Entanglement entropy and determination of an unknown quantum state
Gerardo Aquino, Filippo Giraldi · Physical Review A · 2008
An initial unknown quantum state can be determined with a single measurement apparatus by letting it interact with an auxiliary, ancilla, system as proposed by Allahverdyan, Balian, and Nieuwenhuizen [Phys. Rev. Lett. 92, 120402 (2004)]. In the case of two qubits, this procedure allows one to reconstruct the initial state of the qubit of interest, $S$, by measuring three commuting observables and therefore by means of a single apparatus, for the total system $S+A$ at a later time. The determinant of the matrix of the linear transformation connecting the measurements of three commuting observables at time $t>0$ to the components of the polarization vector of $S$ at time $t=0$ is used as an indicator of the reconstructability of the initial state of the system $S$. We show that a connection between the entanglement entropy of the total system $S+A$ and such a determinant exists, and that for a pure state a vanishing entanglement individuates, without a need for any measurement, those intervals of time for which the reconstruction procedure is least efficient. This property remains valid for a generic dimension of $S$. In the case of a mixed state, this connection is lost.