75,598 research outputs found
An approach to nonstandard quantum mechanics
We use nonstandard analysis to formulate quantum mechanics in
hyperfinite-dimensional spaces. Self-adjoint operators on
hyperfinite-dimensional spaces have complete eigensets, and bound states and
continuum states of a Hamiltonian can thus be treated on an equal footing. We
show that the formalism extends the standard formulation of quantum mechanics.
To this end we develop the Loeb-function calculus in nonstandard hulls. The
idea is to perform calculations in a hyperfinite-dimensional space, but to
interpret expectation values in the corresponding nonstandard hull. We further
apply the framework to non-relativistic quantum scattering theory. For
time-dependent scattering theory, we identify the starting time and the
finishing time of a scattering experiment, and we obtain a natural separation
of time scales on which the preparation process, the interaction process, and
the detection process take place. For time-independent scattering theory, we
derive rigorously explicit formulas for the M{\o}ller wave operators and the
S-Matrix
Fundamental quantum limits to waveform detection
Ever since the inception of gravitational-wave detectors, limits imposed by
quantum mechanics to the detection of time-varying signals have been a subject
of intense research and debate. Drawing insights from quantum information
theory, quantum detection theory, and quantum measurement theory, here we prove
lower error bounds for waveform detection via a quantum system, settling the
long-standing problem. In the case of optomechanical force detection, we derive
analytic expressions for the bounds in some cases of interest and discuss how
the limits can be approached using quantum control techniques.Comment: v1: first draft, 5 pages; v2: updated and extended, 5 pages +
appendices, 2 figures; v3: 8 pages and 3 figure
Direct detection of quantum entanglement
Quantum entanglement, after playing a significant role in the development of
the foundations of quantum mechanics, has been recently rediscovered as a new
physical resource with potential commercial applications such as, for example,
quantum cryptography, better frequency standards or quantum-enhanced
positioning and clock synchronization. On the mathematical side the studies of
entanglement have revealed very interesting connections with the theory of
positive maps. The capacity to generate entangled states is one of the basic
requirements for building quantum computers. Hence, efficient experimental
methods for detection, verification and estimation of quantum entanglement are
of great practical importance. Here, we propose an experimentally viable,
\emph{direct} detection of quantum entanglement which is efficient and does not
require any \emph{a priori} knowledge about the quantum state. In a particular
case of two entangled qubits it provides an estimation of the amount of
entanglement. We view this method as a new form of quantum computation, namely,
as a decision problem with quantum data structure.Comment: 4 pages, 1 eps figure, RevTe
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