Quantum Trajectory Sensing

• Quantum trajectory sensing is a family of methods that analyze path-dependent quantum evolution to extract information beyond static observables.
• It encompasses techniques such as qubit-array discrimination, continuous monitoring using stochastic master equations, and matched sensing in time–frequency domains.
• Applications range from precision metrology with entangled sensors to enhanced state recovery in non-Hermitian and open quantum systems.

Quantum trajectory sensing denotes a family of quantum-sensing paradigms in which a trajectory is either the object to be identified or the organizing variable of the measurement protocol. In the recent literature, the term is used in at least four technically distinct senses: the single-shot discrimination of an incident particle’s path through a qubit array; the reconstruction and exploitation of stochastic conditional state trajectories under continuous measurement; matched sensing of signal trajectories in the time–frequency plane; and the engineering of parameter loops in non-Hermitian control space for sensing and state transfer (Chin et al., 2024, Rossi et al., 2018, Hegde et al., 2024, Wu et al., 25 Mar 2026). Across these settings, the central theme is that trajectory-dependent quantum dynamics carries information that is not captured by static observables alone.

1. Conceptual scope and unifying structure

The literature does not attach a single, exclusive meaning to “quantum trajectory sensing.” In continuously monitored open systems, a quantum trajectory is the measurement-conditioned evolution of a conditional state $\rho_c(t)$, generated by a stochastic measurement record and analyzed by SMEs, Bayesian filters, or path-integral methods (Rossi et al., 2018, Weber et al., 2014). In qubit-array trajectory sensors, by contrast, the trajectory is a discrete spatial subset $T\subseteq[n]$ specifying which sensor qubits an incoming particle perturbs, and the sensing problem is a multi-hypothesis quantum channel discrimination task (Chin et al., 2024, Chin et al., 2024). In fractional-Fourier sensing, the “trajectory” is a line of slope $-q$ through the $(t,f)$ plane along which a nonstationary signal is matched-filtered (Hegde et al., 2024). In non-Hermitian two-level systems, it is a closed control loop $\Gamma$ in parameter space, whose topology relative to an exceptional point governs sensitivity, selectivity, and chirality (Wu et al., 25 Mar 2026).

A plausible synthesis is that these works treat sensing as inference on path-dependent quantum evolution rather than on a single static parameter snapshot. The path may be a physical particle track, a stochastic conditional state history, a signal trajectory in time–frequency space, or a control trajectory in parameter space.

Trajectory notion	Representative task	Representative source
Spatial subset of qubits	One-shot path discrimination	(Chin et al., 2024)
Conditional state history	Filtering, smoothing, force/parameter estimation	(Rossi et al., 2018)
Time–frequency line	Matched detection of chirps and sweeps	(Hegde et al., 2024)
Parameter-space loop	EP-enabled sensing and state transfer	(Wu et al., 25 Mar 2026)

This multiplicity is not merely terminological. It reflects different operational regimes: exact discrimination versus statistical estimation, discrete trajectory sets versus continuous path spaces, and unitary channel identification versus continuously monitored open-system inference.

2. Single-shot discrimination of incident-particle trajectories

In the qubit-array formulation, a sensor consists of $n$ qubits labeled $1,\dots,n$. A candidate particle trajectory is a subset $T$ of qubit indices, and the particle acts by rotating each qubit in $T$ by a fixed angle $\theta$ about the $T\subseteq[n]$0 axis. The local and trajectory unitaries are

$T\subseteq[n]$1

Perfect one-shot discrimination requires mutual orthogonality of all post-interaction outputs,

$T\subseteq[n]$2

so that a single projective measurement can identify the unknown $T\subseteq[n]$3 with zero error (Chin et al., 2024).

Two trajectory families are emphasized. The symmetric set $T\subseteq[n]$4 contains all $T\subseteq[n]$5-element subsets of $T\subseteq[n]$6. The cyclic set $T\subseteq[n]$7 contains only contiguous modulo-$T\subseteq[n]$8 $T\subseteq[n]$9-tuples, $-q$0, where $-q$1 is the cyclic permutation (Chin et al., 2024).

