7,916 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
Biorthogonal systems on unit interval and zeta dilation operators
An elementary 'quantum-mechanical' derivation of the conditions for a system
of functions to form a Reisz basis of a Hilbert space on a finite interval is
presented.Comment: 4 pages, 1 figur
Shapes of Quantum States
The shape space of k labelled points on a plane can be identified with the
space of pure quantum states of dimension k-2. Hence, the machinery of quantum
mechanics can be applied to the statistical analysis of planar configurations
of points. Various correspondences between point configurations and quantum
states, such as linear superposition as well as unitary and stochastic
evolution of shapes, are illustrated. In particular, a complete
characterisation of shape eigenstates for an arbitrary number of points is
given in terms of cyclotomic equations.Comment: Submitted to Proc. R. Statist. So
Modelling election dynamics and the impact of disinformation
Complex dynamical systems driven by the unravelling of information can be
modelled effectively by treating the underlying flow of information as the
model input. Complicated dynamical behaviour of the system is then derived as
an output. Such an information-based approach is in sharp contrast to the
conventional mathematical modelling of information-driven systems whereby one
attempts to come up with essentially {\it ad hoc} models for the outputs. Here,
dynamics of electoral competition is modelled by the specification of the flow
of information relevant to election. The seemingly random evolution of the
election poll statistics are then derived as model outputs, which in turn are
used to study election prediction, impact of disinformation, and the optimal
strategy for information management in an election campaign.Comment: 20 pages, 5 figure
Entanglement Induced Phase Transitions
Starting from the canonical ensemble over the space of pure quantum states,
we obtain an integral representation for the partition function. This is used
to calculate the magnetisation of a system of N spin-1/2 particles. The results
suggest the existence of a new type of first order phase transition that occurs
at zero temperature in the absence of spin-spin interactions. The transition
arises as a consequence of quantum entanglement. The effects of internal
interactions are analysed and the behaviour of the magnetic susceptibility for
a small number of interacting spins is determined.Comment: 4 pages, 2 figure
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
Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian
The differential-equation eigenvalue problem associated with a
recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of
the Riemann zeta function, is analyzed using Fourier and WKB analysis. The
Fourier analysis leads to a challenging open problem concerning the formulation
of the eigenvalue problem in the momentum space. The WKB analysis gives the
exact asymptotic behavior of the eigenfunction
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