Reduced Dynamics

Reduced dynamics is the time evolution assigned to a subsystem after the degrees of freedom outside that subsystem are ignored.

The central formula is:

$$\rho_S(t) = \operatorname{Tr}_E \left[ U_{SE}(t,t_0)\rho_{SE}(t_0)U_{SE}^\dagger(t,t_0) \right].$$

The total system $S+E$ may evolve unitarily. The subsystem $S$ can still evolve nonunitarily because information, coherence, and energy can move into correlations with $E$.

For the standard microscopic starting point that generates $U_{SE}$, see System–Bath Hamiltonians. For the terminology behind environments, baths, reservoirs, noise sources, and records, see Baths, Reservoirs, and Environments.

Exact Reduced State

Let the total Hilbert space be

$$\mathcal H_{SE} = \mathcal H_S\otimes\mathcal H_E.$$

If the total state at time $t$ is $\rho_{SE}(t)$, then the reduced state of $S$ is

$$\rho_S(t) = \operatorname{Tr}_E\rho_{SE}(t).$$

If the total evolution is unitary,

$$\rho_{SE}(t) = U_{SE}(t,t_0)\rho_{SE}(t_0)U_{SE}^\dagger(t,t_0).$$

Combining these equations gives the exact reduced dynamics formula:

$$\rho_S(t) = \operatorname{Tr}_E \left[ U_{SE}(t,t_0)\rho_{SE}(t_0)U_{SE}^\dagger(t,t_0) \right].$$

This equation is always true when the total unitary model and initial total state are specified. It is not yet a closed equation for $\rho_S(t)$ alone.

Factorized Initial State

A common starting assumption is an initially uncorrelated system-environment state:

$$\rho_{SE}(t_0) = \rho_S(t_0)\otimes\eta_E,$$

where $\eta_E$ is a fixed environment reference state.

Then the reduced state can be written as a map on the initial system state:

$$\rho_S(t) = \Phi_{t,t_0}\bigl(\rho_S(t_0)\bigr),$$

with

$$\Phi_{t,t_0}(\rho) = \operatorname{Tr}_E \left[ U_{SE}(t,t_0)(\rho\otimes\eta_E) U_{SE}^\dagger(t,t_0) \right].$$

This map is completely positive and trace preserving. It is a quantum channel.

The factorized-state assumption is not a theorem. It is a modeling assumption that can fail when the system and environment are already correlated before the time interval of interest.

Kraus Form from the Environment

The channel form can be made explicit. Diagonalize the environment reference state:

$$\eta_E = \sum_r q_r |r\rangle\langle r|, \qquad q_r\ge0, \qquad \sum_r q_r=1.$$

Choose an orthonormal environment basis $\{|\mu\rangle\}$ for the final partial trace. Define operators on the system by

$$K_{\mu r}(t,t_0) = \sqrt{q_r}\, \langle\mu| U_{SE}(t,t_0) |r\rangle.$$

The bracket is taken only over the environment Hilbert space, leaving a system operator. Then

$$\Phi_{t,t_0}(\rho) = \sum_{\mu,r} K_{\mu r}(t,t_0)\rho K_{\mu r}^\dagger(t,t_0).$$

The trace-preserving condition follows from unitarity:

$$\sum_{\mu,r} K_{\mu r}^\dagger K_{\mu r} = I_S.$$

This derivation is one of the cleanest ways to see why reduced dynamics is described by completely positive maps when the initial state is factorized.

What Nonunitarity Means

The reduced map $\Phi_{t,t_0}$ usually does not have the form

$$\rho_S \longmapsto U_S\rho_SU_S^\dagger.$$

It may shrink Bloch vectors, suppress off-diagonal elements, transfer population, increase entropy, or drive the system toward a steady state.

This does not contradict unitarity of the larger model. The subsystem map is nonunitary because the subsystem is not closed.

