A Lindblad equation does not determine trajectory entanglement

The fracture

Given an initial state and a time-independent Lindblad generator, the master equation fixes the unconditional density matrix $\rho (t)$.

It does not fix which pure-state trajectories produced that average.

Two observers can monitor the same Markovian output channel in different ways. After discarding their records, both assign the same $\rho (t)$. Conditioned on the records, they can assign different states and different entanglement.

The distinction is

versus

where $f$ is nonlinear. Entanglement entropy is one such $f$.

This is the small algebraic fact behind a large part of monitored many-body physics.

1. What the master equation fixes

Consider

with

For a specified $\mathcal{L}$ and $\rho (0)$, the solution is

It fixes every single-copy linear expectation value:

It also fixes unconditional multitime correlations once the Markovian model, operator insertions, and regression assumptions are specified.

The eigenvalues alone do not contain all of that information. Eigenoperators, overlaps, Jordan blocks, and nonnormal transient growth can matter even for unconditional relaxation.

Two nonuniquenesses

A Lindblad generator admits many stochastic unravelings. Its displayed jump operators are also nonunique: unitary mixing and standard gauge shifts can leave $\mathcal{L}$ unchanged.

An unraveling adds a classical record $\xi$ and a conditional state ${\rho }_{\xi }$. The record may describe photon counts, a homodyne current, heterodyne data, or known classical noise.

The average must recover the channel:

This equality constrains a first moment. It does not determine the probability law over conditional states.

2. The Bell-pair test

Take two qubits with $H=0$, initially

and let each qubit dephase:

Equation (1) is an exact statement inside the assumed time-homogeneous Markovian model. It is not a microscopic derivation of that model.

Only the Bell coherence evolves. Set

For either qubit,

Each dissipator contributes $-2\gamma X$, so

The unconditional state is

Its purity is

The reduced state of either qubit remains $I/2$. Its entropy stays $\ln 2$, but that entropy is not an entanglement measure once the joint state is mixed.

For this two-qubit state, concurrence and negativity are

These are properties of $\rho (t)$. They agree for every unraveling of Eq. (1).

3. Unraveling A: known random phase flips

Choose jump operators

Because

each jump occurs at a state-independent Poisson rate. The waiting time carries no information about the qubit state.

A trajectory is

It is always $|{\mathrm{\Phi }}^{+}⟩$ or

Both states contain one ebit. The trajectory entropy of either qubit is therefore

for every record and every time.

The sum $n_1+n_2$ is Poisson with rate $2\gamma$. Its parity average is

which reproduces Eq. (2).

This unraveling can represent a monitored bath whose observed events cause known phase flips. It can also represent classical stochastic control noise whose realization is logged.

Calling it a “measurement of $Z$” would be misleading. The record tells us which phase kick occurred; it does not reveal whether the system occupied $|0⟩$ or $|1⟩$.

4. Unraveling B: informative continuous monitoring

Now monitor each $Z_j$ diffusively with unit efficiency. In one common convention,

where

The independent Wiener increments satisfy

A compatible measurement current is

Coefficients move between $\gamma$, $\mathcal{H}$, and the current in other conventions. Equations (1), (5), and (6) use one internally consistent choice.

Averaging Eq. (5) removes the innovation term and returns Eq. (1).

Conditioning does something different. In the Bell subspace, both records gather evidence about the alternatives $|00⟩$ and $|11⟩$.

An efficient trajectory remains pure and can be written as

The posterior weight $p_t=|a_t{|}^2$ approaches $0$ or $1$ almost surely as the integrated signal-to-noise ratio grows.

Thus

and

The two records are partly redundant in this subspace, but both favor the same branch. Their information rates add.

5. No contradiction

The random-flip ensemble gives

The informative ensemble approaches

Both average to Eq. (2).

The equality

does not imply

Nor does it imply that two different ensembles have the same average of $f$.

For entropy, concavity gives

That inequality concerns the entropy of the full conditional state. Trajectory entanglement instead applies entropy after reducing each pure trajectory to a subsystem.

The two operations must not be conflated.

