Trace distance between fermionic Gaussian states from a truncation method Jiaju Zhang1and M. A. Rajabpour2 1Center for Joint Quantum Studies and Department of Physics School of Science Tianjin University

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Trace distance between fermionic Gaussian states from a truncation method

Jiaju Zhang1, ∗and M. A. Rajabpour2, †

1Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University,

135 Yaguan Road, Tianjin 300350, China

2Instituto de Fisica, Universidade Federal Fluminense,

Av. Gal. Milton Tavares de Souza s/n, Gragoatá, 24210-346, Niterói, RJ, Brazil

In this paper, we propose a novel truncation method for determining the trace distance between

two Gaussian states in fermionic systems. For two fermionic Gaussian states, characterized by

their correlation matrices, we consider the von Neumann entropies and dissimilarities between their

correlation matrices and truncate the correlation matrices to facilitate trace distance calculations.

Our method exhibits notable eﬃcacy in two distinct scenarios. In the ﬁrst scenario, the states have

small von Neumann entropies, indicating ﬁnite or logarithmic-law entropy, while their correlation

matrices display near-commuting behavior, characterized by a ﬁnite or gradual nonlinear increase in

the trace norm of the correlation matrix commutator relative to the system size. The second scenario

encompasses situations where the two states are nearly orthogonal, with a maximal canonical value

diﬀerence approaching 2. To evaluate the performance of our method, we apply it to various

compelling examples. Notably, we successfully compute the subsystem trace distances between low-

lying eigenstates of Ising and XX spin chains, even for signiﬁcantly large subsystem sizes. This is

in stark contrast to existing literature, where subsystem trace distances are limited to subsystems

of approximately ten sites. With our truncation method, we extend the analysis to subsystems

comprising several hundred sites, thus expanding the scope of research in this ﬁeld.

I. INTRODUCTION

Quantitative diﬀerentiation of quantum states is es-

sential in quantum information theory [1,2], and also

plays important roles in quantum many-body systems,

quantum ﬁeld theories and gravity [3–34]. Various quan-

tities, such as relative entropy, Schatten distances, trace

distance, and ﬁdelity, can be employed for this purpose,

and in this study, our focus is speciﬁcally on the trace

distance. The trace distance, denoted by D(ρ, σ), quan-

tiﬁes the distinguishability between two density matrices

ρand σand is deﬁned as half of the trace norm of their

diﬀerence [1,2], i.e.,

D(ρ, σ) = 1

2tr|ρ−σ|.(1)

In the context of extended quantum systems, the trace

distance oﬀers distinct advantages compared to other

measures of distinguishability. It is not only a well-

deﬁned mathematical distance, but in the scaling limit,

it can discern states that remain indistinguishable us-

ing other measures [24]. Furthermore, the trace dis-

tance D, deﬁned as D=D(ρ, σ)between two states ρ

and σ, serves as an upper bound for the diﬀerence be-

tween their von Neumann entropies S(ρ) = −tr(ρlog ρ)

and S(σ) = −tr(σlog σ), as established by the Fannes-

Audenaert inequality [35,36]

|S(ρ)−S(σ)| ≤ Dlog(d−1)−Dlog D−(1−D) log(1−D),

(2)

∗jiajuzhang@tju.edu.cn

†mohammadali.rajabpour@gmail.com

where drepresents the dimension of the Hilbert space. By

employing a speciﬁc case of Hölder’s inequality, the trace

distance also provides an upper bound on the diﬀerence

in expectation values of an operator

|tr[(ρ−σ)O]| ≤ 2smax(O)D(ρ, σ),(3)

where smax denotes the maximal singular value of the

operator O. This property of the trace distance proves

highly valuable for deﬁning the eigenstate thermalization

hypothesis (ETH), initially proposed in terms of local op-

erator expectation values [37–39], and subsequently gen-

eralized to the subsystem ETH in relation to the subsys-

tem trace distance [10,13]. Moreover, the average sub-

system trace distance of neighboring states in the spec-

trum can also be utilized as a novel signature to diﬀerenti-

ate between chaotic and integrable many-body quantum

systems [40]. However, calculating the trace distance for

large systems poses a notorious challenge due to the ex-

ponential growth of the Hilbert space dimension with the

number of qubits.

