arXiv:1610.01815 · Physical review. B./Physical review. B

Entanglement distance between quantum states and its implications for a density-matrix renormalization group study of degenerate ground states

Mohammad-Sadegh Vaezi, Abolhassan Vaezi

Published Oct 16, 2017• 1 citation

Abstract

We study the concept of entanglement distance between two quantum states, which quantifies the amount of information shared between their reduced density matrices (RDMs). Using analytical arguments combined with density-matrix renormalization group (DMRG) and exact diagonalization (ED) calculations, we show that for gapless systems the entanglement distance has power law dependence on the energy separation and subsystem size, with ${\ensuremath{\alpha}}_{E}$ and ${\ensuremath{\alpha}}_{\ensuremath{\ell}}$ exponents, respectively. Using conformal field theory (CFT) we find ${\ensuremath{\alpha}}_{E}=2$ and ${\ensuremath{\alpha}}_{\ensuremath{\ell}}=4$ for Abelian theories with $c=1$, as in the case of free fermions. For non-Abelian CFTs ${\ensuremath{\alpha}}_{E}=0$, and ${\ensuremath{\alpha}}_{\ensuremath{\ell}}$ is twice the conformal dimension of the thermal primary fields. For instance, for ${Z}_{3}$ parafermion CFT ${\ensuremath{\alpha}}_{E}=1$ and ${\ensuremath{\alpha}}_{\ensuremath{\ell}}=4/5$. For gapped 1+1 dimensional (1+1D) fermion systems, we show that the entanglement distance divides the low energy excitations into two branches with different values of ${\ensuremath{\alpha}}_{E}$ and ${\ensuremath{\alpha}}_{\ensuremath{\ell}}$. These two branches are related to momentum transfers near zero and $\ensuremath{\pi}$. We also demonstrate that the entanglement distance reaches its maximum for degenerate states related through nonlocal operators such as Wilson loops. For example, degenerate ground states (GSs) of 2+1D topological states have maximum entanglement distance. In contrast, degenerate GSs related through confined anyon excitations such as genons have minimum entanglement distance. Various implications of this concept for quantum simulations are discussed. Finally, based on the ideas developed we discuss the computational complexity of DMRG algorithms that are capable of finding all degenerate GSs.

DOI: 10.1103/physrevb.96.165129

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arXiv:1610.01815 · Physical review. B./Physical review. B

Entanglement distance between quantum states and its implications for a density-matrix renormalization group study of degenerate ground states

Mohammad-Sadegh Vaezi, Abolhassan Vaezi

Published Oct 16, 2017• 1 citation

arXiv HTML PDF

Abstract

{\ensuremath{\alpha}}_{E}

and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}

exponents, respectively. Using conformal field theory (CFT) we find

{\ensuremath{\alpha}}_{E}=2

and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}=4

for Abelian theories with

c=1

, as in the case of free fermions. For non-Abelian CFTs

{\ensuremath{\alpha}}_{E}=0

, and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}

is twice the conformal dimension of the thermal primary fields. For instance, for

{Z}_{3}

parafermion CFT

{\ensuremath{\alpha}}_{E}=1

and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}=4/5

. For gapped 1+1 dimensional (1+1D) fermion systems, we show that the entanglement distance divides the low energy excitations into two branches with different values of

{\ensuremath{\alpha}}_{E}

and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}

. These two branches are related to momentum transfers near zero and

\ensuremath{\pi}

. We also demonstrate that the entanglement distance reaches its maximum for degenerate states related through nonlocal operators such as Wilson loops. For example, degenerate ground states (GSs) of 2+1D topological states have maximum entanglement distance. In contrast, degenerate GSs related through confined anyon excitations such as genons have minimum entanglement distance. Various implications of this concept for quantum simulations are discussed. Finally, based on the ideas developed we discuss the computational complexity of DMRG algorithms that are capable of finding all degenerate GSs.

DOI: 10.1103/physrevb.96.165129

Related papers

Entanglement distance between quantum states and its implications for DMRG study of degenerate ground-states

