366 research outputs found
Mutual information of excited states and relative entropy of two disjoint subsystems in CFT
In this paper, we first study mutual information of excited states in the
small subsystem size limit in generic conformal field theory. We then discuss
relative entropy of two disjoint subsystems in the same limit.Comment: 18 page
Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields
We study the entanglement entropy and the modular Hamiltonian of slightly
excited states reduced to a ball shaped region in generic conformal field
theories. We set up a formal expansion in the one point functions of the state
in which all orders are explicitly given in terms of integrals of multi-point
functions along the vacuum modular flow, without a need for replica index
analytic continuation. We show that the quadratic order contributions in this
expansion can be calculated in a way expected from holography, namely via the
bulk canonical energy for the entanglement entropy, and its variation for the
modular Hamiltonian. The bulk fields contributing to the canonical energy are
defined via the HKLL procedure. In terms of CFT variables, the contribution of
each such bulk field to the modular Hamiltonian is given by the OPE block
corresponding to the dual operator integrated along the vacuum modular flow.
These results do not rely on assuming large or other special properties of
the CFT and therefore they are purely kinematic.Comment: 40 pages, 2 figures. v3: some typos corrected, references added,
extended discussion on convergence and holographic interpretatio
Towards an Entanglement Measure for Mixed States in CFTs Based on Relative Entropy
Relative entropy of entanglement (REE) is an entanglement measure of
bipartite mixed states, defined by the minimum of the relative entropy
between a given mixed state and an
arbitrary separable state . The REE is always bounded by the
mutual information because
the latter measures not only quantum entanglement but also classical
correlations. In this paper we address the question of to what extent REE can
be small compared to the mutual information in conformal field theories (CFTs).
For this purpose, we perturbatively compute the relative entropy between the
vacuum reduced density matrix on disjoint subsystems
and arbitrarily separable state in the limit where two subsystems
A and B are well separated, then minimize the relative entropy with respect to
the separable states. We argue that the result highly depends on the spectrum
of CFT on the subsystems. When we have a few low energy spectrum of operators
as in the case where the subsystems consist of a finite number of spins in spin
chain models, the REE is considerably smaller than the mutual information.
However in general our perturbative scheme breaks down, and the REE can be as
large as the mutual information.Comment: 35 pages, 2 figure
Chaos and relative entropy
One characterization of a chaotic system is the quick delocalization of
quantum information (fast scrambling). One therefore expects that in such a
system a state quickly becomes locally indistinguishable from its
perturbations. In this paper we study the time dependence of the relative
entropy between the reduced density matrices of the thermofield double state
and its perturbations in two dimensional conformal field theories. We show that
in a CFT with a gravity dual, this relative entropy exponentially decays until
the scrambling time. This decay is not uniform. We argue that the early time
exponent is universal while the late time exponent is sensitive to the
butterfly effect. This large answer breaks down at the scrambling time,
therefore we also study the relative entropy in a class of spin chain models
numerically. We find a similar universal exponential decay at early times,
while at later times we observe that the relative entropy has large revivals in
integrable models, whereas there are no revivals in non-integrable models.Comment: 34+11 pages, 8 figure
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