In this study, we also examine the ﬁdelity, denoted by

F(ρ, σ), which serves as a benchmark for comparison with

the trace distance. The ﬁdelity provides both an upper

and a lower bound for the trace distance [1,2], given by

1−F(ρ, σ)≤D(ρ, σ)≤p1−F(ρ, σ)2.(4)

For fermionic Gaussian states ρΓ1and ρΓ2, determined

by their respective correlation matrices Γ1and Γ2, the

exact ﬁdelity can be calculated as [41]

F(ρΓ1, ρΓ2) = det 1−Γ1

21/4det 1−Γ2

21/4

×hdet 1 + sr1+Γ1

1−Γ1

1+Γ2

1−Γ2r1+Γ1

1−Γ1i1/2.(5)

arXiv:2210.11865v3 [cond-mat.str-el] 22 Aug 2023

It is important to note that a fermionic Gaussian state

ρand its corresponding correlation matrix Γare related

as described in equation (16). For recent advancements

in calculating the trace distance and ﬁdelity using varia-

tional techniques and quantum computers, refer to [42–

47].

In scenarios where the dimension of the Hilbert space

is excessively large, exact evaluation of the trace distance

becomes impractical, despite the fact that the dominant

contributions to the trace distance often arise from a

signiﬁcantly smaller subspace within the Hilbert space.

This serves as motivation for our development of a trun-

cation method designed to calculate the trace distance

between two Gaussian states in fermionic systems. While

straightforward for two pure states, the task becomes

nontrivial when dealing with two mixed states, which

can manifest as either density matrices of the entire sys-

tem or reduced density matrices (RDMs) of a subsystem.

An ideal situation for the eﬀectiveness of the truncation

method can be characterized as follows: Consider two

states represented by density matrices expressed as

ρ1=eρ⊗eρ1, ρ2=eρ⊗eρ2,(6)

where the dimension of eρis signiﬁcantly larger than that

of eρ1and eρ2. In this ideal scenario, the speciﬁc form of

eρdoes not impact the trace distance calculation, leading

to a simpliﬁed expression:

D(ρ1, ρ2) = D(eρ1,eρ2).(7)

For more general cases involving two distinct states, ρ1

and ρ2, the objective of the truncation method is to iden-

tify eρ1and eρ2with signiﬁcantly smaller dimensions com-

pared to ρ1and ρ2, enabling an approximation

D(ρ1, ρ2)≈D(eρ1,eρ2).(8)

A fermionic Gaussian state can be uniquely determined

by the two-point correlation functions of the Majorana

modes in the system or subsystem, which can be or-

ganized to construct a purely imaginary anti-symmetric

correlation matrix [48,49]. By employing an orthogonal

transformation, the correlation matrix can be brought

into canonical form. The canonical values of the ma-

trix govern the von Neumann entropy of the state, while

the corresponding canonical vectors determine the eﬀec-

tive modes. To truncate the Hilbert space dimension,

we employ diﬀerent strategies to select a limited num-

ber of eﬀective modes. The ﬁrst strategy, which we call

the maximal entropy strategy, involves choosing the ef-

fective modes with the smallest canonical values for the

two relevant states. These modes make the largest con-

tributions to the sum of the von Neumann entropies.

In the second strategy, we compute the canonical val-

ues and canonical vectors of the diﬀerence between the

two correlation matrices and select the eﬀective modes

with the largest canonical values. These chosen modes

contribute the most to the diﬀerences in the two-point

correlation functions of the two states, leading us to name

this method the maximal diﬀerence strategy. Addition-

ally, we adopt a mixed strategy that combines elements

of both the maximal entropy strategy and the maximal

diﬀerence strategy. In this approach, some modes are

chosen according to one strategy, while the remaining

modes are selected using the other strategy. We refer to

this combined method as the mixed strategy. It is worth

noting that the mixed strategy of the truncation method

consistently provides the most accurate estimation of the

trace distance due to the contractive nature of the trace

distance under partial trace.

