Southwest Quantum Information and Technology

Attainability of the quantum information bound in pure state models

Presenting Author: Fabricio Toscano, Instituto de Fisica, Universidade Federal do Rio de Janeiro (UFRJ), Brasil
Contributing Author(s): W. P. Bastos and R. L. de Matos Filho

The attainability of the quantum Cramer-Rao bound, that is the fundamental limit of precision in quantum parameter estimation, involves two steps. The first step is the saturation of the classical Cramer-Rao bound (CCR) associated with the Fisher information associated with the probabilities distributions of a particular positive-operator valued measure (POVM). This saturation depends on the nature of the estimator used to process the data drawn from the set of probabilities in order to estimate the true value of the parameter. Those estimators that saturates the CCR bound are called efficient estimators or asymptotically efficient estimators when the saturation only occurs in the limit of a very large number of measured data (a typical example of this type is the maximum likelihood estimator). The second step is independent on the nature of the estimator and consists in the saturation of the so called quantum information bound (QIB), that occurs when the Fisher information of a suitable POVM coincide with quantum Fisher information associated with the final quantum state where the parameter was imprinted. Braunstein and Caves [1] have shown that the QIB can be always be achieved if the suitable quantum measurement is a von Neumann projective measurement in the eigenvectors basis of an observable called symmetric logarithmic derivative. The problem is that this measurement require the knowledge of the value of the parameter to be estimated. Mainly two approaches have been adopted in order to deal with the fact that the optimal POVM depends on the true value of the parameter. The first one relies on adaptive quantum estimation schemes that could, in principle, asymptotically achieve the QCR bound. The second one looks for the families of density operators where the parameter is imprinted, for which the use of an specific POVM that does not depend on the true value of the parameter leads to the saturation of the QIB. This is known as the search for the global optimal POVM that saturates the QIB independently of the true value of the parameter. For full-rank density operators, Nagaoka [2] showed that saturation of the quantum information bound by using a POVM that does not depend on the true value of the parameter is only possible for the so called quasi-classical family of density operators. He also presented complete characterisation of the quantum measurements that guarantee the saturation for this family. Therefore, the problem of finding the states and the corresponding optimal measurements that lead to the saturation of the QIB, independently of the true value of the parameter, in the case of one-parameter families of full-rank density operators has been already solved. However, for the opposite case of pure states (rank-one density operators), the complete characterisation of the families of states and the corresponding measurements that lead to the saturation of the QIB, independently of the true value of the parameter, is still an open question in the case of arbitrary Hilbert spaces. It is important to remark that inside the families of pure states the QFI reaches its largest values. Here, we consider quantum state families of pure density operators in which the true value of the parameter is imprinted by a unitary evolution whose generator is arbitrary but with discrete spectrum and independent of the true value of the parameter. Thus, we present the complete solution to the problem of which are all the initial states and the corresponding families of global projective measurements that allow the saturation of the QIB, within the pure quantum state families considered. Also, we show that within all the states that saturate the quantum information bound those corresponding to the Heisenberg limit allow the maximum retrieval of information of the parameter in the final state. [1] S. L. Braunstein and C. M. Caves, Physical Review Letters 72, 3439 (1994). [2] H. Nagaoka, in Chapter 9 of ``Asymptotic Theory of Quantum Statistical Inference: Selected Papers'' (2005).

Read this article online: https://arxiv.org/abs/1701.09144

(Session 5 : Thursday from 5:00pm - 7:00pm)