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For a subsystem $A$ of a multipartite system with density operator ${\rho }_{A{E}_1\cdots E_n}$, its reduced density matrix is the unique operatorsuch that $\mathrm{Tr}(M_A{\rho }_A)=\mathrm{Tr}[(M_A\otimes I_E){\rho }_{AE}]$ for every observable $M_A$ on $A$. In an orthonormal basis $\{|e_j⟩\}$ of the other subsystems, the partial trace iswhich is independent of the chosen basis.

Solved by gpt-5.6-sol high.

Write $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. Before any interaction the system is in the pure state $|{\psi }_S⟩=\alpha |0⟩+\beta |1⟩$, soOne interaction leaves the environmental state $|+⟩$ unchanged on the system branch $|0⟩$, while on the branch $|1⟩$ it producesAfter interactions with $n$ distinct environment qubits, the joint state isand the decoherence factor is the product $(\cos ϵ{)}^n$. Taking the partial trace givesFor small $ϵ$, $(\cos ϵ{)}^n\simeq e^{-n{ϵ}^2/2}$: the populations remain fixed while the phase coherence decays. This is environmental decoherence, caused by entanglement with unobserved records rather than by a nonunitary evolution of the complete state.

A measurement in the computational basis cannot reveal the decay because its probabilities remain $|\alpha {|}^2$ and $|\beta {|}^2$. An interference measurement can. For example, measuring in the Hadamard basis givesChanging the measurement phase similarly accesses the imaginary part, so the shrinking interference visibility directly displays decoherence.

If the same environment qubit is reused, the conditional rotation accumulates coherently. After $n$ interactions the overlap is $\cos (nϵ)$ rather than $(\cos ϵ{)}^n$, and the off-diagonal entries oscillate. Coherence vanishes at some times but returns periodically: this finite environment exhibits quantum recoherence rather than effectively irreversible decay.

Solved by gpt-5.6-sol high.