The truncation method demonstrates its eﬀectiveness

in two intriguing scenarios. The ﬁrst scenario encom-

passes the cases where the states have a low von Neu-

mann entropy, indicating either ﬁnite or logarithmic-

law entropy, and their correlation matrices exhibit near-

commuting behavior, which is characterized by a ﬁnite

or slow nonlinear increase of the trace norm of the cor-

relation matrix commutator with respect to the system

size. The second scenario includes the situations where

the states are nearly orthogonal, with a correlation ma-

trix diﬀerence featuring a maximal canonical value ap-

proaching 2. We validate the eﬃcacy of the truncation

method through multiple examples, including the calcu-

lation of eigenstate RDMs in Ising chains, XX chains, and

ground state RDMs in Ising chains with diﬀerent trans-

verse ﬁelds. Notably, we apply the method to compute

subsystem trace distances between low-lying eigenstates

in critical Ising and XX spin chains, with signiﬁcantly

larger subsystem sizes than those considered in previous

works [21,23,34]. The previous studies only obtained

trace distances for relatively small subsystem sizes, with

a maximum size of 7 in [21,23] and 12 in [34]. In con-

trast, this paper presents results with a maximal sub-

system size of 359, surpassing the limitations of previous

studies and enabling trace distance calculations for much

larger subsystem sizes.

The paper is structured as follows: Section II presents

a detailed explanation of the truncated canonicalized cor-

relation matrix method for fermionic Gaussian states.

The conditions under which the truncation method is ef-

fective are discussed in Section III. Examples of its appli-

cation to eigenstate RDMs in the Ising chain and ground

state RDMs in Ising chains with diﬀerent transverse ﬁelds

are examined in Sections IV and V, respectively. The

truncation method is further validated through exam-

ples of low-lying eigenstate RDMs in the critical Ising

chain (Section VI) and half-ﬁlled XX chain (Section VII).

Concluding discussions are provided in Section VIII. Ad-

ditionally, Appendix Apresents a simple example illus-

trating the impossibility of a ﬁnite truncation method

for the trace distance between two orthogonal Gaussian

states in the scaling limit. Appendix Bintroduces the

truncated diagonalized correlation matrix method, a spe-

cialized version of the truncation method, applicable to

Gaussian states in the free fermionic theory where the

number of excited Dirac modes is conserved.

II. TRUNCATED CANONICALIZED

CORRELATION MATRIX METHOD

In this section, we begin by providing a brief overview

of the canonicalized correlation matrix method presented

in [34], which is an exact approach requiring the ex-

plicit construction of density matrices. However, its ef-

ﬁciency is limited to cases involving very small system

sizes. To address this limitation, we introduce the trun-

cated canonicalized correlation matrix method, which al-

lows for proper truncation under speciﬁc conditions. By

constructing eﬀective density matrices within a signiﬁ-

cantly smaller subspace of the full Hilbert space, we can

approximate the trace distance for cases involving much

larger system sizes.

A. Canonicalized correlation matrix method

We consider a system consisting of ℓspinless fermions,

denoted by ajand a†

jwith j= 1,2,··· , ℓ. This system

can represent either the entire system or a subsystem. To

facilitate our analysis, we introduce the Majorana modes

deﬁned as

d2j−1=aj+a†

j, d2j= i(aj−a†

j), j = 1,2,··· , ℓ. (9)

These Majorana modes satisfy the anticommutation re-

lations:

{dm1, dm2}= 2δm1m2, m1, m2= 1,2,··· ,2ℓ. (10)

The system is described by a general Gaussian state with

a density matrix ρ. This state is characterized by a two-

point correlation function matrix Γwith elements given

Γm1m2=⟨dm1dm2⟩ρ−δm1m2,

m1, m2= 1,2,··· ,2ℓ. (11)

All other correlation functions in the Gaussian state can

be derived from the correlation matrix Γusing Wick con-

tractions.

