Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)An ensemble of quantum states can be described by a Hermitian, positive semidefinite and unit trace matrix called density matrix. Thus, the study of methods for optimizing a certain function (energy, entropy) over the set of density matrices has a direct application to important problems in quantum information and computation. We propose a projected gradient method for solving such problems. By exploiting the geometry of the feasible set, which is the intersection of the cone of Hermitian positive semidefinite matrices with the hyperplane defined by the unit trace constraint, we describe an efficient procedure to compute the projection onto this set using the Frobenius norm. Some important applications, such as quantum state tomography, are described and numerical experiments illustrate the effectiveness of the method when compared to previous methods based on fixed-point iterations or semidefinite programming. © 2015 Taylor & Francis.312328341CNPq, Conselho Nacional de Desenvolvimento Científico e TecnológicoConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Sorensen, D., (1999) LAPACK Users' Guide, , 3rd ed., SIAM, Philadelphia, PABarzilai, J., Borwein, J.M., Two-point step size gradient methods (1988) IMA J. Numer. Anal, 8, pp. 141-148Bertsekas, D., (1999) Nonlinear Programming, , 2nd ed., Athena ScientificBertsekas, D., Nedic, A., Ozdaglar, A.E., (2003) Convex Analysis and Optimization, , Athena Scientific, BelmontBirgin, E.G., Martínez, J.M., (2014) Augmented Lagrangian Methods for Constrained Optimization, , SIAM, Philadelphia, PABirgin, E.G., Martínez, J.M., Raydan, M., Nonmonotone spectral projected gradient methods on convex sets (2000) SIAM J. Optimiz, 10, pp. 1196-1211Birgin, E.G., Martínez, J.M., Raydan, M., Inexact spectral projected gradient methods on convex sets (2003) IMA J. Numer. Anal, 23, pp. 539-559Boyd, S., Vandenberghe, L., (2004) Convex Optimization, , Cambridge University Press, New YorkBoyle, J.P., Dykstra, R.L., A method for finding projections onto the intersection of convex sets in Hilbert spaces (1986) Lecture Notes in Statistics, 37, pp. 28-47Brandão, F.G.S.L., Vianna, R.O., Robust semidefinite programming approach to the separability problem (2004) Phys. Rev. A, 70Buzek, V., Drobny, G., Derka, R., Adam, G., Wiedemann, H., Quantum state reconstruction from incomplete data (1999) Chaos Solitons Fractals, 10, pp. 981-1074Causa, A., Raciti, F., A purely geometric approach to the problem of computing the projection of a point on a simplex (2013) J. Optimiz. Theory Appl., 156, pp. 524-528Cuppen, J.J.M., A divide and conquer method for the symmetric tridiagonal eigenproblem (1980) Numer. Math., 36, pp. 177-195Demmel, J., (1997) Applied Numerical Linear Algebra, , SIAM, PhiladelphiaDoherty, A.C., Parrilo, P.A., Spedalieri, F.M., Distinguishing separable and entangled states (2002) Phys. Rev. Lett., 88Duchi, J., Shalev-Shwartz, S., Singer, Y., Chandra, T., Efficient projections onto the l1 -ball for Learning in High Dimensions (2008) Proceedings of the 25th International Conference on Machine Learning, ICML '08, pp. 272-279. , Helsinki, Finland, ACMGlancy, S., Knill, E., Girard, M., Gradient-based stopping rules for maximum-likelihood quantum-state tomography (2012) New J. Phys., 14Golub, G., Van Loan, C., (1996) Matrix Computations, , 3rd ed., Johns Hopkins University Press, BaltimoreGómez, W., Ramírez, H., A filter algorithm for nonlinear semidefinite programming (2010) Comput. Appl. Math., 29, pp. 297-328Gonçalves, D.S., Gomes-Ruggiero, M.A., Lavor, C., Global convergence of diluted iterations in maximum-likelihood quantum tomography (2014) Quant. Inf. Comput, 14, pp. 966-980Gonçalves, D.S., Lavor, C., Gomes-Ruggiero, M.A., Cesário, A.T., Vianna, R.O., Maciel, T.O., Quantum state tomography with incomplete data: Maximum entropy and variational quantum tomography (2013) Phys. Rev. A, 87Grippo, L., Lampariello, F., Lucidi, S., A nonmonotone line search technique for Newton's method (1986) SIAM J. Numer. Anal, 23, pp. 707-716Gu, M., Eisenstat, S.C., A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem (1995) SIAM J. Matrix Anal. Appl., 16, pp. 172-191Horodecki, M., Horodecki, P., Horodecki, R., Separability of mixed states: Necessary and sufficient conditions (1996) Phys. Lett. A, 223, pp. 1-8Hradil, Z., Quantum-state estimation (1997) Phys. Rev. A, 55, pp. R1561-R1564Hradil, Z., Řeháček, J., Fiurášek, J., Ježek, M., Maximum-likelihood methods in quantum mechanics (2004) Lecture Notes in Physics, 649, pp. 163-172. , Quantum State Estimation, Springer, Berlin, HeidelbergJarre, F., An interior method for nonconvex semidefinite programs (2000) Optim. Eng., 1, pp. 347-372De Klerk, E., (2002) Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications, , Kluwer Academic Publishers, DordrechtKočvara, M., Stingl, M., Pennon: A code for convex nonlinear and semidefinite programming (2003) Optim. Methods Softw, 18, pp. 317-333Maciel, T.O., Vianna, R.O., Optimal estimation of quantum processes using incomplete information: Variational quantum process tomography (2012) Quantum Inf. Comput, 12, pp. 442-447Meyer, C., (2001) Matrix Analysis and Applied Linear Algebra, , SIAM, PhiladelphiaMichelot, C., A finite algorithm for finding the projection of a point onto the canonical simplex of ℝn (1986) J. Optimiz. Theory Appl., 50, pp. 195-200Von Neumann, J., Functional Operators Vol. II. The Geometry of Orthogonal Spaces (1950) Annals of Mathematical Studies, 22. , Princeton University Press, PrincetonNielsen, M., Chuang, I., Quantum Computation and Quantum Information (2004) Cambridge Series on Information and the Natural Sciences, , Cambridge University Press, CambridgeNocedal, J., Wright, S.J., (1999) Numerical Optimization, , Springer-Verlag, New YorkParis, M., Rehácek, J., Quantum State Estimation (2004) Lecture Notes in Physics, 649. , Springer, Berlin, HeidelbergPeres, A., Separability criterion for density matrices (1996) Phys. Rev. Lett., 77, pp. 1413-1415Raydan, M., On the Barzilai and Borwein choice of steplength for the gradient method (1993) IMA J. Numer. Anal, 13, pp. 321-326Rehacek, J., Hradil, Z., Maximum Entropy Assisted Maximum Likelihood Inversion (2005) Conference on Lasers and Electro-Optics Europe, 2005 (CLEO/Europe 2005), p. 466Řeháček, J., Hradil, Z., Knill, E., Lvovsky, A.I., Diluted maximum-likelihood algorithm for quantum tomography (2007) Phys. Rev. A, 75Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M., Symmetric informationally complete quantum measurements (2004) J. Math. Phys., 45, pp. 2171-2180Rockafellar, R.T., (1970) Convex Analysis, , Princeton University Press, PrincetonSturm, J.F., (1998) Using SeDuMi 1.02, A Matlab Toolbox for Optimization Over Symmetric ConesTeo, Y.S., Stoklasa, B., Englert, B.G., Rehacek, J., Hradil, Z., Incomplete quantum state estimation: A comprehensive study (2012) Phys. Rev. A, 85Todd, M., Semidefinite optimization (2001) Acta Numer., 10, pp. 515-560Toh, K., Todd, M., Tutuncu, R., SDPT3 - A Matlab software package for semidefinite programming (1999) Optim. Methods Softw, pp. 545-581Toh, K.C., An inexact primal-dual path following algorithm for convex quadratic SDP (2008) Math. Program, 112, pp. 221-254Toh, K.C., Tucuntu, R.H., Todd, M.J., Inexact primal-dual path-following algorithms for a special class of convex quadratic sdp and related problems (2007) Pac. J. Optim, 3, pp. 135-164Vandenberghe, L., Boyd, S., Semidefinite programming (1996) SIAM Rev., 38, pp. 49-95Wolfe, P., Finding the nearest point in a polytope (1976) Math. Program, 11, pp. 128-149Yamashita, H., Yabe, H., Harada, K., A primal-dual interior point method for nonlinear semidefinite programming (2012) Math. Program, 135, pp. 89-121Zhang, H., Hager, W.W., A nonmonotone line search technique and its application to unconstrained optimization (2004) SIAM J. Optimiz, 14, pp. 1043-1056Zyczkowski, K., Bengtsson, I., (2006) Geometry of Quantum States, , Cambridge University Press, New YorkZyczkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M., Volume of the set of separable states (1998) Phys. Rev. A, 58, pp. 883-892Zyczkowski, K., Penson, K.A., Nechita, I., Collins, B., Generating random density matrices (2011) J. Math. Phys., 52