Collision Models

A collision model, also called a repeated-interaction model, represents an environment as a sequence of small ancillary systems. The system interacts with one ancilla for a short time, that ancilla is discarded or moved on, and the next ancilla arrives.

The simplest memoryless version is:

fresh ancilla -> system collision -> discarded ancilla
fresh ancilla -> system collision -> discarded ancilla
fresh ancilla -> system collision -> discarded ancilla

If every ancilla is fresh, uncorrelated, and prepared in the same state, the system evolves by repeated application of the same quantum channel. Correlations, reuse, feedback, or ancilla–ancilla interactions can introduce memory. They provide possible mechanisms, rather than a guarantee of a positive value for every non-Markovianity diagnostic.

Collision models are useful because they make assumptions visible. Instead of saying “the bath forgets quickly,” one can point to the mechanism: every environmental carrier that touched the system is discarded, and the next one arrives uncorrelated.

One-Step Channel

Let $S$ be the system and $A_n$ the $n$th ancilla. Before the $n$th collision, assume the incoming ancilla has state $\eta_n$ and is uncorrelated with the system. A joint unitary $U_n$ acts on $S\otimes A_n$ for a short collision time. After tracing out the ancilla,

$$\rho_n = \Phi_n(\rho_{n-1}) = \operatorname{Tr}_{A_n} \left[ U_n \left( \rho_{n-1}\otimes\eta_n \right) U_n^\dagger \right].$$

This is a completely positive trace-preserving map. It is a finite-dimensional Stinespring dilation with the ancilla playing the environment role.

If $\eta_n=\eta$ and $U_n=U$ for every step, then

$$\rho_n = \Phi^n(\rho_0).$$

The discrete-time evolution is memoryless in the same sense as a homogeneous Markov chain: the next state depends on $\rho_n$ through a fixed channel, not on the earlier trajectory.

Fresh Ancillas and Discrete Markovianity

Fresh, product ancillas mean that the environmental state before the collisions factorizes as

$$\eta_{A_1A_2\cdots A_N} = \eta_1\otimes\eta_2\otimes\cdots\otimes\eta_N.$$

If the system meets each ancilla only once, then after $n$ steps the future carriers have not interacted with the system. They contain no record of earlier system states. The reduced system evolution is therefore a product of channels:

$$\Phi_{n,0} = \Phi_n\Phi_{n-1}\cdots\Phi_1.$$

For any $n\ge m$,

$$\Phi_{n,0} = \Phi_{n,m}\Phi_{m,0}, \qquad \Phi_{n,m} = \Phi_n\cdots\Phi_{m+1}.$$

Because each factor is CPTP, the evolution is CP-divisible in this discrete-time sense. If all steps are identical, it is a discrete semigroup:

$$\Phi_{n+m,0} = \Phi^n\Phi^m.$$

This is the collision-model version of the Markov Approximation.

Continuous-Time Limit

To connect a collision model to a master equation, take a small time step $\tau$ and a family of channels $\Phi_\tau$. If

$$\Phi_\tau = \mathrm{id} + \tau\mathcal L + O(\tau^2),$$

then

$$\frac{\rho(t+\tau)-\rho(t)}{\tau} \to \mathcal L\rho(t)$$

as $\tau\to0$ with $t=n\tau$ fixed. The limit gives

$$\frac{d\rho}{dt} = \mathcal L\rho.$$

With suitable scaling of the interaction strength and ancilla state, $\mathcal L$ can be a Lindblad–GKSL generator. Collision models therefore provide a constructive route from microscopic repeated interactions to Markovian master equations.

The scaling is not automatic. A fixed strong unitary repeated infinitely often can fail to have a smooth generator. One chooses the interaction, collision time, and ancilla preparation so that the first-order term in $\tau$ is finite and the limiting generator is physical.

Example: Partial-Swap Amplitude Damping

Let both the system and each ancilla be qubits. The ancilla enters in the ground state $\lvert g\rangle_A$. During one collision, use the excitation-exchange unitary

$$U_\theta = \exp \left[ -i\theta \left( \sigma_+^{(S)}\sigma_-^{(A)} + \sigma_-^{(S)}\sigma_+^{(A)} \right) \right].$$

Restrict $0\le\theta\le\pi/2$, so $\cos\theta=\sqrt q$ below; outside this interval the coherence retains the sign of $\cos\theta$. In the one-excitation subspace,

$$\lvert e,g\rangle \longmapsto \cos\theta\,\lvert e,g\rangle -i\sin\theta\,\lvert g,e\rangle.$$

Tracing out the ancilla gives an amplitude-damping channel. If $q=\cos^2\theta$, then

$$\rho_{ee}' = q\rho_{ee}, \qquad \rho_{eg}' = \sqrt q\,\rho_{eg}.$$

After $n$ identical fresh collisions,

$$\rho_{ee}^{(n)} = q^n\rho_{ee}^{(0)}, \qquad \rho_{eg}^{(n)} = q^{n/2}\rho_{eg}^{(0)}.$$

