5,307 research outputs found

### Elementary solution to the time-independent quantum navigation problem

A quantum navigation problem concerns the identification of a time-optimal Hamiltonian that realizes a required quantum process or task, under the influence of a prevailing ‘background’ Hamiltonian that cannot be manipulated. When the task is to transform one quantum state into another, finding the solution in closed form to the problem is nontrivial even in the case of timeindependent Hamiltonians. An elementary solution, based on trigonometric analysis, is found here when the Hilbert space dimension is two. Difficulties arising from generalizations to higher-dimensional systems are discussed

### Note on exponential families of distributions

We show that an arbitrary probability distribution can be represented in
exponential form. In physical contexts, this implies that the equilibrium
distribution of any classical or quantum dynamical system is expressible in
grand canonical form.Comment: 5 page

### Thermalisation of Quantum States

An exact stochastic model for the thermalisation of quantum states is
proposed. The model has various physically appealing properties. The dynamics
are characterised by an underlying Schrodinger evolution, together with a
nonlinear term driving the system towards an asymptotic equilibrium state and a
stochastic term reflecting fluctuations. There are two free parameters, one of
which can be identified with the heat bath temperature, while the other
determines the characteristic time scale for thermalisation. Exact expressions
are derived for the evolutionary dynamics of the system energy, the system
entropy, and the associated density operator.Comment: 8 pages, minor corrections. To appear in JM

### Information Content for Quantum States

A method of representing probabilistic aspects of quantum systems is
introduced by means of a density function on the space of pure quantum states.
In particular, a maximum entropy argument allows us to obtain a natural density
function that only reflects the information provided by the density matrix.
This result is applied to derive the Shannon entropy of a quantum state. The
information theoretic quantum entropy thereby obtained is shown to have the
desired concavity property, and to differ from the the conventional von Neumann
entropy. This is illustrated explicitly for a two-state system.Comment: RevTex file, 4 pages, 1 fi

### The Quantum Canonical Ensemble

The phase space of quantum mechanics can be viewed as the complex projective
space endowed with a Kaehlerian structure given by the Fubini-Study metric and
an associated symplectic form. We can then interpret the Schrodinger equation
as generating a Hamiltonian dynamics. Based upon the geometric structure of the
quantum phase space we introduce the corresponding natural microcanonical and
canonical ensembles. The resulting density matrix for the canonical ensemble
differs from density matrix of the conventional approach. As an illustration,
the results are applied to the case of a spin one-half particle in a heat bath
with an applied magnetic field.Comment: 8 pages, minor corrections. to appear in JMP vol. 3

### Random Hamiltonian in thermal equilibrium

A framework for the investigation of disordered quantum systems in thermal
equilibrium is proposed. The approach is based on a dynamical model--which
consists of a combination of a double-bracket gradient flow and a uniform
Brownian fluctuation--that `equilibrates' the Hamiltonian into a canonical
distribution. The resulting equilibrium state is used to calculate quenched and
annealed averages of quantum observables.Comment: 8 pages, 4 figures. To appear in DICE 2008 conference proceeding

### On optimum Hamiltonians for state transformations

For a prescribed pair of quantum states |psi_I> and |psi_F> we establish an
elementary derivation of the optimum Hamiltonian, under constraints on its
eigenvalues, that generates the unitary transformation |psi_I> --> |psi_F> in
the shortest duration. The derivation is geometric in character and does not
rely on variational calculus.Comment: 5 page

### Entropy and Temperature of a Quantum Carnot Engine

It is possible to extract work from a quantum-mechanical system whose
dynamics is governed by a time-dependent cyclic Hamiltonian. An energy bath is
required to operate such a quantum engine in place of the heat bath used to run
a conventional classical thermodynamic heat engine. The effect of the energy
bath is to maintain the expectation value of the system Hamiltonian during an
isoenergetic expansion. It is shown that the existence of such a bath leads to
equilibrium quantum states that maximise the von Neumann entropy. Quantum
analogues of certain thermodynamic relations are obtained that allow one to
define the temperature of the energy bath.Comment: 4 pages, 1 figur

### Information geometry of density matrices and state estimation

Given a pure state vector |x> and a density matrix rho, the function
p(x|rho)= defines a probability density on the space of pure states
parameterised by density matrices. The associated Fisher-Rao information
measure is used to define a unitary invariant Riemannian metric on the space of
density matrices. An alternative derivation of the metric, based on square-root
density matrices and trace norms, is provided. This is applied to the problem
of quantum-state estimation. In the simplest case of unitary parameter
estimation, new higher-order corrections to the uncertainty relations,
applicable to general mixed states, are derived.Comment: published versio

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