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Alternatively to the full reconstruction of an unknown quantum process, theso-called selective and efficient quantum process tomography (SEQPT) allowsestimating, individually and up to the required accuracy, a given element ofthe matrix that describes such an operation with a polynomial amount ofresources. The implementation of this protocol has been carried out withsuccess to characterize the evolution of a quantum system that is welldescribed by a trace preserving quantum map. Here, we deal with a more generaltype of quantum process that does not preserve the trace of the input quantumstate, which naturally arises in the presence of imperfect devices andsystem-environment interactions, in the context of quantum information scienceor quantum dynamics control. In that case, we show that with the aid of {\it apriori} information on the losses structure of the quantum channel, the SEQPTreconstruction can be adapted to reconstruct the non-trace-preserving map. Weexplicitly describe how to implement the reconstruction in an arbitrary Hilbertspace of finite dimension $d$. The method is experimentally verified on asuperconducting quantum processor of the IBM Quantum services, by estimatingseveral non trace-preserving quantum processes in dimensions up to $d=6$. Ourresults show that it is possible to efficiently reconstruct nontrace-preserving processes, with high precision, and with significantly higherfidelity than when the process is assumed to be trace-preserving.

Selective and efficient quantum process tomography for non-trace preserving maps: a superconducting quantum processor implementation