Mohammad-Sadegh Vaezi, Abolhassan Vaezi

We introduce the novel concept of entanglement distance between two quantum states which quantifies the amount of information shared between their reduced density matrices (RDMs). Using analytical arguments combined with density matrix renormalization group (DMRG) and exact diagonalization (ED) calculations, we show that for gapless systems the entanglement distance has power law dependence on the energy separation and subsystem size with $\alpha_E$ and $\alpha_{\ell}$ exponents, respectively. Using conformal field theory (CFT) we find $\alpha_E = 2$ and $\alpha_{\ell} = 4$ for Abelian theories with $c=1$ such as free fermions. For non-Abelian CFTs $\alpha_E = 0$ , and $\alpha_{\ell}$ is twice the conformal dimension of the thermal primary fields. For instance for $Z_3$ parafermion CFT $\alpha_E = 1$ and $\alpha_{\ell} = 4/5$ . For gapped 1+1D fermion systems, we show that the entanglement distance divides the low energy excitations into two branches with different values of $\alpha_E$ and $\alpha_{\ell}$ . These two branches are related to momentum transfers near zero and $\pi$ . We also demonstrate that the entanglement distance reaches its maximum for degenerate states related through nonlocal operators such as Wilson loops. For example, degenerate ground-states (GSs) of 2+1 D topological states have maximum entanglement distance. On the contrary, degenerate GSs related through confined anyon excitations such as genons have minimum entanglement distance. Various implications of this concept for quantum simulations are discussed. Finally, based on the ideas developed we introduce two modified DMRG algorithms that are capable of finding all degenerate GSs.We introduce the novel concept of entanglement distance between two quantum states which quantifies the amount of information shared between their reduced density matrices (RDMs). Using analytical arguments combined with density matrix renormalization group (DMRG) and exact diagonalization (ED) calculations, we show that for gapless systems the entanglement distance has power law dependence on the energy separation and subsystem size with $\alpha_E$ and $\alpha_{\ell}$ exponents, respectively. Using conformal field theory (CFT) we find $\alpha_E = 2$ and $\alpha_{\ell} = 4$ for Abelian theories with $c=1$ such as free fermions. For non-Abelian CFTs $\alpha_E = 0$ , and $\alpha_{\ell}$ is twice the conformal dimension of the thermal primary fields. For instance for $Z_3$ parafermion CFT $\alpha_E = 1$ and $\alpha_{\ell} = 4/5$. For gapped 1+1D fermion systems, we show that the entanglement distance divides the low energy excitations into two branches with different values of $\alpha_E$ and $\alpha_{\ell}$. These two branches are related to momentum transfers near zero and $\pi$. We also demonstrate that the entanglement distance reaches its maximum for degenerate states related through nonlocal operators such as Wilson loops. For example, degenerate ground-states (GSs) of 2+1 D topological states have maximum entanglement distance. On the contrary, degenerate GSs related through confined anyon excitations such as genons have minimum entanglement distance. Various implications of this concept for quantum simulations are discussed. Finally, based on the ideas developed we introduce two modified DMRG algorithms that are capable of finding all degenerate GSs.

Oct 6, 20160 citations

Subsystem trace distance in low-lying states of (1 + 1)-dimensional conformal field theories

Jiaju Zhang, Paola Ruggiero, Pasquale Calabrese

Abstract We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero and P. Calabrese, Phys. Rev. Lett.122(2019) 141602 ], of which this work is a completion. The trace distance between two reduced density matricesρAandσAis obtained from the moments tr(ρA− σA)nand taking the limitn →1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of lengthℓembedded in the low lying eigenstates of a one-dimensional critical system of lengthL, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be ascale invariant universal functionofℓ/L. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schattenn-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers ofℓ/L. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results in two-dimensional CFT are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.Abstract We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero and P. Calabrese, Phys. Rev. Lett.122(2019) 141602 ], of which this work is a completion. The trace distance between two reduced density matricesρAandσAis obtained from the moments tr(ρA− σA)nand taking the limitn →1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of lengthℓembedded in the low lying eigenstates of a one-dimensional critical system of lengthL, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be ascale invariant universal functionofℓ/L. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schattenn-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers ofℓ/L. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results in two-dimensional CFT are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.

