Indefinite causal order strategy does not improve the estimation of group action
Masahito Hayashi
TL;DR
The paper analyzes estimation of an unknown unitary drawn from a projective unitary representation of a compact group $G$ under a covariant error function. It introduces generalized POVMs (GPOVMs) and extends the Hunt-Stein theorem to covariant GPOVMs, proving that the optimal performance is achieved by a covariant GPOVM, which can be realized by a parallel strategy. A detailed decomposition of the representation space shows covariant measurements correspond to a positive operator $T$ on a subspace $\mathcal{H}_0$, with $P_0 M(B) P_0=\int_B f(g)^{\dagger} T f(g)\,\mu(dg)$ and $\mathrm{Tr} \!\ T \rho_\mu=1$. Moreover, the authors demonstrate that any covariant GPOVM optimality can be simulated by a parallel strategy, reducing the problem to a pure-state optimization $\min_{|\psi\rangle} D_w(|\psi\rangle)$. Consequently, indefinite causal order and adaptive strategies provide no advantage in this setting, clarifying the role of symmetry in quantum metrology and channel estimation. The framework also connects to prior finite-group results and opens questions for noisy channels.
Abstract
We consider estimation of unknown unitary operation when the set of possible unitary operations is given by a projective unitary representation of a compact group. We show that neither indefinite causal order strategy nor adaptive strategy improves the performance of this estimation when error function satisfies group covariance. That is, the optimal parallel strategy gives the optimal performance even under indefinite causal order strategy and adaptive strategy. To study this problem, we newly introduce the concept of generalized positive operator valued measure (GPOVM), and its convariance condition. Using these concepts, we show the above statement.