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Process estimation in the presence of time-invariant memory effects

Any repeated use of a fixed experimental instrument is subject
to memory effects. We design an estimation method uncovering
the details of the underlying interaction between the system
and the internal memory without having any experimental access
to memory degrees of freedom. In such case, by definition, any
memoryless quantum process tomography (QPT) fails because the
observed data sequences do not satisfy the elementary condition
of statistical independence. However, we show that the
randomness implemented in certain QPT schemes is sufficient to
guarantee the emergence of observable “statistical” patterns
containing complete information on the memory channels. We
demonstrate the algorithm in detail for the case of qubit
memory channels with two-dimensional memory. Interestingly, we
find that for the arbitrary estimation method, the memory
channels generated by controlled unitary interactions are
indistinguishable from memoryless unitary channels.