Relationship Between Quantum Walk and Relativistic Quantum Mechanics
arXiv:1003.4656 · doi:10.1103/PhysRevA.81.062340
Abstract
Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This paper revisits the relationship between relativistic quantum mechanics and the quantum walks. We show the similarities of the mathematical structure of the decoupled and coupled form of the discrete-time quantum walk to that of the Klein-Gordon and Dirac equations, respectively. In the latter case, the coin emerges as an analog of the spinor degree of freedom. Discrete-time quantum walk as a coupled form of the continuous-time quantum walk is also shown by transforming the decoupled form of the discrete-time quantum walk to the Schrodinger form. By showing the coin to be a means to make the walk reversible, and that the Dirac-like structure is a consequence of the coin use, our work suggests that the relativistic causal structure is a consequence of conservation of information. However, decoherence (modelled by projective measurements on position space) generates entropy that increases with time, making the walk irreversible and thereby producing an arrow of time. Lieb-Robinson bound is used to highlight the causal structure of the quantum walk to put in perspective the relativistic structure of quantum walk, maximum speed of the walk propagation and the earlier findings related to the finite spread of the walk probability distribution. We also present a two-dimensional quantum walk model on a two state system to which the study can be extended.
12 pages and 1 figure, Published version
References in corpus (16)
- Environment-Assisted Quantum Walks in Photosynthetic Energy Transfer
- Lieb-Robinson bounds and the generation of correlations and topological quantum order
- Quantum Walk in Position Space with Single Optically Trapped Atoms
- Realization of quantum walks with negligible decoherence in waveguide lattices
- Discrete single-photon quantum walks with tunable decoherence
- Lieb-Robinson Bounds and the Exponential Clustering Theorem
- Characterizing quantum theory in terms of information-theoretic constraints
- Propagation of Correlations in Quantum Lattice Systems
- General entanglement scaling laws from time evolution
- Optimizing the discrete time quantum walk using a SU(2) coin
- Information Invariance and Quantum Probabilities
- Implementing the one-dimensional quantum (Hadamard) walk using a Bose-Einstein Condensate
- Quantum phase transition using quantum walks in an optical lattice
- Quantum walks and orbital states of a Weyl particle
- Quantum random walks using quantum accelerator modes
- The quantum measurement problem and physical reality: a computation theoretic perspective