PHYSICAL REVIEW RESEARCH 2, 023392 (2020) Uncertainty and symmetry bounds for the quantum total detection probability

We investigate a generic discrete quantum system prepared in state |ψin〉 under repeated detection attempts, aimed to find the particle in state |d〉, for example, a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability Pdet is unity. Due to destructive interference, one may find initial states |ψin〉 with Pdet < 1. We first obtain an uncertainty relation which yields insight on this deviation from classical behavior, showing the relation between Pdet and energy fluctuations: P Var[Ĥ ]d |〈d|[Ĥ, D̂]|ψin〉|, where P = Pdet − |〈ψin|d〉| and D̂ = |d〉〈d| is the measurement projector. Secondly, exploiting symmetry we show that Pdet 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 bound works well even in large systems, and we show how to tighten the lower bound in this case.

The exact solution relies on decomposing the Hilbert space into mutually orthogonal dark and bright subspaces. These are examples of Zeno subspaces [41,42,45], a dynamical separation of the total Hilbert space, which usually appears in the presence of singular coupling or in rapidly measured systems. Here, they are relevant for arbitrary detection frequencies possibly far away from the regime of the quantum Zeno effect [46,47]. This formal solution requires a full diagonalization of the Hamiltonian, involving considerable effort. Therefore, we also present bounds on P det , that give physical insight into the problem. The lower bound is an uncertainty relation and the upper bound exploits symmetry.
Heisenberg's uncertainty relation is probably the most profound signature of quantum reality's deviations from classical Newtonian mechanics [48]. Here we show something very different: how P det of a quantum walk deviates from the corresponding probability of detecting a classical random walk, which is unity. Our uncertainty relation connects this deviation with energy fluctuations and the commutator ofĤ and the measurement projector. It follows from the collapse postulate.
Symmetry and degeneracy play an important part in the physics of dark states and are a crucial mechanism leading to P det < 1 [3]. Consider an initial state which is a superposition of two energy eigenstates, |ψ in = N ( d|E |E − d|E |E ). When |E and |E belong to the same energy level, i.e., H |E = E |E andĤ |E = E |E , the time evolution of |ψ in is a simple phase factor e −itE (hereh = 1). It follows that d|e −itĤ |ψ in = 0 forever. Hence this is a dark state. Importantly, degeneracy is a signature ofĤ 's symmetry, so dark states are deeply connected to the symmetry of the problem. Below we exploit this to find a simple bound on P det .
Model. We consider quantum systems with discrete states, e.g., quantum walks on finite graphs. The system is initially in state |ψ in . We use projective stroboscopic measurements at times τ, 2τ, . . . in an attempt to detect the particle in state |d ; see Fig. 1 and Refs. [5,12,13]. The detection could be performed on a node of the graph, though any state |d is acceptable. Between the measurement attempts the evolution is unitary, described withÛ (τ ) = exp(−iτĤ ). The string of measurements yields a sequence no, no, . . . and in the nth attempt a yes. The time nτ marks the first detected arrival time in state |d . In some measurement sequences, the particle FIG. 1. A quantum walker resides on the nodes of a graph and moves unitarily along its edges, here a ring (left). Every τ time units a detector tests whether the particle is at node |d collapsing the wave function (right). The first successful detection attempt (click) defines an arrival time and stops the protocol. P det is the probability that the detector clicks at all. Here, the initial state is localized.
is not detected at all (n = ∞). Each measurement satisfies the collapse postulate [49]: if the wave function is |ψ right before measurement, the amplitude of detection is d|ψ . Successful detection terminates the experiment. Unsuccessful detection zeros the amplitude d|ψ , the wave function is renormalized, and the unitary evolution continues until the next measurement. Mathematically, the measurement is described by the projectorD = |d d| [see Eq. (8) below]. Repeating this protocol many times, P det is the fraction of runs in which the detector clicked yes at all. It is closely related to the Pólya number and the recurrence of the walk [5,29,50]. If P det < 1, the mean n diverges and the corresponding search problem is ill posed. Furthermore, the time-of-arrival problem [51][52][53][54][55][56] can also be tackled using stroboscopic measurements. General quantum walks have been experimentally realized photonically [21,28,57], with trapped ions [23] and in optical lattices [22,58]. Reference [59] reported the measurement of P det in a photonic walk.