Choosing

$$q=e^{-\gamma\tau}$$

and setting $t=n\tau$ gives the continuous exponential law

$$\rho_{ee}(t) = e^{-\gamma t}\rho_{ee}(0),$$

which is the population decay of the Markovian amplitude-damping master equation. Since $\cos^2\theta=1-\theta^2+O(\theta^4)$, this limit uses $\theta\simeq\sqrt{\gamma\tau}$. An exchange frequency $g_\tau=\theta/\tau$ therefore scales as $\tau^{-1/2}$. Keeping $g_\tau$ fixed instead makes the dissipative change per unit time vanish in this example.

Thermalizing Collision Models

If incoming ancillas are prepared in thermal states and the collision unitary conserves total energy, repeated interactions can drive the system toward a thermal fixed point. This gives a discrete operational picture of relaxation:

system repeatedly samples small thermal carriers

For example, a qubit system coupled by excitation exchange to thermal qubit ancillas experiences both upward and downward transitions. The ratio of the transition probabilities is set by the ancilla temperature when the model satisfies the relevant detailed-balance condition.

This is close in spirit to Thermal Operations Preview, where thermal ancillas and energy-preserving unitaries define an operational thermodynamic framework. Collision models emphasize the time-ordered repeated dynamics rather than only the allowed state transformations.

How Memory Enters

Collision models become non-Markovian when the future collision is not independent of the past. Common mechanisms are:

Mechanism	Memory carrier
Reused ancilla	The same ancilla returns after storing information from an earlier collision.
Correlated ancillas	The next ancilla is initially correlated with previous ancillas.
Ancilla–ancilla interactions	Environmental carriers interact before or after system collisions.
Explicit memory system	A mediator $M$ interacts with $S$ and with fresh ancillas.
Conditional feedback	Measurement outcomes from earlier ancillas influence later collisions.

In these cases, the system state $\rho_n$ alone may not determine the next state. One may need the state of a memory carrier, a history of outcomes, or correlations between the system and unobserved ancillas.

Correlated Ancillas

Suppose the incoming ancillas are not initially in a product state:

$$\eta_{A_1A_2} \ne \eta_{A_1}\otimes\eta_{A_2}.$$

The first collision can create correlations between $S$ and the still-unvisited $A_2$ through the initial $A_1$–$A_2$ correlations. An unconditioned operation on $S+A_1$ does not change the marginal state of $A_2$; there is no signaling to an untouched ancilla. The relevant memory lies in the joint state of $S+A_2$, which affects their later collision.

The reduced dynamics from $\rho_1$ to $\rho_2$ may not be representable by a single channel acting on all possible system states. This is the same conceptual issue that appears in Initial Correlations: once the system is correlated with degrees of freedom outside the reduced description, a universal reduced map may fail to exist on the full state space.

Ancilla–Ancilla Collisions

Another common construction allows environmental carriers to interact:

S collides with A1
A1 collides with A2
S collides with A2
A2 collides with A3
...

The $A_1$-$A_2$ interaction transfers a record of the first system collision into the second ancilla before it meets the system. The next carrier is no longer fresh. In a continuum limit, this kind of mechanism can approximate environments with finite propagation time, memory kernels, or structured correlations.

This construction is useful because the model is still built from ordinary unitary gates and partial traces. Memory is not inserted by hand as a negative rate; it is carried by explicit degrees of freedom.

Memory Register Form

Many non-Markovian collision models can be made Markovian by enlarging the system to include a memory register $M$. At each step, $S+M$ evolves by a channel

$$\rho_{SM}^{(n)} = \Xi_n \left( \rho_{SM}^{(n-1)} \right),$$

where $\Xi_n$ is CPTP. The observed system state is

$$\rho_S^{(n)} = \operatorname{Tr}_M \rho_{SM}^{(n)}.$$

The reduced $S$ dynamics can show memory, but the enlarged $S+M$ dynamics is Markovian at the level of channels. This mirrors Pseudomode Methods and Reaction-Coordinate Mapping: memory often becomes simpler when the correct carrier is promoted into the system boundary.

Relation to Divisibility

For fresh ancillas, the discrete intermediate map

$$\Phi_{n,m} = \Phi_n\cdots\Phi_{m+1}$$

is CPTP. The evolution is CP-divisible at the collision times.

With correlated or recycled ancillas, an apparent intermediate map

$$\rho_m\mapsto\rho_n$$

may depend on hidden correlations not contained in $\rho_m$. If one reconstructs a family of maps from a restricted preparation set, intermediate maps can fail complete positivity. That failure is not a violation of total unitary dynamics; it is a warning that the reduced state is not a complete state of knowledge for future collisions.

For diagnostics, use CP Divisibility, Information Backflow, and Non-Markovianity Measures.

