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arXiv:1402.5092 · Physical Review A

Entangling power of disordered quantum walks

Rafael Vieira, Edgard P. M. Amorim, Gustavo Rigolin

Published Apr 9, 2014• 54 citations
arXivHTMLPDF

Abstract

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 (QRWs). Disorder is modeled by adding another random feature to QRWs, 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 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; and 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.

DOI
10.1103/physreva.89.042307

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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 (QRWs). Disorder is modeled by adding another random feature to QRWs, 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 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; and 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.
We show that the entanglement between the internal (spin) and external (position) degrees of freedom of a qubit in a random (dynamically disordered) one-dimensional discrete time quantum random walk (QRW) achieves its maximal possible value asymptotically in the number of steps, outperforming the entanglement attained by using ordered QRW. The disorder is modeled by introducing an extra random aspect to QRW, a classical coin that randomly dictates which quantum coin drives the system's time evolution. We also show that maximal entanglement is achieved independently of the initial state of the walker, study the number of steps the system must move to be within a small fixed neighborhood of its asymptotic limit, and propose two experiments where these ideas can be tested.
We show that the entanglement between the internal (spin) and external (position) degrees of freedom of a qubit in a random (dynamically disordered) one-dimensional discrete time quantum random walk (QRW) achieves its maximal possible value asymptotically in the number of steps, outperforming the entanglement attained by using ordered QRW. The disorder is modeled by introducing an extra random aspect to QRW, a classical coin that randomly dictates which quantum coin drives the system's time evolution. We also show that maximal entanglement is achieved independently of the initial state of the walker, study the number of steps the system must move to be within a small fixed neighborhood of its asymptotic limit, and propose two experiments where these ideas can be tested.
We investigate one-dimensional discrete-time quantum walk on the line where the links between neighboring sites are randomly broken. Two link-broken ways, static percolation and dynamical percolation, are considered. The former means that the broken links are fixed in position space at each time step, while the latter is that broken links are varied with time step. Our attention focuses on the effects of these disorders on two physical quantities, the probability distribution and the entanglement between the coin degree of freedom and position degree of freedom. Choosing Hadamard coin operator and assuming the walker to start from the position eigenstate|0〉and attach itself to a coherent coin state 1/√2 (|↑〉+ i|↓〉), we give the statistical average results after making numerical calculations many times. The choices of coin operator and initial state, resulting in a symmetric probability distribution about origin in the ideal case, is helpful in comparing with different cases in different disorder strengths. It is shown that the probability distribution of static percolation quantum walk can change from a coherent behavior at short time to Anderson localization at longer time, while the dynamical percolation quantum walk can change to a classical diffusive behavior. With the decrease of the percolation probability, these transitions become faster. The entanglement for ideal case without disorder reaches a constant value after a short time evolution. The static percolation makes the entanglement less than that of ideal case and fluctuate irregularly around a certain value. The situation is very different for the dynamical percolation:the entanglement increases smoothly with the time step and can exceed the constant value in the ideal case at some time. Both of entanglements for two types of percolations decrease with reducing percolation probability. As a striking characteristic, the entanglement in dynamical case can tend to maximum regardless of percolation probability in long time limit, while the static case cannot. In the model for our study, the randomized unitary operations, induced by the static and dynamical percolations, can lead to some noticeable effects on the transport and entanglement of discrete time quantum walk. The results about the interplay between disorder and entanglement not only assist quantum information processing, but also give more options to further explore and understand disorder physical processes in nature.
We study the entanglement between the internal (coin) and the external (position) degrees of freedom in the dynamic and the static deterministic aperiodic quantum walks (QWs). For the dynamic (static) aperiodic QWs, the coin depends on the time (position) and takes two coins C ( α ) and C ( β ) arranged in the two classes of generalized Fibonacci (GF) and the Thue–Morse (TM) sequences. We found that for the dynamic QWs, the entanglement of three kinds of the aperiodic QWs are close to the maximal value, which are all much larger than that of the homogeneous QWs. Further, the first class of GF (1st GF) QWs can achieve the maximum entangled state, which is similar to that of the dynamic disordered QWs. And the entanglement of 1st GF QWs is greater than that of the TM QWs, being followed closely by the entanglement of the second class of GF (2nd GF) QWs. For the static QWs, the entanglement of three kinds of the aperiodic QWs are also close to the maximal value and 1st GF QWs can achieve the maximum entangled state. The entanglement of the TM QWs is between 1st GF QWs and 2nd GF QWs. However, the entanglement of the static disordered QWs is less than that of three kinds of the aperiodic QWs. This is different from those of the dynamic QWs. From these results, we can conclude that the dynamic and static 1st GF QWs can also be considered as maximal entanglement generators.
Quantum Walk (QW) has very different transport properties to its classical counterpart due to interference effects. Here we study the discrete-time quantum walk (DTQW) with on-site static/dynamic phase disorder following either binary or uniform distribution in both one and two dimensions. For one dimension, we consider the Hadamard coin; for two dimensions, we consider either a 2-level Hadamard coin (Hadamard walk) or a 4-level Grover coin (Grover walk) for the rotation in coin-space. We study the transport properties e.g. inverse participation ratio (IPR) and the standard deviation of the density function (σ) as well as the coin-position entanglement entropy (EE), due to the two types of phase disorders and the two types of coins. Our numerical simulations show that the dimensionality, the type of coins, and whether the disorder is static or dynamic play a pivotal role and lead to interesting behaviors of the DTQW. The distribution of the phase disorder has very minor effects on the quantum walk.
We investigate the relation between the ballistic spreading and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following a delta-like (local) and Gaussian distributions. We find out that the knowledge of long-time expected position together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while the expected position with the quantum coin phases give it for highly delocalized states.