents the two-site reduced density matrix of A 1 and A j , which are two lattice qubits separated by a distance r = j − 1.Compared to the single-site entropy used in Eq.( 1), E v 2 (ρ r ) only include one more qubit in the reduced density matrix used for calculation, but is more effective in detecting QPTs [20].The QD can be written as where is the conditional entropy, and the minimum is achieved from a complete set of projective measures {A j k } on site A j [26,52,53].The projectors here can be written as where {|k } is the standard basis {|↑ , |↓ } of any two selected spins, and the transform matrix V is parameterized as [54] V = cos θ e −iϕ sin θ e iϕ sin θ − cos θ .
Then the minimum of the conditional entropy S(ρ r ) can be obtained by traversing the θ and ϕ.
To calculate the above detectors, we use the exact diagonalization (ED) techniques to simulate the systems.To strictly construct the matrix form of the Hamiltonian, we choose the S z representation and work in the whole S z space.Periodic boundary conditions are considered for all the calculated systems to reduce the influence of the boundary.In addition, we point out that, to calculate τ S EF , it is necessary to calculate the EOF between the selected qubit and all other qubits.This is undoubtedly a challenge for large systems, but we found that due to the periodic boundary conditions, the EOF between corresponding lattice sites located on either side of the selected lattice site has a symmetric characteristic.Therefore, only the situation on one side needs to be calculated.Moreover, it was found that the formation of entanglement rapidly decreases with the increase of qubits spacing.Therefore, in practical calculations, it is usually only necessary to calculate some lattice sites with close distances.This has important practical significance for large-scale systems, such as the possible future consideration of using the density matrix renormalization group (DMRG) numerical calculation method to calculate larger systems.

III. RESULTS AND DISCUSSIONS
A. The BKT-type QPT in the XXZ model The Hamiltonian for the XXZ spin model is defined as follows: where ∆ describes the anisotropy of the spin-spin interaction in the z direction, and σ j represents the usual Pauli matrices at site j.It is well known that the model is in a critical phase for −1 ≤ ∆ ≤ 1, an antiferromagnetic phase at ∆ ≥ 1, and ferromagnetic phase at ∆ < −1.
The critical point (CP) at ∆ = 1 belongs to the continuous BKT type, while the CP at ∆ = −1 is a fist-order transition caused by the ground state level crossing [55].The τ SEF and E v 2 results from the exact diagonalization for different system sizes N with periodic boundary conditions are shown in Fig. 1.The first order CP at ∆ = −1 is clearly detected by both detectors, while the BKT-type CP at ∆ = 1 can only be reflected by the minimum of τ SEF (see Fig. 1(a)).The location of the minimum of τ SEF does not change with N , and it will exist even in the thermodynamic limit: the value of the minimum is linearly related to 1/N 2 .As N tends to infinity, meaning that 1/N 2 tends to 0, it approaches a fixed value of 0.36 (8) (see the size scaling behavior in the inset of Fig. 1(a)).However, for this CP, E v 2 exhibits a smooth curve behavior, as shown in Fig. 1(b), and its derivative does not exhibit any singularity (see the inset of Fig. 1(b)); thus, it cannot reflect the BKT-type transition.This result is in agreement with the results of the TMRG in Ref. [30].
We must emphasize that both quantum discord (QD) and entanglement of formation (EOF) can indeed reflect phase transitions in the XXZ model, and the corresponding findings have been published in Refs.[30,56].However, our primary objective was to demonstrate the broader applicability of residual entanglement as a detector for quantum phase transitions compared to other detection methods.Therefore, our focus was on examining whether residual entanglement can overcome the limitations of other detectors.Given that enumerating the corresponding results for QD and EOF would not align with our main theme, we opted not to include them here.

