Subsystem distances between quasiparticle excited states

We investigate the subsystem Schatten distance, trace distance and fidelity between the quasiparticle excited states of the free and the nearest-neighbor coupled fermionic and bosonic chains and the ferromagnetic phase of the spin-1/2 XXX chain. The results support the scenario that in the scaling limit when one excited quasiparticle has a large energy it decouples from the ground state and when two excited quasiparticles have a large momentum difference they decouple from each other. From the quasiparticle picture, we get the universal subsystem distances that are valid when both the large energy condition and the large momentum difference condition are satisfied, by which we mean each of the excited quasiparticles has a large energy and the momentum difference of each pair of the excited quasiparticles is large. In the free fermionic and bosonic chains, we use the subsystem mode method and get efficiently the subsystem distances, which are also valid in the coupled fermionic and bosonic chains if the large energy condition is satisfied. Moreover, under certain limit the subsystem distances from the subsystem mode method are even valid in the XXX chain. We expect that the results can be also generalized for other integrable models.


Introduction
In quantum information theory and quantum many-body systems, it is important to distinguish quantitatively two different states [1][2][3][4][5][6][7]. To differentiate two states with density matrices ρ and ρ , one may compare the expectations values of some specific local or nonlocal operator δ O = O ρ − O ρ . For two different states ρ = ρ , there must exists some operator O so that δ O = 0, but in practice it may be difficult to find a proper operator. One may also calculate the differences of some nonlocal quantities such as the Rényi and entanglement entropies δS A,ρ and δS A = S A,ρ − S A,ρ . The Rényi and entanglement entropies of a subsystem A in the total system in state ρ is defined as follows. The Hilbert space of the total system is divided into that of the subsystem A and that of its complement B.
One integrates out the degrees of freedom of the subsystem B and obtains the reduced density matrix (RDM) ρ A = tr B ρ of the subsystem A. The Rényi entropy of the RDM is defined as 1) and the entanglement entropy is the von Neumann entropy of the RDM The entanglement entropy could be calculated as the n → 1 limit of the Rényi entropy. The Rényi and entanglement entropies in various extended quantum systems have been investigated for the ground state  and the excited states . In this way, the compared quantities are solely defined in terms of the RDMs, but there are still potential problems. Two states with different Rényi and entanglement entropies must be different, however, two different states may well have the same Rényi and entanglement entropies. It is intriguing to investigate other quantities to distinguish quantitatively two different states in extended quantum many-body systems.
There are various quantities that characterize the dissimilarity, or equivalently the similarity, of two states. For two pure states |ψ 1 and |ψ 2 , one could just calculate the overlap | ψ 1 |ψ 2 | 2 . For two mixed states with density matrices ρ 1 and ρ 2 , one may also calculate the overlap tr(ρ 1 ρ 2 ). Furthermore, one could calculate other quantities that are not simply related to the overlap. This is especially true for the RDMs of a subsystem in the total system in various states. When the total system is in a pure state, the RDM of a subsystem is often in a mixed state. For example, the quantities could be the Schatten distance, trace distance, fidelity, relative entropy and other information metrics. These quantities have been investigated in various extended systems in for example . In this paper we will investigate the subsystem Schatten distance, trace distance and fidelity in the fermionic, bosonic and spin-1/2 XXX chains. For two RDMs ρ A and σ A , the subsystem Schatten distance with index n ≥ 1 is defined For convenience in this paper, we introduce a normalization state λ A and write the normalized subsystem Schatten distance as For an even integer n, it is just (1.5) (1.6) which is independent of the normalization state, i.e. that D 1 (ρ A , σ A ; λ A ) = D 1 (ρ A , σ A ). The trace distance could be calculated from the replica trick proposed in [77,79]. One first calculates the Schatten distance with the index n being a general even integer and then takes the analytical continuation n → 1.
The fidelity of two RDMs ρ A and σ A is Though it is not apparent by definition, the fidelity is symmetric to its two arguments F (ρ A , σ A ) = F (σ A , ρ A ). As the case of the trace distance, we do not need to introduce a normalization state for the fidelity. Note that the Schatten and trace distances denote the dissimilarity of two configurations, while the fidelity denotes the two configurations' similarity.
In extended quantum many-body systems, it is interesting to investigate universal behaviors of the Rényi and entanglement entropies. Recently, a new universal behavior of the Rényi entropy and the entanglement entropy in quasiparticle excited states of integrable models was discovered in [43,44,46,47] (one could also see earlier partial results in [34,36,41]). The universal differences of the quasiparticle excited state Rényi and entanglement entropies with those in the ground state are independent of the models and the values of the quasiparticle momenta. To obtain the universal Rényi and entanglement entropies, one has to take the limit that each of the relevant quasiparticle is highly excited above the ground state and each pair of the excited quasiparticle has a large momentum difference, which we will call respectively the large energy condition and the large momentum difference condition. The universal excess Rényi and entanglement entropies could be written out by a simple semiclassical quasiparticle picture with the quantum effects of distinguishability and indistinguishability of the excited quasiparticles. The same universal formulas could be obtained in the classical limit of a one-dimensional quantum gas in presence of an external potential [62]. By relaxing the limit that quasiparticle momentum differences are large, we have obtained additional contributions to the Rényi and entanglement entropies in [56,58]. The results were further formulated into three conjectures for the Rényi and entanglement entropies in [61], and these conjectures were also checked extensively therein.
In this paper, we generalize the results of quasiparticle excited state Rényi and entanglement entropies [43,44,56,58,61] to the subsystem Schatten and trace distances and fidelity. Some preliminary results in the two-dimensional non-compact bosonic theory have been presented in [54,59], and in this paper we will show more systematic details. From the quasiparticle picture, we obtain universal Schatten and trace distances and fidelity that are independent of the models and the explicit values of the quasiparticle momenta. The universal Schatten and trace distances and fidelity are valid when both the large energy condition and large momentum difference condition are satisfied. By relaxing the large momentum difference condition, we obtain additional corrections to the universal results that are different in different models and dependent on the momentum differences of the excited quasiparticles.
We formulate the results of the Schatten and trace distances and fidelity into three conjectures, check these conjectures extensively in the fermionic and bosonic chains and spin-1/2 XXX chains, and obtain consistent results.
The universal Rényi and entanglement entropies in [43,44,46,47] and the universal Schatten and trace distances and fidelity in this paper are just special cases of the three conjectures for the Rényi and entanglement entropies in [61] and the three corresponding conjectures for the subsystem distances in this paper. The universal formulas for the Rényi and entanglement entropies and the Schatten and trace distances and fidelity are obtained from the assumption that when the momentum of each pair of different excited quasiparticles is large all the different excited quasiparticles decouple from each other.
The three conjectures for the Rényi and entanglement entropies and the Schatten and trace distances and fidelity are based on the scenario that in the scaling limit when one excited quasiparticle has a large energy it decouples from the ground state and when two excited quasiparticles have a large momentum difference they decouple from each other. We consider the subsystem with successive sites A on a circular chain with L sites in the scaling limit L → +∞, → +∞ with fixed ratio x ≡ L . We take the ground state |G , single-particle state |k , and double-particle state |kk as examples. In the condition that the energy ε k of the excited quasiparticle with momentum k is large, the quasiparticle decouples from the ground state, and there are universal excess entanglement entropy and trace distance The RHS of (1.8) is nothing but the Shannon entropy of the probability distribution {x, 1 − x}. The RHS of (1.9) is just the classical trace distance between the probability distributions {x, 1 − x} and {0, 1}. In the condition that the energy ε k of the excited quasiparticle with momentum k is large and the momentum difference |k − k | of the two excited quasiparticles with momentum k and k is large, the quasiparticle with momentum k decouples from not only the ground state but also the quasiparticle with momentum k , and there are universal excess entanglement entropy and trace distance We will give more examples and details in the main text of the paper.
The remaining part of the paper is arranged as follows: In section 2 we review the three conjectures for the Rényi and entanglement entropies in [61] and formulate the corresponding three conjectures for the Schatten and trace distances and fidelity. In section 3 we calculate the Schatten and trace distances and fidelity in the free fermionic chain from the subsystem mode method and check the results from various variations of the correlation matrix method. In section 4 we check the three conjectures for the subsystem distances in the nearest-neighbor coupled fermionic chains using the correlation matrix method. In section 5 we calculate the Schatten and trace distances and fidelity in the free bosonic chain from the subsystem mode method and check the Schatten distance with an even index from the wave function method. In section 6 we check the three conjectures for the Schatten distance with an even index in the nearest-neighbor coupled bosonic chains using the wave function method. In section 7 we formulate the three conjectures for the trace distance and fidelity among the ground state and magnon excited states of the ferromagnetic phase of the spin-1/2 XXX chain and check these conjectures from the local mode method. We conclude with discussions in section 8. In appendix A we present an efficient procedure to calculate the subsystem distances for density matrices in a nonorthonormal basis.
In appendix B we give the derivation of a formula that is useful for the recursive correlation matrix method in the free fermionic chain.

