It has been shown that the optimization operator, in the objective function, can be assumed to commute with the symmetry operator. This result leads to an alternative formulation of the dual problem, that presents a reduced number of variables and constraints. The obtained dual statement is convenient to find analytical solutions to the discrimination problem, as we showed with an illus- trative example. On the other hand, the new formulation also permits a computationally efficient numerical solu- tion by means of SDP methods. This property is partic- ularly useful to study the performance limits of quantum mechanical systems described by geometrically uniform quantum states on Hilbert spaces of large dimensions, such as modulated coherent states in optical communi- cations. The authors would like to thank M. Sasaki for encour- aging comments. This work was supported in part by the Q-FUTURE project (prot. STPD08ZXSJ), University of Padua. Acknowledgments 5 [15] G. Cariolaro, R. Corvaja, and G. Pierobon, (to be pub- blished). [9] Y.C. Eldar, A. Megretski, and G.C. Verghese, IEEE Trans. Inf. Theory 69 , 1007 (2003). [8] S. Boyd and L. Vandenberghe, Convex Optimization , 7-th Ed., (Cambridge Univ. Press, Cambridge, U.K., 2009). [7] K. Kato, M. Sasaki, and O. Hirota, IEEE Trans. Com- mun., 47 , 248 (1999). [6] Y.C. Eldar, A. Megretski, and G.C. Verghese, IEEE Trans. Inf. Theory 68 , 1198 (2004). [5] M. Ban, K. Kurokawa, R. Momose, and O. Hirota, Int. J. Theor. Phys 36 , 1269 (1997). [4] Y.C. Eldar and G.D. Forney, IEEE Trans. Inf. Theory 47 , 858 (2001). [3] H.P. Yuen, R.S. Kennedy, and M. Lax, IEEE Trans. Inf. Theory IT-21 , 125 (1975). [2] A.S. Holevo, J. Multivariate Anal. 3 , 337 (1973) [1] C.W. Helstrom, Proc. of the IEEE 58 , 1578 (1970). [10] L.A. Sadun, Applied Linear Algebra: The Decoupling Principle (American Mathematical Society, Rhode Is- land, USA, 2008). [14] G. Cariolaro and G. Pierobon, IEEE Trans. Commun. (to appear), arXiv:0911.2633. [13] H. Hemmati, Deep Space Optical Communications (J. Wiley & Sons, Inc., Hoboken, NJ, 2006). [12] C.-L. Chou and L.Y. Hsu, Phys. Rev. A 68 , 042305 (2003). [11] M. Grant and S. Boyd. CVX: Matlab software for dis- ciplined convex programming (web page and software). http://stanford.edu/ ∼ boyd/cvx, June 2009.