In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.

We develop a notion of stochastic quantum trajectories. First, we construct a basis set of trajectories, called elementary trajectories, and go on to show that any quantum dynamical process, including those that are non-Markovian, can be expressed as a linear combination of this set. We then show that the set of processes divide into two natural classes: those that can be expressed as convex mixture of elementary trajectories and those that cannot be. The former are shown to be entanglement breaking processes (in each step), while the latter are dubbed coherent processes. This division of processes is analogous to separable and entangled states. In the second half of the paper, we show, with an information theoretic game, that when a process is non-Markovian, coherent trajectories allow for decoupling from the environment while preserving arbitrary quantum information encoded into the system. We give explicit expressions for the temporal correlations (quantifying non-Markovianity) and show that, in general, there are more quantum correlations than classical ones. This shows that non-Markovian quantum processes are indeed fundamentally different from their classical counterparts. Furthermore, we demonstrate how coherent trajectories (with the aid of coherent control) could turn non-Markovianity into a resource. In the final section of the paper we explore this phenomenon in a geometric picture with a convenient set of basis trajectories. arXiv:1802.03190v3 [quant-ph] 19 Sep 2018Non-Markovian quantum control as coherent stochastic trajectories

We show theoretically how a correlation of multiple measurements on a qubit undergoing pure dephasing can be expressed as environmental noise filtering. Measurement of such correlations can be used for environmental noise spectroscopy, and the family of noise filters achievable in such a setting is broader than the one achievable with a standard approach, in which dynamical decoupling sequences are used. We illustrate the advantages of this approach by considering a case of noise spectrum with sharp features at very low frequencies. We also show how appropriately chosen correlations of a few measurements can detect the non-Gaussian character of certain environmental noises, particularly the noise affecting the qubit at the so-called optimal working point.

We consider a qubit coupled to another system (its environment), and discuss the relationship between the effects of subjecting the qubit to either a dynamical decoupling sequence of unitary operations, or a sequence of projective measurements. We give a formal statement concerning equivalence of a sequence of coherent operations on a qubit, precisely operations from a minimal set {1 1Q,σx}, and a sequence of projective measurements ofσx observable. Using it we show that when the qubit is subjected to n such successive projective measurements at certain times, the expectation value of the last measurement can be expressed as a linear combination of expectation values ofσx observed after subjecting the qubit to dynamical decoupling sequences of π pulses, with k ≤ n of them applied at subsets of these times. Performing a sequence of measurements on the qubit gives then the same information about qubit decoherence and dynamics of environment as that contained in dynamical decoupling signal. Analysing the latter has been widely used to gain information about the environmental dynamics (perform so-called noise spectroscopy), so our result shows how all the resuts obtained with dynamical decoupling based protocols are related to those that can be obtained just by performing multiple measurements on the qubit. We also discuss in more detail the application of the general result to the case of the qubit undergoing pure dephasing, and outline possible extensions to higher-dimensional (a qudit or multiple qubits) systems.arXiv:1907.05165v2 [quant-ph]