Relationship between subjecting the qubit to dynamical decoupling and to a sequence of projective measurements

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 $\left\{\mathbf{I}_Q,\hat{\sigma}_x\right\}$, and a sequence of projective measurements of $\hat{\sigma}_x$ 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 {\it last} measurement can be expressed as a linear combination of expectation values of $\hat{\sigma}_x$ observed after subjecting the qubit to dynamical decoupling sequences of $\pi$ pulses, with $k\leq n$ of them applied at subsets of these times. Performing a sequence of measurements on the qubit gives then access to the same properties of the environment and qubit-environment coupling that are affecting the coherence observed in a dynamical decoupling experiment. Analysing the latter has been widely used to characterize the environmental dynamics (perform so-called noise spectroscopy), so our result shows how the results 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.