Family	Candidate trajectories	Existence bound for a TS state
$-q$2	All $-q$3-subsets of $-q$4	$-q$5 sufficient in general; also necessary when $-q$6
$-q$7, $-q$8	Contiguous modulo-$-q$9 $(t,f)$0-tuples	$(t,f)$1 sufficient

The principal result is that entanglement lowers the interaction-strength threshold for perfect discrimination. For $(t,f)$2, an entangled trajectory-sensing state exists already for $(t,f)$3, while product states cannot achieve perfect discrimination for any $(t,f)$4 (Chin et al., 2024). The minimal two-qubit example exhibits the mechanism explicitly: $(t,f)$5 distinguishes “rotate qubit 1” from “rotate qubit 2” only at $(t,f)$6, whereas the entangled state $(t,f)$7 achieves orthogonal outputs already at $(t,f)$8 (Chin et al., 2024). The paper proves a general necessity theorem: for arbitrary $(t,f)$9 and any trajectory set with $\Gamma$0, a fully unentangled TS state exists if and only if $\Gamma$1 (Chin et al., 2024).

Within the interval $\Gamma$2 where entangled TS states exist, the contrast with product sensors is operationally sharp. Entangled sensors achieve zero-error discrimination in one shot. Any unentangled sensor must fail with nonzero single-shot probability and, if repeated $\Gamma$3 times and aggregated by majority vote, requires $\Gamma$4 repetitions to reach error probability $\Gamma$5, consistent with the bound

$\Gamma$6

for single-shot error $\Gamma$7 (Chin et al., 2024).

The 2024 follow-up paper reformulates the same problem in group-theoretic terms and shows that symmetry can reduce the search space exponentially. If the trajectory set is transitive under a permutation group $\Gamma$8, one can restrict the search to the $\Gamma$9-invariant subspace, where $n$0 and $n$1, via the projector

$n$2

The orthogonality constraints then collapse to an LP-feasibility problem $n$3, $n$4, on orbit amplitudes rather than on all $n$5 computational-basis coefficients (Chin et al., 2024). This framework also makes explicit the link to QEC: TS states are one-dimensional codes satisfying a Knill–Laflamme-type orthogonality condition for the trajectory operators, and known stabilizer constructions such as subcodes of the $n$6 CSS code and toric-code states can function as trajectory sensors (Chin et al., 2024).

The same literature already moves beyond uniform couplings. For a Gaussian-profile laser pulse with $n$7, a four-atom $n$8 sensor retains an entanglement advantage even in the weak and wide beam limit: $n$9 for the entangled state, versus $1,\dots,n$0 for an unentangled sensor (Chin et al., 2024).

3. Continuously monitored quantum trajectories as conditional sensors

In the continuous-measurement tradition, a quantum trajectory is the conditioned evolution of an open quantum system under weak monitoring. The measurement record incrementally localizes the state in Hilbert or phase space, and the resulting stochastic conditional state becomes the optimal estimator of future measurement outcomes or external perturbations under the assumed model (Rossi et al., 2018, Kumar et al., 2018).

A central experimental realization is cavity optomechanics. In the mechanical-resonator experiment, continuous optical monitoring of the slow quadratures $1,\dots,n$1 and $1,\dots,n$2 produces records

$1,\dots,n$3

with conditional first moments $1,\dots,n$4 and isotropic covariance $1,\dots,n$5 obeying

$1,\dots,n$6

The experiment reached $1,\dots,n$7, verified a conditional purity $1,\dots,n$8, and inferred an equivalent residual occupation $1,\dots,n$9, thereby demonstrating “cooling by measurement” without feedback (Rossi et al., 2018).

Circuit-QED experiments established the same paradigm in superconducting qubits. In dispersive readout, the interaction Hamiltonian is

$T$0

and a phase-sensitive parametric amplifier allows the observer to choose which cavity quadrature is measured. Monitoring the quadrature carrying qubit-state information yields trajectories confined to a Bloch-sphere meridian; monitoring the photon-number quadrature yields equatorial diffusion. The measurement strength is parameterized by

$T$1

with residual decoherence

$T$2

In the reported device, $T$3, and tomography confirmed that the trajectories reconstructed from the measurement record were faithful to the actual conditioned evolution (Murch et al., 2013).

