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

Mohammad-Sadegh Vaezi, Abolhassan Vaezi

Published Oct 6, 2016• 0 citations
OpenAlex

Abstract

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.

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

Mohammad-Sadegh Vaezi, Abolhassan Vaezi

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.

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arXiv:1612.00659•Journal of High Energy Physics•Feb 1, 2017•49 citations
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 αE\alpha_EαE​ and αℓ\alpha_{\ell}αℓ​ exponents, respectively. Using conformal field theory (CFT) we find αE=2\alpha_E = 2αE​=2 and αℓ=4\alpha_{\ell} = 4αℓ​=4 for Abelian theories with c=1c=1c=1 such as free fermions. For non-Abelian CFTs αE=0\alpha_E = 0αE​=0 , and αℓ\alpha_{\ell}αℓ​ is twice the conformal dimension of the thermal primary fields. For instance for Z3Z_3Z3​ parafermion CFT αE=1\alpha_E = 1αE​=1 and αℓ=4/5\alpha_{\ell} = 4/5αℓ​=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 αE\alpha_EαE​ and αℓ\alpha_{\ell}αℓ​. These two branches are related to momentum transfers near zero and π\piπ. 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 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=1c=1c=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 Z3{Z}_{3}Z3​ 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.
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.
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 ρA\rho_AρA​ and σA\sigma_AσA​ is obtained from the moments tr(ρA−σA)n\textrm{tr} (\rho_A-\sigma_A)^ntr(ρA​−σA​)n and taking the limit n→1n\to1n→1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length ℓ\ellℓ embedded in the low lying eigenstates of a one-dimensional critical system of length LLL, 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 ℓ/L\ell/Lℓ/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 nnn-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 ℓ/L\ell/Lℓ/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.
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.
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 study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy S(ρ 1∥ρ 0) between two given reduced density matrices ρ 1 and ρ 0 of a quantum field theory, we employ the replica trick which relies on the path integral representation of Tr(ρ 1 ρ 0 − 1 ) and define a set of Rényi relative entropies S n (ρ 1∥ρ 0). We compute these quantities for integer values of the parameter n and derive via the replica limit the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator i∂ϕ, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.