Abstract: Finitely correlated states (FCSs) are quantum states on a chain generated by linear dynamics of a finite-dimensional memory system, possibly described by a generalized probabilistic theory. I will present two recent results:
-A special subclass of FCSs is given by classical states, or equivalently, stochastic processes. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory state is a finite-dimensional quantum state, and at each step it evolves according to a completely positive map. In (1) we showed that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics, by exhibiting explicit examples.
-Translation invariant FCSs can be reconstructed from the knowledge of a marginal of appropriate size. In (2) we showed that marginals of subchains of length t of any translation invariant FCS can be learned, in trace distance, with O(t^2) copies -- with an explicit dependence on local dimension, memory dimension and the condition number of a certain map constructed from the state. The algorithm finds an estimate of a set of parameters of the FCS and reconstructs the desired marginal as a matrix product operator. We can bound the error of the reconstruction with an approach based on the theory of operator systems. A refined error bound can be proven for states generated by quantum memories. We can also obtain an analogous error bound for a class of non-homogenous finitely correlated states reconstructible by local marginals.
(1) Fanizza, M., Lumbreras, J. & Winter, A. Quantum Theory in Finite Dimension Cannot Explain Every General Process with Finite Memory. Commun. Math. Phys. 405, 50 (2024). https://doi.org/10.1007/s00220-023-04913-4
(2) Fanizza, M., Galke, N., Lumbreras, J., Rouzé, C., & Winter, A. Learning finitely correlated states: stability of the spectral reconstruction. arXiv:2312.07516.