Three conjectures for the entropies and distances
In integrable models, one may use the set of the momenta K = {k 1 , · · · , k r } of the excited quasiparticles to denote the state of the total system as |K . In this paper, we consider circular quantum chains of L sites. The quantity k that we call momentum is actually the number of waves of the total system, and the actual momentum p is related to k as p = 2πk L . In the free fermionic and bosonic chains the momenta may be integers or half-integers depending on whether the boundary conditions are periodic or antiperiodic. In the spin-1/2 XXX chain the momenta k may not necessarily be integers or half-integers and could be real and even complex numbers.
In [61], we have made three conjectures for the Rényi and entanglement entropies of the subsystem A = [1, ] in quasiparticle excited states. The first conjecture is that in the large energy condition In other words, one could write the RDM as The second conjecture is that for the set of momenta K satisfying the large energy condition and the sets K and K satisfying the large momentum difference condition there are the differences of the Rényi and entanglement entropies 8) and the effective RDM with the same effective RDMρ A,K as the one in (2.4). In certain limit, the excited quasiparticles decouple from the background. In the effective RDM (2.4) the ground state RDM ρ A,G is viewed as the background, while in (2.9) the RDM ρ A,K is viewed as the background.
The third conjecture is that for the two sets of momenta K and K satisfying the large momentum difference condition there are relations For the third conjecture, we do not necessarily have general analytical expressions on the RHS of (2.11) and (2.12).
The essence of the above three conjectures for the Rényi and entanglement entropy is the scenario that a set of quasiparticles satisfying the large energy condition decouple from the ground state and two sets of quasiparticles satisfying the large momentum difference condition decouple from each other.
Based on this scenario, we formulate the corresponding three conjectures for the Schatten and trace distances and fidelity as follows.
• The first conjecture is that for two states |K 1 and |K 2 both satisfying the large energy condition there are the normalized Schatten distance, and trace distance and fidelity 14) • The second conjecture is that for two momentum sets K 1 and K 2 both satisfying the large energy condition (2.5) and the large momentum difference condition with respect to the momentum set there are the Schatten and trace distances and fidelity • The third conjecture is that for two sets K 1 and K 2 both satisfying the large momentum difference condition with the momentum set K (2.10), i.e. that The three conjectures for the Rényi and entanglement entropies have been checked extensively in [56,58,61]. In this paper we will check the corresponding three conjectures for the Schatten and trace distances and fidelity in the fermionic, bosonic and XXX chains. Some preliminary results in the two-dimensional non-compact bosonic theory have been presented in [54,59].

Free fermionic chain
In this section, we consider the free fermionic chain. We calculate the Schatten and trace distances and fidelity from the subsystem mode method. We also check the results from various variations of the correlation matrix method, among which the diagonalized truncated correlation matrix method is the most efficient one.

Quasiparticle excited states
The translation invariant free fermionic chain of L sites has the Hamiltonian with the spinless fermions a j , a † j . The quasiparticle modes are Fourier transformations of the local modes Here p k is the actual momentum and k is the total number of waves, which is an integer and a halfinteger depending on the boundary conditions of the spinless fermions a j , a † j . Note that p k ∼ = p k + 2π and k ∼ = k + L. As we mentioned in section 2, we just call k momentum. We only consider the case that L is an even integer. For the states in the Neveu-Schwarz (NS) sector, i.e. antiperiodic boundary conditions for the spinless fermions a L+1 = −a 1 , a † L+1 = −a † 1 , we have the half-integer momenta For the states in the Ramond (R) sector, i.e. periodic boundary conditions for the spinless fermions The ground state of the Hamiltonian is annihilated by all the local and global lowering modes The ground state |G in the free fermionic chain is the ground state of both the NS sector and the R sector, i.e. that |G = |G NS = |G R . The general excited state in the NS sector is generated by applying the raising operators b † k ∈ NS on the ground state The general excited state in the R sector is generated by applying the raising operators b † k ∈ R on the ground state One may consider the subsystem difference of two states in the same sector or two states in different sectors. Note that one state in the NS sector and another state in the R sector with the same energy may not necessary be orthogonal.

