arXiv subjects

Itay Mualem

Publications and source records attributed to Itay Mualem.

4 recordsLinked to original sources

arXiv2021

Uncertainty relation between detection probability and energy fluctuations

A classical random walker starting on a node of a finite graph will always reach any other node since the search is ergodic, namely it is fully exploring space, hence the arrival probability is unity. For quantum walks, destructive interference may induce effectively non-ergodic features in such search processes. Under repeated projective local measurements, made on a target state, the final detection of the system is not guaranteed since the Hilbert space is split into a bright subspace and an orthogonal dark one. Using this we find an uncertainty relation for the deviations of the detection probability from its classical counterpart, in terms of the energy fluctuations.

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arXiv2020

Dark states of quantum search cause imperfect detection

We consider a quantum walk where a detector repeatedly probes the system with fixed rate $1/τ$ until the walker is detected. This is a quantum version of the first-passage problem. We focus on the total probability, $P_{\mathrm{det}}$, that the particle is eventually detected in some target state, for example on a node $r_{\mathrm{d}}$ on a graph, after an arbitrary number of detection attempts. Analyzing the dark and bright states for finite graphs, and more generally for systems with a discrete spectrum, we provide an explicit formula for $P_{\mathrm{det}}$ in terms of the energy eigenstates which is generically $τ$ independent. We find that disorder in the underlying Hamiltonian renders perfect detection: $P_{\mathrm{det}}=1$, and then expose the role of symmetry with respect to sub-optimal detection. Specifically, we give a simple upper bound for $P_{\mathrm{det}}$ that is controlled by the number of equivalent (with respect to the detection) states in the system. We also extend our results to infinite systems, for example the detection probability of a quantum walk on a line, which is $τ$-dependent and less than half, well below Polya's optimal detection for a classical random walk.

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arXiv2019

Uncertainty and symmetry bounds for the quantum total detection probability

We investigate a generic discrete quantum system prepared in state $|ψ_\text{in}\rangle$, under repeated detection attempts aimed to find the particle in state $|d\rangle$, for example a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability $P_\text{det}$ is unity. Due to destructive interference, one may find initial states $|ψ_\text{in}\rangle$ with $P_\text{det}<1$. We first obtain an uncertainty relation which yields insight on this deviation from classical behavior, showing the relation between $P_\text{det}$ and energy fluctuations: $ ΔP \,\mathrm{Var}[\hat{H}]_d \ge | \langle d| [\hat{H}, \hat{D}] | ψ_\text{in} \rangle |^2$ where $ΔP = P_\text{det} - |\langleψ_\text{in}|d\rangle |^2$, and $\hat{D} = |d\rangle\langle d|$ is the measurement projector. Secondly, exploiting symmetry we show that $P_\text{det}\le 1/ν$ where the integer $ν$ is the number of states equivalent to the initial state. These bounds are compared with the exact solution for small systems, obtained from an analysis of the dark and bright subspaces, showing the usefulness of the approach. The upper bounds works well even in large systems, and we show how to tighten the lower bound in this case.

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arXiv2019

Quantum total detection probability from repeated measurements II. Exploiting symmetry

A quantum walker on a graph, prepared in the state $| ψ_{\rm in} \rangle$, e.g. initially localized at node $r_{\rm in}$, is repeatedly probed, with fixed frequency $1/τ$, to test its presence at some target node $r_{\rm d}$ until the first successful detection. This is a quantum version of the first-passage problem. We investigate the total detection probability $P_{\rm det}$, i.e. the probability to eventually detect the particle after an arbitrary number of detection attempts. It is demonstrated that this total detection probability is less than unity in symmetric systems, where it is possible to find initial states which are shielded from the detector by destructive interference, so-called dark states. The identification of physically equivalent initial states yields an upper bound for $P_{\rm det}$ in terms of the reciprocal of the number $ν$ of physically equivalent states. The relevant subgroup of the system's symmetry operations is found to be the stabilizer of the detection state. Using this, we prove that all bright, i.e. surely detectable, states are symmetric with respect to the stabilizer. This implies that $P_{\rm det}$ can be obtained from a diagonalization of the "symmetrized" Hamiltonian, instead of having to find all eigenstates of the Hamiltonian.

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