Asymptotic metrology with quantum-controlled causal order

Problem

Does quantum control of causal order yield a persistent asymptotic metrological advantage over parallel access for some regular, finite-dimensional channel family? Let \(\Lambda_\theta:\mathcal L(A)\to\mathcal L(B)\) be a smooth one-parameter family on an open interval. A parameter value \(\theta\) is regular here if the Choi rank is constant in a neighborhood of \(\theta\); the family then admits a differentiable minimal Kraus representation locally. Let \(\mathcal F_{\mathrm{PAR}}^{(N)}(\theta)\) and \(\mathcal F_{\mathrm{QCQC}}^{(N)}(\theta)\) be the largest output-state symmetric-logarithmic-derivative quantum Fisher information attainable from \(N\) black-box uses by, respectively, a parallel strategy and a quantum circuit with quantum control of causal order (QC-QC). Both classes allow arbitrary finite noiseless ancillary systems. In a QC-QC strategy, a coherent control can dynamically select which unused black-box call occurs next; this is more general than coherently superposing fixed causal orders. At every regular parameter value with \(\mathcal F_{\mathrm{PAR}}^{(1)}(\theta)>0\), decide whether every such family satisfies

\begin{equation} \limsup_{N\to\infty} \frac{\mathcal F_{\mathrm{QCQC}}^{(N)}(\theta)} {\mathcal F_{\mathrm{PAR}}^{(N)}(\theta)} =1. \tag{1} \end{equation}

Equivalently, either prove Eq. (1) or construct a noisy family in this regular domain and physical QC-QC strategies with a persistent constant-factor advantage or a larger scaling exponent.

Source

Mothe, Branciard, and Abbott explicitly ask whether the finite-use QC-QC advantage that they establish can persist asymptotically [MBA24].

Progress

Kurdziałek, Górecki, Albarelli, and Demkowicz-Dobrzański prove that adaptive strategies and coherent superpositions of fixed causal orders are asymptotically equivalent to parallel strategies for repeated estimation of a finite-dimensional channel [KGAD23]. Their recursion does not cover the dynamically controlled QC-QC class in Eq. (1).
Mothe, Branciard, and Abbott exhibit noisy channel families for which QC-QC strictly outperforms parallel, sequential, and causal-superposition strategies at three uses [MBA24]. This proves a finite-resource separation but neither a nonunit asymptotic ratio nor a different scaling exponent.
Abbott, Mhalla, and Pocreau show that QC-QC gives no query advantage for Boolean-function tasks formed by repeated access to one unitary [AMP24]. The restriction to unitary queries does not address the noisy metrological families relevant to Eq. (1).
Salzger and Vilasini show that higher-order processes satisfying their spacetime, Acting Once, and Local Order assumptions reduce operationally to QC-QC processes [SV26]. This identifies QC-QC as the physical target class under those assumptions but supplies no asymptotic Fisher information bound.
Wei, Li, and Pang’s preprint establishes an upper bound for general indefinite-causal-order processes in Theorem 3 and Supplementary Section V [WLP26]. For a differentiable minimal Kraus representation, write \(\alpha=\sum_i\dot K_i^\dagger\dot K_i\) and \(\beta=\sum_i\dot K_i^\dagger K_i\). Combining that upper bound with the parallel attainability results of Zhou and Jiang, Theorems 2 and 3 [ZJ21], gives

\begin{equation} \lim_{N\to\infty}\frac{\mathcal F_{\mathrm{Gen}}^{(N)}(\theta)}{N^s} =\lim_{N\to\infty}\frac{\mathcal F_{\mathrm{PAR}}^{(N)}(\theta)}{N^s} =c>0, \tag{2} \end{equation}

where \(s=1\) and \(c=4\min_{\beta=0}\lVert\alpha\rVert\) if a Kraus gauge with \(\beta=0\) exists; otherwise \(s=2\) and \(c=4\min\lVert\beta\rVert^2\). All minima are over Kraus gauges at the parameter value in question. In the linear regime, product inputs give \(c\ge\mathcal F_{\mathrm{PAR}}^{(1)}(\theta)>0\); in the quadratic regime, positivity follows from the nonzero minimum defining that regime. Since \(\mathrm{PAR}\subseteq\mathrm{QCQC}\subseteq\mathrm{Gen}\), Eq. (2) proves Eq. (1) by squeezing the ratio.

Comment

The asymptotic-ratio question in the explicitly regular, ancilla-assisted domain above is resolved by Wei, Li, and Pang’s Theorem 3, together with the parallel attainability theorem. The resolving Wei–Li–Pang result is a preprint, arXiv:2609.05355v1, rather than a peer-reviewed publication. Rank-changing parameter values are outside the regular domain defined in the statement. Finite-use advantages and subleading corrections can persist without changing the limiting ratio in Eq. (1).

References

[MBA24]: R. Mothe, C. Branciard, and A. A. Abbott, “Reassessing the Advantage of Indefinite Causal Orders for Quantum Metrology,” Physical Review A 109, 062435 (2024).DOI arXiv
[KGAD23]: S. Kurdziałek, W. Górecki, F. Albarelli, and R. Demkowicz-Dobrzański, “Using Adaptiveness and Causal Superpositions Against Noise in Quantum Metrology,” Physical Review Letters 131, 090801 (2023).DOI arXiv
[AMP24]: A. A. Abbott, M. Mhalla, and P. Pocreau, “Quantum Query Complexity of Boolean Functions under Indefinite Causal Order,” Physical Review Research 6, L032020 (2024).DOI arXiv
[SV26]: M. Salzger and V. Vilasini, “Higher-Order Quantum Processes Respecting Closed Labs in a Spacetime Have Quantum-Controlled Causal Order,” arXiv:2605.08351 (2026).arXiv
[WLP26]: W. Wei, Y. Li, and S. Pang, “Fundamental Limits of Quantum Metrology Beyond Fixed Causal Order,” arXiv:2609.05355v1 (2026).arXiv
[ZJ21]: S. Zhou and L. Jiang, “Asymptotic Theory of Quantum Channel Estimation,” PRX Quantum 2, 010343 (2021).DOI arXiv

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