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

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

Spectral Statistics of the Two-Body Random Ensemble Revisited

Using longer spectra we re-analyze spectral properties of the two-body random ensemble studied thirty years ago. At the center of the spectra the old results are largely confirmed, and we show that the non-ergodicity is essentially due to the variance of the lowest moments of the spectra. The longer spectra allow to test and reach the limits of validity of French's correction for the number variance. At the edge of the spectra we discuss the problems of unfolding in more detail. With a Gaussian unfolding of each spectrum the nearest neighbour spacing distribution between ground state and first exited state is shown to be stable. Using such an unfolding the distribution tends toward a semi-Poisson distribution for longer spectra. For comparison with the nuclear table ensemble we could use such unfolding obtaining similar results as in the early papers, but an ensemble with realistic splitting gives reasonable results if we just normalize the spacings in accordance with the procedure used for the data.Comment: 11 pages, 7 figure

Efficient generation of random multipartite entangled states using time optimal unitary operations

We review the generation of random pure states using a protocol of repeated two qubit gates. We study the dependence of the convergence to states with Haar multipartite entanglement distribution. We investigate the optimal generation of such states in terms of the physical (real) time needed to apply the protocol, instead of the gate complexity point of view used in other works. This physical time can be obtained, for a given Hamiltonian, within the theoretical framework offered by the quantum brachistochrone formalism. Using an anisotropic Heisenberg Hamiltonian as an example, we find that different optimal quantum gates arise according to the optimality point of view used in each case. We also study how the convergence to random entangled states depends on different entanglement measures.Comment: 5 pages, 2 figures. New title, improved explanation of the algorithm. To appear in Phys. Rev.

Biorthogonal quantum mechanics

The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems. © 2014 IOP Publishing Ltd

Intruder States and their Local Effect on Spectral Statistics

The effect on spectral statistics and on the revival probability of intruder states in a random background is analysed numerically and with perturbative methods. For random coupling the intruder does not affect the GOE spectral statistics of the background significantly, while a constant coupling causes very strong correlations at short range with a fourth power dependence of the spectral two-point function at the origin.The revival probability is significantly depressed for constant coupling as compared to random coupling.Comment: 18 pages, 10 Postscript figure