3,937 research outputs found
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
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
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
Design of continuous attractor networks with monotonic tuning using a symmetry principle
Neurons that sustain elevated firing in the absence of stimuli have been found in many neural systems. In graded persistent activity, neurons can sustain firing at many levels, suggesting a widely found type of network dynamics in which networks can relax to any one of a continuum of stationary states. The reproduction of these findings in model networks of nonlinear neurons has turned out to be nontrivial. A particularly insightful model has been the "bump attractor," in which a continuous attractor emerges through an underlying symmetry in the network connectivity matrix. This model, however, cannot account for data in which the persistent firing of neurons is a monotonic-rather than a bell-shaped-function of a stored variable. Here, we show that the symmetry used in the bump attractor network can be employed to create a whole family of continuous attractor networks, including those with monotonic tuning. Our design is based on tuning the external inputs to networks that have a connectivity matrix with Toeplitz symmetry. In particular, we provide a complete analytical solution of a line attractor network with monotonic tuning and show that for many other networks, the numerical tuning of synaptic weights reduces to the computation of a single parameter
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