Flux control of measurement back-action and Leggett-Garg correlations in chiral quantum walks

Abstract: Gauge-invariant fluxes control interference in chiral continuous-time quantum walks. We investigate how they affect sequential measurements at a single vertex, using a dichotomic observable that distinguishes return to that vertex from occupation of its complement. For a walker initially localized at the measured vertex, the complete two-time statistics, including measurement back-action and Leggett--Garg correlators, are determined exactly by the return amplitude, connecting temporal correlations to the local spectral measure and gauge-invariant closed-walk interference. At short times, the leading disturbance is independent of the Peierls phases, whereas flux sensitivity enters at higher orders through interference among closed walks. We further identify a graph-independent sufficient mechanism for saturating the Lüders bound: flux can reduce the rooted dynamics to a balanced two-dimensional Krylov subspace with equal spectral weights, yielding a constructive flux-engineering criterion for attaining the Lüders bound of $3/2$ at finite times. The mechanism is realized exactly on a two-flux diamond graph, where destructive interference renders additional rooted modes dark and the local back-action depends on relative combinations of the two independent fluxes. For flux-threaded cycles, an exact winding-number expansion reveals a parity-dependent onset: the leading flux contrast occurs generically at order $t<sup>N$ for even cycles and $t<sup>{2N}$ for odd cycles. Across the cycles examined, flux can either enhance the maximal Leggett--Garg violation or shift strong violations to earlier measurement times, with half flux driving the four-site cycle to the Lüders bound. These results establish gauge-invariant flux as a resource for engineering local measurement back-action and temporal quantum correlations.

• The paper develops an exact return-amplitude framework showing that sequential measurement statistics and back-action in finite chiral quantum walks are determined by rooted return amplitudes and gauge-invariant cycle fluxes.
• Flux sensitivity emerges through rooted closed-walk interference, with leading observable effects at order t^N for even cycles and t^{2N} for odd cycles, while graph symmetries can delay or cancel lower-order contributions.
• Balanced two-dimensional rooted Krylov dynamics can saturate the Lüders bound L₃ = 3/2, and flux can accelerate strong Leggett–Garg violations—by about 3.1 times for C₁₀—without necessarily breaking time-reversal symmetry.

Overview

This paper develops an exact theory of sequential single-vertex measurements in chiral continuous-time quantum walks (CTQWs), where Peierls phases realize synthetic gauge fields whose only physical content is the set of gauge-invariant cycle fluxes $\boldsymbol{\Phi}$ (2608.12519). The central object is a dichotomic return observable $Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$ measured via a Lüders instrument at one vertex, applied to a walker initially localized there. Two complementary diagnostics are analyzed: the Kolmogorov inconsistency $K_{\nu,\boldsymbol{\Phi}}(s,t)$, quantifying measurement back-action on the final return probability, and the three-time Leggett–Garg functional $L_3$. The main achievements are (i) an exact reduction of all two-time statistics to the rooted return amplitude $A_\nu(t;\boldsymbol{\Phi})$; (ii) a parity-dependent short-time onset of flux sensitivity organized by gauge-invariant sums of rooted closed walks; and (iii) a graph-independent sufficient criterion—balanced two-dimensional rooted Krylov dynamics—for saturating the Lüders bound $L_3 = 3/2$, engineered by flux.

Exact statistics from the rooted return amplitude

For any finite graph and time-independent Hamiltonian, the complete joint distribution $P(q_2,t_2;q_1,t_1)$ of dichotomic outcomes is determined by $A_\nu(t)$. Three of the four joint probabilities factorize through return probabilities; the nontrivial branch is $P_{-+} = |A_\nu(t_2) - A_\nu(\tau)A_\nu(t_1)|^2$, an interference quantity between uninterrupted return over $t_2$ and coherent concatenation over the two intervals. Consequently,

$Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$0

i.e., back-action measures exactly the mismatch between incoherent and coherent concatenation of return amplitudes. All quantities are invariant under simultaneous reversal of all fluxes, so the protocol cannot distinguish $Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$1 from $Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$2; however, in multicycle graphs it can detect relative-flux configurations such as partial reversals, since these are not related by global reversal. This distinction matters: flux sensitivity here does not certify global time-reversal-symmetry breaking, only relative interference among closed walks exploring different cycles.

