RLD Fisher Information Bound for Multiparameter Estimation of Quantum Channels

TL;DR

This work introduces the amortized right logarithmic derivative (RLD) Fisher information as a tool to bound multiparameter quantum channel estimation. A core result is a chain-rule inequality for the RLD FI that leads to an amortization collapse, meaning catalyst inputs cannot increase the FI and that sequential and parallel strategies share the same single-letter bound. Consequently, if the RLD FI is finite, Heisenberg scaling is not attainable in multiparameter channel estimation, yielding a computable lower bound on estimation error that scales as $1/(n I_F)$. The authors validate the framework by applying it to the generalized amplitude damping channel, deriving explicit FI expressions and showing the RLD bound is close to the SLD bound, illustrating practical relevance for noisy quantum devices.

Abstract

One of the fundamental tasks in quantum metrology is to estimate multiple parameters embedded in a noisy process, i.e., a quantum channel. In this paper, we study fundamental limits to quantum channel estimation via the concept of amortization and the right logarithmic derivative (RLD) Fisher information value. Our key technical result is the proof of a chain-rule inequality for the RLD Fisher information value, which implies that amortization, i.e., access to a catalyst state family, does not increase the RLD Fisher information value of quantum channels. This technical result leads to a fundamental and efficiently computable limitation for multiparameter channel estimation in the sequential setting, in terms of the RLD Fisher information value. As a consequence, we conclude that if the RLD Fisher information value is finite, then Heisenberg scaling is unattainable in the multiparameter setting.

Figures (2)

• Figure 1: Processing $n$ uses of channel $\mathcal{N}^{\bm{\theta}}$ in a sequential or adaptive manner is the most general approach to channel parameter estimation. The $n$ uses of the channel are interleaved with $n-1$ quantum channels $\mathcal{S}^{1}$ through $\mathcal{S}^{n-1}$, which can also share memory systems with each other. The final measurement's outcome is then used to obtain an estimate of the unknown parameter vector $\bm{\theta}$.
• Figure 2: Logarithms of the SLD quantity \ref{['eq:sld-quantity-gadc']} and the inverse of the RLD Fisher information value \ref{['eq:gadc-rld-info-value']} versus loss $\gamma$ with fixed noise $N$. In (a), $N=0.2$, and in (b), $N=0.3$.

Theorems & Definitions (10)

• Remark 1
• Theorem 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Theorem 7
• Theorem 8
• Lemma 9