Phase diagram of quantum critical system via local convertibility of ground state

We investigate the relationship between two kinds of ground-state local convertibility and quantum phase transitions in XY model. The local operations and classical communications (LOCC) convertibility is examined by the majorization relations and the entanglement-assisted local operations and classical communications (ELOCC) via Rényi entropy interception. In the phase diagram of XY model, LOCC convertibility and ELOCC convertibility of ground-states are presented and compared. It is shown that different phases in the phase diagram of XY model can have different LOCC or ELOCC convertibility, which can be used to detect the quantum phase transition. This study will enlighten extensive studies of quantum phase transitions from the perspective of local convertibility, e.g., finite-temperature phase transitions and other quantum many-body models.

Many developments in quantum information processing (QIP) 1 unveiling the rich structure of quantum states and the nature of entanglement have offered many insights into quantum many-body physics 2 . Because the ground-state wavefunction undergoes qualitative changes at a quantum phase transition, it is important to understand how its genuine quantum aspects evolve throughout the transition 3,4 . For the spin chain model, this question has been widely discussed 4,5 . Many alternative indictors of quantum phase transitions, for example, entanglements measured by concurrence 3 , negativity 6 , geometric entanglement 7 , and von Neumann entropy 8,9 have been investigated in several critical systems, which have become a focus of attention in detecting a number of critical points. From the viewpoint of quantum correlations, mutual information 10 , quantum discord 11 and global quantum discord [12][13][14] have been used for detecting quantum phase transitions. There are investigations on entanglement spectra [15][16][17] , fidelity 18 and local quantum thermal susceptibility 19 showing their abilities in exploring numerous phase transition points in various critical systems as well.
With the concepts of QIP, these studies have achieved great success in understanding the deep nature of the different phase transitions. The reverse, however, is still unclear and deserves further investigations. Recently, from a new point of view, Cui et al. find that the systems undergoing quantum phase transition will also show different operational properties from the perspective of QIP both analytically 20 and numerically 21,22 . For several models, they reveal that the entanglement-assisted local operations and classical communications (ELOCC) convertibility decided by the Rényi entropy 23,24 suddenly changes nearly at the critical point. In ref. 25, the local operations and classical communications (LOCC) convertibility decided by the majorization relation 26 has also been investigated in the one-dimensional transverse field Ising model 27 . These significant results suggest that not only are the methods of QIP useful as alternative signatures of quantum phase transitions, but also the study of quantum phase transitions can offer interesting insight into QIP.
In this paper, we study the relationship between two kinds of ground-state local convertibility and quantum phase transitions in XY model. The majorization relation studied here is necessary and sufficient for the LOCC convertibility 26 and Rényi entropy interception is a necessary and sufficient condition for the ELOCC convertibility 28 . In the phase diagram of XY model, LOCC convertibility and ELOCC convertibility of ground-states are presented and compared. Since the one-dimensional transverse field Ising model can be seen as a special case of our model, we promote and widen the results in previous literature. It is shown that different phases in the phase diagram of XY model can have different LOCC or ELOCC convertibility and both of them changes around the phase transition point, which have successfully help us understand the deep nature of the different phases in XY

