Square-root measurement for pure states

Peculiarities of qubit initial-state preparation by nonselective measurements on an overcomplete basis

“…The dephasing model (15) has two distinctive features. First, the operator σ 3 commutes with the Hamiltonian and, consequently, the average populations of the qubit states do not depend on time.…”

“…In such a case, the system purification occurs only via the re-distribution of the levels population, which is time-independent, because σ 3 is the integral of motion, see Eq. (15).…”

“…The definition (41) of the square error R has to be considered in terms of the Hilbert-Schmidt norm of some specific operators constructed on the measurement state vectors |ψ ′ n and |ϕ n (see Ref. [15] and the references herein for more details). Such schemes with the effects F ′ n = |ψ ′ n ψ ′ n |, which realize the least-square root R, are known as the square root measurements (SRM) and are the subject of thorough investigation in the quantum measurement theory [15,24,25].…”

“…On the other hand, it is interesting to investigate the initial system preparation not only from the viewpoint of the subsequent qubit dynamics. Though these problems were studied usually from pure mathematical points of view [14,15], novel trends and achievements in the quantum measurements and control [12,16,17] dictate new challenges for physicists concerning special postmeasurement system preparation. In particular, some questions that could be imposed, sound as follows: i) what factors influence the system purity (or mixedness); ii) what "risks" should be avoided when manipulating the open quantum system at the initial stage of its preparation; iii) how to separate the factors contributing to the system purification from those leading to undesirable mixedness.…”

“…We consider a special kind of the nonselective measurements [10], where the number N of outcomes is larger than the rank n of the initial density matrix (for the qubits n = 2). Unlike the well-known von Neumann-Lüders projection measurements studied earlier [9], this more general scheme is associated with the notion of the positive operator-valued measure (POVM) [14,15]. In this scheme, the basis of measurement state turns out to be overcomplete, and the measurement state vectors can no longer be orthonormal.…”