Quantum Trajectories: Dynamics & Methods

Updated 13 December 2025

Quantum trajectories are stochastic or deterministic evolutions conditioned on measurement records that unravel ensemble-averaged quantum dynamics.
They employ methods like jump and diffusive unraveling to simulate measurement-induced processes in open quantum systems and many-body setups.
Deterministic techniques such as imaginary-time evolution and phase-space formulations enable scalable exploration of conditional quantum measurements and feedback.

Quantum trajectories are stochastic or deterministic evolutions of quantum states that resolve the dynamics of individual measurement records or conditioning processes, contrasting with ensemble-averaged (master equation) descriptions. Quantum trajectory frameworks are central in open quantum systems, quantum measurement theory, many-body monitored circuits, quantum optics, quantum hydrodynamics, phase-space quantum mechanics, and foundational interpretations of quantum mechanics.

1. Foundational Formulations and Physical Meaning

Prominent families of quantum trajectory formalisms map the evolution of the system's state to stochastic processes conditioned on measurement results or environmental degrees of freedom. In monitored open quantum systems, the density matrix $\rho$ evolves via the Lindblad master equation: $\dot\rho = -i[H, \rho] + \sum_\ell (L_\ell \rho L_\ell^\dagger - \frac{1}{2} \{L_\ell^\dagger L_\ell, \rho\})$ Quantum trajectories "unravel" this equation into stochastic realizations of pure states $|\psi_c(t)\rangle$ driven by discrete (jump) or continuous (diffusive) measurement records. Their ensemble average recovers the deterministic $\rho(t)$ (Daley, 2014).

In quantum measurement and feedback, each run yields a record (e.g., list of detection events, homodyne current) and a corresponding conditional evolution. These trajectories reflect the deepest link between information extraction, quantum stochastic processes, and state reduction protocols.

In quantum hydrodynamics and the de Broglie–Bohm framework, trajectories are defined via the probability current: $v(x,t) = \frac{1}{m} \nabla S(x,t)$ , with $S$ the phase of the wavefunction in the polar decomposition $\Psi(x,t) = R(x,t) \exp[i S(x,t)/\hbar]$ (Gosson et al., 2016, Schild, 2019).

Phase-space formulations, encompassing the Moyal–Weyl product and non-commutative flows on observables, generalize classical Hamiltonian flow, introducing quantum trajectories as deformations of classical phase-space dynamics (Blaszak et al., 2012, Hiley et al., 2016).

2. Stochastic and Deterministic Quantum Trajectories

2.1. Stochastic Master Equations and Unravelings

Quantum trajectory dynamics can be numerically and analytically described using stochastic Schrödinger equations (SSEs):

Jump unraveling: Discrete measurement records (e.g., photon detection) generate piecewise-deterministic processes, punctuated by quantum jumps with probabilities matching the measurement statistics. The evolution for pure states, in Itô form, is

$d|\psi_c\rangle = -i H_{\mathrm{eff}} |\psi_c\rangle dt + \sum_\ell \left(\frac{L_\ell}{\|L_\ell|\psi_c\rangle\|} - 1\right) dN_\ell(t) |\psi_c\rangle$

with $H_{\mathrm{eff}} = H - i/2 \sum_\ell L_\ell^\dagger L_\ell$ , and $dN_\ell(t)$ is a Poisson increment (Daley, 2014).

Diffusive (quantum state diffusion) unraveling: Continuous measurement (e.g., homodyne detection) yields stochastic evolution with real Wiener processes: $d|\psi\rangle = \Big(-i H - \frac{1}{2}\sum_\ell L_\ell^\dagger L_\ell + \sum_\ell \langle L_\ell^\dagger \rangle L_\ell - \frac{1}{2}\sum_\ell |\langle L_\ell \rangle|^2\Big) |\psi\rangle dt + \sum_\ell \left(L_\ell - \langle L_\ell \rangle\right) |\psi\rangle dW_\ell(t)$ (Daley, 2014).

