Fundamental limits of parameter estimation with heralded optical non-Gaussian states generated from Gaussian resources

In optical quantum sensing, non-Gaussian states are often generated probabilistically via conditional measurements (heralding) on Gaussian resources. Because these states can exhibit high quantum Fisher information (QFI), they are frequently proposed as superior probes for quantum metrology. However, this approach incurs a significant overhead due to the low success probability of state generation. This paper addresses whether the metrological gain from these heralded states outweighs the cost of their probabilistic generation.

The authors analyze single-parameter phase estimation using heralded non-Gaussian states. They utilize the property of photon-number conservation in passive linear optical systems to map the heralding process (which occurs before parameter encoding) to an equivalent postselection process (which occurs after parameter encoding). By introducing an Effective Quantum Fisher Information (EQFI)—defined as the success-probability-weighted QFI of the heralded outputs—the authors create a framework for a resource-fair comparison between probabilistic non-Gaussian probes and deterministic Gaussian probes.

The study proves that the EQFI of heralded non-Gaussian states is strictly upper-bounded by the QFI of the original Gaussian input states. The physical intuition is that the high QFI observed in successful heralded events is not an amplification of information, but rather a redistribution of the information already present in the input Gaussian state. As a concrete example, the authors demonstrate that for heralded photon-number states generated via spontaneous parametric down-conversion (SPDC), the averaging of the QFI over all possible detection outcomes results in a sensitivity that is significantly lower than that of the original squeezed vacuum input, effectively reducing to the shot noise limit.