12,142 research outputs found
Stone and double Stone algebras: Boolean and Rough Set Representations, 3-valued and 4-valued Logics
Moisil in 1941, while constructing the algebraic models of n-valued
{\L}ukasiewicz logic defined the set ,where is a Boolean algebra
and `n' being a natural number. Further it was proved by Moisil himself the
representations of n-valued {\L}ukasiewicz Moisil algebra in terms of
. In this article, structural representation results for Stone, dual
Stone and double Stone algebras are proved similar to Moisil's work by showing
that elements of these algebras can be looked upon as monotone ordered tuple of
sets. 3-valued semantics of logic for Stone algebra, dual Stone algebras and
4-valued semantics of logic for double Stone algebras are proposed and
established soundness and completeness results.Comment: arXiv admin note: text overlap with arXiv:1511.0716
Observability of relative phases of macroscopic quantum states
After a measurement, to observe the relative phases of macroscopically
distinguishable states we have to ``undo'' a quantum measurement. We generalise
an earlier model of Peres from two state to N-state quantum system undergoing
measurement process and discuss the issue of observing relative phases of
different branches. We derive an inequality which is satisfied by the relative
phases of macroscopically distinguishable states and consequently any desired
relative phases can not be observed in interference setups. The principle of
macroscopic complementarity is invoked that might be at ease with the
macroscopic world. We illustrate the idea of limit on phase observability in
Stern-Gerlach measurements and the implications are discussed.Comment: Latex file, no figures, 12 pages, submitted to Phys. Lett.
Violation of Invariance of Entanglement Under Local PT Symmetric Unitary
Entanglement is one of the key feature of quantum world and any entanglement
measure must satisfy some basic laws. Most important of them is the invariance
of entanglement under local unitary operations. We show that this is no longer
true with local symmetric unitary operations. If two parties
share a maximally entangled state, then under local symmetric
unitary evolution the entropy of entanglement for pure bipartite states does
not remain invariant. Furthermore, we show that if one of the party has access
to -symmetric quantum world, then a maximally entangled state in
usual quantum theory appears as a non-maximally entangled states for the other
party. This we call as the "entanglement mismatch" effect which can lead to the
violation of the no-signaling condition.Comment: 5 pages, Latex, No fi
Distinguishing two preparations for same pure state leads to signalling
Pure state of a physical system can be prepared in an infinite number of
ways. Here, we prove that given a pure state of a quantum system it is
impossible to distinguish two preparation procedures. Further, we show that if
we can distinguish two preparation procedures for the same pure state then that
can lead to signalling. This impossibility result is different than the no
measurement without disturbance and the no-cloning. Extending this result for a
pure bipartite entangled state entails that the impossibility of distinguishing
two preparation procedures for a mixed state follows from the impossibility of
distinguishing two preparations for a pure bipartite state.Comment: Two and half pages, Comments welcom
Deterministic Inequalities for Smooth M-estimators
Ever since the proof of asymptotic normality of maximum likelihood estimator
by Cramer (1946), it has been understood that a basic technique of the Taylor
series expansion suffices for asymptotics of -estimators with
smooth/differentiable loss function. Although the Taylor series expansion is a
purely deterministic tool, the realization that the asymptotic normality
results can also be made deterministic (and so finite sample) received far less
attention. With the advent of big data and high-dimensional statistics, the
need for finite sample results has increased. In this paper, we use the
(well-known) Banach fixed point theorem to derive various deterministic
inequalities that lead to the classical results when studied under randomness.
In addition, we provide applications of these deterministic inequalities for
crossvalidation/subsampling, marginal screening and uniform-in-submodel results
that are very useful for post-selection inference and in the study of
post-regularization estimators. Our results apply to many classical estimators,
in particular, generalized linear models, non-linear regression and cox
proportional hazards model. Extensions to non-smooth and constrained problems
are also discussed.Comment: 49 page
Fast quantum search algorithm and Bounds on it
We recast Grover's generalised search algorithm in a geometric language even
when the states are not approximately orthogonal. We provide a possible search
algorithm based on an arbitrary unitary transformation which can speed up the
steps still further. We discuss the lower and upper bounds on the transition
matrix elements when the unitary operator changes with time, thereby implying
that quantum search process can not be too fast or too slow. This is a
remarkable feature of quantum computation unlike classical one. Quantum
mechanical uncertainty relation puts bounds on search process. Also we mention
the problems of perturbation and other issues in time-dependent search
operation.Comment: Latex file, Two column, 4 pages, no figure
Probabilistic exact cloning and probabilistic no-signalling
We show that non-local resources cannot be used for probabilistic signalling
even if one can produce exact clones with the help of a probabilistic quantum
cloning machine (PQCM). We show that PQCM cannot help to distinguish two
statistical mixtures at a remote location. Thus quantum theory rules out the
possibility of sending superluminal signals not only deterministically but also
probabilistically. We give a bound on the success probability of producing
multiple clones in an entangled system.Comment: Latex file, 6 pages, minor correction
Physical Cost of Erasing Quantum Correlation
Erasure of information stored in a quantum state requires energy cost and is
inherently an irreversible operation. If quantumness of a system is physical,
does erasure of quantum correlation as measured by discord also need some
energy cost? Here, we show that change in quantum correlation is never larger
than the total entropy change of the system and the environment. The entropy
cost of erasing correlation has to be at least equal to the amount of quantum
correlation erased. Hence, quantum correlation can be regarded as genuinely
physical. We show that the new bound leads to the Landauer erasure. The
physical cost of erasing quantum correlation is well respected in the case of
bleaching of quantum information, thermalization, and can have potential
application for any channel leading to erasure of quantum correlation.Comment: Latex, no figs, 5 page
Measuring Electromagnetic Vector Potential via Weak Value
Electromagnetic vector potential has physical significance in quantum
mechanics as revealed by the Aharonov-Bohm effect for charged particles.
However, till date it is thought that we cannot measure the vector potential
directly as this is not a gauge invariant quantity. Contrary to this belief,
here we show that one can indeed measure the electromagnetic vector potential
using the notion of weak value. We show that it is simply the difference
between the weak value of the canonical momentum of a charged particle in the
presence and absence of magnetic field. This suggests that the vector potential
is not only a physical entity but can be measured directly in the experiment.Comment: Latex, 5 pages, Comments and suggestions welcom
Potent Value and Potent Operator with Pre- and Post-selected Quantum Systems
We introduce a novel concept which we call as potent value of system
observable for pre- and post-selected quantum states. This describes, in
general, how a quantum system affects the state of the apparatus during the
time between two strong measurements corresponding to pre- and post-selections.
The potent value can be realized for any interaction strength and for arbitrary
coupling between the system and the apparatus observables. Most importantly,
potent values generalize and unify the notion of the weak values and modular
values of observables in quantum theory. Furthermore, we define a potent
operator which describes the action of one system on the another and show that
superposition of time-evolutions and time-translation machines are potent
operators. These concepts may find useful applications in quantum information
processing and can lead to technological benefits
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