The global state may retain purity and phase information, while the local reduced state appears mixed:

$$\text{global unitary evolution} \quad\not\Rightarrow\quad \text{subsystem unitary evolution}.$$

Reduced Dynamics and Decoherence

Decoherence is a special reduced-dynamics pattern. In a two-branch model,

$$\left( c_0|0\rangle+c_1|1\rangle \right) |E_{\mathrm{ready}}\rangle \longrightarrow c_0|0\rangle|E_0(t)\rangle + c_1|1\rangle|E_1(t)\rangle.$$

Tracing out $E$ gives

$$\rho_{01}(t) = \langle E_1(t)|E_0(t)\rangle\rho_{01}(0).$$

The environment overlap becomes the decoherence factor. This is reduced dynamics: the full state evolves coherently, but the subsystem loses locally accessible interference.

For the conceptual treatment, see What Is Decoherence?. For environmental monitoring, scattering, and rate estimates, see Environment-Induced Decoherence.

Reduced Dynamics and Dissipation

Energy relaxation is another reduced-dynamics pattern. If an atom emits into field modes that are not retained, the atomic reduced state may relax from an excited state toward a lower state. In an effective qubit model this can become amplitude damping.

The important point is that dephasing and dissipation are both reduced dynamics, but they are different reduced dynamics:

dephasing:     phase information moves into correlations
dissipation:  energy or excitation moves into the environment

The same microscopic Hamiltonian can produce both effects, depending on the coupling operators, the bath spectrum, and the energy scales.

Conditional Versus Unconditional Dynamics

Reduced dynamics usually means the environment is ignored. If instead some environmental record is observed, the state of $S$ should be conditioned on that record.

For example:

• ignoring emitted photons gives an unconditional master equation;
• detecting emitted photons gives quantum jumps;
• ignoring a detector readout gives a channel;
• retaining the readout gives a quantum instrument or trajectory.

The unconditional state is an average over records. The conditional state is the state assigned after a particular record is known. Mixing these descriptions is a common source of confusion.

Initial Correlations

If the initial total state is not factorized,

$$\rho_{SE}(t_0)\ne \rho_S(t_0)\otimes\eta_E,$$

then the reduced state at time $t$ still exists:

$$\rho_S(t) = \operatorname{Tr}_E \left[ U_{SE}\rho_{SE}(t_0)U_{SE}^\dagger \right].$$

But there may be no single channel $\Phi$ that maps every possible $\rho_S(t_0)$ to $\rho_S(t)$ independently of the hidden correlations. Two different total states can have the same reduced state on $S$ and later lead to different reduced states after the same joint unitary.

One may sometimes define a map on a restricted compatibility domain, or introduce an assignment map that reconstructs $\rho_{SE}$ from $\rho_S$. Such maps require care and need not be completely positive on arbitrary system states. The canonical treatment is Initial Correlations.

This is why initial correlations are not a technical footnote. They determine whether reduced dynamics can be represented as a universal channel on the system alone.

Time Locality and Memory

The existence of maps $\Phi_{t,t_0}$ does not imply Markovian dynamics. A reduced state can depend on information that flowed into the environment at earlier times and later returned.

A Markovian semigroup has the special form

$$\Phi_t=e^{t\mathcal L}, \qquad t\ge0,$$

with a generator $\mathcal L$ of Lindblad form under the usual assumptions. This is a much stronger structure than the exact reduced-state formula.

More generally, reduced dynamics may require:

• time-dependent generators;
• memory kernels;
• enlarged effective environments;
• non-Markovian process descriptions;
• explicit system-environment simulations.

Open-system modeling is therefore a hierarchy:

exact S+E unitary
  -> exact reduced state
  -> channel for factorized initial state
  -> time-local master equation if justified
  -> Markovian Lindblad equation under stronger assumptions

For a system field coupled to environmental fields, the Schwinger–Keldysh Bridge shows how the same partial trace becomes an influence functional with causal dissipation and fluctuation kernels.

Diagnostic Questions

Before using a reduced-dynamics model, ask:

• What is the boundary between system and environment?
• Is the initial state factorized, correlated, or only approximately factorized?
• Is the environment state fixed independently of $\rho_S$?
• Is the environment ignored, measured, or fed back?
• Is the resulting map intended to be CPTP on all system states?
• Is a time-local master equation justified, or does the environment retain memory?
• Are the approximations compatible with positivity and trace preservation?

