53 research outputs found
Entanglement at a Scale and Renormalization Monotones
We study the information content of the reduced density matrix of a region in
quantum field theory that cannot be recovered from its subregion density
matrices. We reconstruct the density matrix from its subregions using two
approaches: scaling maps and recovery maps. The vacuum of a scale-invariant
field theory is the fixed point of both transformations. We define the
entanglement of scaling and the entanglement of recovery as measures of
entanglement that are intrinsic to the continuum limit. Both measures increase
monotonically under the renormalization group flow. This provides a unifying
information-theoretic structure underlying the different approaches to the
renormalization monotones in various dimensions. Our analysis applies to
non-relativistic quantum field theories as well the relativistic ones, however,
in relativistic case, the entanglement of scaling can diverge
Modular Hamiltonian of Excited States in Conformal Field Theory
We present a novel replica trick that computes the relative entropy of two
arbitrary states in conformal field theory. Our replica trick is based on the
analytic continuation of partition functions that break the replica Z_n
symmetry. It provides a method for computing arbitrary matrix elements of the
modular Hamiltonian corresponding to excited states in terms of correlation
functions. We show that the quantum Fisher information in vacuum can be
expressed in terms of two-point functions on the replica geometry. We perform
sample calculations in two-dimensional conformal field theories.Comment: 5 pages, 1 figur
Canonical Energy is Quantum Fisher Information
In quantum information theory, Fisher Information is a natural metric on the
space of perturbations to a density matrix, defined by calculating the relative
entropy with the unperturbed state at quadratic order in perturbations. In
gravitational physics, Canonical Energy defines a natural metric on the space
of perturbations to spacetimes with a Killing horizon. In this paper, we show
that the Fisher information metric for perturbations to the vacuum density
matrix of a ball-shaped region B in a holographic CFT is dual to the canonical
energy metric for perturbations to a corresponding Rindler wedge R_B of
Anti-de-Sitter space. Positivity of relative entropy at second order implies
that the Fisher information metric is positive definite. Thus, for physical
perturbations to anti-de-Sitter spacetime, the canonical energy associated to
any Rindler wedge must be positive. This second-order constraint on the metric
extends the first order result from relative entropy positivity that physical
perturbations must satisfy the linearized Einstein's equations.Comment: 26 pages, 1 figur
Perturbation Theory for the Logarithm of a Positive Operator
In various contexts in mathematical physics one needs to compute the
logarithm of a positive unbounded operator. Examples include the von Neumann
entropy of a density matrix and the flow of operators with the modular
Hamiltonian in the Tomita-Takesaki theory. Often, one encounters the situation
where the operator under consideration, that we denote by , can be
related by a perturbative series to another operator , whose
logarithm is known. We set up a perturbation theory for the logarithm . It turns out that the terms in the series possess remarkable algebraic
structure, which enable us to write them in the form of nested commutators plus
some "contact terms."Comment: 30 page
Universality of Quantum Information in Chaotic CFTs
We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal
field theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute
the reduced density matrix of a ball-shaped subsystem of finite size in the
infinite volume limit when the full system is an energy eigenstate. This
reduced density matrix is close in trace distance to a density matrix, to which
we refer as the ETH density matrix, that is independent of all the details of
an eigenstate except its energy and charges under global symmetries. In two
dimensions, the ETH density matrix is universal for all theories with the same
value of central charge. We argue that the ETH density matrix is close in trace
distance to the reduced density matrix of the (micro)canonical ensemble. We
support the argument in higher dimensions by comparing the Von Neumann entropy
of the ETH density matrix with the entropy of a black hole in holographic
systems in the low temperature limit. Finally, we generalize our analysis to
the coherent states with energy density that varies slowly in space, and show
that locally such states are well described by the ETH density matrix.Comment: 43 page
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