Clocks and fisher information

Abstract

In a broad sense, any parametric family of quantum states can be viewed as a quantum clock. The time, which is the parameter,
is encoded in the corresponding quantum states. The quality of such a clock depends on how precisely we can distinguish the
states or, equivalently, estimate the parameter. In view of the quantum Cramér—Rao inequalities, the quality of quantum clocks
can be characterized by the quantum Fisher information. We address the issue of quantum clock synchronization in terms of
quantum Fisher information and demonstrate its fundamental difference from the classical paradigm. The key point is the superadditivity
of Fisher information, which always holds in the classical case but can be violated in quantum mechanics. The violation can
occur for both pure and mixed states. Nevertheless, we establish the superadditivity of quantum Fisher information for any
classical-quantum state. We also demonstrate an alternative form of superadditivity and propose a weak form of superadditivity.
The violation of superadditivity can be exploited to enhance quantum clock synchronization.

KeywordsFisher information–quantum estimation–superadditivity–classical-quantum state–clock synchronization