Subsystem mode method
The subsystem mode method was used in [56,58,61] to calculate the Rényi and entanglement entropies in the quasiparticle excited states of the free fermionic and bosonic chains. Especially the subsystem mode method was formulated systematically in [61], and one could see details therein. In this subsection, we give a brief and self-consistent review of the subsystem mode method and further adapt the method for the calculation of the subsystem distances.
We choose the subsystem A = [1, ] and its complement B = [ + 1, L]. We focus on the scaling limit that L → +∞ and → +∞ with fixed ratio x ≡ L . The ground state |G could be written as a direct product form |G = |G A ⊗ |G B with a j |G A = 0 for all j ∈ A and a j |G B = 0 for all j ∈ B.
We divide the quasiparticle modes as sums The subsystem modes satisfy the nontrivial anti-commutation relations with the definitions of the factors There is β 0 = 1 − α 0 , and for k ∈ Z and k = 0 there is β k = −α k . In this paper, we will also mention a bit the case that the momentum differences are not integers.
For an arbitrary ordered set of momenta K = {k 1 , · · · , k r } with k 1 < · · · < k r , we define the products of subsystem modes The excited state of the total system could be written as with K\K being the complement set of K contained in K. We have defined the factor with sig[K , K\K ] denoting the signature of the two ordered sets K and K\K joining together without changing the orders of the momenta in each of them.
We get the RDM in the nonorthonormal basis b † A,K |G A with K ⊆ K written as with the entries of the 2 |K| × 2 |K| matrix P A,K We have used |K| to denote the number of quasiparticles in the set K. We need to evaluate the expec- The |K 1 | × |K 2 | matrices A K 1 K 2 and B K 1 K 2 have the entries For later convenience, we also define the |K| × |K| matrices A K ≡ A KK and B K ≡ B KK .
For two sets of momenta K 1 and K 2 , we have the union set K 1 ∪ K 2 in which each of the repeated momenta appears only once. For example, for with the entries of the 2 |K 1 ∪K 2 | × 2 |K 1 ∪K 2 | matrices P A,K 1 and P A,K 2 We also define the 2 With the 2 |K 1 ∪K 2 | × 2 |K 1 ∪K 2 | matrices P A,K 1 , P A,K 2 and Q A,K 1 ∪K 2 , we may follow the procedure in appendix A and calculate the Schatten and trace distances and fidelity. Note that the matrices P A,K 1 , P A,K 2 and Q A,K 1 ∪K 2 are block diagonal with |K 1 ∪ K 2 | + 1 blocks.
In the above strategy, we need to use the matrices with sizes that grow exponentially with the number of the excited quasiparticles, while the calculation complexity does not depend on the Schatten index n. There is another strategy to calculate the Schatten distance with an even index n = 2, 4, · · · .
The quantity tr A (ρ A,K 1 − ρ A,K 2 ) n could be evaluated by binomial expansion. For example, to calculate the second Schatten distance D 2 (ρ A,K 1 , ρ A,K 2 ) we need to evaluate 22) and to calculate the fourth Schatten distance D 4 (ρ A,K 1 , ρ A,K 2 ) we need to evaluate We evaluate each term in the binomial expansion following In the second strategy, the sizes of the relevant matrices grow lineally with the number of the excited quasiparticles, the calculation complexity also grows with the Schatten index n.
When the numbers of the excited particles are small, the subsystem mode method is efficient for analytical calculations. When the numbers of the excited particles are not so large, the subsystem mode method is still efficient for numerical evaluations.

Recursive correlation matrix method
To verify the results from the subsystem mode method, we calculate numerically the subsystem distances using the correlation matrix method [15,[17][18][19]. We use the × correlation matrix C A,K with the There is the function We use ρ C to denote the RDM corresponding to the × correlation matrix C. For example, we have To calculate the Schatten distance with an even integer index from the correlation matrices, we use the recursive formula We show the derivation of recursive formula (3.27) in appendix B.
To calculate the fidelity from the correlation matrices, we use the formula [88] F In the recursive correlation matrix method, we only need to use the matrices with sizes increasing algebraically with respect to the size of the subsystem , and so it is very efficient. The drawback is that from this method one could only calculate the Schatten distance with an even integer index and the fidelity.

Contracted correlation matrix method
From correlation matrix, one may construct the numerical RDM explicitly [17,19]. In the subsystem A = [1, ], we use the complete basis of the operators It is easy to check We get the RDM written as The expectation value O † i 1 i 2 ···i K could be evaluated using the anticommutation relations of the local modes a j , a † j and the determinant formula from the Wick contractions Note that the orders of the sets J and J are important, otherwise there would appear possible minus sign. With the explicit numerical RDMs, in principle we could calculate everything defined from the RDMs.
In the contracted correlation matrix method, we need to process the matrices with sizes increase exponentially with respect to the size of the subsystem , and so it is usually not so efficient. We could only consider the subsystem with a rather small size, say 8.

Diagonalized correlation matrix method
The RDM could be written in terms of the modular Hamiltonian [16,18] with the matrix The correlation matrix C is Hermitian and could be diagonalized as with the diagonal matrixC = diag(µ 1 , µ 2 , · · · , µ ) and the unitary matrix U = (u 1 , u 2 , · · · , u ) constructed with the eigenvectors and eigenvalues of the matrix C Cu j = µ j u j , j = 1, 2, · · · , . (3.36) The matrix H is also diagonal under the same basis We define the new modesã in terms of which the RDM takes the form In this way, we construct the explicit numerical RDMs and in principle could calculate everything defined from the RDMs. To calculate the fidelity, it is convenient to use the square root of the RDM The diagonalized correlation matrix method is more efficient than contracted correlation matrix method, but it is not efficient enough, as we still need to construct the explicit RDM with size increasing exponentially with the subsystem size . Explicitly, from this method we could consider the subsystem with size 12.

