Stability Verification of Quantum non-i.i.d. sources

We introduce the problem of stability verification of quantum sources which are non-i.i.d.. The problem consists in ascertaining whether a given quantum source is stable or not, in the sense that it produces always a desired quantum state or if it suffers deviations. Stability is a statistical notion related to the sparsity of errors. This problem is closely related to the problem of quantum verification first proposed by Pallister et. al. [1], however, it extends the notion of the original problem. We introduce a family of states that come from these non-i.i.d. sources which we call a Markov state. These sources are more versatile than the i.i.d. ones as they allow statistical deviations from the norm instead of the more coarse previous approach. We prove in theorem 1 that the Markov states are not well described with tensor products over a changing source. In theorem 2 we further provide a lower bound on the trace distance between two Markov states, or conversely, an upper bound on the fidelity between these states. This is a bound on the capacity of determining the stability property of the source, which shows that it is exponentially easier to ascertain this with respect to n, the number of outcomes from the source.