Multiistochastic operations and convolution channels for quantum states

The notion of convolution of two probability vectors can be extended to operationsdetermined by multistochastic tensors, to describe Markov chains of a higher order. Onthe other hand, the idea of convolution lies in the centre of machine learningalgorithms for image processing: Convolutional Neural Networks.

In my talk, I will firstpresent the characterization of the probability eigenvectors of multi-stochastictensors, corresponding to fixed points of generalized Markov chains. Similar results willbe also obtained in the quantum case for multi-stochastic channels acting on mixedquantum states.

Next, I propose a quantum analogue of the convolution, based oncoherifications of tristochastic tensors, defined for two arbitrary density matrices ofthe same size. Finally, I will discuss possible applications of this notion to constructschemes of error mitigation or as building blocks in quantum Convolutional NeuralNetworks..