This paper asks whether allowing quantum operations to act in no fixed order — a situation called indefinite causal order (ICO) — can give a fundamental, long-run improvement in how precisely a single parameter can be estimated. The authors study the case where the parameter is encoded in N identical uses of a finite-dimensional quantum channel and measure precision using the quantum Fisher information (QFI), a standard quantity that sets the best possible variance in the local frequentist approach. Their main finding is that ICO does not improve the leading, asymptotic precision: it cannot beat the best fixed-order (parallel) strategies in the limit of many channel uses.

To reach this conclusion the authors work in the process-matrix formalism, which describes the most general way quantum operations can be arranged, including arrangements without a definite time order. They prove a universal upper bound that forbids any ICO strategy from achieving better than Heisenberg scaling in the large-N limit. Heisenberg scaling means the estimation uncertainty shrinks like 1/N with N uses of the channel, whereas the more classical standard quantum limit (SQL) gives an uncertainty that shrinks only like 1/√N. For unitary channels (no noise), the authors show the optimal QFI under any ICO strategy matches exactly the QFI achievable by the best parallel strategies. In plain terms: for ideal unitary channels, ICO does not improve asymptotic precision.

The paper also treats noisy (nonunitary) channels. The authors derive a refined structural bound showing that any channel that is stuck at the SQL when probed in parallel remains stuck at the SQL even when ICO is allowed. They go further and prove an “asymptotically tight” bound that fixes the leading coefficient of the QFI: in both the SQL and Heisenberg regimes, the leading term that governs long-run precision is the same for general ICO and for optimal parallel strategies. Thus ICO cannot change the leading scaling or the leading coefficient of precision in the asymptotic limit.