System–environment entanglement and decoherence

Take orthogonal system states $|0\rangle,|1\rangle$ and an initially uncorrelated environment in $|e\rangle$. A controlled interaction sends each system basis state to itself while attaching a normalized conditional environment state. Linear extension determines the evolution of any superposition. The system populations do not change in this model, so if this basis diagonalizes its Hamiltonian, its mean energy is unchanged. Nevertheless the interaction can change all interference experiments involving superpositions of these states.

The conditional environment states need not be orthogonal. Their inner product tells us how well the environment can distinguish the two alternatives. Parallel states leave no distinguishable record; orthogonal states carry a perfectly distinguishable record in principle, whether anyone reads it or not. Such an interaction can be embedded in a unitary transformation because the two specified input vectors are orthogonal and their output vectors remain orthogonal through the system factor. One must also complete the action on the remaining composite basis.

Expand the projector onto the joint final vector before tracing. A diagonal term contains the norm of a conditional environment state, which is one. An off-diagonal term contains the overlap $\eta=\langle e_1|e_0\rangle$. Thus local coherence is suppressed without reducing the joint state purity. Cauchy–Schwarz guarantees an overlap magnitude at most one and thereby keeps the reduced matrix positive. A complex overlap can both reduce contrast and shift the phase of an interference fringe.

For an equal initial superposition, measuring along a transverse direction gives fringe visibility equal to the overlap magnitude. The reduced eigenvalues are half of one plus or minus that magnitude. This is a useful distinction between a coherent phase rotation and decoherence: rotation moves a pure Bloch vector around the sphere, whereas entanglement can shorten the reduced vector. Equal computational-basis populations cannot distinguish the two situations, so a population-only experiment is insufficient to diagnose lost coherence.

With access to the full composite, applying the inverse joint unitary restores the initial product and its interference. Access only to the system generally prevents this recovery by a system unitary, because a unitary cannot change its eigenvalues. Measuring the environment and conditioning on an outcome creates a different, selective description. Averaging those conditioned states reproduces the reduced density operator. The same local mixed state may also arise from a classical mixture, so its preparation history is not recoverable from local tomography alone.

A finite environment can return information and produce revivals. Irreversible exponential decay instead needs further assumptions about many modes, timescales, and accessible controls. Decoherence is also basis dependent: here the interaction selects the displayed basis, and another coupling can select different robust states. Calling the final diagonal matrix a complete solution of the measurement problem goes beyond the calculation. What has been established is suppression of local interference and the presence of correlations, without selecting an observed result.