73,677 research outputs found
Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy
A short and elementary proof of the joint convexity of relative entropy is
presented, using nothing beyond linear algebra. The key ingredients are an
easily verified integral representation and the strategy used to prove the
Cauchy-Schwarz inequality in elementary courses. Several consequences are
proved in a way which allow an elementary proof of strong subadditivity in a
few more lines. Some expository material on Schwarz inequalities for operators
and the Holevo bound for partial measurements is also included.Comment: The proof given here is short and more elementary that in either
quant-ph/0404126 or quant-ph/0408130. The style is intended to be suitable to
classroom presentation. For a Much More Complicated approach, see Section 6
of quant-ph/050619
Relative entropy for compact Riemann surfaces
The relative entropy of the massive free bosonic field theory is studied on
various compact Riemann surfaces as a universal quantity with physical
significance, in particular, for gravitational phenomena. The exact expression
for the sphere is obtained, as well as its asymptotic series for large mass and
its Taylor series for small mass. One can also derive exact expressions for the
torus but not for higher genus. However, the asymptotic behaviour for large
mass can always be established-up to a constant-with heat-kernel methods. It
consists of an asymptotic series determined only by the curvature, hence common
for homogeneous surfaces of genus higher than one, and exponentially vanishing
corrections whose form is determined by the concrete topology. The coefficient
of the logarithmic term in this series gives the conformal anomaly.Comment: 20 pages, LaTeX 2e, 2 PS figures; to appear in Phys. Rev.
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
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