Quantum decoherence

Remarks

Example: let $\rho_E = \lvert \chi \rangle\langle \chi \rvert$ and let $U(t)(e_i \otimes \chi) = e_i \otimes \chi_i(t)$ with unit vectors $\chi_i(t) \in \mathcal{H}_E$. Then $\langle e_i, \rho_S(t) e_j \rangle = \langle e_i, \rho_S e_j \rangle \langle \chi_j(t), \chi_i(t) \rangle$, so clause 1 holds. Clause 2 holds exactly when the overlaps $\langle \chi_j(t), \chi_i(t) \rangle$ tend to $0$, that is, when the environment records which $e_i$ the system is in. Non-example: if $U(t) = V(t) \otimes W(t)$ is a product of unitaries, then $\rho_S(t) = V(t)\rho_S V(t)^*$ stays pure for pure $\rho_S$. A pure state with fixed diagonal entries $p_i$ has off-diagonal entries of modulus $\sqrt{p_i p_j}$, so clause 2 fails. Let $\mathcal{H}_E$ be finite-dimensional and $U(t) = e^{-itH}$ for a fixed Hamiltonian $H$. Then the entries of $\rho_S(t)$ are almost periodic in $t$, so they cannot tend to $0$ unless they vanish, and clause 2 fails. Exact decay therefore needs an environment with infinitely many degrees of freedom, and in physical models it is often exponential, with a rate called the decoherence rate. This is one common way to make the notion precise. In physics, the word is often used more loosely, for the approximate suppression of off-diagonal entries.