Relation to Memory Kernels

A collision model is naturally discrete. Eliminating the ancillas can produce a recurrence relation

$$\rho_{n+1} = \sum_{k=0}^{n} \mathcal K_{n,k}\rho_k + \mathcal J_n,$$

where the kernels encode how earlier reduced states influence the next one. In a continuum limit, this can become a time-nonlocal master equation. The exact form depends on how ancillas are correlated or coupled to one another.

This is the discrete counterpart of Memory Kernels. Collision models are often easier to simulate because the memory is stored in explicit auxiliary systems rather than in a dense history integral.

What Collision Models Are Good For

Collision models are especially useful for:

• deriving simple channels from explicit unitary interactions;
• constructing discrete approximations to Lindblad semigroups;
• modeling quantum thermodynamic operations step by step;
• building non-Markovian toy models with controllable memory carriers;
• simulating waveguide, cascaded, or delayed-feedback intuition in a finite circuit picture;
• separating genuine memory from artifacts of a poor master-equation approximation;
• teaching open-system ideas without starting from continuum bath integrals.

They are also a natural bridge between open-system physics, quantum circuits, and quantum information.

Limitations

Collision models are not automatically realistic. The modeler must justify:

• the ancilla Hilbert space;
• the collision time;
• the unitary interaction;
• the initial ancilla state and correlations;
• whether carriers are really fresh;
• whether a continuous-time limit exists;
• whether the stroboscopic time step resolves the phenomena of interest.

Different collision models can produce the same reduced channel. The microscopic story is therefore not unique unless additional physical information fixes the carriers and interactions.

Exercises

One-step CPTP map

Show that

$$\Phi(\rho) = \operatorname{Tr}_A \left[ U(\rho\otimes\eta)U^\dagger \right]$$

is trace preserving and completely positive.

Solution

The map is a composition of three completely positive maps: tensoring with a fixed positive state $\eta$, conjugating by a unitary $U$, and tracing out $A$. Therefore it is completely positive.

For trace preservation,

$$\operatorname{Tr}_S\Phi(\rho) = \operatorname{Tr}_{SA} \left[ U(\rho\otimes\eta)U^\dagger \right] = \operatorname{Tr}_{SA}(\rho\otimes\eta) = \operatorname{Tr}_S\rho\,\operatorname{Tr}_A\eta = \operatorname{Tr}_S\rho.$$

Thus $\Phi$ is CPTP.

Partial-swap decay

In the partial-swap example, show that the excited-state population after $n$ fresh collisions is $q^n\rho_{ee}^{(0)}$.

Solution

One collision maps

$$\rho_{ee}^{(k+1)} = q\rho_{ee}^{(k)}.$$

Iterating gives

$$\rho_{ee}^{(n)} = q^n\rho_{ee}^{(0)}.$$

If $q=e^{-\gamma\tau}$ and $t=n\tau$, then

$$q^n = e^{-\gamma n\tau} = e^{-\gamma t}.$$

Reusing one ancilla

A qubit collides twice with the same ground-state ancilla, using a full SWAP each time. Compare initial system states $\lvert e\rangle$ and $\lvert g\rangle$. What are their trace distances after zero, one, and two collisions?

Solution

The first SWAP puts either system in $\lvert g\rangle$ and transfers the original state into the ancilla. The second SWAP restores that state. The distances are therefore $1,0,1$. No channel acting only on the identical system states after the first collision can produce different states after the second. The ancilla is the missing memory. Replacing it by a fresh ground-state ancilla would keep both system states identical.

Correlated ancillas

Explain why initial correlations between $A_1$ and $A_2$ can make the second collision depend on the first.

Solution

The first collision can establish $S$–$A_2$ correlations even while the unconditioned marginal $\eta_{A_2}$ remains unchanged. The second collision acts on this correlated joint state, so its output need not be determined by $\rho_S$ and $\eta_{A_2}$ separately. Conditioning on measured outcomes is a different operation and may also change the conditional state of $A_2$.

References

• S. Attal and Y. Pautrat, “From repeated to continuous quantum interactions,” Annales Henri Poincaré 7, 59–104 (2006).
• H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, “Colloquium: Non-Markovian dynamics in open quantum systems,” Reviews of Modern Physics 88, 021002 (2016).
• F. Ciccarello, G. M. Palma, and V. Giovannetti, “Collision-model-based approach to non-Markovian quantum dynamics,” Physical Review A 87, 040103(R) (2013).
• F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma, “Quantum collision models: Open system dynamics from repeated interactions,” Physics Reports 954, 1–70 (2022).
• J. Rau, “Relaxation Phenomena in Spin and Harmonic Oscillator Systems,” Physical Review 129, 1880–1888 (1963).
• Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quantum non-Markovianity: characterization, quantification and detection,” Reports on Progress in Physics 77, 094001 (2014).