Abstract
We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.
Highlights
Quantum metrology aims at exploiting all possible features of quantum systems, such as coherence or entanglement, in order to boost the precision of measurements beyond that achievable by metrological schemes that operate within classical or semi-classical paradigms [1,2,3,4,5,6,7,8,9]
The theory allows for a quick identification of the most promising quantum metrological models and provides a clear recipe for designing the optimal adaptive schemes based on appropriately tailored quantum error correction (QEC) protocols
In case of scenarios when HNLS is satisfied, we developed an efficient numerical algorithm to find the optimal QEC protocol, including the optimal input state, QEC codes and measurements
Summary
Quantum metrology aims at exploiting all possible features of quantum systems, such as coherence or entanglement, in order to boost the precision of measurements beyond that achievable by metrological schemes that operate within classical or semi-classical paradigms [1,2,3,4,5,6,7,8,9]. The most persuasive promise of quantum metrology is the possibility of obtaining the so-called Heisenberg scaling (HS), which manifests itself in the quadratically improved scaling of precision as a function of number of elementary probe systems involved in the experiment [10,11,12,13,14,15,16,17,18,19] or the total interrogation time of a probe system [20] In either of these cases, the presence of decoherence typically restricts the quadratic improvement to a small particle number or a short-time regime, whereas in the asymptotic regime the quantum-enhancement amounts to constant factor improvements [21,22,23,24,25,26] even under the most general adaptive schemes [27].
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