Diagonalized truncated correlation matrix method
We generalize the diagonalized correlation matrix method to the cases of a much larger subsystem size. Note that the correlation matrix C A,K in state |K has rank min( , |K|), and when > |K| we may truncate it into a |K| × |K| matrix. The subspace after truncation is nothing but the subspace generated by the subsystem modes c † A,k with k ∈ K. We consider two states |K 1 and |K 2 with the correlation matrices C A,K 1 and C A,K 2 , and we denote r = |K 1 ∪ K 2 |. When > |K 1 ∪ K 2 |, we may truncate the correlation matrices C A,K 1 and C A,K 2 into r × r matrices. Firstly, we collect all the r 1 = |K 1 | eigenvectors of C A,K 1 with nonvanishing eigenvalues u 1 , u 2 , · · · , u r 1 and all the r 2 = |K 2 | eigenvectors of C A,K 2 with nonvanishing eigenvalues u r 1 +1 , u r 1 +2 , · · · , u r 1 +r 2 . All the r 1 + r 2 vectors u 1 , u 2 , · · · , u r 1 +r 2 form a r-dimensional complex linear space, in which we find r orthonormal basis v 1 , v 2 , · · · , v r . Note that there is r < r 1 +r 2 if K 1 ∩K 2 = ∅.
Then we construct the × r matrix with v 1 , v 2 , · · · , v r viewed as -component column vectors. We define the new truncated correlation matrices of size r × rC Finally, we construct the 2 r × 2 r truncated RDMsρ A,K 1 andρ A,K 2 from the r × r truncated correlation matricesC A,K 1 andC A,K 2 using the diagonalized correlation matrix method in the above section. With the truncated RDMsρ A,K 1 andρ A,K 2 , we calculate the Schatten and trace distances and fidelity.
The diagonalized truncated correlation matrix method in the free fermionic chain is an exact method, and no approximation has been used. For a large subsystem, the size of the truncated RDMs only depends on the number of excited quasiparticles and the method is very efficient when the number of excited quasiparticles is not so large, say |K 1 ∪ K 2 | 12.

Schatten and trace distances
We first give the universal Schatten and trace distances from the quasiparticle picture, which are

Universal Schatten and trace distances
The effective RDM of the subsystem A in the ground state is where |0] = |G A denoting the ground state of the subsystem A, i.e. that state with no quasiparticle in it. The single-particle state effective RDM takes the form where |k] denotes the state of the subsystem A with one quasiparticle of momentum k. In the limit that the momentum difference of each pair of the excited quasiparticle is large, all the excited quasiparticles are independent and the RDM in a general state |k 1 · · · k r takes a universal form We consider two general states |k 1 · · · k r k 1 · · · k r and |k 1 · · · k r k 1 · · · k r with r overlapping excited quasiparticles. From the universal RDM (3.45) and assumption that different quasiparticles are independent, we get the universal Schatten and trace distances The overlapping excited quasiparticles could be viewed as the background, and we obtain the normalized Remember that there is no need to normalize the trace distance. Both the normalized Schatten distance and the trace distance is independent of the background ρ A,k 1 ···kr . A special case of the universal Schatten and trace distances (3.46) and (3.47) are We emphasize that the validity of the universal Schatten and trace distances (3.46), (3.47), (3.48), (3.49) and (3.50) requires that all the momentum differences among the excited quasiparticles are large.
In figure 1, we see that the Schatten and trace distances approach the universal Schatten and trace distances in the large momentum difference condition.
The RDM ρ A,G = |G A G A | is a pure state and the two RDMs ρ A,G and ρ A,K commutes. We get the exact Schatten and trace distances from the subsystem mode method where we have Remember that the |K| × |K| matrix A K ≡ A KK is defined following (3.18). In the limit that all the momentum differences are large, we have Explicitly, we get for the special case r = 1 55) and the special case r = 2    with the shorthand α 12 ≡ α k 1 −k 2 and the definition of α k (3.10). The universal version of these results

(ρ
We compare the analytical results (3.56) and (3.57) with the corresponding numerical ones from the diagonalized truncated correlation matrix method in the first column of figure 1. There are perfect matches between the analytical and the numerical results.
From the subsystem mode method, we get the Schatten and trace distances Note that it is independent of the index n. In the large momentum difference condition, there is the universal Schatten and trace distances We compare these analytical results of the Schatten and trace distances with the corresponding numerical ones as well as the universal Schatten and trace distances in the second column of figure 1.
The results (3.61) also apply to the case that one state |k 1 is in the NS sector and another state |k 2 is in the R sector, i.e. that k 1 is a half integer, k 2 is an integer, and so k 1 − k 2 is a half integer. For finite half-integer k 1 − k 2 in the scaling limit, the two states |k 1 and |k 2 are not orthogonal and we From the subsystem mode method, we get The corresponding universal version are We compare these analytical results of the Schatten and trace distances with the numerical ones and the universal ones in the third column of figure 1.

ρ
For general states with more quasiparticles, it is difficult to obtain the analytical results, but still we may get the numerical results efficiently from the subsystem mode method and the diagonalized truncated correlation matrix method. There are perfect matches between the results obtained from different methods. We will not show the results here.

A conjecture for trace distance
From the trace distances (3.55) and (3.64), it is tempting to conjecture the trace distance in the free We have checked it numerically for extensive examples, which we will not show here. It would be interesting to derive it rigorously.

Universal short interval expansion
It is interesting to look into the behavior of the Schatten and trace distances in short interval expansion. The leading order of the result is independent of the Schatten index n. Note that |r − r | is just the difference of the excited quasiparticle numbers of the two states. We do not know how to derive it for general states, but we have checked it extensively using the numerical realization of the subsystem mode method, which we will not show here.

Fidelity
We present the universal subsystem fidelity from the semiclassical quasiparticle picture and examples of the analytical fidelity from the subsystem mode methods. We also check the analytical fidelity numerically using the diagonalized truncated correlation matrix method.