arXiv:1907.04332Journal of High Energy PhysicsOct 1, 201935 citations

Scaling of entanglement entropy in some condensed matter systems

Xiao Dong Chen

The following thesis focuses on the scaling of entanglement entropy in lower dimensions and is divided into three main parts. Chapter 2 studies the thermal reduced density matrices in fermion and spin systems on ladders. Chapter 3 studies the many-body localization phase transition in a Rokhsar-Kivelson type wave function. Chapter 4 studies the subleading correction term of entanglement entropy in 2+1 dimensional scale invariant systems. Chapter 5 studies the bulk-boundary correspondence in 3 + 1 dimensional topological phases and its entanglement entropy. In chapter 2, we investigate the reduced density matrices for a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in 1/2 spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1+1 dimensions) is discussed using the replica trick and scaling arguments. We show that 1) for a system with a bulk gap, the reduced density matrix has the form of a thermal density matrix, 2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, 3) the von Neumann entropy equals to the thermodynamic entropy of one chain, and 4) the bulk gap plays the role of effective temperature. In chapter 3, we construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the Random Energy Model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with subsystem size: scaling with entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a sub-extensive scaling between these limits. Near the phase transition point, the fluctuations of the Renyi entropies are non-Gaussian. We find that Renyi entropies with different Renyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior. In chapter 4, we study the universal scaling behavior of the entanglement entropy of critical theories in 2+1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from a) the quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz model, and c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision. In chapter 5, we discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level K, and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus T3 parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under SL(3, Z) modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular S and T matrices computed from the boundary field theories with those computed in the bulk. We propose the three-loop braiding statistics can be studied by constructing the modular S and T matrices from an appropriate boundary field theory. We also study the EE for 3 + 1- dimensional topological phase with or without three-loop braiding phase.The following thesis focuses on the scaling of entanglement entropy in lower dimensions and is divided into three main parts. Chapter 2 studies the thermal reduced density matrices in fermion and spin systems on ladders. Chapter 3 studies the many-body localization phase transition in a Rokhsar-Kivelson type wave function. Chapter 4 studies the subleading correction term of entanglement entropy in 2+1 dimensional scale invariant systems. Chapter 5 studies the bulk-boundary correspondence in 3 + 1 dimensional topological phases and its entanglement entropy. In chapter 2, we investigate the reduced density matrices for a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in 1/2 spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1+1 dimensions) is discussed using the replica trick and scaling arguments. We show that 1) for a system with a bulk gap, the reduced density matrix has the form of a thermal density matrix, 2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, 3) the von Neumann entropy equals to the thermodynamic entropy of one chain, and 4) the bulk gap plays the role of effective temperature. In chapter 3, we construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the Random Energy Model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with subsystem size: scaling with entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a sub-extensive scaling between these limits. Near the phase transition point, the fluctuations of the Renyi entropies are non-Gaussian. We find that Renyi entropies with different Renyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior. In chapter 4, we study the universal scaling behavior of the entanglement entropy of critical theories in 2+1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from a) the quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz model, and c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision. In chapter 5, we discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level K, and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus T3 parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under SL(3, Z) modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular S and T matrices computed from the boundary field theories with those computed in the bulk. We propose the three-loop braiding statistics can be studied by constructing the modular S and T matrices from an appropriate boundary field theory. We also study the EE for 3 + 1- dimensional topological phase with or without three-loop braiding phase.

Aug 12, 20160 citations

Subsystem trace distance in low-lying states of conformal field theories

Jiaju Zhang, Paola Ruggiero, Pasquale Calabrese

We report on a systematic replica approach to calculate the subsystem trace distance for a generic quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero, P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019)], of which this work is a completion. The trace distance between two reduced density matrices $\rho_A$ and $\sigma_A$ is obtained from the moments $\textrm{tr} (\rho_A-\sigma_A)^n$ and taking the limit $n\to1$ of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length $\ell$ embedded in the low lying eigenstates of a one-dimensional critical system of length $L$ , a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a {\it scale invariant universal function} of $\ell/L$ . Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten $n$ -distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of $\ell/L$ . The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.We report on a systematic replica approach to calculate the subsystem trace distance for a generic quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero, P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019)], of which this work is a completion. The trace distance between two reduced density matrices $\rho_A$ and $\sigma_A$ is obtained from the moments $\textrm{tr} (\rho_A-\sigma_A)^n$ and taking the limit $n\to1$ of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length $\ell$ embedded in the low lying eigenstates of a one-dimensional critical system of length $L$, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a {\it scale invariant universal function} of $\ell/L$. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten $n$-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of $\ell/L$. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.

Jul 9, 20191 citation

Quantum entanglement and thermal reduced density matrices in fermion and spin systems on ladders

Xiao Chen, Eduardo Fradkin

Numerical studies of the reduced density matrix of a gapped spin-1/2 Heisenberg antiferromagnet on a two-leg ladder find that it has the same form as the Gibbs density matrix of a gapless spin-1/2 Heisenberg antiferromagnetic chain at a finite temperature determined by the spin gap of the ladder. We investigate this interesting result by considering a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. a cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1 + 1 dimensions) are discussed using the replica trick and scaling arguments. We show that (1) for a system with a bulk gap the reduced density matrix has the form of a thermal density matrix, (2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, (3) the von Neumann entropy equals the thermodynamic entropy of one chain, and (4) the bulk gap plays the role of effective temperature.Numerical studies of the reduced density matrix of a gapped spin-1/2 Heisenberg antiferromagnet on a two-leg ladder find that it has the same form as the Gibbs density matrix of a gapless spin-1/2 Heisenberg antiferromagnetic chain at a finite temperature determined by the spin gap of the ladder. We investigate this interesting result by considering a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. a cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1 + 1 dimensions) are discussed using the replica trick and scaling arguments. We show that (1) for a system with a bulk gap the reduced density matrix has the form of a thermal density matrix, (2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, (3) the von Neumann entropy equals the thermodynamic entropy of one chain, and (4) the bulk gap plays the role of effective temperature.