2.2. Trajectory Techniques in Many-Body and Measurement-Driven Systems

Quantum trajectory techniques are crucial in simulating open many-body systems with strong interactions and engineered dissipation (e.g., optical lattice gases under photon scattering, two-body/three-body loss, reservoir engineering, or topological state stabilization) (Daley, 2014). Advanced methods combine trajectory simulation with matrix-product-state (MPS) techniques to address scaling in Hilbert space dimension and growth of entanglement entropy.

Measurement-induced phase transitions in monitored quantum circuits manifest in the properties of individual quantum trajectories associated with fixed measurement records. For instance, area-law vs. volume-law entanglement scaling emerges only upon post-selection on particular measurement strings (Mittal et al., 31 Mar 2025).

3. Deterministic Quantum Trajectory Synthesis via Imaginary Time Evolution

Quantum trajectories characterized by a specific sequence of measurement outcomes are exponentially suppressed in direct experiments—the post-selection barrier—as the likelihood for any fixed trajectory is $\sim 2^{-M}$ for $M$ measurement events (Mittal et al., 31 Mar 2025).

"Deterministic quantum trajectory via imaginary time evolution" (Mittal et al., 31 Mar 2025) introduces a protocol that eliminates exponential overhead in preparing such trajectories:

The key is to design a Hamiltonian $H$ whose unique ground state is the desired post-measurement state associated with a specific outcome sequence.
Imaginary-time evolution projects any initial state onto this ground state: $|\psi(\tau)\rangle \propto e^{-\tau H} |\psi(0)\rangle$ , converging exponentially fast with rate set by the spectral gap $\Delta$ .
In practice, the non-unitary evolution $e^{-\beta H}$ is implemented via a Deterministic Quantum Imaginary Time Evolution (DQITE) algorithm, combined with local tomography to construct system-specific unitaries on small regions.
The runtime per measurement scales polynomially in system size and inverse error, provided the system exhibits exponential clustering of correlations (i.e., area-law entanglement).
This approach is provably efficient in the area-law regime but cannot be universally applied to all quantum trajectories—volume-law (highly entangled) circuits violate the exponential clustering condition.

This method unlocks experimental access to post-selection-dependent phenomena, including the direct measurement of conditional entropies, correlation functions, and teleportation fidelities, enabling scalable exploration of individual quantum trajectories for moderately large quantum devices (Mittal et al., 31 Mar 2025).

4. Mathematical Structures and Limit Theorems in Quantum Trajectory Processes

Quantum trajectories can also be formulated as Markov processes on complex projective Hilbert space, where state updates correspond to random application of quantum channels or measurement outcomes (Benoist et al., 6 Feb 2024). The associated Markov transition operator $P$ exhibits:

Quasi-compactness, admitting a finite-rank peripheral spectral decomposition with a nontrivial spectral gap $\gamma>0$ , ensuring exponential mixing towards a unique invariant measure under mild irreducibility and purification conditions.
Analytic perturbations ("tiltings") of $P$ yield cumulant generating functions for empirical observables. This structure enables the derivation of central limit theorems (CLTs), Berry-Esseen bounds (optimal convergence rates), and large deviation principles (LDPs) for time-averaged empirical process observables and Lyapunov exponents of random matrix products.
These limit theorems provide a rigorous basis for the long-time, statistical behavior of quantum trajectories generated by iterated quantum measurements (Benoist et al., 6 Feb 2024).

5. Quantum Trajectories in Phase Space and Hydrodynamical/Bohmian Contexts

Phase-space quantum mechanics (the Moyal formalism) offers an alternative to the Hilbert-space operator language, defining quantum trajectories as flows generated by quantum Hamiltonian equations in the non-commutative algebra of phase-space functions: $\dot A(t) = \{A(t), H \}_M$ where $\{\cdot,\cdot\}_M$ is the Moyal bracket, and the star-product deforms classical mechanics by introducing $\hbar$ -dependent corrections (Blaszak et al., 2012, Hiley et al., 2016). For quadratic Hamiltonians, quantum and classical trajectories coincide; for nonlinear systems, they become genuinely quantum, requiring twisted pullbacks and automorphisms to relate Heisenberg evolution to phase-space flows.