These questions often determine whether a channel, master equation, trajectory, or explicit composite model is appropriate.

Common Mistakes

• Thinking nonunitary reduced dynamics violates unitary quantum mechanics.
• Forgetting the partial trace: $\rho_S(t)$ is not obtained by simply deleting terms from the total state vector.
• Assuming every open-system map is Markovian.
• Assuming every reduced evolution is a channel on arbitrary $\rho_S$ even with initial correlations.
• Treating the environment as ignored in one equation and observed in the next without changing the state-update rule.
• Confusing a phenomenological fitted channel with a microscopic derivation.
• Inferring a unique environment model from a reduced channel.

References

• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012).
• I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Reviews of Modern Physics 89, 015001 (2017).
• F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, “Non-Markovian quantum processes: complete framework and efficient characterization,” Physical Review A 97, 012127 (2018).

Exercises

1. Kraus operators from a unitary dilation. Starting from

$$\Phi(\rho) = \operatorname{Tr}_E \left[ U(\rho\otimes\eta_E)U^\dagger \right],$$

with $\eta_E=\sum_rq_r|r\rangle\langle r|$, derive the Kraus operators $K_{\mu r}=\sqrt{q_r}\langle\mu|U|r\rangle$.

Solution

Insert the spectral decomposition of $\eta_E$:

$$\Phi(\rho) = \sum_r q_r \operatorname{Tr}_E \left[ U(\rho\otimes|r\rangle\langle r|)U^\dagger \right].$$

Now evaluate the partial trace in the basis $\{|\mu\rangle\}$:

$$\Phi(\rho) = \sum_{\mu,r}q_r \langle\mu|U|r\rangle \rho \langle r|U^\dagger|\mu\rangle.$$

Defining

$$K_{\mu r} = \sqrt{q_r}\langle\mu|U|r\rangle$$

gives

$$\Phi(\rho) = \sum_{\mu,r}K_{\mu r}\rho K_{\mu r}^\dagger.$$

2. Trace preservation. Prove that the Kraus operators in the previous exercise satisfy $\sum_{\mu,r}K_{\mu r}^\dagger K_{\mu r}=I_S$.

Solution

Using the definition,

$$\sum_{\mu,r}K_{\mu r}^\dagger K_{\mu r} = \sum_{\mu,r} q_r \langle r|U^\dagger|\mu\rangle \langle\mu|U|r\rangle.$$

The sum over $\mu$ gives the identity on the environment:

$$\sum_\mu|\mu\rangle\langle\mu|=I_E.$$

Thus

$$\sum_{\mu,r}K_{\mu r}^\dagger K_{\mu r} = \sum_r q_r \langle r|U^\dagger U|r\rangle = \sum_r q_r I_S =I_S.$$

3. Initial correlations can matter. Let $S$ and $E$ be qubits. Define two initial total states

$$\rho_{SE}^{(A)} = \frac12|00\rangle\langle00| + \frac12|11\rangle\langle11|,$$

and

$$\rho_{SE}^{(B)} = \frac12|01\rangle\langle01| + \frac12|10\rangle\langle10|.$$

Both have $\rho_S=I/2$. Let $U$ flip $S$ if $E=1$ and do nothing if $E=0$. Compute the final reduced state of $S$ for each initial total state.

Solution

The unitary acts as

$$|s,e\rangle \longmapsto |s\oplus e,e\rangle.$$

For $\rho_{SE}^{(A)}$, the components transform as

$$|00\rangle\to|00\rangle, \qquad |11\rangle\to|01\rangle.$$

Both final components have $S=0$, so

$$\rho_S^{(A)}(t) = |0\rangle\langle0|.$$

For $\rho_{SE}^{(B)}$,

$$|01\rangle\to|11\rangle, \qquad |10\rangle\to|10\rangle.$$

Both final components have $S=1$, so

$$\rho_S^{(B)}(t) = |1\rangle\langle1|.$$

The same initial reduced state $\rho_S=I/2$ led to different final reduced states because the hidden initial correlations were different. Therefore no map of $\rho_S$ alone can describe both cases.