Closed vs Open Quantum Systems

A closed-system model evolves the retained quantum state unitarily. An open-system model describes a subsystem influenced by degrees of freedom outside it. The same atom can be open when radiation is ignored and part of a closed model when the relevant radiation field is included. What matters is the chosen system and the information used for predictions, rather than whether the laboratory apparatus looks isolated.

Required background. Use Unitary Time Evolution for closed dynamics, Density Operators for general states, and Partial Trace for subsystem predictions. One Interaction, Different Measurement Records connects these descriptions to measurements.

Unitary evolution and the system boundary

For a specified self-adjoint Hamiltonian with a well-defined unitary propagator,

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

In finite dimension, this preserves the eigenvalues, purity, and von Neumann entropy of the state. A time-dependent Hamiltonian can still produce unitary evolution. For example, a spin in a prescribed classical magnetic field may exchange work with the source while remaining a closed quantum model. “Closed” in this sense does not imply constant energy or a thermodynamically isolated apparatus.

The approximation is that fluctuations and backaction of the source can be neglected. If those matter, include the source explicitly or model their effect on the spin. Hamiltonians as Generators explains the closed dynamics; Driven Open Systems adds environmental effects.

Reduced dynamics can lose and regain coherence

For a joint state $\rho_{SE}$, the system state is always $\rho_S=\operatorname{Tr}_E\rho_{SE}$. If the environment is initialized in the same state $\eta_E$ for every input and initially uncorrelated with the system, then

$$\Phi_t(\rho)=\operatorname{Tr}_E \!\left[U_t(\rho\otimes\eta_E)U_t^\dagger\right]$$

is a channel on arbitrary system inputs. The fixed-environment assumption is essential to that statement. A known correlated initial joint state still has perfectly meaningful reduced dynamics, but its reduced state alone need not determine the future. See Initial Correlations.

An open system need not become steadily more mixed. Consider two qubits with

$$\begin{gathered} H_{SE}=\hbar g\,Z_S\otimes Z_E,\\ |\psi(0)\rangle=|+\rangle_S|+\rangle_E. \end{gathered}$$

Here $g>0$ is a frequency. The system has populations $1/2$ and off-diagonal element $\cos(2gt)/2$. Consequently,

$$\operatorname{Tr}\rho_S(t)^2 =\frac{1+\cos^2(2gt)}2.$$

At $t=\pi/(4g)$ it is maximally mixed; at $t=\pi/(2g)$ it is pure again. The joint state remains pure throughout. A small environment stores correlations and later returns coherence, so tracing it out does not imply permanent irreversibility. A large bath with rapidly dispersing correlations can behave very differently.

A retained record changes the prediction

Description	State used for predictions
Joint system and apparatus	$\rho_{SE}$, retaining their correlations
System with environment ignored	$\operatorname{Tr}_E\rho_{SE}$
System after a known outcome $r$	$\mathcal I_r(\rho)/p_r$, provided $p_r>0$
System after outcomes are averaged	$\sum_r\mathcal I_r(\rho)$

Discarding a record changes what can be predicted conditionally. It does not require a second physical disturbance after the measurement has already occurred. Conversely, ignoring a measurement outcome does not undo the interaction that produced the record. Selective and Nonselective Measurements gives concrete examples.

A phenomenological noise model supplies a channel or generator directly, often from calibrated data. Its successful predictions need not identify a unique microscopic environment. An explicit system–bath model instead exposes couplings and correlation times, but still needs approximations when reduced to a simple evolution law.

Two distinctions that survive changing the boundary

Open versus Markovian. “Open” describes the subsystem relationship. A semigroup or a particular definition of Markovianity places additional conditions on evolution over time. The two-qubit example is open but has revivals. A time-local equation can also describe some non-Markovian dynamics; locality in the time variable alone does not establish absence of memory.

Unitary versus conditional no-click evolution. A non-Hermitian effective Hamiltonian can describe survival in a subspace or evolution conditioned on seeing no photon. Its decreasing squared norm is a probability. Renormalizing the surviving state gives a conditional prediction, not a closed unitary evolution. The omitted outcomes must be included to recover the full ensemble; see Quantum Jump Trajectories.

Exercises

An open system can become purer. In the two-qubit example, find the reduced-state eigenvalues and identify when the system is entangled with the environment.

Solution

The eigenvalues are $(1\pm\cos 2gt)/2$, with their ordering interchanged when the cosine changes sign. Since the joint state is pure, it is entangled exactly when the system is mixed: $|\cos2gt|<1$. The system becomes pure at $t=n\pi/(2g)$, so subsystem entropy need not increase monotonically.

Uncertain classical control. A spin evolves unitarily under each known field realization. If the field is random and unrecorded, must its averaged state evolve unitarily?

Solution

No. Averaging gives a mixture of unitary maps, $\Phi(\rho)=\sum_j p_jU_j\rho U_j^\dagger$ for discrete realizations. This is a channel, but generally not a single unitary conjugation. For random opposite phase rotations it can reduce the transverse Bloch components. Randomness in the unobserved control can therefore produce an open description even when each realization has unitary dynamics.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Reduced dynamics and memory.
• Nielsen, M. A., and I. L. Chuang. Quantum Computation and Quantum Information. 10th anniversary edition, Cambridge University Press, 2010. Channels, partial trace, and mixed states.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Conditioning and unconditioned evolution.