Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.

Quantum Computation and Quantum Information

THE paradox of Einstein, Podolsky and Rosen [1] was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables These additional variables were to restore to the theory causality and locality [2] In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty There have been attempts [3] to show that even without such a separability or locality requirement no "hidden variable" interpretation of quantum mechanics is possible These attempts have been examined elsewhere [4] and found wanting Moreover, a hidden variable interpretation of elementary quantum theory [5] has been explicitly constructed That particular interpretation has indeed a grossly nonlocal structure This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions

/pdf/on-the-einstein-podolsky-rosen-paradox-y92u378sck.pdf

On the Einstein-Podolsky-Rosen paradox

All our former experience with application of quantum theory seems to say:
{\it what is predicted by quantum formalism must occur in laboratory} But the
essence of quantum formalism - entanglement, recognized by Einstein, Podolsky,
Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a
new resource as real as energy This holistic property of compound quantum systems, which involves
nonclassical correlations between subsystems, is a potential for many quantum
processes, including ``canonical'' ones: quantum cryptography, quantum
teleportation and dense coding However, it appeared that this new resource is
very complex and difficult to detect Being usually fragile to environment, it
is robust against conceptual and mathematical tools, the task of which is to
decipher its rich structure This article reviews basic aspects of entanglement including its
characterization, detection, distillation and quantifying In particular, the
authors discuss various manifestations of entanglement via Bell inequalities,
entropic inequalities, entanglement witnesses, quantum cryptography and point
out some interrelations They also discuss a basic role of entanglement in
quantum communication within distant labs paradigm and stress some
peculiarities such as irreversibility of entanglement manipulations including
its extremal form - bound entanglement phenomenon A basic role of entanglement
witnesses in detection of entanglement is emphasized

Quantum entanglement

Quantum cryptography could well be the first application of quantum mechanics at the individual quanta level. The very fast progress in both theory and experiments over the recent years are reviewed, with emphasis on open questions and technological issues.

Quantum Cryptography

A theorem of Bell, proving that certain predictions of quantum mechanics are inconsistent with the entire family of local hidden-variable theories, is generalized so as to apply to realizable experiments. A proposed extension of the experiment of Kocher and Commins, on the polarization correlation of a pair of optical photons, will provide a decisive test between quantum mechanics and local hidden-variable theories.

Proposed Experiment to Test Local Hidden Variable Theories.

In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.

/pdf/can-quantum-mechanical-description-of-physical-reality-be-46250ivm32.pdf

Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?

The writers investigate the possibility of an atomistic theory of matter and electricity which, while excluding singularities of the field, makes use of no other variables than the ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$ of the general relativity theory and the ${\ensuremath{\phi}}_{\ensuremath{\mu}}$ of the Maxwell theory. By the consideration of a simple example they are led to modify slightly the gravitational equations which then admit regular solutions for the static spherically symmetric case. These solutions involve the mathematical representation of physical space by a space of two identical sheets, a particle being represented by a "bridge" connecting these sheets. One is able to understand why no neutral particles of negative mass are to be found. The combined system of gravitational and electromagnetic equations are treated similarly and lead to a similar interpretation. The most natural elementary charged particle is found to be one of zero mass. The many-particle system is expected to be represented by a regular solution of the field equations corresponding to a space of two identical sheets joined by many bridges. In this case, because of the absence of singularities, the field equations determine both the field and the motion of the particles. The many-particle problem, which would decide the value of the theory, has not yet been treated.

https://link.aps.org/pdf/10.1103/PhysRev.48.73

The Particle Problem in the General Theory of Relativity

The rigorous solution for cylindrical gravitational waves is given. For the convenience of the reader the theory of gravitational waves and their production, already known in principle, is given in the first part of this paper. After encountering relationships which cast doubt on the existence of rigorous solutions for undulatory gravitational fields, we investigate rigorously the case of cylindrical gravitational waves. It turns out that rigorous solutions exist and that the problem reduces to the usual cylindrical waves in euclidean space.

On gravitational waves

An exact solution of the wave equation is found for a form of one-dimensional potential energy which may be of use in discussing polyatomic molecular vibrational energies. An example of its use is given in an analysis of the vibration of the nitrogen in the ammonia molecule. The potential energy for this atom has two minima a distance $2{x}_{m}$ apart, separated by a "hill" of height $H$. The values of ${x}_{m}$ and $H$ are not known directly from band spectral data, and are needed for a full analysis of the spectrum. By joining two potential curves of the sort dealt with in the first part of this paper in a symmetric manner, a curve simulating that for the nitrogen atom in ammonia was formed. It was found that for certain values of the constants fixing this curve, the allowed vibrational energies were the same as the experimentally determined values for ammonia. The corresponding value of ${x}_{m}$ was 0.38A, and that of $H$ was \textonequarter{} electron-volt. These values are probably near the correct values of ${x}_{m}$ and $H$ for ammonia.

On the Vibrations of Polyatomic Molecules

The theory of the double Stern-Gerlach experiment is developed where the rate of rotation has a maximum at $t=0$ and goes gradually to zero at $t=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$. The negative results of the experiments of Phipps and Stern are completely accounted for. The same analysis is applied to those collision phenomena where only two quantum states need be considered, and where their difference of energy, $\ensuremath{\Delta}E$, is so much smaller than the relative kinetic energy of the two systems that the positions of the centers of gravity of the two systems may be regarded as time parameters. If $f(t)=\frac{2\ensuremath{\pi}}{h}$ times the matrix element of the perturbation, and if $A=\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}f(t)\mathrm{dt}$, then the transition probability is ${\left|\frac{sinA}{A}\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}f(t){e}^{2\ensuremath{\pi}i(\frac{\ensuremath{\Delta}E}{h})t}\mathrm{dt}\right|}^{2}$ for the cases investigated, and this probably holds for all experimental cases.

Nathan Rosen

Papers

Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?

The Particle Problem in the General Theory of Relativity

On gravitational waves

On the Vibrations of Polyatomic Molecules

Double Stern-Gerlach Experiment and Related Collision Phenomena