The Hughston-Jozsa-Wootters theorem sharpens the static statement: different measurements on a purifying system can realize different pure-state ensembles of the same density matrix.

A continuous unraveling carries extra causal and hardware constraints, so not every static decomposition defines a realizable monitoring protocol for a given apparatus.

6. What counts as a physical unraveling

A stochastic differential equation can be useful numerically without describing an installed detector.

To assign operational meaning, specify at least:

• a system-environment dilation or input-output model;
• which environmental observable is measured;
• detector efficiency and bandwidth;
• what part of the record is retained;
• whether feedback uses that record.

The same reduced Lindbladian can arise from inequivalent microscopic environments. A mathematically valid unraveling need not be implementable by measuring the output ports of every such realization.

Efficiency matters. If a fraction $\eta <1$ of the output is observed, Eq. (5) acquires $\sqrt{\eta }$ in the stochastic term while the full dissipator remains.

The observer has discarded part of the record. The conditional state is then generally mixed, and “pure-state trajectory entanglement” is no longer the right quantity without a further purification convention.

Different observers can therefore assign different conditional states to the same device because they possess different records. Their unconditional prediction remains the same.

This is observer dependence with an operational cause, not arbitrary subjectivity.

7. Which quantities can distinguish unravelings

Every quantity linear in the single-copy conditional state collapses to the unconditional answer:

Nonlinear trajectory averages retain information about the record-conditioned ensemble.

A useful object is the replicated moment

$M_1=\rho$ obeys the ordinary master equation. For $n>1$, the shared record couples replicas, and the evolution depends on the unraveling.

Rényi entropies can be represented through swap operators acting on these replicas. The von Neumann entropy follows from a replica derivative when the continuation is controlled.

This is why trajectory entanglement requires replica, Keldysh, tensor-network, or postselection-aware methods absent from the one-copy Lindblad equation.

Piñol et al. gave an operational proposal for distinguishing photon-counting and phase-sensitive unravelings through nonlinear trajectory-averaged fluctuations.

The full record statistics can also distinguish monitoring schemes. The unconditional system state cannot reconstruct a record that was never retained.

8. From a Bell pair to a monitored phase

The Bell calculation proves unraveling dependence in a finite system. It does not prove a phase transition.

In a one-dimensional many-body system, an entanglement transition requires a scaling statement after specifying the order of limits. Schematically,

in a volume-law phase, while

in an area-law phase for a finite fraction subsystem.

At criticality, logarithmic or other subvolume scaling can occur. Finite-size crossovers can imitate such scaling over many decades.

Local informative measurements compete with unitary scrambling. Scrambling hides information in nonlocal correlations; the measurement record attempts to learn it locally.

This motivates the quantum-error-correction interpretation of the volume-law phase. In random-circuit models, a reference can remain encoded against low-rate measurements until a threshold is crossed.

See Choi et al. for the channel-capacity formulation.

The analogy is not an identity for every monitored model. Nonlocal measurements, degenerate measured observables, symmetry sectors, adaptive feedback, and dark subspaces can create or preserve entanglement in an area-law state.

“Measurement destroys entanglement” is therefore too crude. The measured operator, retained outcome, scrambling dynamics, and feedback decide what information leaves the system.

9. Fixed Lindbladian, changing trajectory physics

The literature contains explicit same-Lindbladian comparisons.

A 2022 study of a dephasing quantum Ising chain found different entanglement behavior for a quantum-state-diffusion unraveling and a Gaussian-preserving jump unraveling.

This establishes that a master equation does not select one trajectory ensemble. See Piccitto, Russomanno, and Rossini.

A 2024 free-fermion study varied the detected quadrature at fixed dissipative rate and reported an unraveling-induced area-to-logarithmic entanglement transition.

See Eissler, Lesanovsky, and Carollo.

The later infrared analysis changed the interpretation. A 2026 nonlinear-sigma-model treatment with larger-scale numerics found that, for the studied one-dimensional chain, every informative angle ultimately flows to an area law.

The apparent critical regime survives below a large crossover length. Close to the random-unitary endpoint, the asymptotic scale behaves as

Only the exactly no-information endpoint $\phi =\pi /2$ retains the volume law in that model.

