  The purpose of this paper is to discuss the existence of a nontrivial
tradeoff relation for estimating two unitary-shift parameters in a relativistic
spin-1/2 system. It is shown that any moving observer cannot estimate two
parameters simultaneously, even though a parametric model is classical in the
rest frame. This transition from the classical model to a genuine quantum model
is investigated analytically using a one-parameter family of quantum Fisher
information matrices. This paper proposes to use an indicator that can not only
detect the existence of a tradeoff relation but can also evaluate its strength.
Based on the proposed indicator, this paper investigates the nature of the
tradeoff relation in detail.
