Noisy initial-state qubit-channel metrology with additional undesirable noisy evolution

TL;DR

This work analyzes parameter estimation for single-parameter unital qubit channels when initial states are highly mixed, comparing a single-qubit protocol to a correlated-state protocol with spectator qubits. Extending prior results, it incorporates additional noisy spectator evolution and derives algebraic conditions showing when the CS protocol can outperform the SQSC scheme, as well as when noise renders it inferior. A noise-alleviation method, twisting spectator channels, is introduced to align spectator-noise effects with the parameter-bearing channel, potentially recovering CS advantages across several noise models. The framework is instantiated with concrete channels (unitary phase shifts, flips, and depolarizing channels), revealing a nuanced dependence of performance on spectator noise type and providing measurement strategies (generic or tailored) to approach the quantum Cramér-Rao bound, with practical implications for metrology in NMR-like, highly mixed-state settings.

Abstract

We consider protocols for estimating the parameter in a single-parameter unital qubit channel, assuming that the available initial states are highly mixed with very low purity. We compare two protocols, each invoking the channel once, via their quantum Fisher informations. One uses $n$ qubits prepared in a particular correlated input state and subsequently queries the channel on one qubit. The other uses a single qubit. We extend the results of Collins [1] by allowing for additional noisy evolution on the spectator qubits in the $n$-qubit protocol. We provide simple algebraic expressions that will determine when the $n$-qubit protocol is superior. We provide a technique that can alleviate certain types of noise. We show that for certain types of noisy evolution the $n$-qubit protocol will be inferior but for others it will be superior.

Figures (6)

• Figure 1: Possible metrology protocols. The protocol in a) uses $\hat{\Gamma}(\lambda)$ independently on each of $q$ qubits. The protocol in b) uses $n$ qubits with $\hat{\Gamma}(\lambda)$ acting on each of $q$ channel qubits. There are additional spectator qubits, on which the channel does not act. All are subjected to a parameter-independent unitary operation, $\hat{U}_\mathrm{prep}$, prior to querying the channel and a measurement, possibly global, after channel queries.
• Figure 2: Single channel qubit estimation protocols. (a) The SQSC protocol with one qubit and one channel query. (b) The CS protocol queries $\hat{\Gamma}(\lambda)$ once on a single qubit with the remaining spectator qubits assisting. $\hat{\Gamma}_2,\ldots,\hat{\Gamma}_n$ indicate unwanted but unavoidable noisy channels on the spectators.
• Figure 3: Symmetric pairwise correlated protocol preparatory unitary. The circuit element within the blue dashed box a single iteration of $\hat{U}_{\boldsymbol{\mathrm{c}}}$; the solid boxes indicate the two qubits on which the gate acts.
• Figure 4: Symmetric pairwise correlated protocol with twisted spectator channels. Each spectator noisy channel is preceded by a channel-dependent unitary $\hat{U}_j$ and is followed by $\hat{U}_j^\dagger.$
• Figure 5: Generic measurement scheme. The projection operators associated with the measurement are $\hat{\Pi}_\pm = \left( \hat{I} \pm \boldsymbol{\mathrm{r}}_\textrm{0}\cdot \boldsymbol{\hat{\sigma}}\right)/2.$