Uncertainty relation. Reference [1] showed how the detector's action separates the Hilbert space of a finite system into a "bright" and a "dark" subspace: H = H B ⊕ H D . Any initial condition within the bright/dark subspace is detected with probability one/zero, respectively. We present a proof of this fundamental result in Ref. [60]. For the dark, as for the bright space, we can find a basis in terms of the eigenstates ofĤ , denoted {|E B j } and {|E D j }, respectively. The subspaces are thus orthogonal and invariant underĤ andD.
Reference [5] [and Eq. (13) below] showed that the particular state |ψ in = |d is bright. Since |d is bright, it is orthogonal to every dark state, i.e., E D j |d = 0. Therefore, To obtain a useful bound, we create a pair of orthonormal bright states from |d and H s |d : The normalization 2 is related to the energy fluctuations in the detected state. As each term in the sum Eq. (1) is non-negative, a lower bound is reached by omitting some of the bright states: We now define the difference between the probability of detection after repeated measurements from the probability of detection at preparation: Using Eqs. (2) and (3) we find Figure 2 shows this uncertainty bound P det P unc det for several graphs. Some remarks are in place. First, after the successful detection, the system is in its final state |ψ f = |d . 1 This means that we may rewrite the uncertainty principle, say for s = 1, as The fluctuations of energy are actually in the final state of the particle. So, Eqs. (5) and (6) are relations between the initial condition and the finally selected state. Importantly, after the system is projected into its final state, and the detector turned off, the fluctuation of energy Var[Ĥ ] ψ f is a constant of motion.
The energy measurement can be made at any time after the detection. Notably, Eqs. (5) and (6) do not depend on τ .
Path-counting approach. We consider the standard quantum walk withĤ =Â, whereÂ is the adjacency matrix of some graph. Hence there are no on-site energies, and all bonds in the system are identical, namelyĤ ii = 0 andĤ i j = 1 if site i and j are connected and zero otherwise. We are interested in a particle starting on vertex |ψ in = |r and the detection on another vertex |d . Notice that d|Ĥ s |r = N r→d (s) is the number of paths of length s starting on |r and ending at |d . Then using Eq. (5), we find We must choose s here larger or equal to the distance ξ between |r and |d ; otherwise, one gets the trivial P det 0.
Upper bound from symmetry. To complement our lower bound, we use a different approach. The detection probability is by definition P det = ∞ n=1 |ϕ n | 2 , where ϕ n is the amplitude of first detection at the nth attempt [12]. This can be expressed as [8] Reading this right to left, we see that ϕ n is given by the initial condition, followed by steps combining unitary evolution and attempted detection, of which the final, nth detection is successful. It is crucial for our discussion that ϕ n is linear with respect to |ψ in , so it obeys the superposition principle. We are interested in the total detection probability starting from node |r and detecting on another |d . In the system we have a set {|r j } ν j=1 of ν states, which are equivalent to |r and For the lower bound we took s equal to the distance between initial and detection node. In many cases lower and upper bound coincide and P det is determined without tedious calculations. For starred graphs the shell-state method gives exact results. |r 1 = |r . This means that each |r j gives the same amplitude on |d for all times; mathematically d|Û (t )|r = d|Û (t )|r j for 1 j ν. Physically, it is often easy to identify all the states |r j using symmetry arguments. However, even if we miss some of them, the bound derived below is useful though not optimal.
From the equivalent states |r j , we construct a normalized auxiliary uniform state Now, by definition of the detection amplitudes and the equivalence of all {|r j } ν j=1 , we find ϕ n (r j ) = ϕ n (r). It follows from superposition, Eq. (8), that ϕ n (AUS) = √ νϕ n (r).
