Beyond^2 IID 4

Title: Optimized measurements for quantum information processing: A geometric approach

Abstract: In [1] we introduced a geometric approach to solve the longstanding problem of optimum unambiguous discrimination of more than two states. It is an example of a broader class of convex optimization problems. In such problems the constraints imposed by quantum mechanics dene a convex hypersurface while the cost function to be optimized is another hypersurface in the parameter space of the problem. The parameter values for which these two surfaces become tangents determine the optimal solution. The method proved to be quite powerful in delivering solutions to other outstanding problems in measurement optimization, as well. It allowed us to develop a theory of sequential measurements performed by subsequent observers on the same quantum system, bypassing the collapse postulate [2]. In this talk, using probabilistic [3] and deterministic quantum cloning [4], and quantum state separation [5] as illustrative examples we develop a complete geometric solution for nding their optimal success probabilities. In some cases the method delivers analytical results, in others it leads to intuitive and straightforward numerical solutions. We close with the discussion of two works in progress: geometric view of the measurement that maximizes the mutual information and the sequential measurement for the minimum error discrimination of qubit states. This later is the quintessential projective measurement, so a non-destructive implementation and its optimization

are of interest for many applications.

Acknowledgments. This publication was made possible through the support of a Grant from the John Templeton

Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reect the

views of the John Templeton Foundation. Partial nancial support by a Grant from PSC-CUNY is also gratefully

acknowledged.

[1] J. A. Bergou, U. Futschik, and E. Feldman, Optimal unambiguous discrimination of pure quantum states, Phys. Rev. Lett.

108, 250502 (2012).

[2] J. A. Bergou, E. Feldman, and M. Hillery, Extracting information from a qubit by multiple observers: Toward a theory of

sequential state discrimination, Phys. Rev. Lett. 111, 100501 (2013).

[3] V. Yerokhin, A. Shehu, E. Feldman, E. Bagan, and J. A. Bergou, Probabilistically perfect cloning of two pure states:

Geometric approach, Phys. Rev. Lett. 116, 200401 (2016).

[4] V. Yerokhin, A. Shehu, E. Bagan, E. Feldman, and J. A. Bergou, Approximate cloning of quantum states, in preparation.

[5] E. Bagan, V. Yerokhin, A. Shehu, E. Feldman, and J. A. Bergou, A geometric approach to quantum state separation, New