Compare quantum enhancement after noise acts

Consider an explicit erasure model in which each of N probes survives independently with probability $\eta$. An ideal GHZ interrogation retains phase information only if every probe survives; any loss produces a flagged outcome carrying no phase information. The probability of a usable event is therefore $\eta^N$. Since the success flag is independent of the unknown phase, the Fisher information per attempted batch is the success probability times the conditional information, namely $\eta^N N^2$. This is a defined channel model, not a formula for every optical loss process.

Conditioning on successful runs can leave an impressive phase variance, but it does not account for the resources spent on failures. A fair particle budget includes every attempted probe, any heralded preparation failures, and the elapsed time consumed before rejection. Independent equatorial probes in this same erasure model retain information from each surviving particle separately, giving expected information proportional to the number that survive. State which flags are available and what measurement follows a loss before comparing the two input strategies.

Now assume each probe's off-diagonal coherence decays as $e^{-\gamma t}$, with known Markovian rate $\gamma>0$, while a constant angular-frequency offset is encoded. Independent probes have midpoint frequency information $Nt^2e^{-2\gamma t}$ per batch. A GHZ state's coherence decays as $e^{-N\gamma t}$ for independent local dephasing, giving $N^2t^2e^{-2N\gamma t}$. Both formulas assume suitable local readout and neglect preparation and readout overhead. The factor of two comes from squared contrast, not from redefining the coherence lifetime.

For a fixed observation time $T$, the available repetition count is approximately $T/t$. Total independent-probe information is consequently $NTt e^{-2\gamma t}$. Differentiating its logarithm gives an optimum $t=1/(2\gamma)$. The GHZ optimum is shorter by N, at $t=1/(2N\gamma)$. Substitution gives the same optimized total information for these two strategies. Comparing them at one arbitrary common duration would miss this reoptimization and could overstate or understate the actual gap.

Equality of those two optimized examples does not prove that every entangled state is useless. Other states may offer constant improvements, and other noise models can change the asymptotic scaling. Spatial correlations, non-Markovian short-time behavior, monitored environments, ancillary systems, and available control all affect the channel and its resource cost. General limits must specify those assumptions. A result for independent fixed-strength noise should not be extrapolated silently to a protected collective mode or a different encoding Hamiltonian.

Error correction also requires distinguishing signal from noise. A correction that removes the same generator through which the desired parameter acts can erase the signal along with the disturbance. Useful protected sensing needs a physical separation that preserves parameter response while suppressing the relevant errors, plus a budget for controls and ancillas. In practical design, calculate the output-state information or measured likelihood after the proposed protection, include preparation time and failures, and compare against an optimized independent baseline using the same laboratory constraints.