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arXiv:0904.0107 · Journal of Physics A Mathematical and Theoretical

Random unitary dynamics of quantum networks

Jaroslav Novotný, G. Alber, Igor Jex

Published Jun 26, 2009• 23 citations
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Abstract

We investigate the asymptotic dynamics of quantum networks under repeated applications of random unitary operations. It is shown that in the asymptotic limit of large numbers of iterations this dynamics is generally governed by a typically low dimensional attractor space. This space is determined completely by the unitary operations involved and it is independent of the probabilities with which these unitary operations are applied. Based on this general feature analytical results are presented for the asymptotic dynamics of arbitrarily large cyclic qubit networks whose nodes are coupled by randomly applied controlled-NOT operations.

DOI
10.1088/1751-8113/42/28/282003

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The asymptotic dynamics of many-qubit quantum systems is investigated under iteratively and randomly applied unitary transformations. For a one-parameter family of unitary transformations, which entangle pairs of qubits, two main theorems are proved. They characterize completely the dependence of the resulting asymptotic dynamics on the topology of the interaction graph that encodes all possible qubit couplings. These theorems exhibit clearly which aspects of an interaction graph are relevant and which ones are irrelevant to the asymptotic dynamics. On the basis of these theorems, the local entropy transport between an open quantum system and its environment are explored for strong non-Markovian couplings and for different sizes of the environment and different interaction topologies. It is shown that although the randomly applied unitary entanglement operations cannot decrease the overall entropy of such a qubit network, a local entropy decrease or 'cooling' of subsystems is possible for special classes of interaction topologies.
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The stochastic dynamics of a quantum system driven by NNN statistically independent random sudden quenches in a fixed time interval is studied. We reveal that with increasing NNN the system approaches a deterministic limit, indicating self-averaging with respect to its temporal unitary evolution. This phenomenon is quantified by the variance of the unitary matrix governing the time evolution of a finite-dimensional quantum system which, according to an asymptotic analysis, decreases at least as 1/N1/N1/N. For a special class of protocols (when the averaged Hamiltonian commutes at different times), we prove that for finite NNN the distance (according to the Frobenius norm) between the averaged evolution unitary operator generated by the Hamiltonian HHH and the unitary evolution operator generated by the averaged Hamiltonian $\ensuremath{\langle}H\ensuremath{\rangle}$ scales as 1/N1/N1/N. Numerical simulations enlarge this result to a broader class of noncommuting protocols.
Abstract Asymptotic dynamics of a Hadamard walk of two non-interacting quantum particles on a dynamically percolated finite line or a circle is investigated. We construct a basis of the attractor space of the corresponding random-unitary dynamics and prove the completeness of our solution. In comparison to the one-particle case, the structure of the attractor space is much more complex, resulting in intriguing asymptotic dynamics. General results are illustrated on two examples. First, for circles of length not divisible by 4 the boundary conditions reduces the number of attractors considerably, allowing for fully analytic solution. Second, we investigate line of length 4 and determine the asymptotic cycle of reduced coin states and position distributions, focusing on the correlations between the two particles. Our results show that a random unitary evolution, which is a combination of quantum dynamics and a classical stochasticity, leads to correlations between initially uncorrelated particles. This is not possible for purely unitary evolution of non-interacting quantum particles. The shared dynamically percolated graph can thus be considered as a weak form of interaction.
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