Memory return in quantum machines

Abstract

We ask whether a quantum machine can repeatedly transform fresh inputs while returning its memory to its initial state. We obtain three results. First, exact memory return prevents a closed machine from purifying maximally mixed inputs, for every initial memory state. With return tolerance δ, exact purification of n qubits requires n/δ + O(n) memory qubits; independent randomization requires only ⌈n/2⌉ and allows exact return. Second, one use of the coherent quantum homogenizer requires a reservoir of size Θ(log²(1/ϵ)/δ²) for output error ϵ and return error δ. Accurate repeated output marginals can coexist with small return error, but two-output correlations impose ϵ + δ/2 ≥ 1/4 when independent outputs and return after every use are required. Third, the exact repeated dynamics give identical iterated deterioration limits in both directions, while a forward-capable product initializer makes the published first-use capability infimum tend to zero. These results separate the return of a memory’s reduced state from its ability to supply fresh independent randomness.