We now square both sides of this equation, sum over n, and use the obvious P det (AUS) 1 to find the sought after P det (r): Figure 2 shows the upper bound for several graphs. Reference [60] will show that ν = dim S d |ψ in can be determined from the stabilizer S d , the group of all symmetry operations that commute withÛ (τ ) andD.
Ring. Consider a ring with an even number L of identical sites, with localized initial and detection states. The detection site and its opposing site are unique, such that ν = 1. For all other sites, we have one equivalent partner found by reflection symmetry; hence ν = 2. We derived a lower bound from Eq. (7) with s = ξ < L/2 [61]: where ξ is the distance between initial and detection site and where the second binomial must be omitted for odd ξ . For nearest neighbors ξ = 1, we find the exact result P det (d ± 1) = 1/2 from sandwiching. Consider now the detection of the ring's ground state |d = L r=1 |r / √ L. Since |d is a uniform state over the whole ring, each localized initial condition is physically equivalent. The upper bound gives P det (r) 1/L, which is also equal to the exact result.
The exact solution is.
Here the eigenstates |E l,m and the energies E l are defined as usual withĤ |E l,m = E l |E l,m , where l, m are quantum numbers and m = 1, . . . , g l , so g l is the degeneracy of energy level E l . The sum runs over all l for which the denominator does not vanish. Let us briefly outline the derivation of this formula and then discuss its consequences. Sketch of proof. Equation (13) follows directly from the decomposition of the Hilbert space into dark and bright components. Technically, we use the energy basis and consider an energy sector {|E l,m } g l m=1 . This sector yields either one bright state (and g l − 1 dark states) or none at all (and g l dark states). If E l,m |d = 0 for all 1 m g l then clearly all the g l states are dark and the sector has no bright state. Otherwise, there is only one bright state, namely |E B l = N B l g l m=1 |E l,m E l,m |d with appropriate normalization. We need to demonstrate (i) that indeed |E B l is bright and (ii) that the remaining states are dark. The latter is easily shown. Consider, for example, g l = 2. We have |E B l = N B l ( E l,1 |d |E l,1 + E l,2 |d |E l,2 ). It is easy to see that |E D l = N D l ( E l,2 |d |E l,1 − E l,1 |d |E l,2 ) is dark as d|E D l = 0 and E B l |E D l = 0. Similar arguments hold for g l > 2 [60], showing that P det (E B l ) = 1 is involved. For that aim, we analyzed in Ref. [60] the eigenvalues of the operator (1 −D)Û (τ ), which determine the evolution of the measurement process. These eigenvalues lie inside the unit disk. This fact is used to show that |E B l is detected with probability one. Once we have all the bright states, we use Eq. (1) to obtain Eq. (13).
Features of Eq. (13). The exact formula exhibits some remarkable properties. We easily see that the initial states |ψ in = |d and |ψ in = N sĤ s |d yield P det = 1. Hence, as claimed earlier, these states are bright. Furthermore, the total detection probability is τ independent. The only exception, not considered here in depth, is when |E l − E l |τ = 0 mod 2π , for some pairs of energy levels. They are a unique feature of the stroboscopic detection protocol. These special τ 's are isolated, but still of interest since the statistics exhibit gigantic fluctuations and discontinuous behavior in their vicinity [5,62,63], related to partial revivals of the state function. More importantly, Eq. (13)'s τ independence ensures its general validity, even if one tampers with the detection protocol, for example, by sampling with a Poisson process. The reason is that any initial state |ψ in starting in the dark space has zero overlap with the detected state for t 0. No measurement protocol can detect this state. Figure 2 compares our main results, the uncertainty relation (5), and the symmetry bound (11), with the exact result (13) for small systems where the results are apparent to the eye. In some cases both bounds coincide and thus determine the exact results from elementary calculations instead of full diagonalization ofĤ . Experimental measurement of P det as in Ref. [59] requires the system size to be small compared to the achievable decoherence-free observation time. This motivated our focus on small systems.
