# Asymptotic evolution of quantum walks on the N-cycle subject to decoherence on both the coin and position degrees of freedom

## Abstract

Consider a discrete-time quantum walk on the N-cycle subject to decoherence both on the coin and the position degrees of freedom. By examining the evolution of the density matrix of the system, we derive some conclusions about the asymptotic behavior of the system. When N is odd, the density matrix of the system tends, in the long run, to the maximally mixed state, independent of the initial state. When N is even, although the behavior of the system is not necessarily asymptotically stationary, in this case too an explicit formulation is obtained of the asymptotic dynamics of the system. Moreover, this approach enables us to specify the limiting behavior of the mutual information, viewed as a measure of quantum entanglement between subsystems (coin and walker). In particular, our results provide efficient theoretical confirmation of the findings of previous authors, who have arrived at their results through extensive numerical simulations. Our results can be attributed to an important theorem which, for a generalized random unitary operation, explicitly identifies the structure of all of its eigenspaces corresponding to eigenvalues of unit modulus.

- Authors:

- Department of Mathematics, Bowie State University, Bowie, Maryland 20715 (United States)

- Publication Date:

- OSTI Identifier:
- 22038623

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. A

- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 1; Other Information: (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1050-2947

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; DEGREES OF FREEDOM; DENSITY MATRIX; EIGENVALUES; MARKOV PROCESS; MIXED STATES; QUANTUM DECOHERENCE; QUANTUM ENTANGLEMENT; QUANTUM INFORMATION

### Citation Formats

```
Liu Chaobin, and Petulante, Nelson.
```*Asymptotic evolution of quantum walks on the N-cycle subject to decoherence on both the coin and position degrees of freedom*. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVA.84.012317.

```
Liu Chaobin, & Petulante, Nelson.
```*Asymptotic evolution of quantum walks on the N-cycle subject to decoherence on both the coin and position degrees of freedom*. United States. doi:10.1103/PHYSREVA.84.012317.

```
Liu Chaobin, and Petulante, Nelson. Fri .
"Asymptotic evolution of quantum walks on the N-cycle subject to decoherence on both the coin and position degrees of freedom". United States. doi:10.1103/PHYSREVA.84.012317.
```

```
@article{osti_22038623,
```

title = {Asymptotic evolution of quantum walks on the N-cycle subject to decoherence on both the coin and position degrees of freedom},

author = {Liu Chaobin and Petulante, Nelson},

abstractNote = {Consider a discrete-time quantum walk on the N-cycle subject to decoherence both on the coin and the position degrees of freedom. By examining the evolution of the density matrix of the system, we derive some conclusions about the asymptotic behavior of the system. When N is odd, the density matrix of the system tends, in the long run, to the maximally mixed state, independent of the initial state. When N is even, although the behavior of the system is not necessarily asymptotically stationary, in this case too an explicit formulation is obtained of the asymptotic dynamics of the system. Moreover, this approach enables us to specify the limiting behavior of the mutual information, viewed as a measure of quantum entanglement between subsystems (coin and walker). In particular, our results provide efficient theoretical confirmation of the findings of previous authors, who have arrived at their results through extensive numerical simulations. Our results can be attributed to an important theorem which, for a generalized random unitary operation, explicitly identifies the structure of all of its eigenspaces corresponding to eigenvalues of unit modulus.},

doi = {10.1103/PHYSREVA.84.012317},

journal = {Physical Review. A},

issn = {1050-2947},

number = 1,

volume = 84,

place = {United States},

year = {2011},

month = {7}

}