This is a useful correction, not a retreat from unraveling dependence. The trajectory steady states still differ sharply. What failed was the finite-size inference of a generic phase boundary.

See Niederegger et al..

10. Channel transitions and trajectory transitions can separate

An ordinary Liouvillian gap diagnoses relaxation of the averaged channel. A trajectory entanglement transition concerns a replicated, record-conditioned statistical object.

There is no general theorem forcing both singularities to coincide.

A 2026 superconducting-processor experiment combined mid-circuit measurement with low-latency feedback. It observed an absorbing-state transition in the averaged channel and a trajectory entanglement transition at distinct tuning values.

The result gives direct experimental support to the separation of diagnostics. It does not imply that every monitored platform has two transitions.

See Wu et al., Physical Review Letters 137, 010401 (2026).

The safe inference is

Conversely, a Liouvillian critical point need not imply a change in trajectory entanglement scaling.

The deliberately false claim

If two monitoring protocols have the same Lindbladian and the same Liouvillian spectrum, they must have the same measurement-induced entanglement transition.

The premise fixes one-copy unconditional dynamics. Trajectory entanglement depends on higher moments of the conditioned ensemble and on the measurement record.

Even the phrase “same Liouvillian spectrum” is weaker than “same Liouvillian”: eigenvalues without eigenoperators and Jordan structure need not determine the full linear evolution.

Collision: five different statements

1. Master equation: $\rho (t)=e^{t\mathcal{L}}\rho (0)$ fixes the unconditional state.
2. Unraveling: a monitoring rule fixes a distribution over records and conditional states.
3. Finite-system fact: nonlinear averages can differ while every linear system observable agrees.
4. Many-body claim: a phase requires controlled long-time and thermodynamic scaling.
5. Experimental claim: the physical detector, efficiency, feedback, and accessible record select which trajectory ensemble is realized.

Skipping a line changes the question.

Do this now

For Eq. (1) and the initial Bell state:

Target A

Derive Eq. (2) and the purity in Eq. (3).

Target B

For independent phase-flip counts $n_1,n_2$ of rate $\gamma$, compute the probabilities of even and odd total parity and recover Eq. (2).

Target C

Compute the trajectory-averaged single-qubit entropy for the random-jump unraveling.

Target D

Determine its $t\to \mathrm{\infty }$ value for ideal informative $Z$ monitoring.

Target E

Compute the eigenvalues of ${\rho }^{T_2}(t)$ and verify the negativity in Eq. (4). Explain why this unraveling-independent number differs from both trajectory averages.

Oral check 1

Can an unconditional density matrix become mixed while every efficient conditioned trajectory stays pure?

Oral check 2

If detector efficiency is zero, which state does the observer assign?

Hints

• Only $|00⟩⟨11|$ and its adjoint evolve.
• A Poisson variable with mean $\lambda$ satisfies $\mathbb{E}[(-1{)}^n]=e^{-2\lambda }$.
• Partial transpose moves the corner coherence into the $|01⟩,|10⟩$ block.

Solution outline

Each local dissipator multiplies $X$ by $-2\gamma$, so

$$X(t)=e^{-4\gamma t}X(0).$$Copy LaTeX

The populations remain $1/2$, which gives Eq. (2). Squaring its nonzero $2\times 2$ block yields

$$\mathrm{Tr}{\rho }^2=\frac{1}{2}(1+e^{-8\gamma t}).$$Copy LaTeX

The total number of jumps is Poisson with mean $2\gamma t$. Therefore

$$p_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}=\frac{1+e^{-4\gamma t}}{2},p_{\mathrm{o}\mathrm{d}\mathrm{d}}=\frac{1-e^{-4\gamma t}}{2}.$$Copy LaTeX

The averaged state is

$$\rho (t)=p_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}|{\mathrm{\Phi }}^{+}⟩⟨{\mathrm{\Phi }}^{+}|+p_{\mathrm{o}\mathrm{d}\mathrm{d}}|{\mathrm{\Phi }}^{-}⟩⟨{\mathrm{\Phi }}^{-}|,$$Copy LaTeX

which is Eq. (2).

