Non-Markovian memory kernels

Take a system qubit and one environment qubit with interaction Hamiltonian $H_{SE}=\hbar g Z_S\otimes Z_E/2$, and prepare the environment in its plus state. Remove any commuting free Hamiltonians by changing frame. Conditional environment rotations give the overlap $\eta(t)=\cos(gt)$, multiplying the system off-diagonal entries while leaving its populations unchanged. This is exact for the stated finite composite, with no Born or Markov approximation.

An equivalent channel is the equal mixture of system phase rotations generated by the two environment Z eigenvalues. Consequently every initial-time map is a valid channel even when the cosine becomes negative. A negative coherence factor is a phase reversal, not a negative population. The magnitude returns to one after reaching zero, so local purity can revive. A one-qubit environment is not a thermal reservoir causing irreversible decay; its finite information capacity makes recurrence central rather than negligible.

Where the cosine is nonzero, divide its derivative by itself to obtain a time-local dephasing equation. The resulting rate becomes negative during intervals of coherence revival. This does not invalidate the original channel from the initial preparation. It instead signals that intermediate propagation cannot always be represented as a completely positive dephasing channel acting on arbitrary inputs. At the cosine zeros, the reduced map is noninvertible and the time-local rate becomes singular even though the exact composite evolution is smooth.

The convolution description avoids that singular inverse by retaining a history integral. More general projection-operator equations can include an inhomogeneous term from initial correlations, as well as complicated superoperator kernels. Merely choosing a decaying kernel does not guarantee complete positivity; an approximate memory equation must be checked just as carefully as an approximate local equation. Conversely, writing an equation locally in time does not establish that it represents a memoryless physical process.

Completely positive divisibility asks whether each later map factors through an earlier map with a valid intermediate channel. Trace-distance backflow instead asks whether some initially distinct states become more distinguishable over an interval. Contractivity of positive trace-preserving propagation makes an increase a useful witness, but the two criteria are not equivalent for every system dimension and process. Multi-time conditional statistics provide still more information, so a label of non-Markovianity should name the operational or mathematical criterion.

For this finite-bath example, opposite equatorial inputs become the same local state at a zero of the cosine. Later they become distinguishable again. No map on that single identical intermediate state can recover two different outputs without additional information, demonstrating why the environment matters. Resetting the environment at the intermediate time changes the experiment and can eliminate this return. Parameter changes in a teaching instrument likewise should not be interpreted as implementing a continuous physical bath protocol.