56 research outputs found
Mutually Unbiased Measurements, Hadamard Matrices, and Superdense Coding
Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert
spaces, and the corresponding rank-1 projective measurements are ubiquitous in
quantum information theory. In this work, we study a recently introduced
generalization of MUBs called mutually unbiased measurements (MUMs). These
measurements inherit the essential property of complementarity from MUBs, but
the Hilbert space dimension is no longer required to match the number of
outcomes. This operational complementarity property renders MUMs highly useful
for device-independent quantum information processing. It has been shown that
MUMs are strictly more general than MUBs. In this work we provide a complete
proof of the characterization of MUMs that are direct sums of MUBs. We then
proceed to construct new examples of MUMs that are not direct sums of MUBs. A
crucial technical tool for these construction is a correspondence with
quaternionic Hadamard matrices, which allows us to map known examples of such
matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that --
in stark contrast with MUBs -- the number of MUMs for a fixed outcome number is
unbounded. Next, we focus on the use of MUMs in quantum communication. We
demonstrate how any pair of MUMs with d outcomes defines a d-dimensional
superdense coding protocol. Using MUMs that are not direct sums of MUBs, we
disprove a recent conjecture due to Nayak and Yuen on the rigidity of
superdense coding for infinitely many dimensions. The superdense coding
protocols arising in the refutation reveal how shared entanglement may be used
in a manner heretofore unknown.Comment: v2: Added some references and related discussion. v1: 20 pages.
Comments welcome
Random Matrix Theories in Quantum Physics: Common Concepts
We review the development of random-matrix theory (RMT) during the last
decade. We emphasize both the theoretical aspects, and the application of the
theory to a number of fields. These comprise chaotic and disordered systems,
the localization problem, many-body quantum systems, the Calogero-Sutherland
model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions.
The review is preceded by a brief historical survey of the developments of RMT
and of localization theory since their inception. We emphasize the concepts
common to the above-mentioned fields as well as the great diversity of RMT. In
view of the universality of RMT, we suggest that the current development
signals the emergence of a new "statistical mechanics": Stochasticity and
general symmetry requirements lead to universal laws not based on dynamical
principles.Comment: 178 pages, Revtex, 45 figures, submitted to Physics Report
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