A complementary theoretical and experimental analysis showed that the entire trajectory distribution for weak continuous $T$4 measurement can be captured by a single-parameter white-noise stochastic process. In Stratonovich form,

$T$5

with

$T$6

The Born-rule constraint $T$7 removes ensemble bias and yields the Itô equation

$T$8

The associated Fokker–Planck theory reproduced experimentally measured histograms over up to $T$9 trajectories, validating likelihood-based inference on the full trajectory ensemble rather than only on mean signals (Kumar et al., 2018).

4. Path likelihoods, smoothing, and most-likely trajectories

Once a measurement record is available, trajectory sensing naturally becomes a path-inference problem. In superconducting qubits under simultaneous weak measurement and unitary drive, the record $T$0 is used to propagate the conditioned state, but one can also ask for the most probable path between chosen boundary conditions. The least-action construction introduced for this problem writes the joint probability of states and readouts as a constrained path probability and yields a continuous-time action

$T$1

whose extremization produces deterministic ODEs for $T$2 together with the optimal detector signal

$T$3

For $T$4, the most likely path from $T$5 to $T$6 is analytic: $T$7 with $T$8 (Weber et al., 2014).

Retrodictive smoothing extends this logic by incorporating future as well as past records. In the past-quantum-state formalism, the forward state $T$9 is paired with a backward effect matrix $\theta$0, and intermediate outcomes obey

$\theta$1

For the monitored mechanical resonator, comparison of forward prediction and backward retrodiction at a common time $\theta$2 produced

$\theta$3

and the measured $\theta$4 agreed within $\theta$5 with the predicted $\theta$6 after correcting a $\theta$7 demodulation-filter systematic (Rossi et al., 2018). The same work notes that, for Gaussian-Markov processes, past-quantum-state smoothing can yield up to a factor $\theta$8 reduction in variance relative to forward-only filtering (Rossi et al., 2018).

In continuously monitored resonance fluorescence, the path-integral viewpoint becomes even richer. For heterodyne monitoring of a spontaneously emitting qubit, the measured currents are

$\theta$9

and the stochastic action yields not only most-likely paths but also diagrammatic approximations to correlation functions of both system variables and detector outputs (Jordan et al., 2015). The formalism explicitly permits “energy-increasing” trajectories, in which measurement backaction transiently overcomes spontaneous emission.

With phase-sensitive homodyne detection of resonance fluorescence, the same path-integral machinery predicts quantum caustics: regions in boundary-conditioned trajectory space where multiple extrema of the stochastic action exist. In the experiment, a driven superconducting artificial atom with $T\subseteq[n]$00, $T\subseteq[n]$01, and $T\subseteq[n]$02 exhibited two distinct most-likely paths connecting the same initial and final states at $T\subseteq[n]$03, in reasonable quantitative agreement with the Hamiltonian boundary-value equations derived from the action (Naghiloo et al., 2016). This shows that trajectory sensing need not lead to a unique “best” history; under strong drive and finite-time postselection, the path space itself can develop fold and cusp structures analogous to optical caustics.

5. Sensitivity enhancement mechanisms

Several works treat the trajectory record not merely as a reconstruction tool but as the locus where metrological gain is created. One route is dissipative criticality. For a Markovian quantum-optical sensor with continuous monitoring of emitted quanta, the global QFI of the joint system and observed environment can be written in terms of autocorrelators of $T\subseteq[n]$04,

$T\subseteq[n]$05

At a dissipative critical point, this yields universal transient and long-time scaling laws that surpass the SQL. In the open Rabi model, with sensed parameter $T\subseteq[n]$06 and effective size $T\subseteq[n]$07, numerical trajectory simulations give

$T\subseteq[n]$08

while the global bound scales as

$T\subseteq[n]$09

The metrological “sweet spot” is $T\subseteq[n]$10, where the practical bound is approached (Ilias et al., 2021).

A second route is to bias the trajectory ensemble toward high-sensitivity records. In the ancilla-assisted open-Rabi protocol, each time slice is described by Kraus updates

$T\subseteq[n]$11

with trajectory likelihood

$T\subseteq[n]$12

The key engineering idea is to map each informative phonon event to a fluorescence burst. The local sensitivity

$T\subseteq[n]$13

is thus amplified by increasing the number of detected photons per event,

$T\subseteq[n]$14

while the induced phonon dissipation obeys

$T\subseteq[n]$15

The same work reports finite-size scaling

$T\subseteq[n]$16

with data collapse consistent with exponents $T\subseteq[n]$17 and $T\subseteq[n]$18 (Ilias, 23 Apr 2025).

