Decoherence in a one-dimensional quantum walk

Abstract

In this paper we study decoherence in the quantum walk on the line. We generalize the method of decoherent coin quantum walk, introduced by Brun et al [Phys. Rev. A 67, 32304 (2003)]. Our analytical expressions are applicable for all kinds of decoherence. As an example of the coin-position decoherence, we study the broken line quantum walk and compare our results with the numerical one. We also show that our analytical results reduce to the Brun formalism when only the coin is subjected to decoherence.

Chirality asymptotic behavior and non-Markovianity in quantum walks on a line

“…[13] and analyzed in the frame of previous Section in Ref. [12]. This model induces decoherence in both degrees of freedom, coin and position.…”

“…In order to use the affine map approach [11,12], Eq. (9) can be transformed to express the two-by-two matrix as a four-dimensional column vector, obtaining…”

Analytical expression for variance of homogeneous-position quantum walk with decoherent position

“…We restrict our attention to the one dimensional quantum walk with homogeneous position (p n x,l ≡ p n l ) as Annabestani et al assumed in [25], so…”

“…Unfortunately this equation is more complicated than that can be used for analytical calculations, but in some cases it can be simplified using some mathematical tricks. Brun [22] has been derived a compact analytical formula for the first and the second moments of probability distribution of one dimensional quantum walk in the presence of coin only decoherence and Annabestani et al [25] have generalized Brun approach and have found a compact formula for moments in general form of decoherence. In the next section we briefly explain their approach.…”

“…These interactions can eliminate the quantum properties of the system and change it to a classical one, therefore many research topics are focused on decoherency and the effects of noise on quantum walks [21][22][23][24][25]. Especially it is proven that noise in the coin space eliminates the quantum properties of the walk and changes it to a classical random walk [22,26].…”