Fig. 4: Multi-probe scenarios supplemented by control operations.
a Canonical multi-probe scenario of quantum metrology with imperfect measurements. N probes, generally prepared in a entangled state, ρN, undergo identical parameter encoding, \({{{{{{{{\mathcal{U}}}}}}}}}_{\theta }\), and are subject to identical local projective measurements {Πi}, whose outcomes are affected by the noisy detection channel (stochastic map) \({{{{{{{\mathcal{P}}}}}}}}\). In between the encoding and measurement, control operations are applied and optimised in order to compensate for local measurement imperfections, so that the minimal error in estimating θ can be attained. b Control operations in a may always be represented by a global unitary transformation, \({{{{{{{{\mathcal{V}}}}}}}}}_{\vec{\Phi }}\); or be rather constrained to a product of general local unitaries, \({\bigotimes }_{j=1}^{N}{{{{{{{{\mathcal{V}}}}}}}}}_{{\vec{\phi }}_{j}}^{(j)}\). c Single-probe evolution as a quantum-classical channel, denoted as \({\Lambda }_{\theta,\vec{\phi }}\) that transforms the d-dimensional state ρ of the probe into a classical state \({\rho }_{{{{{{{{\rm{cl}}}}}}}}}(\theta,\vec{\phi })\) defined in a fictitious Hilbert space, whose dimension is specified by the number of outcomes of the noisy detection channel \({{{{{{{\mathcal{P}}}}}}}}\).
