Coherence cost for measurement and computation under conservation laws

Nature imposes many restrictions on the operations that we perform. Among such restrictiones, the restriction imposed by conservation laws on two types of the basic operations of quantum information processing, i.e. measurements and unitary operations, has been studied for a particularly long time.
One of the open problems in this field is to clarify the amount of required resource for implementing unitary operations and measurements under conservation laws.
Here, we provide a solution to this open problem.
We derive two asymptotically exact equalities that clarify the necessary and sufficient amount of quantum coherence as a resource to implement an arbitrary unitary operation and a measurement for arbitrary physical quantities within a desired error, respectively.
This work provides an optimal improvement of the Wigner-Araki-Yanase-Ozawa theorem and a proof of the long standing conjecture that WAY-type tradeoff relation holds for the implementation of an arbitrary unitary channel under conservation laws.
It also clarifies the key question of the resource theory of the quantum channels in the region of resource theory of
asymmetry, for the case of unitary channels.
The technical details of this work are in [PRL \textbf{121}, 110403], [arXiv:1906.04076] and [arXiv:1909.02904].