Indefinite causal order strategy does not improve the estimation of group action

TL;DR

This paper proves that indefinite causal order and adaptive strategies do not enhance the estimation of unknown unitary operations within a group-covariant framework, establishing the optimality of parallel strategies.

Contribution

It introduces the concept of GPOVM and its covariance condition to demonstrate the limitations of advanced strategies in group-covariant unitary estimation.

Findings

Indefinite causal order does not improve estimation performance.

Adaptive strategies do not outperform parallel strategies.

Parallel strategies are optimal under group covariance conditions.

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.