Fig. 1: An illustrative summary of process characterisation.

a The state of an open system over time follows a trajectory through state space until some final time at which the state is probed (top). By applying control operations at times t₁ and t₂, an experimenter can anchor and change the trajectory, which can be inferred via a linear combination of trajectories corresponding to basis operations (bottom). b A circuit model showing a sequence of operations \(\{{{\mathcal{A}}}_{j}\}\) interleaved with SE interactions, resulting in a final state ρ_A. c A sequence of operations A_k−1:0 can be expressed as a tensor product of independently chosen operations \({{\mathcal{A}}}_{j}\) at each time step. These can then be individually decomposed into a chosen basis \(\{{{\mathcal{B}}}_{j}^{{\mu }_{j}}\}\) together giving a basis of sequences \(\{{{\bf{B}}}_{k-1:0}^{{\boldsymbol{\mu }}}\}\). d A process can be fully characterised by measuring the output state for a complete set of basis operations at different times. Then, an arbitrary process can be expressed as a linear combination of each basis process; because of the linear construction, the intermediate evolution is completely preserved in the description of the arbitrary process. e The final state density matrix for the process A_k−1:0 can be expressed by tracing over all of the intermediate operations, contracting to a coefficient expansion for the measured density matrices in the basis processes. This is the same density matrix as in b.