70 research outputs found
Quantum Computing and Shor`s Factoring Algorithm
Lectures on quantum computing. Contents: Algorithms. Quantum circuits.
Quantum Fourier transform. Elements of number theory. Modular exponentiation.
Shor`s algorithm for finding the order. Computational complexity of Schor`s
algorithm. Factoring integers. NP-complete problems.Comment: 21 page
Photon Antibunching, Sub-Poisson Statistics and Cauchy-Bunyakovsky and Bell's Inequalities
We discuss some mathematical aspects of photon antibunching and sub-Poisson
photon statistics. It is known that Bell's inequalities for entangled states
can be reduced to the Cauchy-Bunyakovsky inequalities. In this note some
rigorous results on impossibility of classical hidden variables representations
of certain quantum correlation functions are proved which are also based on the
Cauchy-Bunyakovsky inequalities.
The difference K between the variance and the mean as a measure of
non-classicality of a state is discussed. For the classical case K is
nonnegative while for the n-particle state it is negative and moreover it
equals -n. The non-classicality of quantum states discussed here for the
sub-Poisson statistics is different from another non-classicality called
entanglement though both can be traced to the violation of the
Cauchy-Bunyakovsky inequality.Comment: 5 page
Bell's Theorem and Locality in Space
Bell's theorem states that some quantum correlations can not be represented
by classical correlations of separated random variables. It has been
interpreted as incompatibility of the requirement of locality with quantum
mechanics. We point out that in fact the space part of the wave function was
neglected in the proof of Bell's theorem. However this space part is crucial
for considerations of property of locality of quantum system. Actually the
space part leads to an extra factor in quantum correlations and as a result the
ordinary proof of Bell's theorem fails in this case. We present a criterium of
locality in a realist theory of hidden variables. It is argued that predictions
of quantum mechanics for Gaussian wave functions can be consistent with Bell's
inequalities and hence Einstein's local realism is restored in this case.Comment: 7 pages, LaTe
Quantum Information in Space and Time
Many important results in modern quantum information theory have been
obtained for an idealized situation when the spacetime dependence of quantum
phenomena is neglected. However the transmission and processing of (quantum)
information is a physical process in spacetime. Therefore such basic notions in
quantum information theory as the notions of composite systems, entangled
states and the channel should be formulated in space and time. We emphasize the
importance of the investigation of quantum information in space and time.
Entangled states in space and time are considered. A modification of Bell`s
equation which includes the spacetime variables is suggested. A general
relation between quantum theory and theory of classical stochastic processes is
proposed. It expresses the condition of local realism in the form of a {\it
noncommutative spectral theorem}. Applications of this relation to the security
of quantum key distribution in quantum cryptography are considered.Comment: 14 page
Time Irreversibility Problem and Functional Formulation of Classical Mechanics
The time irreversibility problem is the dichotomy of the reversible
microscopic dynamics and the irreversible macroscopic physics. This problem was
considered by Boltzmann, Poincar\'e, Bogolyubov and many other authors and
though some researchers claim that the problem is solved, it deserves a further
study. In this paper an attempt is performed of the following solution of the
irreversibility problem: a formulation of microscopic dynamics is suggested
which is irreversible in time.
A widely used notion of microscopic state of the system at a given moment of
time as a point in the phase space and also a notion of trajectory does not
have an immediate physical meaning since arbitrary real numbers are non
observable. In the approach presented in this paper the physical meaning is
attributed not to an individual trajectory but only to a bunch of trajectories
or to the distribution function on the phase space.
The fundamental equation of the microscopic dynamics in the proposed
"functional" approach is not the Newton equation but the Liouville equation for
the distribution function of a single particle. Solutions of the Liouville
equation have the property of delocalization which accounts for
irreversibility. It is shown that the Newton equation in this approach appears
as an approximate equation describing the dynamics of the average values of the
position and momenta. Corrections to the Newton equation are computed.Comment: 21 page
Quantum Information and Spacetime Structure
In modern quantum information theory one deals with an idealized situation
when the spacetime dependence of quantum phenomena is neglected. However the
transmission and processing of (quantum) information is a physical process in
spacetime. Therefore such basic notions in quantum information theory as qubit,
channel, composite systems and entangled states should be formulated in space
and time. In this paper some basic notions of quantum information theory are
considered from the point of view of quantum field theory and general
relativity. It is pointed out an important fact that in quantum field theory
there is a statistical dependence between two regions in spacetime even if they
are spacelike separated. A classical probabilistic representation for a family
of correlation functions in quantum field theory is obtained. A noncommutative
generalization of von Neumann`s spectral theorem is discussed. We suggest a new
physical principle describing a relation between the mathematical formalism of
Hilbert space and quantum physical phenomena which goes beyond the
superselection rules. Entangled states and the change of state associated with
the measurement process in space and time are discussed including the black
hole and the cosmological spacetime.
