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 $H$ invariant under an irreducible representation of the Poincare group labeled by $[m,s]$ where $m\geq 0$ is mass and $s=0,1/2,1,...$ 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

Quantum Computing, NP-complete Problems and Chaotic Dynamics

An approach to the solution of NP-complete problems based on quantum computing and chaotic dynamics is proposed. We consider the satisfiability problem and argue that the problem, in principle, can be solved in polynomial time if we combine the quantum computer with the chaotic dynamics amplifier based on the logistic map. We discuss a possible implementation of such a chaotic quantum computation by using the atomic quantum computer with quantum gates described by the Hartree-Fock equations. In this case, in principle, one can build not only standard linear quantum gates but also nonlinear gates and moreover they obey to Fermi statistics. This new type of entaglement related with Fermi statistics can be interesting also for quantum communication theory.Comment: 11 pages, 1 figur

On Quantum Capacity and its Bound

The quantum capacity of a pure quantum channel and that of classical-quantum-classical channel are discussed in detail based on the fully quantum mechanical mutual entropy. It is proved that the quantum capacity generalizes the so-called Holevo bound.Comment: 10 page

An Attack to Quantum Cryptography from Space

The promise of secure cryptographic quantum key distribution schemes is based on the use of quantum effects in the spin space. We point out that in fact in many current quantum cryptography protocols the space part of the wave function is neglected. However exactly the space part of the wave function describes the behaviour of particles in ordinary real three-dimensional space. As a result such schemes can be secure against eavesdropping attacks in the abstract spin space but could be insecure in the real three-dimensional space. We discuss an approach to the security of quantum key distribution in space by using Bell's inequality and a special preparation of the space part of the wave function.Comment: 8 pages, LaTe