Closed vs Open Quantum Systems

A closed quantum system is modeled by keeping all degrees of freedom relevant to the prediction, so its state evolves by unitary dynamics. An open quantum system is modeled as a subsystem whose dynamics is affected by degrees of freedom that are ignored, measured, coarse-grained, or represented phenomenologically.

The distinction is not absolute. It is a modeling decision:

$$\text{closed or open} \quad \text{depends on the chosen boundary of the system}.$$

A single laboratory setup can be described as closed if the system, apparatus, and environment are all included; as open if only the system is retained; as conditional if a measurement record is used; or as phenomenological if a noise model is fitted directly.

Closed-System Dynamics

For a closed system with Hamiltonian $H(t)$, pure states obey the Schrödinger equation and density operators obey the Liouville–von Neumann equation:

$$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H(t),\rho].$$

Equivalently,

$$\rho(t) = U(t,t_0)\rho(t_0)U^\dagger(t,t_0),$$

where $U(t,t_0)$ is unitary. This form preserves trace, positivity, eigenvalues of $\rho$, and von Neumann entropy for the closed system.

Closed does not mean “no time-dependent controls.” A driven two-level system with a specified time-dependent Hamiltonian can still be closed at the level of the modeled degrees of freedom. The drive may be an externally prescribed classical field, and the quantum state can still evolve unitarily under $H(t)$.

Open-System Dynamics

An open-system description keeps a subsystem $S$ and discards or conditions on something else. The minimal composite model uses

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

If the total state is $\rho_{SE}(t)$, the reduced system state is

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

Even if $\rho_{SE}(t)$ evolves unitarily, $\rho_S(t)$ need not. Information, phase coherence, and energy can move into correlations with $E$.

For an initially uncorrelated system-environment state,

$$\rho_{SE}(0) = \rho_S(0)\otimes\eta_E,$$

the reduced dynamics can be written as a quantum channel:

$$\rho_S(t) = \Phi_t(\rho_S(0)).$$

A microscopic expression is

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

This formula is the basic bridge between closed dynamics of a larger system and open dynamics of a subsystem.

Four Common Descriptions

Description	What is retained?	Typical object
closed system	all modeled degrees of freedom	unitary $U$ or Hamiltonian $H$
reduced open system	subsystem only; environment ignored	channel $\Phi$ or master equation
conditional measurement	subsystem plus observed record	conditional state $\rho_i$ or trajectory
phenomenological noise model	calibrated effect of ignored details	fitted channel, rate, or Lindblad operators

These descriptions are not competitors. They answer different questions. If a detector record is known, a conditional state is appropriate. If the record is discarded, the unconditional state is appropriate. If the apparatus is explicitly included, the system-apparatus state may evolve unitarily while the system alone evolves nonunitarily.

Examples

Physical situation	Natural first model	Why
isolated spin in a specified magnetic field	closed	unitary spin precession under a Hamiltonian
qubit with unknown phase noise	open	phase information is lost to uncontrolled degrees of freedom
atom coupled to vacuum radiation modes	open for atom, closed for atom plus field	spontaneous emission requires field degrees of freedom
detector outcome is recorded	conditional measurement	state assignment depends on the outcome
detector outcome is produced but ignored	channel	average over the outcome-conditioned maps
damped cavity mode with a Markovian reservoir	master equation	reduced dynamics may be time local

The same physical object can move between columns as the model boundary changes.

What Changes When a System Is Open

Closed unitary evolution preserves inner products between pure states. Open dynamics need not preserve them for the subsystem. A channel may shrink Bloch vectors, remove off-diagonal density-matrix elements, transfer population, or drive states toward a thermal or nonequilibrium steady state.

For example, an ideal pure dephasing map for a qubit has the schematic effect

$$\begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} \longmapsto \begin{pmatrix} \rho_{00} & \lambda\rho_{01} \\ \lambda^*\rho_{10} & \rho_{11} \end{pmatrix}, \qquad |\lambda|\le1.$$

The populations stay fixed, while coherence in the displayed basis is reduced. This is open-system behavior even though it can be derived from unitary dynamics on $S+E$.

