Indefinite causal order for quantum metrology with quantum thermal noise

Abstract: A switched quantum channel with indefinite causal order is studied for the fundamental metrological task of phase estimation on a qubit unitary operator affected by quantum thermal noise. Specific capabilities are reported in the switched channel with indefinite order, not accessible with conventional estimation approaches with definite order. Phase estimation can be performed by measuring the control qubit alone, although it does not actively interact with the unitary process -- only the probe qubit doing so. Also, phase estimation becomes possible with a fully depolarized input probe or with an input probe aligned with the rotation axis of the unitary, while this is never possible with conventional approaches. The present study extends to thermal noise, investigations previously carried out with the more symmetric and isotropic qubit depolarizing noise, and it contributes to the timely exploration of properties of quantum channels with indefinite causal order relevant to quantum signal and information processing.