B. The topological QPT in the SSH model
Then we consider the topological QPT in the onedimensional Su-Schrieffer-Heeger (SSH) model [57].The Hamiltonian is determined as follows: where the operator c j,α destroys a spinless fermion at the unit cell j of type α = A, B (where A and B are the sublattice indices), and η represents the dimerization.
There is a topological phase transition at η = 0.When η transitions from being negative to positive, the system undergoes a transformation from a topological phase to a topological trivial phase [57,58].Different from the general QPTs, a topological phase transition can not be characterized by a local order parameter [16,60].
The multipartite entanglement results of τ SEF are shown in Fig. 2.There is a maximum that appears at η = 0, and it becomes clearer as N increases.The    value of τ SEF at η = 0 increases linearly with decreasing 1/N 2 , reaching a maximum value of approximately 0.925 (8) when N approaches infinity, specifically when 1/N 2 = 0, as shown in the inset of Fig. 2. The maximum clearly indicates the topological QPT of the system.As a comparison, the bipartite entanglement entropy E v 2 changes continuously and smoothly near η = 0, which does not reflect the QPT.To reflect its occurrence, we need to use the derivative dE v 2 /dη, which exhibits a minimum behavior, and the minimum position η m can be considered as a pseudo-critical point [61,62]: as the system size N increases, the minimum becomes more pronounced and moves towards the critical point η c = 0 (see Fig. 3).The performance of the other detectors, such as the QD and EOF, show similar behaviors (see Fig. 4).In order to obtain the accurate critical point, we need to conduct the finite-size scaling analysis.However, scaling analysis of most practical systems at large scales is often difficult.Therefore, it has certain advantages in determining quantum phase transitions for the behavior of the residual entanglement τ SEF , where the indicated critical point is independent of the system size.
Further research finds that when the spacing between the two selected spins r > 1, E v 2 can also exhibit a minimum behavior near the phase transition point η c , as shown in Fig. 3(b).When r is even, the minimum point accurately corresponds to η c .When r is odd, the position of the minimum significantly deviates from η c - the smaller r is, the greater the deviation.Moreover, this deviation does not change significantly as N increases, as shown in the inset of Fig. 3(b).Consequently, to obtain the exact location of the phase transition point, it may be necessary to calculate on a significantly larger scale N or consider r = N/2, which represents the maximum distance between two spins in the system.We believe that the reason for this is related to the structure of the spin-spin interaction in the system.The variation in the interaction of odd and even bonds gives rise to different symmetries in the bond interactions between the selected pair of spins and the remaining spins.If it is symmetrical, it will align with the overall symmetry of the system's ground-state vector.Conversely, a larger size is necessary to minimize the influence of the selected spin positions on the overall symmetry.
C. The QPT in the SSH-XY model The quantum phase transitions mentioned above are well known to researchers.Next, we consider an unknown quantum phase transition that occurs in a novel system.We call this model the SSH-XY model, which is similar to the SSH-XXZ model in Ref. [63], and its Hamiltonian can be written as where σ α represents the Pauli matrices (α = x, y) as those in Eq.( 7).γ 1 and γ 2 describe the anisotropy in the XY directions that arises from the spin-spin interaction between the odd and even bonds, respectively.The results for γ 1 = −0.8 are plotted in Fig. 5. τ SEF displays a sharp peak behavior at γ 2 = 0.8, which indicates a quantum phase transition.To further demonstrate this, we also calculated other detectors, as shown in  2 do not exhibit singularities (see the inset in Fig. 5(b)) but their first-order derivatives exhibit extreme behavior at this location.As the value of N increases, these behaviors become increasingly evident (the τ SEF results for different values of N are presented in the inset in Fig. 5(a)), indicating the existence of a quantum phase transition point at γ 2 = 0.8.QD has similar results, and for EOF and E v 2 , regardless of how large N is, the derivative behavior is necessary to capture the phase transition, so only the results for N = 20 are given here.In addition, the deep behavior near at γ 2 = 0.66 for τ SEF is weakened as N increases, tending towards the critical point γ 2 = 0.8 and thus, is not an indicator of QPT.
The occurrence of this phase transition is actually easy to understand.It is the result of the competition between the X and Y directions from the odd and even coupling bonds.In the process where γ 2 is constantly increasing from −1, the corresponding system will experience an increase in coupling in the X direction and a decrease in coupling in the Y direction.When γ 2 > 0.8, the coupling in the X direction on the even bond dominates the system, resulting in a phase transition similar to the phase transition point at γ = 0 in the XY model.The results remain completely consistent for other values of γ 1 , and we have chosen γ 1 = −0.8 as a representative point for demonstration purposes only.Here, τ SEF , as well as QD, which contains more quantum correlation information compared to two-body entanglement, can accurately reflect the occurrence of phase transitions without requiring derivation.