Boundary Cases

Classical control fields

A prescribed classical control field can make $H(t)$ explicitly time dependent without making the system open in the density-operator sense. The state still evolves by a unitary operator. Energy may not be conserved, but the evolution is still closed for the modeled system.

Effective non-Hermitian Hamiltonians

An effective non-Hermitian Hamiltonian often describes conditional evolution, absorption, decay out of a subspace, or a no-jump trajectory. It is not ordinary closed-system evolution. The missing norm or probability must be accounted for by jumps, reservoirs, detectors, or omitted channels.

Initial correlations

The formula $\rho_{SE}(0)=\rho_S(0)\otimes\eta_E$ is an assumption, not a theorem. If $S$ and $E$ are initially correlated, the reduced dynamics may not be represented by a single channel acting only on arbitrary $\rho_S(0)$ without additional structure.

Markovianity

Open does not mean Markovian. A reduced state can depend on environmental memory. A Markovian master equation is a further approximation or structural result, not the definition of an open system.

Diagnostic Questions

Before choosing a formalism, ask:

1. Which degrees of freedom are included in the state?
2. Which degrees of freedom are traced out, averaged over, or represented by rates?
3. Is there a measurement record, and is it conditioned on?
4. Is the Hamiltonian self-adjoint on the modeled Hilbert space?
5. Is the resulting state map trace preserving?
6. Is complete positivity required because the system may be entangled with a reference?
7. Are memory effects negligible enough for a time-local master equation?

These questions are more reliable than deciding from vocabulary alone.

Common Mistakes

• Saying “open” whenever a Hamiltonian is time dependent.
• Saying “closed” merely because the model has a Hamiltonian.
• Treating an effective non-Hermitian Hamiltonian as ordinary unitary dynamics.
• Forgetting that tracing out an environment can produce mixed reduced states from pure total states.
• Assuming every open-system map is Markovian.
• Using a channel when a known measurement record requires a conditional state.
• Ignoring initial correlations when deriving reduced dynamics.

Cross-Links

• Concept Map
• Unitary Time Evolution
• Hamiltonians as Generators of Time Evolution
• Liouville–von Neumann Equation
• Partial Trace
• Reduced Density Operators
• Decoherence Preview

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer, 2012.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. A spin-$1/2$ particle evolves under $H(t)=-\gamma B(t)S_z$, where $B(t)$ is a prescribed classical control field. Is this automatically an open-system model?

Solution

No. A time-dependent Hamiltonian can still generate unitary closed-system evolution for the modeled spin. The model becomes open only if additional degrees of freedom are ignored, averaged over, measured, or represented by nonunitary dynamics.

2. Let $S$ and $E$ begin in

$$\left(\alpha\lvert0\rangle+\beta\lvert1\rangle\right)\lvert e_0\rangle,$$

and evolve to

$$\alpha\lvert0\rangle\lvert e_0\rangle + \beta\lvert1\rangle\lvert e_1\rangle.$$

What happens to the off-diagonal element of $\rho_S$ after tracing out $E$?

Solution

The reduced state has off-diagonal element

$$(\rho_S)_{01} = \alpha\beta^*\langle e_1|e_0\rangle.$$

If $\langle e_1|e_0\rangle$ is small, coherence between $\lvert0\rangle$ and $\lvert1\rangle$ is suppressed in the reduced state. The total state may still be pure; the apparent loss of coherence comes from ignoring $E$.

3. A detector produces an outcome, but the data file is lost before analysis. Should the later system state be conditioned on a particular outcome?

Solution

No. If the outcome is not available, predictions should use the nonselective state obtained by averaging over the outcome-conditioned maps. Conditioning on a guessed outcome would introduce information not available to the model.