Finite-dimensional Hamiltonian Research Articles

First-detection time of a quantum state under random probing

We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the $n\mathrm{th}$ attempt. We calculate as well the mean and mean square, both of the number of the first successful detection attempt and the time until first detection. We present explicit results for a particle initially localized at a site on a ring of size $L$, probed at some arbitrary given site, in the case when the detection intervals are distributed exponentially. We prove that, for all interval distributions and finite-dimensional Hamiltonians, the mean detection time is equal to the mean attempt number times the mean time interval between attempts. We further prove that for the return problem when the initial and target state are identical, the total detection probability is unity and the mean attempts until detection is an integer, which is the size of the Hilbert space (symmetrized about the target state). We study an interpolation between the fixed time interval case to an exponential distribution of time intervals via the Gamma distribution with constant mean and varying width. The mean arrival time as a function of the mean interval changes qualitatively as we tune the interarrival time distribution from very narrow ($\ensuremath{\delta}$ peaked) to exponential, as resonances are wiped out by the randomness of the sampling.

Universal Qudit Hamiltonians

A family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems):We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms.We find that, for all k geqslant 2 and all local dimensions d geqslant 2, almost all such interactions are universal aside from a simple stoquastic class.We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions d geqslant 2 (spin geqslant 1/2), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for d=3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model.We prove universality of any interaction proportional to the projector onto a pure entangled state.

Finite Dimensional Hamiltonians and Almost-Periodic Solutions for 2+1 Dimensional Three-Wave Equations

For the 2+1 dimensional three-wave equation, by using the known nonlinear constraints from 2+1 dimensions to 1+1 dimensions, we reduce it further to 0+1 dimensional (finite dimensional) Hamiltonian systems with constraints of Neumann type. These Hamiltonian systems are proved to be Liouville integrable by finding a full set of involutive conserved integrals and proving their functional independence. Moreover, almost-periodic solutions of the 2+1 dimensional three-wave equation are obtained by solving these Hamiltonian systems explicitly.

Intrinsic Characterizations of Orthogonal Separability for Natural Hamiltonians with Scalar Potentials on Pseudo-Riemannian Spaces

Orthogonal separability of finite-dimensional Hamiltonians is characterized by using various geometrical concepts, including Killing tensors, moving frames, the Nijenhuis tensor, bi-Hamiltonian and quasi-bi-Hamiltonian representations. In addition, a complete classification of separable metrics defined in two-dimensional locally flat Lorenzian spaces is presented.