Entanglement entropy and the determination of an unknown quantum state

TL;DR

This paper explores how entanglement entropy relates to the ability to reconstruct an unknown quantum state using a single measurement setup involving an auxiliary system, revealing that for pure states, entanglement indicates less efficient reconstruction intervals.

Contribution

It establishes a novel connection between entanglement entropy and the reconstructability of quantum states, extending the analysis to systems of arbitrary dimension.

Findings

Entanglement entropy correlates with the efficiency of quantum state reconstruction.

For pure states, zero entanglement indicates intervals of least reconstruction efficiency.

The connection between entanglement and reconstruction efficiency does not hold for mixed states.

Abstract

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 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.