Algorithm for nondestructive estimation of quantum relative entropy

We propose a new simple algorithm for non-destructive estimation of quantum relative entropy S(rho\sigma), where the density operators rho and sigma are defined on the same finite-dimensional Hilbert space but are otherwise completely unspecified (unknown). Given N copies of a quantum state rho and N copies of a quantum state sigma, the algorithm estimates S(rho\sigma) with arbitrary high accuracy and without demolition of the input for all suitably large N. More precisely, the input of algorithm is the tensor product state rho(circle timesN) circle times sigma(circle timesN) of the 2N-fold quantum system, and the output of the algorithm consists of (i) a state of the 2N-fold quantum system and (ii) a measurement outcome which is an estimate of S (rho\\sigma). We show that, as N goes to infinity, the fidelity between the algorithm's output and input states of the 2N-fold system converges to the unity and that the quantum relative entropy estimate converges to S(rho\\sigma). In order to estimate S(rho\\sigma), the algorithm performs reversible computations followed by so-called "weak" (non-demolition) measurements on the 2N-fold system.