D. XY model with multi-site interactions(XYMI)
The Hamiltonian of the XY spin model in a transverse field with three-and four-spin interactions can be written as follows: where σ α , γ, and λ have the same meaning as those in Eq.( 7); α and β denote the three-spin and fourth-spin interactions, respectively.Using the Jordan-Wigner, Fourier, and Bogoliubov transformations successively [16], the spin model can be mapped into a fermion model, and then the following diagonalized analytical solution can be obtained in momentum space: ) and x k = 2πk/N is the energy spectrum(Ref.[27]).According to Ref. [67,68], the reduced density matrix for two arbitrary sites i and j at arbitrary distance r = j − i can be obtained as follows where the elements can be calculated from the correlation functions The mean magnetization and correlation functions can be written as [64,65]  where the reduced density matrices for a single-site ρ i or two-site ρ ij can be obtained, and all the detectors introduced in Sec.II can be calculated.
We first consider the case of β = 0, where the phase diagram is known [27,31].When |λ| is large enough, all spins become polarized, and the system enters a spinsaturated phase.However, when |λ| is small, there is only one spin liquid phase for small α values, and the phase transition between it and the spin-saturated phase belongs to the Ising-like order-disorder type QPT.As α increases, another type of spin liquid phase appears in the system [27,31,66].The results of τ SEF , QD, and EOF as a function of λ for γ = 0 and α = 0.5 are plotted in the left panel of Fig. 6.One can see that there are three obvious turning points in both the curves of τ SEF and QD.Obviously, it corresponds to the spin-polarized state in the region where λ < −1.12 (II) and λ > 2 (I).Because the magnetic field in these regions is strong enough, all spins become polarized, and the entanglement and correlation between spins disappear.As a result, the values of τ SEF and QD are zero in this phase.The turning point at λ = 0 separates the two spin liquid phases, SL-I and SL-II.Here, τ SEF and QD demonstrate equivalent capabilities in detecting quantum phase transitions, which is more evident in their derivative behavior (see Fig. 6(c) and Fig. 6(d)), where the non-analytical peak positions, which reflect QPTs, are entirely consistent.As a comparison, when λ < 0, EOF is always zero, making it impossible to detect the QPT located at λ = −1.12(seeFig. 6(b)).At the same time, the peak of dQD/dλ that reflects the occurrence of this QPT is also very weak.Therefore, it seems that τ SEF is a more reliable indicator for detecting the QPTs in the system.Further study has confirmed this point.
In Fig. 7, we present the results of the derivatives of τ SEF and QD for various values of γ and α.The twosite entropy E v 2 is also calculated for comparison.When γ = 0.01 and α = 0.25 (left panel in Fig. 7), three detec- tors provide a completely consistent result for the position of the CP that separates the SL-I and SL-II states.However, during the quantum phase transition from the SL-II state to the spin-polarized II state, there is a noticeable peak behavior in dτ SEF /dλ, and the location of the reflected phase transition, λ m ≈ −0.747, closely matches the result obtained from dE v 2 /dλ.In contrast, only a small, less noticeable bump is shown here for dQD/dλ, and its location has an obvious deviation from λ m .
The comparison of the results for γ = α = 0.5 is even more pronounced (see the right panel of Fig. 7).The two CPs indicated by the three detectors, λ c1 ≈ 2 and λ c2 ≈ 0, which separate the SL-I state from the spinpolarized state I and the SL-II state, are completely consistent.However, at the third CP λ c3 , dQD/dλ yields a significantly different result compared to the other two detectors.The peak positions given by dτ SEF /dλ and dE v 2 /dλ that reflect the occurrence of the phase transition at λ c3 = −1 are basically consistent, while dQD/dλ has no obvious singularity in this area, and its slow peak position also differs significantly from the results for dτ SEF /dλ.Moreover, there is no finite-size effect; all the results are less affected by the system sizes, as shown in Figs.7(d) and (e), where all the curves for different N almost collapse into a single one.
When it comes to the QD itself (see Fig. 8), except for the dramatically changing points at λ ≈ 2 and λ ≈ 0, which correspond to the CPs λ c1 and λ c2 , respectively, there is a peak behavior at λ = −1.13 for the case when r = 1.However, the peak position deviates significantly from the CP λ c3 , and there are other non-analyzed points at significantly different λ for cases where r > 1.This is consistent with the conclusion that the singularity of QD does not always correspond to a QPT [58].The possible reason is that the calculations of QD involve the min function procedures in Eq.( 4), which may destroy the analyticity of elements in the density matrix [59].Therefore, compared to QD, multipartite quantum entanglement has a clear advantage in reflecting the phase transition here.
Using the derivative of multipartite entanglement dτ SEF /dλ, we obtain the phase diagram of the system for the case when β = 0.The derivatives dτ SEF /dλ and its contour map for γ = 0 as functions of λ and α are shown in Fig. 9.The sharp peaks and valleys in the figure clearly separate these three phases, dividing the system into four regions.Regions I and II correspond to the spin-polarized state, while regions SL-I and SL-II correspond to two spin liquid states, respectively.Actually, for a comparison, we choose the same parameter conditions as that in Ref. [31].The phase diagram given by dτ SEF /dλ are consistent with those in Refs.[31,66] given by the steered quantum correlation, but τ SEF can avoid the problem of non-analytic points appearing at non-critical points for the steered quantum correlation(see details in Ref. [31]).
We then consider the influence of the four-spin interaction β on the QPTs of the system.The results for dτ SEF /dλ with γ = 0.1 and α = 0.5 are shown in Fig. 10.The sharp peaks or sudden drops in the value of the derivative of the multipartite entanglement clearly indicate the regions of criticality.Except for the already known quantum states I, II, SL-I, and SL-II, which can be easily recognized by their continuous existence from the β = 0 case, a new quantum state named SL-III appears when β > 0.2.As β increases, the SL-III phase expands to a larger parameter range, while the region where the SL-II phase exists gradually shrinks.However, the parameter range where phase SL-I exists is almost unaffected.This indicates that phases SL-II and SL-III are related to the multi-site interactions in the system, and β is beneficial for the stability of quantum phase SL-III.Although the specific physical properties of the new state need to be further studied in the future, the results of multipartite entanglement provide clear evidence of its existence.