The correlation matrix Γpossesses the properties of

being purely imaginary and anti-symmetric, and it can

be converted to the canonical form using the approach

described in [50,51]. In this paper, we adopt the proce-

dure outlined in [51]. Speciﬁcally, we have the equations

Γuj=−iγjvj,Γvj= iγjuj, j = 1,2,··· , ℓ, (12)

where γjrepresents real numbers within the range [0,1]

and ujand vjare real 2ℓ-component vectors. Each index

j= 1,2,··· , ℓ corresponds to a canonical value of Γ, and

the associated vectors ujand vjare referred to as the

canonical vectors. It is important to note that a 2ℓ×2ℓ

correlation matrix has ℓcanonical values and 2ℓcanoni-

cal vectors. The canonical vectors satisfy orthonormality

conditions given by

j1uj2=vT

j1vj2=δj1j2, uT

j1vj2= 0,

j1, j2= 1,2,··· , ℓ. (13)

To transform Γinto the canonical form, we deﬁne the

2ℓ×2ℓorthogonal matrix as

Q= (u1, v1, u2, v2,··· , uℓ, vℓ),(14)

where each vector is represented as a column vector. The

matrix Qfacilitates the transformation of Γaccording to

the equation

QTΓQ=

ℓ

j=1 0 iγj

−iγj0!.(15)

The density matrix of the Gaussian state can be ex-

pressed in terms of the modular Hamiltonian as [52–54]

ρ=rdet 1−Γ

2exp −1

2ℓ

m1,m2=1

Wm1m2dm1dm2,

(16)

where the matrix Wis deﬁned as

W= arctanh Γ.(17)

Similar to Γ, the matrix Wis also purely imaginary and

anti-symmetric and can be transformed as

QTW Q =

ℓ

j=1 0 iδj

−iδj0!,

δj= arctanh γj, j = 1,2,··· , ℓ. (18)

We introduce the eﬀective Majorana modes ˜

dm1, deﬁned

dm1≡

2ℓ

m2=1

Qm2m1dm2, m1= 1,2,··· ,2ℓ, (19)

which also satisfy the anticommutation relations

{˜

dm1,˜

dm2}= 2δm1m2, m1, m2= 1,2,··· ,2ℓ. (20)

The 2ℓeﬀective Majorana modes ˜

dmwith m=

1,2,··· ,2ℓcan be organized into ℓpairs {˜

d2j−1,˜

d2j}with

j= 1,2,··· , ℓ. In the Gaussian state ρcharacterized by

the correlation matrix Γ, each pair of eﬀective Majorana

modes {˜

d2j−1,˜

d2j}with j= 1,2,··· , ℓ decouple from the

other eﬀective Majorana modes.

The density matrix can be expressed as

ρ="ℓ

j=1 p(1 + γj)(1 −γj)

2#exp −i

ℓ

j=1

δj˜

d2j−1˜

d2j

ℓ

j=1

1−iγj˜

d2j−1˜

d2j

2.(21)

To verify the properties of the density matrix, we check

that trρ= 1 and tr(ρ˜

d2j−1˜

d2j) = iγjfor j= 1,2,··· , ℓ.

For calculating the trace distance between two Gaussian

states, we use their explicit density matrices. To calculate

the ﬁdelity, we can utilize the formula

√ρ=

ℓ

j=1 [(p1 + γj+p1−γj)(22)

−i(p1 + γj−p1−γj)˜

d2j−1˜

d2j]/(2√2),

which is derived by considering

√ρ=

ℓ

j=1

(˜αj+˜

βj˜

d2j−1+ ˜γj˜

d2j+˜

δj˜

d2j−1˜

d2j),(23)

and determining the constants ˜αj,˜

βj,˜γj, and ˜

δjby com-

paring √ρ2with ρin (21).

B. Truncated canonicalized correlation matrix

method

To calculate the approximate trace distance and ﬁ-

delity, we employ three strategies for implementing the

truncated canonicalized correlation matrix method. The

approach involves dimension truncation of the Hilbert

space to a subspace that is spanned by a properly se-

lected, limited number of eﬀective Majorana modes.