Results
Local convertibility of ground states of XY model. In this section, given the XY model [29][30][31] in the zero-temperature case, we use majorization relation and Rényi entropy interception to study the relationship between two kinds of local convertibility and quantum phase transitions. The Hamiltonian of our model is as follows 14 : with L being the number of spins in the chain, σ i m the ith spin Pauli operator in the direction m = x, y, z and periodic boundary conditions assumed. The XX model and transverse field Ising model thus correspond to the special cases for this general class of models. For the case that γ → 0, our model reduces to XX model. The quantum criticality of ground and thermal states in XX Model has been studied 32 . When γ = 1, the model reduces to transverse field Ising model 27 . For h = 1, a second-order quantum phase transition takes place for any 0 ≤ γ ≤ 1. In fact, there exists additional structure of interest in phase space beyond the breaking of phase flip symmetry at h = 1. It's worth noting that there exists a circle boundary between Phase 1A and Phase 1B, h 2 + γ 2 = 1, on which the ground state is fully separable. According to the previous literature, this circle can be seen as a boundary between two differing phases which are characterized by the presence and absence of parallel entanglement [33][34][35][36][37] . In fact, for each fixed γ, the system is only locally non-convertible when h 2 + γ 2 > 1. Now we can divide the system into three separate regimes, Phase 1A, Phase 1B and Phase 2, where the ferromagnetic region is now divided into two regions defined by their differential local convertibility. These results are summarized in "phase-diagram" Fig. 1(a).
We study the ground states of this model for different system sizes with the field parameter h varying from 0 to 1.5. The ground states labeled as G h ( ) AB are obtained by exactly diagonalizing the whole Hamiltonian (1). For infinite long spin chain of XY model, the analytical results of the thermal ground states are also discussed, see Methods for details. This proposal is also worth further investigations by other numerical methods such as the Lanczos algorithm and density matrix renormalization group (DMRG), which are not included in this work. ELOCC convertibility via entanglement Rényi entropy. Using this method, we provide a phase diagram which describes the ELOCC convertibility for different phases in XY model. We consider the case that Alice shared a reduced system of two spins.
In Fig. 1(c), for a system size L = 15, we have two different regions that indicate different ELOCC convertibility. In the yellow (light) areas, the sign of ∂ h S α (ρ A (h)) is negative for all α, that is, the two states |G(h)〉 AB and |G(h + Δ )〉 AB can be converted to each other by ELOCC. In the red (dark) areas, the sign of ∂ h S α (ρ A (h)) is positive for all α, there is no ELOCC convertibility in these regions. Similar as Fig. 2(e,f), the obscure areas in Fig. 1(c) reflect the level-crossings that redefine the ground state of the system (which are evident from the spectrum of the model). For infinite long spin chain of XY model, the ELOCC convertibility of the thermal ground state is plotted in Fig. 1(d), see methods for details. Comparing Fig. 1(a) and Fig. 1(c,d), it is obvious that some boundaries between yellow (light) area and red (dark) area can be used to detect the critical line for different phases. Specifically, in Phase 1A, the ELOCC convertibility does not exist; in some area of Phase 1B and the entire area of Phase 2, the ground states |G(h)〉 AB and |G(h + Δ )〉 AB can be converted to each other by ELOCC. Therefore, the transition between Phase 1A and Phase 2 and the one between Phase 1A and Phase 1B can be described precisely by the ELOCC convertibility via entanglement Rényi entropy.

LOCC convertibility via majorization relation.
For the same case that the reduced system shared by Alice has two spins, we only need to consider three largest eigenvalues of the reduced states of any two neighboring spins as λ 1 (h), λ 2 (h), and λ 3 (h) in descending order. To detect the majorization relations between two ground states |G(h)〉 AB and |G(h + Δ )〉 AB given some infinitesimal Δ , we should judge the monotonicities of three functions for the field parameter f 1 (h), f 2 (h) and f 3 (h) defined in Eq. (3). Thus, three cases will be met: (i) when monotonicities of these three functions are all non-increasing, |G(h + Δ )〉 AB can be converted to |G(h)〉 AB by LOCC with 100% certainty; (ii) when monotonicities of these three functions all are non-decreasing, |G(h)〉 AB can be converted to |G(h + Δ )〉 AB by LOCC with 100% certainty; (iii) except for these two cases, no state can be converted to the other via LOCC with certainty. For ease of comparison with ELOCC convertibility, We only show in Fig. 1(e) either the case (ii) or other two cases.
Using the method mentioned above, we can also provide a phase diagram describing the LOCC convertibility for different phase in XY model. In Fig. 1(e), for system sizes N = 15, we have shown two different regions. In the yellow (light) area, |G(h + Δ )〉 AB can be transformed to |G(h)〉 AB by LOCC with 100% probability of success; whilst, in the red (dark) area, |G(h + Δ )〉 AB can not be transformed to |G(h)〉 AB by LOCC with certainty. We should stress that in few parts of red (dark) area in Phase 1A, |G(h)〉 AB can be transformed to |G(h + Δ )〉 AB by LOCC with certainty, which is not shown in Fig. 1(e). For infinite long spin chain of XY model, the LOCC convertibility of the thermal ground state is plotted in Fig. 1(f), see methods for details. In the XY model, there are three regimes, Phase 1A, Phase 1B and Phase 2 as shown in Fig. 1(a). From Fig. 1(e,f), in Phase 1A, no majorization relations can be fulfilled, |G(h + Δ )〉 AB and |G(h)〉 AB can not be converted to each other by LOCC. However, in Phase 1B and Phase 2, there are two areas which have different LOCC convertibility, which have richer and more complex features of LOCC convertibility than those of ELOCC. Similarly, we can use the dividing line between red (dark) and yellow (light) areas with different LOCC convertibility to detect the critical line between Phase 1A and Phase 1B precisely. As the increase of the system size L, our result can be shown to get better using the scaling analysis on the critical point from LOCC convertibility 25 and ELOCC convertibility 21 to get rid of the finite-size effect. Comparing the result with the ELOCC method, we can see that the ELOCC method is superior to LOCC metohd for detecting the critical line between two different quantum phases. In order to detect the phase transitions in our model, the ELOCC method is more accurate than LOCC method in the XY model as well as in the one-dimensional transverse field Ising model 25 . Moreover, we should note that if the LOCC convertibility exists, the ELOCC convertibility must exist. That is, in different phases of quantum critical systems, various and complicated local conversion can be expected, will benefit many applications such as to estimate the computational power of different phases.
To get rid of the finite-size effect, we give a scaling analysis of the critical points h L ( ) c (2) of the second-order phase transition detected by the ELOCC convertibility for γ = 3 /2 in Fig. 3. With the exponential fitting, it is shown that for infinitely large size L → ∞ and γ = 3 /2, the ELOCC convertibility changes at = . h 1 0273 c (2) , which detects the second-order phase transition of the XY model. The scaling analysis of critical points detected by the LOCC convertibility can be found in ref. 25.