IV. SUMMARY
To summarize, we have examined the efficacy and superiority of the residual entanglement τ SEF as a measure of multipartite entanglement in determining quantum phase transitions in various typical physical systems.After comparing several commonly used detectors known for their superior performance in determining quantum phase transitions, it was discovered that multipartite quantum entanglement possesses distinct advantages.
The bipartite quantum entanglement E v 2 cannot characterize the BKT-type QPT in the XXZ model.To accurately detect the location of the topological QPT in the SSH model, apart from the multipartite entanglement τ SEF , other bipartite detectors require the use of derivative behavior and size scaling analysis.For the EOF, the value of some quantum states in the XYMI model will become zero, which cannot reflect the occurrence of phase transitions.The phenomenon of QD in characterizing the SL-II to spin-polarized II phase transition is not readily apparent and it may exhibit singularity at a non-phase transition point.In contrast, the residual entanglement τ SEF completely avoids the issues of these detectors and can unequivocally identify the occurrence of these quantum phase transitions.
We believe that the mechanism is that the residual entanglement τ SEF measures the multipartite entanglement in the system, which incorporates entanglement information concerning both the considered spin and all other spins.This allows it to provide an overall and comprehensive reflection of changes in the entanglement degrees of the system.In contrast, detectors based on bipartite correlation, which reflect the entanglement or correlation information between partial states, are easily affected by the chosen lattice site positions and are not immune to the influence of interactional structures.This is especially true for models such as the SSH and XYMI models, where asymmetry and multi-spin interactions are present.The localized nature of these detectors limits their ability to reflect overall information changes in the system.Hence, multipartite quantum entanglement possesses inherent advantages over bipartite entanglement and correlation in determining quantum phase transitions.Following this line of thought, and considering that other measures of quantum correlations may contain more quantum correlation information than quantum entanglement, we believe that multipartite quantum correlations may have more advantages in characterizing quantum phase transitions.Further research in this area is needed in the future.In addition, experimentally, the multipartite entanglement based on residual entanglement of formation can be obtain via quantum state tomography [69] on one-site and two-site reduced density matrices of the ground state, which can be used to calculate the relevant bipartite entanglements.We hope that this paper offers valuable theoretical insights for future experimental efforts in this field.
Furthermore, by utilizing the derivative of τ SEF , we present the quantum phase diagram of the XYMI model and reveal a new spin liquid phase that emerges from the interaction of four spins.