Universal fidelity
In the limit that the momentum differences of the excited quasiparticles are large, we get the universal fidelity from the semiclassical quasiparticle picture There is the special case In figure 2, we see the fidelity approach the universal fidelity in the large momentum difference condition.

ρ A,G VS ρ A,K
In the free fermionic chain, the RDM of the ground state is a pure state, and there is a simpler result for the fidelity with F fer A,G,K (3.53). With D fer 1 (ρ A,G , ρ A,K ) (3.52) and the fact that 0 ≤ F fer A,G,K ≤ 1, it is easy to see the expected inequality [1, 2] For the special case r = 1 there is and for r = 2 there is Note the universal fidelities We compare the analytical results with the numerical ones and the universal ones in the first panel of We get the fidelity from the subsystem mode method The corresponding universal fidelity is We compare the analytical results with the numerical and universal ones in the second panel of figure 2.

ρ
From the subsystem mode method we get with corresponding universal fidelity We show the results in the third panel of figure 2.

ρ
For more general cases we calculate the fidelity numerically from the subsystem mode method and the diagonalized truncated correlation matrix method. Different methods lead to the same results. We will not show details here.

Nearest-neighbor coupled fermionic chain
We use the correlation matrix method and check the three conjectures for the subsystem distances between the quasiparticle excited states in the nearest-neighbor coupled fermionic chain.

Quasiparticle excited states
We consider the chain of L spinless fermions a j , a † j with the Hamiltonian It could be diagonalized following [89][90][91] Here p k is the actual momentum and k is the total number of waves, which is an integer and a half-integer depending on the boundary conditions. The quasiparticle modes c k , c † k are Bogoliubov transformation of the modes where the angle θ k is determined by As we have mentioned before, we just call k momentum in this paper. In this paper, we only consider the case that L is an even integer. For the states in the NS sector we have the half-integer momenta The general excited state in the NS sector is generated by applying the raising operators c † k ∈ NS on the NS sector ground state The general excited state in the R sector is generated similarly |K = |k 1 · · · k r = c † k 1 · · · c † kr |G R , k 1 , · · · , k r ∈ R. (4.7)

Recursive correlation matrix method
We calculate numerically the results using the correlation matrix method [15,[17][18][19]. In the nearestneighbor coupled fermionic chain, one could define the Majorana modes In the general excited state |K = |k 1 k 2 · · · k r , one defines the 2 × 2 correlation matrix Γ K with entries Γ K m 1 m 2 = d m 1 d m 2 K − δ m 1 m 2 , m 1 , m 2 = 1, 2, · · · , 2 . (4.9) Explicitly, there are with the definitions See the definitions of ε k and p k in (4.2) and the definition of θ k in (4.4). The RDM is fully determined by the correlation matrix, and so one may use ρ Γ to denote the RDM corresponding to the correlation matrix Γ.
To evaluate the Schatten distance with index n being an even integer, we use the recursive formula We calculate the fidelity from the correlation matrices using the formula [94] F fer (ρ Γ 1 , ρ Γ 2 ) = det 1 − Γ 1 2 The correlation matrix Γ often has a lot of eigenvalues equaling or close to one, and we have to introduce a cutoff to regularize the artificial divergence in the above formula (4.13).

Contracted correlation matrix method
To evaluate the trace distance and other Schatten distances with odd integer indices n, we need to construct numerically the explicit RDMs from the correlation functions [17,19] ρ A,K = 1 2 where the multi-point correlation functions are evaluated from the two-point correlation functions by Wick contractions. In the contracted correlation matrix method, we could only consider the subsystem with a rather small size, say 6.

Canonicalized correlation matrix method
The RDM could be written in terms of the modular Hamiltonian as [93] with the 2 ×2 real orthogonal matrix Q satisfying Q T Q = QQ T = 1 and the real numbers γ j ∈ [−1, 1], j = 1, 2, · · · , . We define the new Majorana modes and write the explicit RDM as From the explicit RDM, in principle we could calculate everything. To calculate the fidelity, it is convenient to use the formula (4.20) The canonicalized correlation matrix method is a little more efficient than the contracted correlation matrix method in the previous subsection. With the canonicalized correlation matrix method we could consider the subsystem with size 12.

Free bosonic chain
We calculate the Schatten and trace distances and fidelity in the free bosonic chain from the subsystem mode method. We also obtain the same Schatten distances with even integer indices from the wave function method.         1 (ρ A,K 1 ∪K , ρ A,K 2 ∪K ), the normalized Schatten distance D n ≡ D fer n (ρ A,K 1 ∪K , ρ A,K 2 ∪K ; ρ A,K ) with n = 2, 3, 4, and the fidelity F ≡ F fer (ρ A,K 1 ∪K , ρ A,K 2 ∪K ), among which D 1 and D 3 are from the canonicalized correlation matrix method and D 2 , D 4 and F are from the recursive correlation matrix method. In the first and the second rows, the solid lines are the analytical conjectured results from the subsystem mode method in the free fermionic chain. In the third row, the solid lines are the numerical conjectured results in the nearest-neighbor coupled fermionic chain. In each panel, we give the inset with the results of D 1 and D 3 . We have set γ = λ = 1, (k 1 , k 2 ) = ( 1 2 , 3 2 )+ L

Quasiparticle excited states
We consider the translational invariant chain of L independent harmonic oscillators In terms of the local bosonic modes the Hamiltonian becomes The quasiparticle modes are We only consider the periodic boundary conditions a L+1 = a 1 , a † L+1 = a † 1 with L being an even integer, and so there are integer momenta The ground state |G is defined as A general quasiparticle excited state takes the form with the normalization factor