arXiv:1305.6538Journal of Statistical Mechanics Theory and ExperimentAug 27, 201339 citations

Subsystem Trace Distance in Quantum Field Theory

Jiaju Zhang, Paola Ruggiero, Pasquale Calabrese

We develop a systematic method to calculate the trace distance between two reduced density matrices in 1+1 dimensional quantum field theories. The approach exploits the path integral representation of the reduced density matrices and an ad hoc replica trick. We then extensively apply this method to the calculation of the distance between reduced density matrices of one interval of length ℓ in eigenstates of conformal field theories. When the interval is short, using the operator product expansion of twist operators, we obtain a universal form for the leading order in ℓ of the trace distance. We compute the trace distances among the reduced density matrices of several low lying states in two-dimensional free massless boson and fermion theories. We compare our analytic conformal results with numerical calculations in XX and Ising spin chains finding perfect agreement.We develop a systematic method to calculate the trace distance between two reduced density matrices in 1+1 dimensional quantum field theories. The approach exploits the path integral representation of the reduced density matrices and an ad hoc replica trick. We then extensively apply this method to the calculation of the distance between reduced density matrices of one interval of length ℓ in eigenstates of conformal field theories. When the interval is short, using the operator product expansion of twist operators, we obtain a universal form for the leading order in ℓ of the trace distance. We compute the trace distances among the reduced density matrices of several low lying states in two-dimensional free massless boson and fermion theories. We compare our analytic conformal results with numerical calculations in XX and Ising spin chains finding perfect agreement.

arXiv:1901.10993Physical Review LettersApr 9, 201948 citations

{\ensuremath{\alpha}}_{E}

and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}

exponents, respectively. Using conformal field theory (CFT) we find

{\ensuremath{\alpha}}_{E}=2

and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}=4

for Abelian theories with

c=1

, as in the case of free fermions. For non-Abelian CFTs

{\ensuremath{\alpha}}_{E}=0

, and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}

is twice the conformal dimension of the thermal primary fields. For instance, for

{Z}_{3}

parafermion CFT

{\ensuremath{\alpha}}_{E}=1

and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}=4/5

. For gapped 1+1 dimensional (1+1D) fermion systems, we show that the entanglement distance divides the low energy excitations into two branches with different values of

{\ensuremath{\alpha}}_{E}

and

{\ensuremath{\alpha}}_{\ensuremath{\ell}}

. These two branches are related to momentum transfers near zero and

\ensuremath{\pi}

\alpha_E

and

\alpha_{\ell}

exponents, respectively. Using conformal field theory (CFT) we find

\alpha_E = 2

and

\alpha_{\ell} = 4

for Abelian theories with

c=1

such as free fermions. For non-Abelian CFTs

\alpha_E = 0

, and

\alpha_{\ell}

is twice the conformal dimension of the thermal primary fields. For instance for

Z_3

parafermion CFT

\alpha_E = 1

and

\alpha_{\ell} = 4/5

. For gapped 1+1D fermion systems, we show that the entanglement distance divides the low energy excitations into two branches with different values of

\alpha_E

and

\alpha_{\ell}

. These two branches are related to momentum transfers near zero and

\pi

. We also demonstrate that the entanglement distance reaches its maximum for degenerate states related through nonlocal operators such as Wilson loops. For example, degenerate ground-states (GSs) of 2+1 D topological states have maximum entanglement distance. On the contrary, degenerate GSs related through confined anyon excitations such as genons have minimum entanglement distance. Various implications of this concept for quantum simulations are discussed. Finally, based on the ideas developed we introduce two modified DMRG algorithms that are capable of finding all degenerate GSs.

\rho_A

and

\sigma_A

is obtained from the moments

\textrm{tr} (\rho_A-\sigma_A)^n

and taking the limit

n\to1

of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length

\ell

embedded in the low lying eigenstates of a one-dimensional critical system of length

L

, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a {\it scale invariant universal function} of

\ell/L

. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten

n

-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of

\ell/L

. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.