It is shown that any reasonable state in relativistic quantum field theory
becomes disentangled at large spacelike distances if one makes local
observations. As a result a violation of Bell`s inequalities can be observed
without inconsistency with principles of relativistic quantum theory only if
the distance between detectors is rather small. We suggest a further
experimental study of entangled states in spacetime by studying the dependence
of the correlation functions on the distance between detectors.Comment: 13 pages. Talk at the Conference on Quantum information, Meijo
University, Nagoya, Japan, December 200
Towards Quantum Information Theory in Space and Time
Modern quantum information theory deals with an idealized situation when the
spacetime dependence of quantum phenomena is neglected. However the
transmission and processing of (quantum) information is a physical process in
spacetime. Therefore such basic notions in quantum information theory as qubit,
channel, composite systems and entangled states should be formulated in space
and time. In particlular we suggest that instead of a two level system (qubit)
the basic notion in a relativistic quantum information theory should be a
notion of an elementary quantum system, i.e. an infinite dimensional Hilbert
space invariant under an irreducible representation of the Poincare group
labeled by where is mass and is spin. We
emphasize an importance of consideration of quantum information theory from the
point of view of quantum field theory. We point out and discuss a fundamental
fact that in quantum field theory there is a statistical dependence between two
regions in spacetime even if they are spacelike separated. A classical
probabilistic representation for a family of correlation functions in quantum
field theory is obtained. Entangled states in space and time are considered. It
is shown that any reasonable state in relativistic quantum field theory becomes
disentangled (factorizable) at large spacelike distances if one makes local
observations. As a result a violation of Bell`s inequalities can be observed
without inconsistency with principles of relativistic quantum theory only if
the distance between detectors is rather small. We suggest a further
experimental study of entangled states in spacetime by studying the dependence
of the correlation functions on the distance between detectors.Comment: 16 page
Holographic Photosynthesis
There are successful applications of the holographic AdS/CFT correspondence
to high energy and condensed matter physics. We apply the holographic approach
to photosynthesis that is an important example of nontrivial quantum phenomena
relevant for life which is being studied in the emerging field of quantum
biology. Light harvesting complexes of photosynthetic organisms are many-body
quantum systems, in which quantum coherence has recently been experimentally
shown to survive for relatively long time scales even at the physiological
temperature despite the decohering effects of their environments.
We use the holographic approach to evaluate the time dependence of
entanglement entropy and quantum mutual information in the Fenna-Matthews-Olson
(FMO) protein-pigment complex in green sulfur bacteria during the transfer of
an excitation from a chlorosome antenna to a reaction center. It is
demonstrated that the time evolution of the mutual information simulating the
Lindblad master equation in some cases can be obtained by means of a dual
gravity describing black hole formation in the AdS-Vaidya spacetime. The wake
up and scrambling times for various partitions of the FMO complex are
discussed.Comment: 35 pages, 30 figures, misprints are corrected, one figure is
replaced, refs adde
Local Realism, Contextualism and Loopholes in Bell`s Experiments
It is currently widely accepted, as a result of Bell's theorem and related
experiments, that quantum mechanics is inconsistent with local realism and
there is the so called quantum non-locality. We show that such a claim can be
justified only in a simplified approach to quantum mechanics when one neglects
the fundamental fact that there exist space and time. Mathematical definitions
of local realism in the sense of Bell and in the sense of Einstein are given.
We demonstrate that if we include into the quantum mechanical formalism the
space-time structure in the standard way then quantum mechanics might be
consistent with Einstein's local realism. It shows that loopholes are
unavoidable in experiments aimed to establish a violation of Bell`s
inequalities. We show how the space-time structure can be considered from the
contextual point of view. A mathematical framework for the contextual approach
is outlined
A Note on the Eigenstate Thermalization Hypothesis
Eigenstate thermalization hypothesis (ETH) is discussed. We point out that
the common formulation of the ETH suffers from the mixing of random and
deterministic variables. We suggest a modified formulation of the ETH which
includes only deterministic variables. We also show that this formulation of
the ETH implies thermalization as well
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