Subsystem mode method
The subsystem is A = [1, ] and its complement is B = [ + 1, L]. We divide the quasiparticle modes into the subsystem modes as with α k and β k defined the same as those in (3.10) and (3.11).
For an arbitrary set K = {k r 1 1 , · · · , k rs s }, which we may write for short K = k r 1 1 · · · k rs s when there is no ambiguity, we have the number of excited quasiparticles Then there is the excited state with K\K being the complement of K contained in K and the factor s K,K defined as Then we get the RDM Note the possible momenta repetitions of the sets and subsets used in the bosonic chain.
Then we have the RDM in the form of (A.1) with the entries of the |K| × |K| matrix P A,K We need to evaluate the expectation values c A,K 1 c † A,K 2 G and c B,K 1 c † B,K 2 G , which are just the permanents where the |K 1 | × |K 2 | matrices A K 1 K 2 and B K 1 K 2 have the entries 18) with the definitions of α k and β k in (3.10) and (3.11). We also define the |K|×|K| matrices A K ≡ A KK and B K ≡ B KK for later convenience.
For two sets of momenta K 1 and K 2 , we define the specific union set K 1 ∪ K 2 as follows. Firstly, we write K 1 = k r 1 1 · · · k rs s and K 2 = k r 1 1 · · · k r s s with the s momenta k i , i = 1, · · · , s appearing at least once in K 1 or K 2 and some of the 2s integers r i , r i , i = 1, · · · , s being possibly zero. Then, we define 1 · · · k r s s with r i = max(r i , r i ), i = 1, · · · , s. For example, from K 1 = 1 2 23 = 1 2 234 0 , K 2 = 12 4 4 = 12 4 3 0 4 we get the union set K 1 ∪ K 2 = 1 2 2 4 34, and there are also |K 1 | = 4, |K 2 | = 6, We obtain the RDMs in the nonorthonormal basis c † with the entries of the 2 |K 1 ∪K 2 | × 2 |K 1 ∪K 2 | matrices P A,K 1 and P A,K 2 We also define the 2 With the 2 |K 1 ∪K 2 | × 2 |K 1 ∪K 2 | matrices P A,K 1 , P A,K 2 and Q A,K 1 ∪K 2 , we follow the procedure in appendix A and calculate the Schatten and trace distances and fidelity.
Besides the above strategy, we have another strategy to calculate the Schatten distance with an even index n = 2, 4, · · · . The quantity tr A (ρ A,K 1 − ρ A,K 2 ) n could be evaluated by binomial expansion, and we evaluate each term in the expansion following

Wave function method
We also calculate the Schatten distances with even integer indices from the wave function method [43,44]. One could also see the wave function method in [58,59]. From the wave function method it is easy to get the same permanent formula (5.22) in the free bosonic chain. We will not give details of the derivation of the permanent formula (5.22) from the wave function method. We will review briefly the wave function method in the nearest-neighbor coupled bosonic chain in subsection 6.2.

Schatten and trace distances
We give examples of the Schatten and trace distances in the free bosonic chain from the semiclassical quasiparticle picture and the subsystem mode method.

ρ A,k r VS ρ A,k s
Without loss of generality we require r < s. From the quasiparticle picture and the subsystem mode method, we get the same Schatten and trace distances where i 0 is the largest integer in the range [0, r] that satisfies Note that D bos n (ρ A,k r , ρ A,k s ) = D univ n (ρ A,k r , ρ A,k s ) and D bos 1 (ρ A,k r , ρ A,k s ) = D univ 1 (ρ A,k r , ρ A,k s ).
We show examples of the results in figure 4. It is interesting to note that the derivative of the trace distance D bos 1 (ρ A,k r , ρ A,k s ) with respect to x is not continuous. In the range x ∈ [0, 1], the derivative of the trace distance has min(r, s) discontinuous points.

ρ A,G VS ρ A,K
The universal Schatten and trace distances (3.46) and (3.47) still apply to the RDMs in the bosonic chain, but in the bosonic chain there are more general cases. We just consider the universal distance between the RDMs in the ground state |G and the most general quasiparticle excited state |K = |k r 1 1 · · · k rs s . From the quasiparticle picture, we get the universal Schatten and trace distances Remember the total number of excited quasiparticles R = |K| = s i=1 r i (5.10). In the free bosonic chain, the universal Schatten and trace distances are valid in the condition that all the momentum differences among the excited quasiparticles are large.
In the free bosonic chain, we get the exact Schatten and trace distances from the subsystem mode method D bos n (ρ A,G , ρ A,K ) = where we have One special case of the universal Schatten and trace distances (5.26) and (5.27) are For more general cases, there are corrections to the universal Schatten and trace distances. For example, we obtain the Schatten and trace distances Remember the shorthand α 12 ≡ α k 1 −k 2 with the definition of α k in (3.10). We show the results in figure 5.

ρ A,k 1 VS ρ A,k 2
From the subsystem mode method, we get the Schatten and trace distances We show the results in figure 5.

ρ
From the subsystem mode method, we get the Schatten and trace distances

ρ A,K 1 VS ρ A,K 2
For more general cases, we calculate the Schatten and trace distances numerically, which we will not show here.

Universal short interval expansion
With the above examples, we conjecture that there is universal short interval expansion of the Schatten and trace distances with the definition (5.10), generalizing the result (3.68) in the fermionic chain. In [59] there have been extensive numerical checks for the special case We check extensive examples to support the conjecture (5.39), which we will not show in this paper.

Fidelity
We calculate the fidelity from the subsystem mode method. As the density matrices of the total system and RDMs in the excited states in the bosonic chain are not Gaussian, it is difficult to evaluate the square root of the RDMs, and we could not calculate the general fidelity from the wave function method.

ρ A,k r VS ρ A,k s
From the quasiparticle picture and subsystem mode method, we get the same result of the fidelity Note that F bos (ρ A,k r , ρ A,k s ) = F univ (ρ A,k r , ρ A,k s ).
We show examples of the results in the figure 4.

ρ A,G VS ρ A,K
From the quasiparticle picture, we get the universal fidelity 42) which is valid in the free bosonic chain when the large momentum difference condition is satisfied. More generally, from the subsystem mode method we get the exact fidelity in the free bosonic chain with the definition of F bos A,G,K (5.30). For the single-particle state, there is which is the same as the fidelity (3.73) in the free fermionic chain and the universal fidelity (3.75).
For the double-particle state, there is which is different from the fidelity F fer (ρ A,G , ρ A,k 1 k 2 ) (3.74) in the free fermionic chain. We show it in figure 6.

ρ A,k 1 VS ρ A,k 2
We get the fidelity from the subsystem mode method which is the same as the fidelity F fer (ρ A,k 1 , ρ A,k 2 ) (3.77) in the fermionic chain. We show it in figure 6.

ρ
We also get which is different from the fidelity F fer (ρ A,k 1 , ρ A,k 1 k 2 ) (3.79) in the fermionic chain. The result is shown in figure 6.

