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 smooth finite-dimensional channel family? Let \(\Lambda_\theta:\mathcal L(A)\to\mathcal L(B)\) be a smooth one-parameter family, and let \(\mathcal F_{\mathrm{PAR}}^{(N)}(\theta)\) and \(\mathcal F_{\mathrm{QCQC}}^{(N)}(\theta)\) be the largest output-state 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). 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 for which \(\mathcal F_{\mathrm{PAR}}^{(N)}(\theta)>0\) for all sufficiently large \(N\), decide whether every channel 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 finite-dimensional family 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 leaves open 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.

Comment

The known three-use separation does not determine the leading asymptotic coefficient or scaling exponent. The unresolved alternatives are precisely Eq. (1) for every smooth finite-dimensional family or one noisy counterexample with a persistent QC-QC advantage.

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

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