77 research outputs found
Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice
We study entanglement entropy (EE) for a Maxwell field in 2+1 dimensions. We
do numerical calculations in two dimensional lattices. This gives a concrete
example of the general results of our recent work on entropy for lattice gauge
fields using an algebraic approach. To evaluate the entropies we extend the
standard calculation methods for the entropy of Gaussian states in canonical
commutation algebras to the more general case of algebras with center and
arbitrary numerical commutators. We find that while the entropy depends on the
details of the algebra choice, mutual information has a well defined continuum
limit. We study several universal terms for the entropy of the Maxwell field
and compare with the case of a massless scalar field. We find some interesting
new phenomena: An "evanescent" logarithmically divergent term in the entropy
with topological coefficient which does not have any correspondence with
ultraviolet entanglement in the universal quantities, and a non standard way in
which strong subadditivity is realized. Based on the results of our
calculations we propose a generalization of strong subadditivity for the
entropy on some algebras that are not in tensor product.Comment: 27 pages, 15 figure
Localization of Negative Energy and the Bekenstein Bound
A simple argument shows that negative energy cannot be isolated far away from
positive energy in a conformal field theory and strongly constrains its
possible dispersal. This is also required by consistency with the Bekenstein
bound written in terms of the positivity of relative entropy. We prove a new
form of the Bekenstein bound based on the monotonicity of the relative entropy,
involving a "free" entropy enclosed in a region which is highly insensitive to
space-time entanglement, and show that it further improves the negative energy
localization bound.Comment: 5 pages, 1 figur
Spread of entanglement and causality
We investigate causality constraints on the time evolution of entanglement
entropy after a global quench in relativistic theories. We first provide a
general proof that the so-called tsunami velocity is bounded by the speed of
light. We then generalize the free particle streaming model of
arXiv:cond-mat/0503393 to general dimensions and to an arbitrary entanglement
pattern of the initial state. In more than two spacetime dimensions the spread
of entanglement in these models is highly sensitive to the initial entanglement
pattern, but we are able to prove an upper bound on the normalized rate of
growth of entanglement entropy, and hence the tsunami velocity. The bound is
smaller than what one gets for quenches in holographic theories, which
highlights the importance of interactions in the spread of entanglement in
many-body systems. We propose an interacting model which we believe provides an
upper bound on the spread of entanglement for interacting relativistic
theories. In two spacetime dimensions with multiple intervals, this model and
its variations are able to reproduce intricate results exhibited by holographic
theories for a significant part of the parameter space. For higher dimensions,
the model bounds the tsunami velocity at the speed of light. Finally, we
construct a geometric model for entanglement propagation based on a tensor
network construction for global quenches.Comment: v2: minor improvements; v1: 78 pages, 30 figure
Holographic RG flows, entanglement entropy and the sum rule
We calculate the two-point function of the trace of the stress tensor in
holographic renormalization group flows between pairs of conformal field
theories. We show that the term proportional to the momentum squared in this
correlator gives the change of the central charge between fixed points in d=2
and in d>2 it gives the holographic entanglement entropy for a planar region.
This can also be seen as a holographic realization of the Adler-Zee formula for
the renormalization of Newton's constant. Holographic regularization is found
to provide a perfect match of the finite and divergent terms of the sum rule,
and it is analogous to the regularization of the entropy in terms of mutual
information. Finally, we provide a general proof of reflection positivity in
terms of stability of the dual bulk action, and discuss the relation between
unitarity constraints, the null energy condition and regularity in the interior
of the gravity solution.Comment: v2: 32 pages, 1 figure. Refs. and comments added. Version published
in JHE
The g-theorem and quantum information theory
We study boundary renormalization group flows between boundary conformal
field theories in dimensions using methods of quantum information theory.
We define an entropic -function for theories with impurities in terms of the
relative entanglement entropy, and we prove that this -function decreases
along boundary renormalization group flows. This entropic -theorem is valid
at zero temperature, and is independent from the -theorem based on the
thermal partition function. We also discuss the mutual information in boundary
RG flows, and how it encodes the correlations between the impurity and bulk
degrees of freedom. Our results provide a quantum-information understanding of
(boundary) RG flow as increase of distinguishability between the UV fixed point
and the theory along the RG flow.Comment: 34 pages + appendices, 8 figures. v2. Improved and corrected version
of the proo
Remarks on entanglement entropy for gauge fields
In gauge theories the presence of constraints can obstruct expressing the
global Hilbert space as a tensor product of the Hilbert spaces corresponding to
degrees of freedom localized in complementary regions. In algebraic terms, this
is due to the presence of a center --- a set of operators which commute with
all others --- in the gauge invariant operator algebra corresponding to finite
region. A unique entropy can be assigned to algebras with center, giving place
to a local entropy in lattice gauge theories. However, ambiguities arise on the
correspondence between algebras and regions. In particular, it is always
possible to choose (in many different ways) local algebras with trivial center,
and hence a genuine entanglement entropy, for any region. These choices are in
correspondence with maximal trees of links on the boundary, which can be
interpreted as partial gauge fixings. This interpretation entails a gauge
fixing dependence of the entanglement entropy. In the continuum limit however,
ambiguities in the entropy are given by terms local on the boundary of the
region, in such a way relative entropy and mutual information are finite,
universal, and gauge independent quantities.Comment: 26 pages, 7 figure
Area terms in entanglement entropy
We discuss area terms in entanglement entropy and show that a recent formula
by Rosenhaus and Smolkin is equivalent to the term involving a correlator of
traces of the stress tensor in Adler-Zee formula for the renormalization of the
Newton constant. We elaborate on how to fix the ambiguities in these formulas:
Improving terms for the stress tensor of free fields, boundary terms in the
modular Hamiltonian, and contact terms in the Euclidean correlation functions.
We make computations for free fields and show how to apply these calculations
to understand some results for interacting theories which have been studied in
the literature. We also discuss an application to the F-theorem.Comment: 26 pages, no figures, references adde
Proof of a Quantum Bousso Bound
We prove the generalized Covariant Entropy Bound, , for light-sheets with initial area and final area .
The entropy is defined as a difference of von Neumann entropies of
an arbitrary state and the vacuum, with both states restricted to the
light-sheet under consideration. The proof applies to free fields, in the limit
where gravitational backreaction is small. We do not assume the null energy
condition. In regions where it is violated, we find that the bound is protected
by the defining property of light-sheets: that their null generators are
nowhere expanding.Comment: 19 pages, 3 figures; v2: references adde
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