Exact decoherence and orthogonality of pointer states in the measurement of a qubit observable without an environment

Abstract

We apply the work of E. A. Galapon [EPL 113 60007, (2016)] in the measurement of the z-component of an electron by means of a measuring instrument which is decomposed into a probe and a pointer, with the probe being inaccessible from observation. It is found that if the probe is initially prepared in a momentum-limited state, the coherences of the system-pointer reduced density matrix vanishes identically to zero at a finite time without an environment. Moreover, it is shown that in the case of a momentum-limited initial state of the pointer, the pointer states are exactly orthogonal. In this scheme,  exact decoherence is attained at an earlier time before the pointer states become exactly orthogonal.