Quantum limits on postselected, probabilistic quantum metrology

Abstract

Probabilistic metrology attempts to improve parameter estimation by occasionally reporting an excellent estimate and the rest of the time either guessing or doing nothing at all. Here we show that probabilistic metrology can never improve quantum limits on estimation of a single parameter, both on average and asymptotically in number of trials, if performance is judged relative to mean-square estimation error. We extend the result by showing that for a finite number of trials, the probability of obtaining better estimates using probabilistic metrology, as measured by mean-square error, decreases exponentially with the number of trials. To be tight, the performance bounds we derive require that likelihood functions be approximately normal, which in turn depends on how rapidly specific distributions converge to a normal distribution with number of trials.

Multiple phase estimation in quantum cloning machines

“…Especially for the latter, a post-selection of the measurement results is involved and the role played by post-selection in quantum metrology has attracted a lot of attention recently [68][69][70][71][72]. Mixed-state cloning or broadcasting would be also very interesting in terms of QFI [20].…”

Weak-value amplification: state of play

“…However, unless there is a severe mismatch between the quality of detection in the two variables, imaginary weak values will not provide a signi cant advantage [12]. On the other hand, when case 2 is compared to case 3 in Figure 1, then postselection is merely data rejection, which cannot improve estimation under the most general evolution allowed by quantum theory [17]. This latter analysis therefore covers experiments involving any dened quantities, including imaginary and even complex weak values.…”

“…This fact has motivated a long standing feeling that weak-value ampli cation can help to overcome such problems. However, we have shown that estimation inequalities analogous to (4) hold in the presence of a broad class of noise before, during, and after the weak measurement [12,16,17,21]. Together these articles treat the most prevalent types of noise (including pixelation, detector jitter, and dephasing), and a similar approach can be used to analyse other imperfections.…”

“…However, the probability of a windfall persisting decreases exponentially as more trials are performed [16,17].…”

“…For example keeping all of the data (rather than discarding most of it) will give higher Fisher information [15,16,22]; similarly, choosing an optimal initial state and not performing the second measurement at all will outperform a postselected weak measurement [12]. A strong measurement, if available, provides the highest precision of all [12,[15][16][17][18]. Furthermore, a simple estimator based upon the approximate weak value (rather than the true average of the postselected weak measurement) is not unbiased: there is a systematic error in the estimate in any real experiment.…”