Decoherence and Classicality

Decoherence makes interference between selected alternatives inaccessible to measurements on a subsystem by correlating those alternatives with an environment. It explains a major part of practical classicality. It does not, by itself, replace a pure entangled global state by one definite outcome.

Required background. Density operators: Distinguish a density operator, its coherences, and its outcome probabilities.

Local equivalence and global difference

The technical account is environment-induced decoherence. Its foundational consequence can be seen from

$$|\Psi\rangle=\alpha|0,E_0\rangle+\beta|1,E_1\rangle.$$

The system density operator has off-diagonal entry $\alpha\beta^*\langle E_1|E_0\rangle$. When the environment states are orthogonal, all system-only observables have the same expectations as the mixture

$$\rho_{\rm mix}=|\alpha|^2|0\rangle\langle0| +|\beta|^2|1\rangle\langle1|.$$

The equivalence concerns a specified restricted set of observables. A global observable coupling $|0,E_0\rangle$ to $|1,E_1\rangle$ can still distinguish the entangled pure state from a corresponding global mixture.

How accurate is the local replacement?

For this two-state model, let $\rho_{\rm diag}$ be the state obtained by removing the off-diagonal entries. The trace distance is

$$\frac12\|\rho_S-\rho_{\rm diag}\|_1 =|\alpha\beta|\,|\langle E_1|E_0\rangle|.$$

This bounds the difference in the probability of any system-only yes/no outcome. For equal amplitudes and overlap $10^{-6}$, that maximum difference is $5\times10^{-7}$. The result gives a precise operational meaning to “negligible coherence.”

It does not bound measurements acting jointly on the environment. It also assumes the two alternatives and environment states are those of the model.

Preferred structure is dynamical

Every density operator can be diagonalized. Merely choosing its eigenbasis does not explain a persistent classical record. Robust alternatives must remain distinguishable and stable under the actual interaction and system Hamiltonian.

Pointer states and einselection address this dynamical question. A basis selected at one instant can be unstable immediately afterward.

What interpretations add

Everettian accounts read decohered components as branch-relative records. Bohmian accounts use disjoint environmental branches to justify effective collapse for the occupied configuration. Collapse theories add an actual stochastic modification. Operational accounts use the reduced state for predictions while taking records as given.

The common decoherence calculation does not decide among these additions. Calling the reduced state an “improper mixture” emphasizes that its mixedness can arise from entanglement rather than classical ignorance about a selected pure subsystem state.

Exercises

For $|\alpha|^2=0.9$, $|\beta|^2=0.1$, and environment overlap $0.01$, calculate the trace distance from the diagonal state.

Solution

The distance is $\sqrt{0.9(0.1)}(0.01)=0.003$. Every system-only outcome probability changes by at most that amount. This statement says nothing about which alternative becomes actual, nor about access to a suitable joint system–environment measurement.

References

• M. Schlosshauer, “Decoherence, the measurement problem, and interpretations of quantum mechanics,” Reviews of Modern Physics 76, 1267–1305 (2005), author manuscript.
• W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Reviews of Modern Physics 75, 715–775 (2003), doi:10.1103/RevModPhys.75.715.