18,039 research outputs found
Limitations on quantum dimensionality reduction
The Johnson–Lindenstrauss Lemma is a classic result which implies that any set of n real vectors can be compressed to O(log n) dimensions while only distorting pairwise Euclidean distances by a constant factor. Here we consider potential extensions of this result to the compression of quantum states. We show that, by contrast with the classical case, there does not exist any distribution over quantum channels that significantly reduces the dimension of quantum states while preserving the 2-norm distance with high probability. We discuss two tasks for which the 2-norm distance is indeed the correct figure of merit. In the case of the trace norm, we show that the dimension of low-rank mixed states can be reduced by up to a square root, but that essentially no dimensionality reduction is possible for highly mixed states.European Commission (Grant QESSENCE)United States. Air Force Office of Scientific Research (United States. Defense Advanced Research Projects Agency. Microsystems Technology Office. Quantum Entanglement Science and Technology
The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch
Recent and forthcoming advances in instrumentation, and giant new surveys,
are creating astronomical data sets that are not amenable to the methods of
analysis familiar to astronomers. Traditional methods are often inadequate not
merely because of the size in bytes of the data sets, but also because of the
complexity of modern data sets. Mathematical limitations of familiar algorithms
and techniques in dealing with such data sets create a critical need for new
paradigms for the representation, analysis and scientific visualization (as
opposed to illustrative visualization) of heterogeneous, multiresolution data
across application domains. Some of the problems presented by the new data sets
have been addressed by other disciplines such as applied mathematics,
statistics and machine learning and have been utilized by other sciences such
as space-based geosciences. Unfortunately, valuable results pertaining to these
problems are mostly to be found only in publications outside of astronomy. Here
we offer brief overviews of a number of concepts, techniques and developments,
some "old" and some new. These are generally unknown to most of the
astronomical community, but are vital to the analysis and visualization of
complex datasets and images. In order for astronomers to take advantage of the
richness and complexity of the new era of data, and to be able to identify,
adopt, and apply new solutions, the astronomical community needs a certain
degree of awareness and understanding of the new concepts. One of the goals of
this paper is to help bridge the gap between applied mathematics, artificial
intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in
Astronomy, special issue "Robotic Astronomy
Effective Field Theory of pure Gravity and the Renormalization Group
The general structure of the renormalization group equations for the low
energy effective field theory formulation of pure gravity is presented. The
solution of these equations takes a particular simple form if the mass scale of
the effective theory is much smaller than the Planck mass (a possibility
compatible with the renormalization of the effective theory). A theory with
just one free renormalized parameter is obtained when contributions suppressed
by inverse powers of the Planck mass are neglected.Comment: latex, 9 pages, minor correction, version publishe
Evanescent-wave trapping and evaporative cooling of an atomic gas near two-dimensionality
A dense gas of cesium atoms at the crossover to two-dimensionality is
prepared in a highly anisotropic surface trap that is realized with two
evanescent light waves. Temperatures as low as 100nK are reached with 20.000
atoms at a phase-space density close to 0.1. The lowest quantum state in the
tightly confined direction is populated by more than 60%. The system offers
intriguing prospects for future experiments on degenerate quantum gases in two
dimensions
WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction
WavePacket is an open-source program package for numeric simulations in
quantum dynamics. It can solve time-independent or time-dependent linear
Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions.
Also coupled equations can be treated, which allows, e.g., to simulate
molecular quantum dynamics beyond the Born-Oppenheimer approximation.
Optionally accounting for the interaction with external electric fields within
the semi-classical dipole approximation, WavePacket can be used to simulate
experiments involving tailored light pulses in photo-induced physics or
chemistry. Being highly versatile and offering visualization of quantum
dynamics 'on the fly', WavePacket is well suited for teaching or research
projects in atomic, molecular and optical physics as well as in physical or
theoretical chemistry. Building on the previous Part I which dealt with closed
quantum systems and discrete variable representations, the present Part II
focuses on the dynamics of open quantum systems, with Lindblad operators
modeling dissipation and dephasing. This part also describes the WavePacket
function for optimal control of quantum dynamics, building on rapid
monotonically convergent iteration methods. Furthermore, two different
approaches to dimension reduction implemented in WavePacket are documented
here. In the first one, a balancing transformation based on the concepts of
controllability and observability Gramians is used to identify states that are
neither well controllable nor well observable. Those states are either
truncated or averaged out. In the other approach, the H2-error for a given
reduced dimensionality is minimized by H2 optimal model reduction techniques,
utilizing a bilinear iterative rational Krylov algorithm
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