Simultaneous Heisenberg-Limited Multiparameter Metrology via Indefinite Evolution

Multiparameter quantum metrology is fundamentally limited by two factors: encoding incompatibility, where noncommuting signal generators prevent the quantum Fisher information matrix from reaching the Heisenberg limit for all parameters simultaneously, and measurement incompatibility, where the optimal measurements for different parameters cannot be performed at the same time. This paper investigates whether indefinite evolution (IE)—a framework where control operations are placed in a coherent superposition—can overcome these constraints to achieve simultaneous Heisenberg-limited (SHL) sensing.

The researchers develop an IE-based protocol that uses auxiliary systems to coherently control the probe's evolution. By applying history-dependent compensation operations, the protocol maps different signal contributions into separate degrees of freedom (e.g., the probe and an auxiliary qubit). This separation allows the parameters to be read out using compatible measurements. The authors analyze this framework across several scenarios, including noiseless qubit probes, noisy systems (using error-correction-inspired syndrome extraction), multi-qubit probes, and high-dimensional systems (using subspace projection).

The study demonstrates that IE is a powerful operational resource for multiparameter sensing. For orthogonal signal generators, IE achieves the SHL without requiring signal reversal. For parallel signal generators, the protocol achieves SHL provided that signal reversal is available. In contrast, standard definite evolution (DE) is shown to be incapable of achieving this performance, as it typically erases one signal component while attempting to measure another. The authors also prove that the protocol can be extended to noisy environments by integrating noise-syndrome extraction and to high-dimensional probes by confining dynamics to an optimal two-dimensional encoding subspace.