ρ
For more general cases, we calculate the fidelity numerically and will not show the results in this paper.

Nearest-neighbor coupled bosonic chain
We use the correlation matrix method and check the three conjectures for the Schatten distances with even integer indices in the quasiparticle excited states of the nearest-neighbor coupled bosonic chain.

Quasiparticle excited states
We consider the chain of nearest-neighbor coupled harmonic oscillators with periodic boundary condition q L+1 = q 1 . It could be diagonalized as The ground state |G is defined as A general excited state takes the form with the normalization factor N K = r 1 ! · · · r s !.

Wave function method
We denote the canonical coordinates of A as R = (q 1 , · · · , q ) and the canonical coordinates of B as S = (q +1 , · · · , q L ). For each quasiparticle state |K , we have the wave function R, S|K , which could be found for example in [59]. In the replica trick, there are n copies of the system, and we have the canonical coordinates Q = (R 1 , S 1 , · · · , R n , S n ) with R a = (q a,1 , · · · , q a, ) and S a = (q a, +1 , · · · , q a,L ), a = 1, · · · , n. We get the trace of the product from which we calculate the Schatten distance with an even integer.

Checks of the three conjectures
We have introduced the three conjectures for subsystem distances in section 2. We check the first conjecture (2.14), the second conjecture (2.19), and the third conjecture (2.23) in respectively the first row, second and third rows of figure 7.     x (e)K 1 =k 1 ,K 2 =k 2 ,K ′ =k 3

XXX chain
In the spin-1/2 XXX chain, we focus on the trace distance and fidelity among the ferromagnetic ground state and the magnon excited states.

Magnon excited states
We consider the spin-1/2 XXX chain in positive transverse field h > 0 with the Hamiltonian and periodic boundary conditions σ x,y,z L+1 = σ x,y,z 1 . We focus on the case with the total number of sites L being four times of an integer.
The XXX chain is in the ferromagnetic phase, and the unique ground state is The low-lying excited states are magnon excited states and can be obtained from the coordinate Bethe ansatz [96,97]. We use the Bethe quantum numbers of the excited magnons I = {I 1 , · · · , I m }, which are integers in the range [0, I − 1], to denote the magnon excited states.
A general magnon excited state |I = |I 1 · · · I m takes the form The normalization factor is We use |j 1 · · · j m to denote configuration that the spins on the sites j 1 , · · · , j m are spin downward and all the other L − m sites are spin upward. The ansatz for the wave function is where S m is the permutation group. The phase θ ii is determined by the equation We always use the convention p i = 2πk i L with the actual momenta and the momenta When there is no ambiguity, we will also use the momenta of the excited magnons K = {k 1 , · · · , k m } to denote the same state. Note that nontrivial relation between the Bethe numbers and the momenta (7.8).

Local mode method
We have the subsystem A = [1, ] and its complement B = [ + 1, L] in the state |I (7.3) with m magnons. We define the indices X i = (x 1 , · · · , x i ) to denote the configuration of the subsystem A that the sites at (x 1 , · · · , x i ) are flipped. Similarly, we define Y i = (y i+1 , · · · , y m ) to characterize the configurations of the subsystems B that the sites at (y i+1 , · · · , y m ) are flipped. The tensor U could be We write the magnon excited state |I in the orthonormal basis Then we get the RDM The matrix V i is well-defined only for i in the range and we also define V i = 0 for other values of i.
For another general state |I with m particle, we get the RDM similar to (7.11) with V i defined in the same way as above. We get the Schatten distance and trace distance  trV i , (7.18) which is a little more efficient for numerical evaluations than formula (7.16).

ρ A,G VS ρ A,I
For the subsystem A = [1, ], we get the trace distance and fidelity between the ground state RDM ρ A,G and the RDM in the general magnon excited state |I = |I 1 · · · I m D XXX with the coefficient

ρ A,G VS ρ A,I
The single-magnon state is 7.5 ρ A,G VS ρ A,I 1 I 2 The double-magnon states could be scattering states or bound states and take the form with Bethe numbers I 1 , I 2 satisfying 0 ≤ I 1 ≤ I 2 ≤ L − 1 and U j 1 j 2 = e i(j 1 p 1 +j 2 p 2 + 1 2 θ) + e i(j 1 p 2 +j 2 p 1 − 1 2 θ) .
The normalization factor is N = The two magnons have physical momenta p 1 , p 2 and momenta k 1 , k 2 being related as Note that k 1 , k 2 may not necessarily be integers or half-integers and may be possibly complex numbers for bound states. The total physical momentum, total momentum, and total Bethe number of the state are p = p 1 + p 2 , k = k 1 + k 2 , I = I 1 + I 2 , (7.28) with p = 2πk L , k = I. (7.29) The total Bethe number I is an integer in the range [0, 2L − 2]. The angle θ is determined by the equation To the equation (7.30), there are three classes of solutions [97], which we reorganize into three cases following [61].

Case I state
For the case I state, there are 31) and the state is We get the trace distance In the scaling limit, it is just where D bos 1 (ρ A,G , ρ A,k 2 ) is the trace distance in the bosonic chain, i.e. the r = 2 case of D bos 1 (ρ A,G , ρ A,k r ) (5.33).
We get the trace distance which in the scaling limit interpolates between the results in the fermionic and bosonic chain. We have There are three cases in the scaling limit.
• For all the other cases, there is the large momentum difference lim L→+∞ |k 12 | = +∞. (7.44) The trace distance (7.41) becomes with the possible values of the total Bethe number There is the odd integerĨ ≈ √ L/π. The Bethe numbers of the two magnons are We get the trace distance In the scaling limit, the parameter v is in the range For v = u L with fixed u in the scaling limit, the trace distance (7.49) becomes We summarize the trace distance (7.49) in the table 1. We show the trace distance (7.51) in the left panel of figure 8.