a. Large systems. We start discussing our results in large systems with the example of the B-dimensional hypercube. Here, each node is represented by a string of B bits, e.g., |01011 · · · 0 , and each transition corresponds to flipping one bit. We detect on node |d and start at any node |r ξ with ξ bits different from |d , i.e., ξ is the Hamming distance between the nodes. Remarkably, the upper bound works perfectly here, coinciding with the exact result P det (r ξ ) = 1/ν = 1/ B ξ = P sym det . What about the lower bound? Equation (7) yields [61]: The second term in the denominator has to be omitted when ξ is odd. Several strategies to improve the lower bound are compared in the Supplemental Material (SM) [61]. Exploiting the shell structure of a graph leads to a much better lower bound. b. Shell-state method. When the bright state |d is localized, H |d is supported only on nearest neighbors of |d . We call those nodes the "first shell." Similarly,Ĥ 2 |d is a bright state supported on the next-nearest neighbors, the second shell, as well as |d itself. Since the sth shell is only connected to the (s ± 1)th shells, we can construct a useful bright state |β ξ by the following strategy: we start with the zeroth and first shell states |β 0 := |d and |β 1 :=Ĥ |d . Each subsequent state |β s is obtained from orthogonalizingĤ |β s to |β s and |β s−1 . The procedure is terminated when s = ξ and yields a state with large overlap in the ξ th shell. A lower bound is obtained from P det P shell det := | β ξ |ψ in | 2 . For our nearest-neighbor hoppingĤ on the hypercube, |β ξ = r ξ |r ξ / √ ν, where ν = B ξ . This is the relevant AUS for r ξ . Hence lower and upper bound coincide: P shell det = 1/ν P det 1/ν = P sym det . We also successfully applied the shell-state method to a system with two loops and to one with nonuniform coupling constants [61]. In many situations, the shell-state method gives the exact, simply computed result (e.g., those starred in Fig. 2). As a rule of thumb, P shell det is exact, when |d sits in a symmetry center of the system. More precisely, it is exact when the sequence {|β s } turns out to be fully orthogonalized. We will elaborate on this in [60]. Even when it is not exact, it gives a bound from 2ξ − 1 orthogonalization operations. This is a huge advantage compared to the minimally necessary ξ (ξ + 1)/2 operations in a full Gram-Schmidt procedure [16].
c. Disordered systems. Symmetries lead to tight, possibly exact bounds and remove the necessity of diagonalizingĤ . In their absence, when irregularity has broken all degeneracy, Eq. (13) provides the value for P det . Provided the system's spectrum is nondegenerate and |d has overlap with all eigenstates, the exact result predicts classical behavior: P det = 1. Thus the bounds are not necessary and their performance of rather academic interest. Generically, one obtains ν = 1 giving P det = P sym det = 1 from Eqs. (11) and (13). Assuming the eigenstates are extended and span the whole system, one expects that the uncertainty bound behaves like P unc det ∼ 1/D. In the SM, this is demonstrated for random Hamiltonians from the Gaussian unitary ensemble [61]. We also show numerically that the shell-state bound performs polynomially better P shell det ∼ D −0.75 . To conclude, we have used symmetry and an uncertainty principle to find upper and lower bounds on the detection probability. These bounds show a symmetry-induced restriction to well-posed quantum search which requires P det = 1. We showed that P det is surprisingly almost independent of τ . Our results also apply to discrete-time quantum walks as the symmetries and spectral properties can equivalently be obtained from the evolution operator. 2 Furthermore, they are relevant to non-Hermitian models [7,16,42,55]. We presented the shell-state method, which improves on the uncertainty principle. Our methods perform well in systems with a certain degree of symmetry; exactly those from which one expects a significant quantum computational speedup. The symmetry and uncertainty bounds often allow one to avoid the Hamiltonian's diagonalization and provide a physical interpretation to the value of P det . For very irregular, nondegenerate systems, the answer is immediately provided by the exact formula, making diagonalization again obsolete.