A structurally important exact identity connects the two diagnostics under the equally spaced protocol $Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$3:

$Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$4

with equality structure showing that $Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$5 implies nonzero marginal back-action for this localized preparation. The paper is careful to state that this implication depends on the deterministic outcome at $Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$6 and is not a universal NSIT–LGI relation; moreover, back-action is necessary but not sufficient for violation, since the disturbance must be positive and must exceed $Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$7.

Short-time onset: rooted closed-walk cancellations

The short-time expansion of $Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$8 contains only even powers of $Q_\nu = 2|\nu\rangle\langle\nu| - \mathbb{I}$9. The leading term $K_{\nu,\boldsymbol{\Phi}}(s,t)$0 depends solely on the local coupling strength $K_{\nu,\boldsymbol{\Phi}}(s,t)$1 (the vertex degree for unit-modulus hopping), hence is completely phase-insensitive. Flux sensitivity first enters at fourth order through the centered moment $K_{\nu,\boldsymbol{\Phi}}(s,t)$2, interpreted as a sum over rooted closed walks of length four winding around cycles. Crucially, individual walk contributions being flux-dependent does not guarantee observable flux dependence: parity, graph automorphisms, or destructive interference can cancel entire orders. The observable onset occurs at the lowest even order where the complete gauge-invariant combination survives.

The diamond graph (two triangles sharing an edge, cycle-space dimension two) illustrates this concretely. An isolated triangle has $K_{\nu,\boldsymbol{\Phi}}(s,t)$3, flux-independent, delaying its response to sixth order; embedded in the diamond, the cancellation is lifted and the generic response is quartic, with coefficients depending on both the root and relative combinations like $K_{\nu,\boldsymbol{\Phi}}(s,t)$4. For example, partial flux reversal produces a contrast proportional to $K_{\nu,\boldsymbol{\Phi}}(s,t)$5. Notably, when $K_{\nu,\boldsymbol{\Phi}}(s,t)$6 the outer quadrilateral flux vanishes yet the quartic contrast remains strictly positive at the shared root—the onset is driven by the triangle fluxes themselves, not by the shortest accessible even cycle alone.

Balanced rooted Krylov dynamics and Lüders-bound saturation

The paper derives a graph-independent sufficient mechanism for reaching the maximal quantum value $K_{\nu,\boldsymbol{\Phi}}(s,t)$7 with a fixed dichotomic instrument. If the rooted Krylov space $K_{\nu,\boldsymbol{\Phi}}(s,t)$8 has dimension two, then $K_{\nu,\boldsymbol{\Phi}}(s,t)$9, saturating the bound if and only if the two rooted spectral weights are equal. This condition is equivalent to the operator criterion

$L_3$0

which defines a constructive leakage functional $L_3$1 whose vanishing certifies balanced dynamics. The paper emphasizes that dimension two alone is insufficient—equal weights are equally essential—and that the criterion is not necessary in full generality, since higher-dimensional rooted spectra may approach the bound through special phase relations. Unlike multilevel enhancement schemes that raise the quantum bound above $L_3$2 via finer projective resolution, here the measurement resolution is fixed and flux merely reshapes the rooted spectral measure.

Two exact realizations are given. On the diamond graph at the outer vertex with relative fluxes $L_3$3, destructive interference renders additional modes dark, yielding $L_3$4 and exact saturation at $L_3$5. At zero flux the same vertex gives $L_3$6, so flux enhances the unrestricted maximum by approximately $L_3$7—a change of spectral support, not merely a relocation of optimal timing. For $L_3$8 at half flux, degeneracies collapse the rooted measure to two equally weighted eigenvalues with identical consequences. Both cases demonstrate that sequential return statistics probe rooted spectral equivalence rather than graph topology per se.

