arXiv:2002.09985

Relative entanglement entropy of thermal states of Klein-Gordon and Dirac quantum field theories

Onirban Islam

Published Feb 23, 2020• 0 citations

Abstract

An upper bound of the relative entanglement entropy of thermal states at an inverse temperature $β$ of linear, massive Klein-Gordon and Dirac quantum field theories across two regions, separated by a nonzero distance $d$ in a Cauchy hypersurface of an ultrastatic (spin-)spacetime has been computed. This entanglement measure is bounded by a negative constant times $\ln | \tanh (πd/ 2 β) |$ which signifies power law decay for asymptotic $d$ where the exponent depends on $β< \infty$.

DOI: 10.48550/arxiv.2002.09985

Related papers

Relative entanglement entropy for widely separated regions in curved spacetime

Stefan Hollands, Onirban Islam, Ko Sanders

arXiv:1711.02039Journal of Mathematical PhysicsJun 1, 20185 citations

Long-range entanglement in the Dirac vacuum

Jonathan Silman, Benni Reznik

arXiv:quant-ph/0609212Physical Review AMay 4, 200737 citations

Speed Limits for Entanglement

Thomas Hartman, Nima Afkhami-Jeddi

arXiv:1512.02695Dec 8, 201526 citations

Entanglement entropy after a partial projective measurement in 1 + 1 dimensional conformal field theories: exact results

M. A. Rajabpour

arXiv:1512.03940Journal of Statistical Mechanics Theory and ExperimentJun 1, 201637 citations

Studies of Entanglement Entropy, and Relativistic Fluids for Thermal Field Theories

Michael C. Spillane

Jan 1, 20160 citations

Universal corrections to the entanglement entropy in gapped quantum spin chains: A numerical study

Emanuele Levi, Olalla A. Castro-Alvaredo, Benjamin Doyon

arXiv:1304.6874Physical Review BSep 30, 201316 citations

arXiv:2002.09985

Relative entanglement entropy of thermal states of Klein-Gordon and Dirac quantum field theories

Onirban Islam

Published Feb 23, 2020• 0 citations

arXiv HTML PDF

Abstract

An upper bound of the relative entanglement entropy of thermal states at an inverse temperature

β

of linear, massive Klein-Gordon and Dirac quantum field theories across two regions, separated by a nonzero distance

d

in a Cauchy hypersurface of an ultrastatic (spin-)spacetime has been computed. This entanglement measure is bounded by a negative constant times

\ln | \tanh (πd/ 2 β) |

which signifies power law decay for asymptotic

d

where the exponent depends on

β< \infty

DOI: 10.48550/arxiv.2002.09985

Related papers

Relative entanglement entropy for widely separated regions in curved spacetime

Stefan Hollands, Onirban Islam, Ko Sanders

We give an upper bound of the relative entanglement entropy of the ground state of a massive Dirac-Majorana field across two widely separated regions A and B in a static slice of an ultrastatic Lorentzian spacetime. Our bound decays exponentially in dist(A, B) at a rate set by the Compton wavelength and the spatial scalar curvature. The physical interpretation of our result is that, on a manifold with positive spatial scalar curvature, one cannot use the entanglement of the vacuum state to teleport one classical bit from A to B if their distance is of the order of the maximum of the curvature radius and the Compton wavelength or greater.We give an upper bound of the relative entanglement entropy of the ground state of a massive Dirac-Majorana field across two widely separated regions A and B in a static slice of an ultrastatic Lorentzian spacetime. Our bound decays exponentially in dist(A, B) at a rate set by the Compton wavelength and the spatial scalar curvature. The physical interpretation of our result is that, on a manifold with positive spatial scalar curvature, one cannot use the entanglement of the vacuum state to teleport one classical bit from A to B if their distance is of the order of the maximum of the curvature radius and the Compton wavelength or greater.

arXiv:1711.02039Journal of Mathematical PhysicsJun 1, 20185 citations

Long-range entanglement in the Dirac vacuum

Jonathan Silman, Benni Reznik

Recently, there have been a number of works investigating the entanglement properties of distinct noncomplementary parts of discrete and continuous bosonic systems in ground and thermal states. The relativistic fermionic case, however, has yet to be expressly addressed. In this paper we investigate the entanglement between a pair of far-apart regions of the $(3+1)$ -dimensional massless Dirac vacuum via a previously introduced distillation protocol [Reznik et al., Phys. Rev. A 71, 042104 (2005)]. We show that entanglement persists over arbitrary distances, and that as a function of $L∕R$ , where $L$ is the distance between the regions and $R$ is their typical scale, it decays no faster than $\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}(L∕R{)}^{2}]$. We discuss the similarities with and differences from analogous results obtained for the massless Klein-Gordon vacuum.Recently, there have been a number of works investigating the entanglement properties of distinct noncomplementary parts of discrete and continuous bosonic systems in ground and thermal states. The relativistic fermionic case, however, has yet to be expressly addressed. In this paper we investigate the entanglement between a pair of far-apart regions of the $(3+1)$-dimensional massless Dirac vacuum via a previously introduced distillation protocol [Reznik et al., Phys. Rev. A 71, 042104 (2005)]. We show that entanglement persists over arbitrary distances, and that as a function of $L∕R$, where $L$ is the distance between the regions and $R$ is their typical scale, it decays no faster than $\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}(L∕R{)}^{2}]$. We discuss the similarities with and differences from analogous results obtained for the massless Klein-Gordon vacuum.

