Exact quantum noise deconvolution with partial knowledge of noise

We introduce a new quantum noise deconvolution technique that does not rely on the complete knowledge of noise and does not require partial noise tomography. In this new method, we construct a set of observables with completely correctable expectation values despite our incomplete knowledge of noise. This task is achieved just by classical post-processing without extra quantum resources. We show that the number of parameters in the subset of observables with correctable expectation values is the same for all unitary quantum channels. For random unitary channels and the assumption that the probability distribution of unitary errors is unknown, we instruct the construction of the set of observables with correctable expectation values. For a particular case where the random unitary channel is made of just two random unitary operators acting on $d$-dimensional Hilbert-space, we show that the observable with correctable expectation value belongs to a set with at least $d$ parameters. We extend our method by considering observables for which the partial recovery of the expectation value is possible, at the cost of having partial knowledge about the noise-free initial state.