A third route exploits nonclassical probe pulses and full trajectory likelihoods. In the cascaded-SME treatment of travelling radiation pulses, the observed output operator is

$T\subseteq[n]$19

and for homodyne detection the conditional state obeys

$T\subseteq[n]$20

with current

$T\subseteq[n]$21

The norm of the unnormalized conditional state supplies $T\subseteq[n]$22 for Bayesian discrimination between hypotheses $T\subseteq[n]$23, allowing direct comparison between coherent, squeezed, Fock, and cat-like input pulses. The reported conclusion is strategy-dependent: homodyne is optimal for dispersive phase sensing with squeezed probes, whereas photon counting is especially informative for absorption or emission patterns and for Fock-state pulses with known total photon number (Khanahmadi et al., 2022).

6. Generalized trajectory spaces: non-Markovian memory, time–frequency lines, and non-Hermitian loops

One extension replaces Markovian SMEs with generalized stochastic Schrödinger equations having random operator coefficients. In this framework,

$T\subseteq[n]$24

with the norm process used as a Girsanov density to define the physical measure and the a posteriori state. The associated instruments remain completely positive, while the output spectra and counting statistics become sensitive to colored noise, random couplings, and delays. In the noisy-oscillator example, the heterodyne PSD directly reflects the environmental spectrum $T\subseteq[n]$25 through cavity and detection windows, and the analysis of fluorescence can in principle reveal non-Markovian features and the spectra of noises affecting the dynamics (Barchielli et al., 2012).

Another extension uses “trajectory” in the time–frequency plane. In fractional-Fourier-domain sensing, the qubit accumulates phase

$T\subseteq[n]$26

under a chirped dynamical-decoupling filter

$T\subseteq[n]$27

Its instantaneous modulation frequency

$T\subseteq[n]$28

defines a straight trajectory in the $T\subseteq[n]$29 plane, and the axis coordinate is $T\subseteq[n]$30. In the matched-response limit,

$T\subseteq[n]$31

Using NV-center ensembles, the reported experiments obtained two-order-of-magnitude lower MSE in estimating a chirped signal’s center frequency $T\subseteq[n]$32 at large $T\subseteq[n]$33 than unchirped DD, and near-zero MAP error rates in a binary test $T\subseteq[n]$34 versus $T\subseteq[n]$35 across the tested chirp range (Hegde et al., 2024).

A third extension concerns non-Hermitian control loops. For the time-modulated two-level Hamiltonian

$T\subseteq[n]$36

the exceptional-point condition is $T\subseteq[n]$37, and the loop topology is quantified by

$T\subseteq[n]$38

Loops with $T\subseteq[n]$39 yield symmetric, direction-independent transfer; loops encircling the EP have $T\subseteq[n]$40 and produce chiral transfer together with singular susceptibilities. The paper distinguishes eigenvalue-based sensing, with susceptibilities such as

$T\subseteq[n]$41

from eigenstate-based sensing based on

$T\subseteq[n]$42

The reported conclusion is that eigenstate-based sensing can achieve full parameter selectivity unattainable with eigenvalue-based methods, and that trajectory topology controls sensitivity amplitude, time window, and robustness (Wu et al., 25 Mar 2026).

Taken together, these developments show that quantum trajectory sensing has become a broad research domain rather than a single protocol class. In one branch it is an exact discrimination problem solved by entangled sensor states and symmetry reduction; in another it is a stochastic-estimation framework based on SMEs, likelihoods, and path integrals; in yet others it refers to matched sensing along structured trajectories in abstract spaces such as time–frequency planes or non-Hermitian control manifolds. This suggests that future unification will likely proceed not by enforcing a single definition of “trajectory,” but by comparing how different trajectory spaces encode information, how measurement backaction reshapes those spaces, and when trajectory-aware sensing outperforms static-observable strategies (Chin et al., 2024, Ilias et al., 2021).