Quantum theory: Concepts and methods

The Theory of Groups and Quantum Mechanics

Quantum trajectory theory, developed largely in the quantum optics community to describe open quantum systems subjected to continuous monitoring, has applications in many areas of quantum physics. I present a simple model, using two-level quantum systems (q-bits), to illustrate the essential physics of quantum trajectories and how different monitoring schemes correspond to different “unravelings” of a mixed state master equation. I also comment briefly on the relationship of the theory to the consistent histories formalism and to spontaneous collapse models.

A simple model of quantum trajectories

We investigate the average bipartite entanglement, over all possible divisions of a multipartite system, as a useful measure of multipartite entanglement We expose a connection between such measures and quantum-error-correcting codes by deriving a formula relating the weight distribution of the code to the average entanglement of encoded states The multipartite entangling power of quantum evolutions is also investigated

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Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions

This is a collection of references (papers, books, preprints, book reviews, Ph. D. thesis, patents, web sites, etc.), sorted alphabetically and (some of them) classified by subject, on foundations of quantum mechanics and quantum information. Specifically, it covers hidden variables (``no-go'' theorems, experiments), interpretations of quantum mechanics, entanglement, quantum effects (quantum Zeno effect, quantum erasure, ``interaction-free'' measurements, quantum ``non-demolition'' measurements), quantum information (cryptography, cloning, dense coding, teleportation), and quantum computation.

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Bibliographic guide to the foundations of quantum mechanics and quantum information

We derive a simple closed form for the matrix elements of the quantum baker's map that shows that the map is an approximate shift in a symbolic representation based on discrete phase space. We use this result to give a formal proof that the quantum baker's map approaches a classical Bernoulli shift in the limit of a small effective Planck's constant.

Classical limit in terms of symbolic dynamics for the quantum baker's map

Consider a joint quantum state of a system and its environment. A measurement on the environment induces a decomposition of the system state. Using algorithmic information theory, we define the preparation information of a pure or mixed state in a given decomposition. We then define an optimal decomposition as a decomposition for which the average preparation information is minimal. The average preparation information for an optimal decomposition characterizes the system-environment correlations. We discuss properties and applications of the concepts introduced above and give several examples.

Preparation information and optimal decompositions for mixed quantum states

Hypothesis elimination is a special case of Bayesian updating, where each piece of new data rules out a set of prior hypotheses. We describe how to use Grover's algorithm to perform hypothesis elimination for a class of probability distributions encoded on a register of qubits, and establish a lower bound on the required computational resources.

Hypothesis elimination on a quantum computer

Hypothesis elimination is a special case of Bayesian updating, where each piece of new data rules out a set of prior hypotheses. We describe an algorithm to perform hypothesis elimination for a class of probability distributions encoded on a register of qubits, and establish a lower bound on the computational resources it requires.

Andrei N. Soklakov

Papers

Classical limit in terms of symbolic dynamics for the quantum baker's map

Preparation information and optimal decompositions for mixed quantum states

Hypothesis elimination on a quantum computer

Preparation information and optimal decompositions for mixed quantum states

Hypothesis Elimination on a Quantum Computer