A mixed-unitary channel applies a random unitary to a quantum state — flip the qubit with probability p, leave it alone with probability 1-p. Mixed-unitary channels are the simplest form of quantum noise: classical randomness choosing between deterministic quantum operations. Every such channel preserves the maximally mixed state (it’s unital), but not every unital channel is mixed-unitary.

The distinction matters because mixed-unitary noise is classically correctable in principle — if you knew which unitary was applied, you could undo it. Non-mixed-unitary noise involves genuinely quantum irreversibility that no classical knowledge can fix. Characterizing which channels are mixed-unitary has been an open problem with implications for error mitigation.

The connection to memory: quantum channels with non-Markovian dynamics — channels where the system remembers its history through correlations with its environment — cannot be mixed-unitary. Memory and classical randomness are incompatible. If the dynamics have genuine quantum memory (detected through the process tensor formalism), then the noise cannot be decomposed as a probabilistic mixture of unitaries.

This yields a practical witness. The process tensor encodes the full multi-time statistics of the dynamics. A hierarchy of semidefinite programs, applied to the process tensor, efficiently detects non-mixed-unitary behavior — outperforming existing criteria that check matrix-theoretic conditions one channel at a time.

Quantum memory is the signature of noise that no classical strategy can undo. The system that remembers is the system that cannot be saved by forgetting which error occurred.