Dark states of quantum search cause imperfect detection

Abstract

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 det that the particle is eventually detected in some target state, for example, on a node r 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 det in terms of the energy eigenstates which is generically τ independent. We find that disorder in the underlying Hamiltonian renders perfect detection, P det = 1, and then expose the role of symmetry with respect to suboptimal detection. Specifically, we give a simple upper bound for P 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.

Uncertainty Relation between Detection Probability and Energy Fluctuations

“…However, there is no symmetry between dark and bright states, in the sense that, every system has at least one bright state , but not every system necessarily has a dark state. As we showed in [ 25 ], totally bright systems have non-degenerate energy levels and all the energy eigenstates have a finite overlap with the detected state. Still, let us assume that we find a state , which is dark, we can then apply nearly the same strategy as before to find an upper bound for .…”

“…A bright state is an initial condition that is eventually detected with probability one and , while a dark state is never detected, and . Following [ 24 , 25 ], it is not difficult to show that if is a normalised bright state then is bright as well. We will soon present simple arguments to explain this statement, but first let us point out its usefulness.…”

“…Less well known is that we may construct a complete set of stationary eigenstates of H which are either bright or dark [ 25 ]. In [ 25 ], we presented a formula for the stationary dark and bright states, making this statement more explicit, but for now all we need to know is that the finite Hilbert space can be divided into these two orthogonal subspaces denoted (bright) and (dark). Let be a specific stationary dark state, and j is an index enumerating this family of states, and the corresponding energy.…”

“…Clearly here we assume that the dark state is not a stationary state of the system, since otherwise is not defined. Analogous to Equation ( 6 ), the detection probability is [ 25 ] Here the summation is over a basis of the subspace . Since all the terms in the sum are clearly non-negative For simplicity, assume that .…”

“…A formal solution for the detection probability was found in [ 8 , 24 ] and obtained explicitly for a few examples [ 25 ]. Since is non-trivial, as compared with its classical counterpart, we will find its bound.…”