Discussion
In conclusion, we have investigated the relationship between local convertibility of ground states and quantum phase transitions in XY model. We study the LOCC convertibility by examining the majorization relations and the ELOCC convertibility via Rényi entropy interception. It shows that the boundary of areas which have different LOCC or ELOCC convertibility can be used to detect the critical line for different phases. In Phase 1A, both the LOCC and ELOCC convertibility do not exist. In Phase 1B, the situation is slightly more complicated. There are two areas which has different LOCC convertibility. In most area of Phase 1B, the ELOCC convertibility exists. In Phase 2, there is no LOCC convertibility in the area that nearly the critical point. In the remaining area, both the LOCC and ELOCC convertibility exist. It is obvious that if the LOCC convertibility exists, the ELOCC convertibility must exist. We believe that our method should be applicable in other systems similar as in the XY model. The LOCC proposal in detecting quantum phase transition can play a complementary role to the ELOCC method, which will help to exploit the rich features of local conversion of ground states of quantum critical systems. This study will enlighten extensive studies of quantum phase transitions from the perspective of local convertibility, e.g., finite-temperature phase transitions and other quantum many-body models.
We remark that the majorization relation is necessary and sufficient for the LOCC convertibility 28 and Renyi entropy interception is a necessary and sufficient condition for the ELOCC convertibility 38,39 between two bipartite pure quantum states. In applying those relations to a ground state of many-body system, it is important to divide the system into bipartitions to obtain the bipartite pure states. In our study of phase diagram of the spin chains, we can show that the local convertibility identified by majorization relation can demonstrate clearly different quantum phases by some bipartitions of the ground state.

Methods
Criterions for LOCC and ELOCC convertibility. As showed in ref. 22, we consider a system with an adjustable external parameter h, which partitioned into two parties, Alice and Bob, and operated by ELOCC or LOCC. Let |G(g)〉 AB be the ground state of the system when the parameter is set as h. Given the infinitesimal Δ , the necessary and sufficient condition for ELOCC between |G(h)〉 AB and |G(h + Δ )〉 AB is given by the inequality  a state with higher entanglement can be locally transferred to the lower entanglement one by ELOCC 21 . These results can be applied to study the quantum critical phenomena, i.e. when a quantum phase transition occurs, the local transformation property of the ground state wave function changes as well as the different quantum phases boundaries can be determined by the behavior of entanglement Rényi entropies of ground states 21,22 . Referring the scaling analysis in ref. 21 phase transition point obtained by this proposal tends to 0.9940.
On the other hand, the majorization relations provide a necessary and sufficient condition for the LOCC convertibility. Using the Schmidt decomposition theorem 40 , the ground state can be written as Then, we have that if and only if the majorization relations The eigenvalues of the reduced matrix (4) against parameter h for γ = 0.6 are shown in Fig. 1(b). With the analytical results of the thermal ground state for the infinite long spin chain, the ELOCC convertibility and LOCC convertibility of the ground state of the XY model at zero temperature are presented in Fig. 1(d,f), respectively.