arXiv:quant-ph/0609212Physical Review AMay 4, 200737 citations

Speed Limits for Entanglement

Thomas Hartman, Nima Afkhami-Jeddi

We show that in any relativistic system, entanglement entropy obeys a speed limit set by the entanglement in thermal equilibrium. The bound is derived from inequalities on relative entropy with respect to a thermal reference state. Thus the thermal state constrains far-from-equilibrium entanglement dynamics whether or not the system actually equilibrates, in a manner reminiscent of fluctuation theorems in classical statistical mechanics. A similar shape-dependent bound constrains the full nonlinear time evolution, supporting a simple physical picture for entanglement propagation that has previously been motivated by holographic calculations in conformal field theory. We discuss general quantum field theories in any spacetime dimension, but also derive some results of independent interest for thermal relative entropy in 1+1d CFT.We show that in any relativistic system, entanglement entropy obeys a speed limit set by the entanglement in thermal equilibrium. The bound is derived from inequalities on relative entropy with respect to a thermal reference state. Thus the thermal state constrains far-from-equilibrium entanglement dynamics whether or not the system actually equilibrates, in a manner reminiscent of fluctuation theorems in classical statistical mechanics. A similar shape-dependent bound constrains the full nonlinear time evolution, supporting a simple physical picture for entanglement propagation that has previously been motivated by holographic calculations in conformal field theory. We discuss general quantum field theories in any spacetime dimension, but also derive some results of independent interest for thermal relative entropy in 1+1d CFT.

arXiv:1512.02695Dec 8, 201526 citations

Entanglement entropy after a partial projective measurement in 1 + 1 dimensional conformal field theories: exact results

M. A. Rajabpour

Abstract We calculate analytically the Rényi bipartite entanglement entropy S α of the ground state of 1 + 1 dimensional conformal field theories (CFT) after performing a projective measurement in part of the system. We show that the entanglement entropy in this setup is dependent on the central charge and the operator content of the system. When the measurement region A separates the two parts B and B ¯ , the entanglement entropy between B and B ¯ decreases like a power-law with respect to the characteristic distance between the two regions with an exponent which is dependent on the rank α of the Rényi entanglement entropy and the smallest scaling dimension present in the system. We check our findings by making numerical calculations on the Klein–Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators. We also comment on the post-measurement entanglement entropy in the massive quantum field theories.Abstract We calculate analytically the Rényi bipartite entanglement entropy S α of the ground state of 1 + 1 dimensional conformal field theories (CFT) after performing a projective measurement in part of the system. We show that the entanglement entropy in this setup is dependent on the central charge and the operator content of the system. When the measurement region A separates the two parts B and B ¯ , the entanglement entropy between B and B ¯ decreases like a power-law with respect to the characteristic distance between the two regions with an exponent which is dependent on the rank α of the Rényi entanglement entropy and the smallest scaling dimension present in the system. We check our findings by making numerical calculations on the Klein–Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators. We also comment on the post-measurement entanglement entropy in the massive quantum field theories.

arXiv:1512.03940Journal of Statistical Mechanics Theory and ExperimentJun 1, 201637 citations

Studies of Entanglement Entropy, and Relativistic Fluids for Thermal Field Theories

Michael C. Spillane

In this dissertation we consider physical consequences of adding a finite temperature to quantum field theories. At small length scales entanglement is a critically important feature. It is therefore unsurprising that entanglement entropy and Renyi entropy are useful tools in studying quantum phase transition, and quantum information. In this thesis we consider the corrections to entanglement and Renyi entropies due to addition of a finite temperature. n this dissertation we consider physical consequences of adding a finite temperature to quantum field theories. At small length scales entanglement is a critically important feature. It is therefore unsurprising that entanglement entropy and Renyi entropy are useful tools in studying quantum phase transition, and quantum information. In this thesis we consider the corrections to entanglement and Renyi entropies due to addition of a finite temperature. More specifically, we investigate the entanglement entropy of a massive scalar field in 1+1 dimensions at nonzero temperature. In the small mass (m) and temperature (T) limit, we put upper and lower bounds on the two largest eigenvalues of the covariance matrix used to compute the entanglement entropy. We argue that the entanglement entropy has exp(-m/T) scaling in the limit TIn this dissertation we consider physical consequences of adding a finite temperature to quantum field theories. At small length scales entanglement is a critically important feature. It is therefore unsurprising that entanglement entropy and Renyi entropy are useful tools in studying quantum phase transition, and quantum information. In this thesis we consider the corrections to entanglement and Renyi entropies due to addition of a finite temperature. n this dissertation we consider physical consequences of adding a finite temperature to quantum field theories. At small length scales entanglement is a critically important feature. It is therefore unsurprising that entanglement entropy and Renyi entropy are useful tools in studying quantum phase transition, and quantum information. In this thesis we consider the corrections to entanglement and Renyi entropies due to addition of a finite temperature. More specifically, we investigate the entanglement entropy of a massive scalar field in 1+1 dimensions at nonzero temperature. In the small mass (m) and temperature (T) limit, we put upper and lower bounds on the two largest eigenvalues of the covariance matrix used to compute the entanglement entropy. We argue that the entanglement entropy has exp(-m/T) scaling in the limit T

