49,425 research outputs found
Quality of a Which-Way Detector
We introduce a measure Q of the "quality" of a quantum which-way detector,
which characterizes its intrinsic ability to extract which-way information in
an asymmetric two-way interferometer. The "quality" Q allows one to separate
the contribution to the distinguishability of the ways arising from the quantum
properties of the detector from the contribution stemming from a-priori
which-way knowledge available to the experimenter, which can be quantified by a
predictability parameter P. We provide an inequality relating these two sources
of which-way information to the value of the fringe visibility displayed by the
interferometer. We show that this inequality is an expression of duality,
allowing one to trace the loss of coherence to the two reservoirs of which-way
information represented by Q and P. Finally, we illustrate the formalism with
the use of a quantum logic gate: the Symmetric Quanton-Detecton System (SQDS).
The SQDS can be regarded as two qubits trying to acquire which way information
about each other. The SQDS will provide an illustrating example of the
reciprocal effects induced by duality between system and which-way detector.Comment: 10 pages, 5 figure
Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
We investigate the set a) of positive, trace preserving maps acting on
density matrices of size N, and a sequence of its nested subsets: the sets of
maps which are b) decomposable, c) completely positive, d) extended by identity
impose positive partial transpose and e) are superpositive. Working with the
Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds
for the volumes of all five sets. A sample consequence is the fact that, as N
increases, a generic positive map becomes not decomposable and, a fortiori, not
completely positive.
Due to the Jamiolkowski isomorphism, the results obtained for quantum maps
are closely connected to similar relations between the volume of the set of
quantum states and the volumes of its subsets (such as states with positive
partial transpose or separable states) or supersets. Our approach depends on
systematic use of duality to derive quantitative estimates, and on various
tools of classical convexity, high-dimensional probability and geometry of
Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision
On the noise-resolution duality, Heisenberg uncertainty and Shannon's information
Several variations of the Heisenberg uncertainty inequality are derived on
the basis of "noise-resolution duality" recently proposed by the authors. The
same approach leads to a related inequality that provides an upper limit for
the information capacity of imaging systems in terms of the number of imaging
quanta (particles) used in the experiment. These results can be useful in the
context of biomedical imaging constrained by the radiation dose delivered to
the sample, or in imaging (e.g. astronomical) problems under "low light"
conditions
Integral and measure-turnpike properties for infinite-dimensional optimal control systems
We first derive a general integral-turnpike property around a set for
infinite-dimensional non-autonomous optimal control problems with any possible
terminal state constraints, under some appropriate assumptions. Roughly
speaking, the integral-turnpike property means that the time average of the
distance from any optimal trajectory to the turnpike set con- verges to zero,
as the time horizon tends to infinity. Then, we establish the measure-turnpike
property for strictly dissipative optimal control systems, with state and
control constraints. The measure-turnpike property, which is slightly stronger
than the integral-turnpike property, means that any optimal (state and control)
solution remains essentially, along the time frame, close to an optimal
solution of an associated static optimal control problem, except along a subset
of times that is of small relative Lebesgue measure as the time horizon is
large. Next, we prove that strict strong duality, which is a classical notion
in optimization, implies strict dissipativity, and measure-turnpike. Finally,
we conclude the paper with several comments and open problems
BROJA-2PID: A robust estimator for bivariate partial information decomposition
Makkeh, Theis, and Vicente found in [8] that Cone Programming model is the
most robust to compute the Bertschinger et al. partial information decompostion
(BROJA PID) measure [1]. We developed a production-quality robust software that
computes the BROJA PID measure based on the Cone Programming model. In this
paper, we prove the important property of strong duality for the Cone Program
and prove an equivalence between the Cone Program and the original Convex
problem. Then describe in detail our software and how to use it.\newline\inden
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