1. Maximal entropy strategy

Using a similarity transformation, we can change the

corresponding density matrix ρinto a speciﬁc form for

the correlation matrix Γ(as given by Equation (15)).

This transformation is described in references [48,49]

and yields

ρ∼

ℓ

j=1 1−γj

21+γj

2!.(24)

The von Neumann entropy of ρis simply the sum of the

Shannon entropies of each eﬀective probability distribu-

tion {1−γj

2,1+γj

2}with j= 1,2,··· , ℓ

S(ρ) =

ℓ

j=1 −1−γj

2log 1−γj

2−1 + γj

2log 1 + γj

2.

(25)

For low-rank states ρ, the entropy S(ρ)is not large, al-

lowing us to truncate the eﬀective probability distribu-

tion {1−γj

2,1+γj

2}with j= 1,2,··· , ℓ to only include

a few values with the smallest γj. Applying the same

idea to two Gaussian states ρ1and ρ2, we truncate the

density matrices and calculate their trace distance. It’s

important to note that for high-rank states ρ, where the

entropy S(ρ)is large, imposing truncation is generally in-

eﬃcient and the maximal entropy strategy introduced in

this subsection is expected to fail. The ﬂowchart in Fig-

ure 1illustrates the evaluation of the trace distance using

start

canonicalize and

construct and from and

calculate

stop

choose vectors

with maximal contributions to

orthonormalize vectors

and obtain vectors

truncate , and obtain ,

FIG. 1. The ﬂowchart illustrates the evaluation of the trace

distance using the maximal entropy strategy within the trun-

cated canonicalized correlation matrix method.

the maximal entropy strategy of the truncated canonical-

ized correlation matrix method.

We consider two general Gaussian states ρ1and ρ2with

correlation matrices Γ1and Γ2. The canonical values

and vectors (γj, uj, vj)with j= 1,2,··· , ℓ correspond

to the matrix Γ1, while the canonical values and vectors

(γj, uj, vj)with j=ℓ+ 1, ℓ + 2,··· ,2ℓcorrespond to the

matrix Γ2. To choose 2teﬀective Majorana modes whose

corresponding canonical values contribute the most to

the entropy sum of the two states, given by

S(ρ1) + S(ρ2) =

2ℓ

j=1 −1−γj

2log 1−γj

−1 + γj

2log 1 + γj

2,(26)

we sort the 2ℓsets of canonical values and vectors

(γj, uj, vj)with j= 1,2,··· ,2ℓin ascending order based

on the values of γj∈[0,1]. Then, we select the ﬁrst

2tcanonical vectors (uj, vj)corresponding to the ﬁrst t

canonical values γj, and denote these selected vectors as

wi, where i= 1,2,··· ,2t.

The 2treal vectors wiwith i= 1,2,··· ,2tare not

generally orthogonal and may not be independent. To

orthogonalize and normalize, a.k.a. orthonormalize, these

vectors, we diagonalize the 2t×2tmatrix V, whose en-

tries are given by Vi1i2=wT

i1wi2for i1, i2= 1,2,··· ,2t.

If the 2tvectors wiare not independent, the 2t×2tma-

trix Vwill have eigenvalues close to zero. It’s important

to consider numerical errors that can lead to very small

eigenvalues. We sort the eigenvalues and orthonormal

eigenvectors (αi, ai)with i= 1,2,··· ,2tof Vbased on

the values of αiin descending order, discarding those

smaller than a certain cutoﬀ (e.g., 10−9). In cases where

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TracedistancebetweenfermionicGaussianstatesfromatruncationmethodJiajuZhang1,∗andM.A.Rajabpour2,†1CenterforJointQuantumStudiesandDepartmentofPhysics,SchoolofScience,TianjinUniversity,135YaguanRoad,Tianjin300350,China2InstitutodeFisica,UniversidadeFederalFluminense,Av.Gal.MiltonTavaresdeSouzas/n,Grago...

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Trace distance between fermionic Gaussian states from a truncation method Jiaju Zhang1and M. A. Rajabpour2 1Center for Joint Quantum Studies and Department of Physics School of Science Tianjin University

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