FIG. 1 :
FIG. 1: (Color online) (a) Residual entanglement τSEF and (b) two-site entropy E v 2 as functions of ∆ for the XXZ model, with different system sizes N .The inset in (a) shows the finite size-scaling behavior of the minimum value of τSEF at δ = 1.0.

FIG. 2 :
FIG. 2: (Color online) Residual entanglement τSEF as a function of η under different system sizes N for the SSH model.The inset at the bottom of the figure illustrates the finitesize scaling behavior of the maximum value of τSEF at η = 0, while the inset in the upper right corner of the figure provides an enlarged view of the curves of τSEF near η = 0 for better clarity.

FIG. 3 :
FIG. 3: (Color online) (a) E v 2 and dE v 2 /dη as a function of η for r = 1 and different N in the SSH model.The arrows indicate the corresponding vertical axis of these curves.(b) E v 2 as a function of η for different values of r when N = 24.The dashed circles indicate the minimum points of the curves, and the inset demonstrates that the valley position for r = 3 is less affected by the system size N .

FIG. 4 :
FIG. 4: (Color online) (a) QD and EOF as a function of η for r = 1 under different N for the SSH model.The arrows indicate the corresponding vertical axis of these curves.(b) and (c) show the derivatives of QD and EOF in (a), respectively, where the peak or deep position does not directly locate at the accurate QPT point.In order to obtain the accurate critical point, we need to conduct the finite-size scaling analysis.

FIG. 5 :
FIG. 5: (Color online) (a) QD and residual entanglement τSEF and (b) the γ2 derivatives of EOF and E v 2 as a function of γ2 under γ1 = −0.8 and N = 20.The inset in (a) shows the peak portion of τSEF for different values of N , while the behaviors of EOF and E v 2 are displayed as an inset in (b).The arrows indicate the vertical axis of the corresponding curves.

Fig. 5 .
Fig.5.The QD displays a non-analytical turning point at the same parameter position, whereas the EOF and von Neumann entropy E v 2 do not exhibit singularities (see the inset in Fig.5(b)) but their first-order derivatives exhibit extreme behavior at this location.As the value of N increases, these behaviors become increasingly evident (the τ SEF results for different values of N are presented in the inset in Fig.5(a)), indicating the existence of a quantum phase transition point at γ 2 = 0.8.QD has similar results, and for EOF and E v 2 , regardless of how large N is, the derivative behavior is necessary to capture the phase transition, so only the results for N = 20 are given here.In addition, the deep behavior near at γ 2 = 0.66 for τ SEF is weakened as N increases, tending towards the critical point γ 2 = 0.8 and thus, is not an indicator of QPT.The occurrence of this phase transition is actually easy to understand.It is the result of the competition between the X and Y directions from the odd and even coupling bonds.In the process where γ 2 is constantly increasing from −1, the corresponding system will experience an increase in coupling in the X direction and a decrease in coupling in the Y direction.When γ 2 > 0.8, the coupling in the X direction on the even bond dominates the system, resulting in a phase transition similar to the phase transition point at γ = 0 in the XY model.The results remain completely consistent for other values of γ 1 , and we have chosen γ 1 = −0.8 as a representative point for demonstration purposes only.Here, τ SEF , as well as QD, which contains more quantum correlation information compared to two-body entanglement, can accurately

FIG. 6 :
FIG. 6: (Color online) (a) τSEF and (b) QD and EOF as functions of λ under γ = 0, α = 0.5, β = 0, and N = 5001 for the XYMI model.The derivatives of τSEF and QD under the same conditions are shown in (c) and (d), respectively.Dashed lines indicate the positions of the peaks and separate the system into four states: I, II, SL-I, and SL-II.

FIG. 7 : 2 ,
FIG. 7: (Color online) The derivative behaviors of τSEF , E v 2 , and QD as a function of λ.The left panel represents the case where γ = 0.01, α = 0.25, and β = 0 with N = 1001, while the right panel represents the case where γ = α = 0.5 and β = 0 with different values of N .The position of the peak in dE v 2 /dλ is indicated by the dashed line, which marks the phase transition point from SL-II to IV.

I
FIG. 10: (Color online) The derivative dτSEF /dλ and its contour map as functions of β and λ for γ = 0.1, α = 0.5, and N = 1001.The dashed lines serve as a guide for the points of extremes in the curves.