In quantum hydrodynamics, particle trajectories arise from the probability current and the quantum Hamilton–Jacobi formalism; in the Bohm–de Broglie viewpoint, these "quantum trajectories" are defined by

$\frac{dx}{dt} = \frac{1}{m} \nabla S(x,t)$

with dynamics subject to a "quantum potential" $Q(x,t)$ . Weak-measurement and hydrodynamical approaches have enabled experimental reconstruction of average quantum trajectories in interferometric contexts (Schild, 2019, Gosson et al., 2016, Mori et al., 2014).

6. Non-Markovian and Novel Quantum Trajectory Frameworks

Quantum trajectories can extend to non-Markovian environments and non-Gaussian input fields. For example, quantum systems probed by propagating Fock states or single-photon wavepackets yield coupled sets of stochastic master equations involving auxiliary density matrices $\rho_{m,n}(t)$ tracking photon-number sectors (Baragiola et al., 2017, Dabrowska et al., 2017). In such cases, dynamical equations retain memory of the environmental correlations, and the resulting conditional evolution is fundamentally non-Markovian—averaged dynamics do not reduce to standard Lindblad form.

Trajectories in collisional models with entangled environment qubit pairs exhibit new types of stochastic processes (e.g., nonlocal jump maps, correlated "no-jump" events), enabling, for instance, jump-induced entanglement creation in remote atomic systems (Daryanoosh et al., 2022).

Entropic dynamics offers a distinct ontological framework, positing particles with real but unknown positions, and trajectories as stochastic differential equations interpolating between diffusive (Nelson) and deterministic (Bohmian) limits, derived from entropic inference principles (Carrara, 2019).

7. Applications, Experimental Realizations, and Deterministic Protocols

Quantum trajectory techniques underpin real-time monitoring and feedback in superconducting qubits, enabling continuous state tracking and entanglement generation, with strong agreement between Bayesian trajectory reconstructions and quantum tomography (Weber et al., 2015, Chantasri et al., 2016).

Experimental quantum communication schemes exploit quantum superpositions of spatial trajectories, enabling capacity activation even for entanglement-breaking channels. These protocols utilize Mach–Zehnder interferometers, controlled unitaries, and channel order superpositions to implement trajectory-based error filtration and quantum switch operations in quantum-optical networks (Rubino et al., 2020).

In quantum cosmology and the Wheeler–DeWitt framework (mini-superspace models), trajectory formulations enable singularity resolution and detailed cosmological evolution modeling via quantum-corrected Hamilton–Jacobi equations and pilot-wave guiding laws (Peter, 2019).

Recent advances have leveraged imaginary-time evolution and input-dependent unitaries to deterministically synthesize highly specified quantum trajectories with polynomial efficiency in the "area law" entanglement phase, directly overcoming the exponential post-selection cost and enabling systematic exploration of trajectory-resolved many-body physics (Mittal et al., 31 Mar 2025).

References:

(Daley, 2014) (Quantum trajectories and open many-body quantum systems)
(Mittal et al., 31 Mar 2025) (Deterministic quantum trajectory via imaginary time evolution)
(Benoist et al., 6 Feb 2024) (Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems)
(Blaszak et al., 2012, Hiley et al., 2016, Schild, 2019) (Phase-space and hydrodynamical/Bohmian approaches)
(Weber et al., 2015, Chantasri et al., 2016, Rubino et al., 2020) (Experimental quantum trajectories in superconducting qubits, quantum communication)
(Carrara, 2019) (Entropic dynamics)
(Daryanoosh et al., 2022, Baragiola et al., 2017, Dabrowska et al., 2017) (Non-Markovian and Fock-state trajectory frameworks)