Entanglement dynamics and quasi-periodicity in discrete quantum walks

Abstract

We study the entanglement dynamics of discrete time quantum walks acting on bounded finite sized graphs. We demonstrate that, depending on system parameters, the dynamics may be monotonic, oscillatory but highly regular, or quasi-periodic. While the dynamics of the system are not chaotic since the system comprises linear evolution, the dynamics often exhibit some features similar to chaos such as high sensitivity to the system's parameters, irregularity and infinite periodicity. Our observations are of interest for entanglement generation, which is one primary use for the quantum walk formalism. Furthermore, we show that the systems we model can easily be mapped to optical beamsplitter networks, rendering experimental observation of quasi-periodic dynamics within reach.

Keywords:

• quantum walk
• quantum information theory
• quantum computation
• quantum optics (inc. quantum information)

Acknowledgements

We thank Gerard Milburn and Cathy Holmes for helpful discussions. This research was conducted by the Australian Research Council Centre for Excellence for Quantum Computation and Communication Technology (Project number CE110001027).

Notes

Notes

1. As opposed to the interpretation that position states are positions in Hilbert space.

2. Historically, there has been some debate as to whether a single walker (e.g. photon) state can in fact be entangled. We take the current view that it can be, since if we couple each mode in the system to an atom (i.e. local operations) we can create a superposition of a single excitation across multiple atoms, which is entangled.

3. The neighbourhood of x is defined as the set of vertices connected to x via an edge (an outgoing edge in the case of a directed graph).

4. In present day experiments with a walker on a line the photon may have a coin state via the polarisation degree of freedom. However, polarisation is a two-level Hilbert space. Thus, for higher order graphs where the number of available coin states is |G|, polarisation is insufficient for use as a coin state. Orbital angular momentum modes are a candidate for use as coin states 42, however this is experimentally challenging.

5. If p _j /q _j is already in lowest denominator form then we simply have t _j  = q _j .

6. That is, the Hamiltonian describing the evolution of the system has non-linear terms.