Jan 1, 20160 citations

Universal corrections to the entanglement entropy in gapped quantum spin chains: A numerical study

Emanuele Levi, Olalla A. Castro-Alvaredo, Benjamin Doyon

We carry out a numerical study of the bipartite entanglement entropy in the gapped regime of two paradigmatic quantum spin chain models: the Ising chain in an external magnetic field and the antiferromagnetic XXZ model. The universal scaling limit of these models is described by the massive Ising field theory and the $SU(2)$ -Thirring (sine-Gordon) model, respectively. We may therefore exploit quantum field theoretical results to predict the behavior of the entropy. We numerically confirm that in the scaling limit, corrections to the saturation of the entropy at large region size are proportional to a modified Bessel function of the first kind, ${K}_{0}(2mr)$ , where $m$ is a mass scale (the inverse correlation length) and $r$ the length of the region under consideration. The proportionality constant is simply related to the number of particle types in the universal spectrum. This was originally predicted by J. L. Cardy, O. A. Castro-Alvaredo, and B. Doyon [J. Stat. Phys. 130, 129 (2008)] and B. Doyon [Phys. Rev. Lett. 102, 031602 (2009)] for two-dimensional quantum field theories. Away from the universal region our numerics suggest an entropic behavior following quite closely the quantum field theory prediction, except for extra dependencies on the correlation length.We carry out a numerical study of the bipartite entanglement entropy in the gapped regime of two paradigmatic quantum spin chain models: the Ising chain in an external magnetic field and the antiferromagnetic XXZ model. The universal scaling limit of these models is described by the massive Ising field theory and the $SU(2)$-Thirring (sine-Gordon) model, respectively. We may therefore exploit quantum field theoretical results to predict the behavior of the entropy. We numerically confirm that in the scaling limit, corrections to the saturation of the entropy at large region size are proportional to a modified Bessel function of the first kind, ${K}_{0}(2mr)$, where $m$ is a mass scale (the inverse correlation length) and $r$ the length of the region under consideration. The proportionality constant is simply related to the number of particle types in the universal spectrum. This was originally predicted by J. L. Cardy, O. A. Castro-Alvaredo, and B. Doyon [J. Stat. Phys. 130, 129 (2008)] and B. Doyon [Phys. Rev. Lett. 102, 031602 (2009)] for two-dimensional quantum field theories. Away from the universal region our numerics suggest an entropic behavior following quite closely the quantum field theory prediction, except for extra dependencies on the correlation length.

arXiv:1304.6874Physical Review BSep 30, 201316 citations

An upper bound of the relative entanglement entropy of thermal states at an inverse temperature

β

of linear, massive Klein-Gordon and Dirac quantum field theories across two regions, separated by a nonzero distance

d

in a Cauchy hypersurface of an ultrastatic (spin-)spacetime has been computed. This entanglement measure is bounded by a negative constant times

\ln | \tanh (πd/ 2 β) |

which signifies power law decay for asymptotic

d

where the exponent depends on

β< \infty

(3+1)

-dimensional massless Dirac vacuum via a previously introduced distillation protocol [Reznik et al., Phys. Rev. A 71, 042104 (2005)]. We show that entanglement persists over arbitrary distances, and that as a function of

L∕R

, where

L

is the distance between the regions and

R

is their typical scale, it decays no faster than $\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}(L∕R{)}^{2}]$. We discuss the similarities with and differences from analogous results obtained for the massless Klein-Gordon vacuum.

SU(2)

-Thirring (sine-Gordon) model, respectively. We may therefore exploit quantum field theoretical results to predict the behavior of the entropy. We numerically confirm that in the scaling limit, corrections to the saturation of the entropy at large region size are proportional to a modified Bessel function of the first kind,

{K}_{0}(2mr)

, where

m

is a mass scale (the inverse correlation length) and

r

the length of the region under consideration. The proportionality constant is simply related to the number of particle types in the universal spectrum. This was originally predicted by J. L. Cardy, O. A. Castro-Alvaredo, and B. Doyon [J. Stat. Phys. 130, 129 (2008)] and B. Doyon [Phys. Rev. Lett. 102, 031602 (2009)] for two-dimensional quantum field theories. Away from the universal region our numerics suggest an entropic behavior following quite closely the quantum field theory prediction, except for extra dependencies on the correlation length.