One-dimensional quantum walk with unitary noise

Shapira, Daniel; Biham, Ofer; Bracken, A. J.; Hackett, Michelle

doi:10.1103/physreva.68.062315

Cited by 64 publications

(71 citation statements)

References 26 publications

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“…These authors considered pure dephasing as a decoherence mechanism and they could explore the quantum to classical transition by means of tunable decoherence. In other theoretical studies [10,16,[20][21][22], the authors analyzed discrete-time and continuous-time QW in a random environment, and they could also study the quantum-classical transition. In [31] the authors study QW with decoherence by analyzing a non-unitary evolution in QW.…”

Section: Introductionmentioning

confidence: 99%

“…Interestingly, the concept of Quantum Walk (QW), borrowed from classical statistics [11][12][13][14], has the same properties as a tight-binding free particle [14,15]. Two kinds of QW are considered in the literature: discrete-time quantum coined walk [7,9,11,13,[16][17][18][19][20][21][22][23] and continuous-time QW [10,14,15,[24][25][26][27]. In the former (proposed by Aharonov et al [11]), a two-level state, the so-called "coin", rules the unitary discrete-time evolution of a particle moving in a lattice.…”

Section: Introductionmentioning

confidence: 99%

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Non-equilibrium transition from dissipative quantum walk to classical random walk

Nizama¹,

Cáceres²

2012

J. Phys. A: Math. Theor.

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We have investigated the time-evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson's boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann's entropy and quantum purity. In addition, we have studied Wigner's function. The time-dependent behavior of the quantum entanglement between a free particle -in the lattice-and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time-dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-equilibrium transition from dissipative quantum walk to classical random walk

Nizama¹,

Cáceres²

2012

J. Phys. A: Math. Theor.

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“…Ordered QRW has also proved useful in a possible implementation of quantum search algorithms [17], of a universal quantum computer [18,19], and in the simulation of Dirac-like Hamiltonians [20]. For disordered QRW , where the coin and/or the displacement operator is randomly chosen at each lattice site and/or time step, only the transport properties were studied in some depth [21][22][23][24][25][26][27][28][29][30][31][32]. In this scenario, depending on the type of disorder, we can have diffusive or ballistic transport or Anderson localization.…”

Section: Introductionmentioning

confidence: 99%

Entangling power of disordered quantum walks

2014

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We investigate how the introduction of different types of disorder affects the generation of entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional quantum random walks (QRW ). Disorder is modeled by adding another random feature to QRW , i.e., the quantum coin that drives the system's evolution is randomly chosen at each position and/or at each time step, giving rise to either dynamic, fluctuating, or static disorder. The first one is position-independent, with every lattice site having the same coin at a given time, the second has time and position dependent randomness, while the third one is time-independent. We show for several levels of disorder that dynamic disorder is the most powerful entanglement generator, followed closely by fluctuating disorder. Static disorder is the less efficient entangler, being almost always less efficient than the ordered case. Also, dynamic and fluctuating disorder lead to maximally entangled states asymptotically in time for any initial condition while static disorder has no asymptotic limit and, similarly to the ordered case, has a long time behavior highly sensitive to the initial conditions.

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“…Moreover, if the coin operator is chosen at random at each time step, which yields a non autonomous random dynamical system, then, typically, the averaged motion is diffusive, see e.g. [24], [25], see also [32].…”

Section: Introductionmentioning

confidence: 99%

Dynamical Localization of Quantum Walks in Random Environments

2010

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The dynamics of a one dimensional quantum walker on the lattice with two internal degrees of freedom, the coin states, is considered. The discrete time unitary dynamics is determined by the repeated action of a coin operator in U (2) on the internal degrees of freedom followed by a one step shift to the right or left, conditioned on the state of the coin. For a fixed coin operator, the dynamics is known to be ballistic.We prove that when the coin operator depends on the position of the walker and is given by a certain i.i.d. random process, the phenomenon of Anderson localization takes place in its dynamical form. When the coin operator depends on the time variable only and is determined by an i.i.d. random process, the averaged motion is known to be diffusive and we compute the diffusion constants for all moments of the position.

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One-dimensional quantum walk with unitary noise

Cited by 64 publications

References 26 publications

Non-equilibrium transition from dissipative quantum walk to classical random walk

Non-equilibrium transition from dissipative quantum walk to classical random walk

Entangling power of disordered quantum walks

Dynamical Localization of Quantum Walks in Random Environments

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