Flux-threaded cycles and parity-dependent onset

For $L_3$9 threaded by uniform flux $A_\nu(t;\boldsymbol{\Phi})$0, translation invariance yields an exact Bessel-function expansion in winding sectors $A_\nu(t;\boldsymbol{\Phi})$1: $A_\nu(t;\boldsymbol{\Phi})$2. The leading winding contribution enters the amplitude at order $A_\nu(t;\boldsymbol{\Phi})$3, but its observability is parity dependent:

Cycle parity	Leading flux contrast in $A_\nu(t;\boldsymbol{\Phi})$4
Even $A_\nu(t;\boldsymbol{\Phi})$5	order $A_\nu(t;\boldsymbol{\Phi})$6, coefficient $A_\nu(t;\boldsymbol{\Phi})$7
Odd $A_\nu(t;\boldsymbol{\Phi})$8	order $A_\nu(t;\boldsymbol{\Phi})$9, coefficient $L_3 = 3/2$0

For odd cycles, $L_3 = 3/2$1 is purely imaginary, so linear interference with the real zero-winding sector cancels in probabilities. Additionally, odd cycles enjoy a half-flux symmetry making return statistics $L_3 = 3/2$2-periodic in $L_3 = 3/2$3 and exactly equivalent at $L_3 = 3/2$4 and $L_3 = 3/2$5. Finite-time analysis shows flux reorganizes the temporal structure of disturbance rather than rescaling it, with alternating enhancement/suppression regions (illustrated for $L_3 = 3/2$6).

On the Leggett–Garg side, the results split cleanly into strength versus accessibility. Half flux drives $L_3 = 3/2$7 to the Lüders bound. For $L_3 = 3/2$8 and $L_3 = 3/2$9 within finite windows, the optimized violation varies by less than one percent over the flux range; the dominant effect is temporal. Defining a hitting-time speedup $P(q_2,t_2;q_1,t_1)$0 for reaching a fraction $P(q_2,t_2;q_1,t_1)$1 of the zero-flux optimized excess, the representative case gives $P(q_2,t_2;q_1,t_1)$2: half flux relocates a strong violation from $P(q_2,t_2;q_1,t_1)$3 to $P(q_2,t_2;q_1,t_1)$4. A survey over $P(q_2,t_2;q_1,t_1)$5 finds speedups of order $P(q_2,t_2;q_1,t_1)$6–$P(q_2,t_2;q_1,t_1)$7 for $P(q_2,t_2;q_1,t_1)$8. Strikingly, the optimal flux for $P(q_2,t_2;q_1,t_1)$9 is $A_\nu(t)$0, which is itself time-reversal invariant, so the speedup stems from rearrangement of spectral gaps and recurrences rather than genuine symmetry breaking. The rooted Krylov counts ($A_\nu(t)$1 at $A_\nu(t)$2 generic flux) confirm that neither dimension nor violation strength is determined by the count of distinct rooted eigenvalues alone; commensurabilities and multiplicities matter.

Limitations and open questions

The paper states several caveats explicitly. The implication $A_\nu(t)$3 holds only for the localized preparation with deterministic first outcome. Global flux-reversal invariance limits the protocol's ability to witness time-reversal-symmetry breaking directionally. The balanced-Krylov criterion guarantees saturation but is sufficient rather than necessary, and its realization requires specific fine-tuned flux values. The temporal-speedup results are numerical, restricted to specific windows and thresholds, and their advantage under decoherence—which is known to constrain maximal Leggett–Garg violations—is not assessed; an explicit open-system analysis is deferred. Finally, whether Leggett–Garg violations connect to quantum Fisher information for the nonstationary localized preparation considered here, enabling flux estimation from local return statistics, is left as an open question.

Conclusion

The paper establishes gauge-invariant flux as a controllable parameter for local measurement back-action and temporal quantum correlations in chiral CTQWs, grounded in exact identities relating all two-time statistics to the rooted return amplitude and its spectral measure. Its principal analytical contributions are the even-power short-time structure with closed-walk-mediated flux onset (order $A_\nu(t)$4 versus $A_\nu(t)$5 by cycle parity), and the constructive balanced-rooted-Krylov criterion for Lüders-bound saturation realized exactly on the diamond graph and $A_\nu(t)$6 at half flux. Numerically, flux acts primarily on the timing rather than the magnitude of violations in larger cycles, achieving speedups near $A_\nu(t)$7 for $A_\nu(t)$8. The framework applies to arbitrary finite graphs and motivates extensions to decoherence robustness and flux metrology via local dichotomic readout.