Coin Degrees Of Freedom Research Articles

Multiparameter quantum metrology with discrete-time quantum walks

We address multi-parameter quantum estimation for one-dimensional discrete-time quantum walks and its applications to quantum metrology. We use the quantum walker as a probe for unknown parameters encoded on its coin degrees of freedom. We find an analytic expression of the quantum Fisher information matrix for the most general coin operator, and show that only two out of the three coin parameters can be accessed. We also prove that the resulting two-parameter coin model is asymptotically classical i.e. the Uhlmann curvature vanishes. Finally, we apply our findings to relevant case studies, including the simultaneous estimation of charge and mass in the discretized Dirac model.

Induced on-demand revival in coined quantum walks on infinite d -dimensional lattices

The study of recurrences and revivals in quantum systems has attracted a great deal of interest because of its importance in the control of quantum systems and its potential use in developing new technologies. In this work, we introduce a protocol to induce full-state revivals in a huge class of quantum walks on a $d$-dimensional lattice governed by a $c$-dimensional coin system. The protocol requires two repeated interventions in the coin degree of freedom. We also present a characterization of the walks that admits such a protocol. Moreover, we modify the quantity known as P\'olya number, typically used in the study of recurrences in classical random walks and quantum walks, to create a witness of the first revival of the walk.

Analysis of Decoherence in Linear and Cyclic Quantum Walks

We theoretically analyze the case of noisy Quantum walks (QWs) by introducing four qubit decoherence models into the coin degree of freedom of linear and cyclic QWs. These models include flipping channels (bit flip, phase flip and bit-phase flip), depolarizing channel, phase damping channel and generalized amplitude damping channel. Explicit expressions for the probability distribution of QWs on a line and on a cyclic path are derived under localized and delocalized initial states. We show that QWs which begin from a delocalized state generate mixture probability distributions, which could give rise to useful algorithmic applications related to data encoding schemes. Specifically, we show how the combination of delocalzed initial states and decoherence can be used for computing the binomial transform of a given set of numbers. However, the sensitivity of QWs to noisy environments may negatively affect various other applications based on QWs.

Attractors and asymptotic dynamics of open discrete-time quantum walks on cycles

Open quantum walks often lead to a classical asymptotic behavior. Here, we look for a simple open quantum walk whose asymptotic behavior can be non-classical. We consider a discrete-time quantum walk on n-cycle subject to a random coin-dependent phase shift at a single position. This finite system, whose evolution is described by only two Kraus operators, can exhibit all kinds of asymptotic behavior observable in quantum Markov chains: it either evolves towards a maximally mixed state, or partially mixed state, or tends to an oscillatory motion on an asymptotic orbit. We find that the asymptotic orbits do not have a product structure, therefore the corresponding states can manifest entanglement between the position and the coin degrees of freedom, even if the system started in a product state.

Experimental realization of a momentum-space quantum walk

We report on a discrete-time quantum walk that uses the momentum of ultra-cold rubidium-87 atoms as the walk space and two internal atomic states as the coin degree of freedom. Each step of the walk consists of a coin toss (a microwave pulse) followed by a unitary shift operator (a resonant ratchet pulse). We carry out a comprehensive experimental study on the effects of various parameters, including the strength of the shift operation, coin parameters, noise, and initialization of the system on the behavior of the walk. The walk dynamics can be well controlled in our experiment; potential applications include atom interferometry and engineering asymmetric walks.

Properties of distribution and entanglement in discrete-time quantum walk with percolation

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.

Time exponents of asymptotic entanglement of discrete quantum walk in one dimension

We report a systematic study of asymptotic behavior of entanglement between the position and coin degrees of freedom for a one-dimensional discrete quantum walk. Effects of coin bias, initial coin asymmetry and phase difference and non-locality of initial position state on the fluctuations and asymptotic value of entanglement are investigated by using Fourier approach. Fluctuations in entropy are found to die out as power law in time as t-α with α = 1/2,3/2 and 5/2 depending on the initial state parameters of the system.

Measurement-Induced Generation of Spatial Entanglement in a Two-Dimensional Quantum Walk with Single-Qubit Coin

One of the proposals for the exploitation of two-dimensional quantum walks has been the efficient generation of entanglement. Unfortunately, the technological effort required for the experimental realization of standard two-dimensional quantum walks is significantly demanding. In this respect, an alternative scheme with less challenging conditions has been recently studied, particularly in terms of spatial-entanglement generation [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. Here, we extend the investigation to a scenario where a measurement is performed on the coin degree of freedom after the evolution, allowing a further comparison with the standard two-dimensional Grover walk.

On the von Neumann entropy of certain quantum walks subject to decoherence

In this paper, we consider a discrete-time quantum walk on theN-cycle governed by the condition that at every time step of the walk, the option persists, with probabilityp, of exercising a projective measurement on the coin degree of freedom. For a bipartite quantum system of this kind, we prove that the von Neumann entropy of the total density operator converges to its maximum value. Thus, when influenced by decoherence, the mutual information between the two subsystems corresponding to the space of the coin and the space of the walker must eventually diminish to zero. Put plainly, any level of decoherence greater than zero forces the system to become completely ‘disentangled’ eventually.

Quantum walks on theN-cycle subject to decoherence on the coin degree of freedom

For a discrete time quantum walk (QW) on the N-cycle, allowing for decoherence on the coin, we derive a number of results, including an explicit formula for the position probability distribution. For a QW of this type, we show that the mixing behavior tends, in the long run, to a uniform distribution regardless of the initial state of the system and irrespective of the parity of the number of nodes N. These results confirm the findings of previous authors who arrived at similar conclusions through extensive numerical simulations. In particular, we infer that the mixing time M(epsilon) for the time-averaged probability distribution is of order no greater than O(N2/).

Quantum walk on the line with quantum rings

We propose a scheme to implement the one-dimensional coined quantum walk with electrons transported through a two-dimensional network of spintronic semiconductor quantum rings. The coin degree of freedom is represented by the spin of the electron while the discrete position of the walker corresponds to the label of the rings in one of the spatial directions in the network. We assume that Rashba-type spin-orbit interaction is present in the rings, the strength of which can be tuned by an external electric field. The geometry of the device, together with the appropriate spin-orbit interaction strengths, ensure the realization of the coin toss (i.e., spin flip) and the step operator.

Generic quantum walk using a coin-embedded shift operator

The study of quantum walk processes has been widely divided into two standard variants, the discrete-time quantum walk (DTQW) and the continuous-time quantum walk (CTQW). The connection between the two variants has been established by considering the limiting value of the coin operation parameter in the DTQW, and the coin degree of freedom was shown to be unnecessary [F. W. Strauch, Phys. Rev. A 74, 030301(R) (2006)]. But the coin degree of freedom is an additional resource which can be exploited to control the dynamics of the QW process. In this paper we present a generic quantum walk model using a quantum coin-embedded unitary shift operation ${U}_{C}$. The standard version of the DTQW and the CTQW can be conveniently retrieved from this generic model, retaining the features of the coin degree of freedom in both variants.