Relationship between quantum walks and relativistic quantum mechanics

Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This article revisits the relationship between relativistic quantum mechanics and the quantum walks. We show the similarities of the mathematical structure of the decoupled a...

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Bibliographic Details
Published in:Physical review. A, Atomic, molecular, and optical physics Atomic, molecular, and optical physics, 2010-06, Vol.81 (6), Article 062340
Main Authors: Chandrashekar, C. M., Banerjee, Subhashish, Srikanth, R.
Format: Article
Language:English
Subjects:
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
COMPUTERS
DEGREES OF FREEDOM
DIFFERENTIAL EQUATIONS
DIRAC EQUATION
ENERGY RANGE
ENTROPY
EQUATIONS
FIELD EQUATIONS
INFORMATION
KLEIN-GORDON EQUATION
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
PHYSICAL PROPERTIES
PROBABILITY
QUANTUM COMPUTERS
QUANTUM INFORMATION
QUANTUM MECHANICS
RELATIVISTIC RANGE
SPACE
THERMODYNAMIC PROPERTIES
TWO-DIMENSIONAL CALCULATIONS
WAVE EQUATIONS
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Summary:Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This article revisits the relationship between relativistic quantum mechanics and the quantum walks. We show the similarities of the mathematical structure of the decoupled and coupled forms 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 Schroedinger 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 (modeled by projective measurements on position space) generates entropy that increases with time, making the walk irreversible and thereby producing an arrow of time. The Lieb-Robinson bound is used to highlight the causal structure of the quantum walk to put in perspective the relativistic structure of the quantum walk, the maximum speed of walk propagation, and 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.
ISSN:1050-2947
1094-1622
DOI:10.1103/PhysRevA.81.062340