Case IIIb states
For case IIIb states there are with the possible values of the total Bethe number  Figure 8: The subsystem trace distances between the ground state and the case IIIa double-particle bound states D XXX 1 (ρ A,G , ρ A,I 1 I 2 ) (7.51) (left) and the trace distance between the ground state and the case IIIb double-particle bound states D XXX 1 (ρ A,G , ρ A,I 1 I 2 ) (7.57) (right) in the XXX chain. The horizonal axes u is defined as v = u L . In the left and right panels, the dotted lines are the lower bound of the trace distance D 1 (ρ A,G , ρ A,k ) = x. In the right panel, the dashed lines are the upper bound D bos We get the analytical trace distance In the scaling limit, the parameter v is in the range For v = u L with fixed u in the scaling limit, the trace distance (7.55) becomes

Check of three conjectures
As the XXX chain we consider has a finite positive transverse field, the model is finitely gapped, and the excited magnons always have large energies in the scaling limit, i.e. the large energy condition is always satisfied. We reformulate the three conjectures for the subsystem distances among the ferromagnetic ground state and low-lying magnon scattering states in the ferromagnetic XXX chain.
• For state |I = |I 1 · · · I m with finite number of excited magnons m in the scaling limit, we follow [61] and group the magnons into α clusters according to the scaled Bethe numbers The magnons with ι i = 0 and the magnons with ι i = 1 are grouped in the same cluster, and other magnons with the same ι i ∈ (0, 1) are grouped in the same cluster. We have I = α a=1 I a with I a = {I ab |b = 1, 2, · · · , β a } and m = α a=1 β a . We also denote the same state by the momenta |K = |k 1 · · · k m with the momenta related to the Bethe numbers as (7.8). The momenta are grouped into α clusters in the same way for the Bethe numbers K = α a=1 K a . We conjecture the trace distance and fidelity For the cluster K a with ι a = 0 or ι a = 1, there is For a cluster K a with ι a ∈ (0, 1), there is Here F bos A,G,Ka and F fer A,G,Ka are results in respectively the free fermionic and bosonic chains.
• For two states |K and |K∪K denoted by the momenta satisfying the large momentum difference condition |k − k | → +∞, ∀k ∈ K, ∀k ∈ K , (7.66) we conjecture the trace distance and fidelity The RHS of the conjecture (7.67) and (7.68) could be further simplified according the first conjecture (7.61) and (7.62).
• For two states |K 1 ∪ K and |K 2 ∪ K denoted by the momenta satisfying the condition we conjecture the trace distance and fidelity We check the above three conjectures in the XXX chain in figure 9.

Conclusion and discussion
We have calculated the subsystem Schatten distance, trace distance and fidelity in the quasiparticle excited states of free and coupled fermionic and bosonic chains and the ferromagnetic phase of the spin-1/2 XXX chain from various methods and found consistency for the results. In the free fermionic and bosonic chains, we obtained the subsystem distances from the subsystem mode method, which are still valid in the coupled fermionic and bosonic chains and the XXX chain under certain limit. We followed the universal Rényi and entanglement entropies in [43,44,46,47] and obtained the universal Rényi and entanglement entropies in the large energy and large momentum difference limit. More generally, we followed the three conjectures for the Rényi and entanglement entropies in [61] and formulated three conjectures for subsystem distances and checked the conjectures in the coupled fermionic and bosonic chains and XXX chain. The results in this paper support the scenario that quasiparticles with large energies decouple from the ground state and two sets of quasiparticles with large momentum differences decouple from each other. In particular, we think that the same kind of phenomena should be valid in other integrable models too. Most notably, following the ideas in [61] combined with the results of current paper, calculating the universal subsystem trace distances and their corrections in the XXZ chain is straightforward.
The trace distance is usually difficult to evaluate. For the cases with a few quasiparticles excited in the free fermionic and bosonic chain, we could calculate the trace distance using the subsystem mode  Figure 9: Checks of the first conjecture (7.61) and (7.62) (the first row), the second conjecture (7.67) and (7.68) (the second row), and the third conjecture (7.70) and (7.71) (the third row) in the ferromagnetic phase of the spin-1/2 XXX chain. The symbols in each panel are numerical results for the trace distance D 1 ≡ D XXX 1 (ρ A,K 1 ∪K , ρ A,K 2 ∪K ) and fidelity F ≡ F XXX (ρ A,K 1 ∪K , ρ A,K 2 ∪K ), which are from the local mode method. The solid lines are the analytical conjectured results from the subsystem mode method in the free fermionic and bosonic chains. We have used the Bethe numbers of the excited magnons to denote the states. We have set the Bethe numbers (I 1 , I 2 , I 3 , I 4 , I 5 ) = (1, 3, L 4 , L 4 + 2, L 2 ). For the analytical results we have set L = +∞, and for numerical results we have set L = 128. method. To calculate the trace distance directly in the coupled fermionic chain, we need to construct the explicit RDMs and this method is unfortunately only applicable for a subsystem a very small number of sites. It is worse in the coupled bosonic chain, and we do not have a direct way to calculate the trace distance, even for a small subsystem. In the coupled bosonic chain, it is also difficult to calculate the fidelity. We hope to come back to these problems in the future.

A Calculations for states in nonorthonormal basis
In this appendix, we give an efficient procedure to calculate the Schatten and trace distances and fidelity for density matrices in a general nonorthonormal basis, similar to the calculations of the Rényi and entanglement entropies in [61].
We consider the general density matrix Here ρ P could be the density matrix of the total system or the RDM of a subsystem. For two density matrices ρ P , ρ P , there are the Schatten and trace distances D n (ρ P , ρ P ) = 1 2 1/n (tr|R − R | n ) 1/n , Then we get the fidelity of two density matrices ρ P , ρ P F (ρ P , ρ P ) = tr[(S 1/2 S S 1/2 ) 1/2 ]. (A.10) Noting that S = Λ 1/2 U † RU Λ −1/2 and S = Λ 1/2 U † R U Λ −1/2 we obtain the fidelity calculated as F (ρ P , ρ P ) = tr[(R 1/2 R R 1/2 ) 1/2 ]. (A.11) When the matrices P, Q, R, S are block diagonal We further write the Schatten and trace distances and fidelity as In this appendix, we give a derivation of the recursive formula (3.27), following the derivation of (4.12) in [93].
For the interval A = [1, ], the RDM ρ C corresponding to the × correlation matrix C is [16,18] ρ C = det(1 − C)e −c † Hc , (B.1) with the relation H = log 1−C C and the shorthand Note that the RDM has been properly normalized trρ C = 1. From we get which is just With some simple algebra, we get Then we obtain the trace Then the recursive formula (3.27) is derived.