There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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 field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It has moved into a central place in condensed matter studies. Statistical physics, and more specifically, the theory of transitions between states of matter, more or less defines what we know about 'everyday' matter and its transformations. The major aim of this serial is to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments.

Phase Transitions and Critical Phenomena

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state $\ensuremath{\rho}$ is the minimum average entanglement of an ensemble of pure states that represents \ensuremath{\rho}. An earlier paper conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula for special states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.

Entanglement of Formation of an Arbitrary State of Two Qubits

A spin-1 Ising model, which simulates the thermodynamic behavior of ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ mixtures along the $\ensuremath{\lambda}$ line and near the critical mixing point, is introduced and solved in the mean-field approximation. For reasonable values of the parameters of the model the phase diagram is qualitatively similar to that observed experimentally and the phase separation appears as a consequence of the superfluid ordering. Changing the parameters produces many different types of phase diagram, including as features $\ensuremath{\lambda}$ lines, critical points, tricritical points, and triple points. Certain thermodynamic features which differ from the ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ experiments may be artifacts of the mean-field theory.

Ising Model for the λ Transition and Phase Separation in He 3 - He 4 Mixtures

The usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times, called “consistent histories,” through a criterion which ensures that, within limits which are explicitly defined within the formalism, classical rules for probabilities are satisfied The interpretive scheme which results is applicable to closed (isolated) quantum systems, is explicitly independent of the sense of time (ie, past and future can be interchanged), has no need for wave function “collapse,” makes no reference to processes of measurement (though it can be used to analyze such processes), and can be applied to sequences of microscopic or macroscopic events, or both, as long as the mathematical condition of consistency is satisfied When applied to appropriate macroscopic events it appears to yield the same answers as other interpretative schemes for standard quantum mechanics, though from a different point of view which avoids the conceptual difficulties which are sometimes thought to require reference to conscious observers or classical apparatus

Consistent histories and the interpretation of quantum mechanics

The thermodynamics of critical points in multicomponent systems, more generally systems with more than two independent variables (including binary fluid mixtures, the helium $\ensuremath{\lambda}$ transition, order-disorder transitions in alloys, and antiferromagnetism) are discussed from a unified geometrical point of view, in analogy with one-component (liquid-vapor and simple-ferromagnetic) systems. It is shown that, from a few simple postulates, the qualitative behavior near the critical point of quantities such as compressibilities, susceptibilities, and heat capacities, with different choices of the variables held fixed, can be easily predicted. A number of seemingly exceptional cases (such as critical azeotropy), which arise when critical or coexistence surfaces bear an "accidental" geometrical relationship with the thermodynamic coordinate axes, are explained in terms of the same postulates. The predicted results are compared with several theoretical models and experimental data for a variety of systems.

Critical Points in Multicomponent Systems

1. Introduction 2. Wave functions 3. Linear algebra in Dirac notation 4. Physical properties 5. Probabilities and physical variables 6. Composite systems and tensor products 7. Unitary dynamics 8. Stochastic histories 9. The Born rule 10. Consistent histories 11. Checking consistency 12. Examples of consistent families 13. Quantum interference 14. Dependent (contextual) events 15. Density matrices 16. Quantum reasoning 17. Measurements I 18. Measurements II 19. Coins and counterfactuals 20. Delayed choice paradox 21. Indirect measurement paradox 22. Incompatibility paradoxes 23. Singlet state correlations 24. EPR paradox and Bell inequalities 25. Hardy's paradox 26. Decoherence and the classical limit 27. Quantum theory and reality Bibliography.

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Consistent Quantum Theory

An inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannot exhibit a spontaneous magnetization at temperatures above the mean-field approximation to the Curie or “critical” point. (As a consequence, the corresponding “lattice gas” cannot undergo a first order phase transition in density (condensation) above this temperature.) The mean-field susceptibility in zero magnetic field at high temperatures is shown to be an upper bound for the exact result.

Robert B. Griffiths

Papers

Ising Model for the λ Transition and Phase Separation in He 3 - He 4 Mixtures

Consistent histories and the interpretation of quantum mechanics

Critical Points in Multicomponent Systems

Consistent Quantum Theory

Correlations in Ising Ferromagnets. I