8 research outputs found
Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces
Columns of d^2 x N matrices are shown to create different sets of N operators
acting on -dimensional Hilbert space. This construction corresponds to a
formalism of the star-product of operator symbols. The known bases are shown to
be partial cases of generic formulas derived by using d^2 x N matrices as a
source for constructing arbitrary bases. The known examples of the SIC-POVM,
MUBs, and the phase-space description of qubit states are considered from the
viewpoint of the developed unified approach. Star-product schemes are
classified with respect to associated d^2 x N matrices. In particular, unitary
matrices correspond to self-dual schemes. Such self-dual star-product schemes
are shown to be determined by dequantizers which do not form POVM.Comment: 12 pages, 1 figure, 1 table, to appear in Journal of Russian Laser
Researc
Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
In three spacetime dimensions, general relativity drastically simplifies,
becoming a ``topological'' theory with no propagating local degrees of freedom.
Nevertheless, many of the difficult conceptual problems of quantizing gravity
are still present. In this review, I summarize the rather large body of work
that has gone towards quantizing (2+1)-dimensional vacuum gravity in the
setting of a spatially closed universe.Comment: 61 pages, draft of review for Living Reviews; comments, criticisms,
additions, missing references welcome; v2: minor changes, added reference
Entanglement entropy of black holes
The entanglement entropy is a fundamental quantity which characterizes the
correlations between sub-systems in a larger quantum-mechanical system. For two
sub-systems separated by a surface the entanglement entropy is proportional to
the area of the surface and depends on the UV cutoff which regulates the
short-distance correlations. The geometrical nature of the entanglement entropy
calculation is particularly intriguing when applied to black holes when the
entangling surface is the black hole horizon. I review a variety of aspects of
this calculation: the useful mathematical tools such as the geometry of spaces
with conical singularities and the heat kernel method, the UV divergences in
the entropy and their renormalization, the logarithmic terms in the
entanglement entropy in 4 and 6 dimensions and their relation to the conformal
anomalies. The focus in the review is on the systematic use of the conical
singularity method. The relations to other known approaches such as 't Hooft's
brick wall model and the Euclidean path integral in the optical metric are
discussed in detail. The puzzling behavior of the entanglement entropy due to
fields which non-minimally couple to gravity is emphasized. The holographic
description of the entanglement entropy of the black hole horizon is
illustrated on the two- and four-dimensional examples. Finally, I examine the
possibility to interpret the Bekenstein-Hawking entropy entirely as the
entanglement entropy.Comment: 89 pages; an invited review to be published in Living Reviews in
Relativit