Conditional probability distributions of finite absorbing quantum walks

Kuklinski, Parker

doi:10.1103/physreva.101.032309

Cited by 11 publications

(6 citation statements)

References 36 publications

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“…Here we used the fact, well known from linear algebra, that the left and right eigenvalues are equal; hence they are denoted with ξ. These are complex numbers, and for the problem at hand, it was shown that these eigenvalues are all on the unit disk |ξ| ≤ 1 [60,61]. We will use normalized states…”

Section: The Survival Operatormentioning

confidence: 99%

Driving quantum systems with repeated conditional measurements

Liu,

Ziegler,

Kessler

et al. 2021

Preprint

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We investigate the effect of conditional null measurements on a quantum system and find a rich variety of behaviors. Specifically, quantum dynamics with a time independent H in a finite dimensional Hilbert space are considered with repeated strong null measurements of a specified state. We discuss four generic behaviors that emerge in these monitored systems. The first arises in systems without symmetry, along with their associated degeneracies in the energy spectrum, and hence in the absence of dark states as well. In this case, a unique final state can be found which is determined by the largest eigenvalue of the survival operator, the non-unitary operator encoding both the unitary evolution between measurements and the measurement itself. For a three-level system, this is similar to the well known shelving effect. Secondly, for systems with builtin symmetry and correspondingly a degenerate energy spectrum, the null measurements dynamically select the degenerate energy levels, while the non-degenerate levels are effectively wiped out. Thirdly, in the absence of dark states, and for specific choices of parameters, two or more eigenvalues of the survival operator match in magnitude, and this leads to an oscillatory behavior controlled by the measurement rate and not solely by the energy levels. Finally, when the control parameters are tuned, such that the eigenvalues of the survival operator all coalesce to zero, one has exceptional points that corresponds to situations that violate the null measurement condition, making the conditional measurement process impossible.

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Section: The Survival Operatormentioning

confidence: 99%

Driving quantum systems with repeated conditional measurements

Liu,

Ziegler,

Kessler

et al. 2021

Preprint

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“…We can notice that once a quantum walker penetrates into the interior, the situation for this quantum walker is the same as that of the absorption problem of quantum walks [14]. The detailed analysis on the absorption problem especially the spectral analysis can be seen, for example, [15]. Then, for a convergence time to the stationary state, such a spectral analysis on the truncated finite matrix of the time evolution operator of our model, which is non-unitary, will be useful.…”

Section: Where Is the Comfortable Place For Quantum Walkers On The Pa...mentioning

confidence: 97%

Comfortable place for quantum walkers on finite path

Yoshihiro¹,

Konno²,

Morioka³

et al. 2022

Preprint

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We consider the stationary state of a quantum walk on the finite path, where the sink and source are set at the left and right boundaries. The quantum coin is uniformly placed at every vertex of the path graph. At every time step, a new quantum walker penetrates into the internal from the left boundary and also some existing quantum walkers in the internal goes out to the sinks located in the left and right boundaries. The square modulus of the stationary state at each vertex is regarded as the comfortability for a quantum walker to this vertex in this paper. We show the weak convergence theorem for the scaled limit distribution of the comfortability in the limit of the length of the path.

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“…Note that since this quantum walk can be converted to a quantum walk with absorption walls, the problem is reduced to analysis on a finite matrix E M , which is obtained by picking up from the total unitary time evolution operator with respect to the perturbed region Γ M . See [16] for a precise spectral results on E M . This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

A Discontinuity of the Energy of Quantum Walk in Impurities

Higuchi

Komatsu²,

Konno

et al. 2021

Symmetry

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We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

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Conditional probability distributions of finite absorbing quantum walks

Cited by 11 publications

References 36 publications

Driving quantum systems with repeated conditional measurements

Driving quantum systems with repeated conditional measurements

Comfortable place for quantum walkers on finite path

A Discontinuity of the Energy of Quantum Walk in Impurities

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