Every random-jump trajectory is maximally entangled, so

$${\bar{S}}_1^{\mathrm{j}\mathrm{u}\mathrm{m}\mathrm{p}}=\ln 2.$$Copy LaTeX

Ideal informative monitoring eventually identifies one branch, hence

$${\bar{S}}_1^{\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{o}}⟶0.$$Copy LaTeX

The partial-transpose eigenvalues are

$$\frac{1}{2},\frac{1}{2},\frac{1}{2}{e}^{-4\gamma t},-\frac{1}{2}{e}^{-4\gamma t}.$$Copy LaTeX

Thus

$$\mathcal{N}(t)=\frac{1}{2}{e}^{-4\gamma t}.$$Copy LaTeX

Negativity is calculated from the unconditional density matrix. It quantifies entanglement left after the record is discarded.

The trajectory averages ask how entangled the conditional pure state is for an observer who keeps a specified record. They answer a different operational question.

Exit ticket

A simulation finds logarithmic trajectory entanglement up to size $L=512$, while the one-copy Liouvillian gap stays finite.

What must be checked before calling this an unraveling-induced critical phase?

A strong answer should include larger-size flow, crossover scaling, order of limits, detector efficiency, record definition, and a replicated or information-theoretic diagnostic.

Research checks worth doing next

1. Record audit. State exactly which environmental quadrature or event stream is measured.
2. Efficiency sweep. Track how lost records convert pure trajectories into mixed conditional states.
3. Replica observable. Measure or simulate $M_2=\mathbb{E}[{\rho }_c^{\otimes 2}]$ instead of inferring trajectory physics from $\rho$.
4. Crossover collapse. Test competing algebraic and exponential length scales before fitting a critical exponent.
5. Reference test. Couple the system to an external reference and measure recoverable information, not entropy alone.
6. Feedback separation. Compare passive conditioning with active feedback driven by the same record.
7. Dilation check. Verify that the proposed unraveling can be implemented by the actual output channels of the device.

Further reading

• Wave-function approach to dissipative processes in quantum optics develops the Monte Carlo wave-function picture of quantum jumps.
• Complete classification of quantum ensembles having a given density matrix gives the static ensemble theorem behind environment-assisted decompositions.
• Measurement-Induced Phase Transitions in the Dynamics of Entanglement establishes volume-law and area-law phases in monitored circuit models.
• Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition connects the transition to channel capacity.
• Entanglement transitions in the quantum Ising chain compares two unravelings of one dephasing Lindbladian.
• Telling different unravelings apart via nonlinear quantum-trajectory averages proposes an operational nonlinear diagnostic.
• Unraveling-induced entanglement phase transition in diffusive trajectories reports the finite-size free-fermion transition later reassessed in the infrared.
• Absence of measurement- and unraveling-induced entanglement transitions in continuously monitored one-dimensional free fermions identifies the exponentially large crossover and the special random-unitary endpoint.
• Measurement- and Feedback-Driven Nonequilibrium Phase Transitions on a Quantum Processor experimentally separates a channel transition from a trajectory entanglement transition.

What to retain

• A master equation fixes the unconditional density matrix, given its initial condition and model assumptions.
• The displayed Lindblad operators and the stochastic unraveling are not unique.
• Random phase-flip records can preserve Bell entanglement along every trajectory while the average state dephases.
• Informative $Z$ records localize the same Bell pair and remove its conditional entanglement.
• Unconditional concurrence or negativity is not trajectory-averaged pure-state entanglement.
• A physical unraveling requires a detector model, accessible output channel, and efficiency.
• Quantities linear in one conditional copy cannot distinguish unravelings of the same master equation.
• Replicated moments retain record-dependent information.
• A finite Bell-pair example proves unraveling dependence, not a many-body phase transition.
• A finite Liouvillian gap does not rule out trajectory criticality.
• Large crossover lengths can masquerade as unraveling-induced phases.

Next: derive the two-replica stochastic generator for dephasing and show where the shared measurement record couples replicas.