Distilling common randomness from bipartite quantum states
Igor Devetak, Andreas Winter
ISIT 2003
Two very basic transformations in multivariate statistics are those of a p×q matrix X to a p×q matrix Y defined by Y=AXB (where A and B are matrices of constants) and of a p×p nonsingular matrix X to a p×p matrix W defined by W=X-1. The Jacobians of these transformations are known to be |A|q|B|p and (-1)p|X|-2p, respectively, or |A|p+1 and (-1)p(p+1)/2|X|-(p+1), respectively, depending on whether X is unrestricted or X is symmetric and B=A′. The derivation of these formulas is greatly facilitated by the introduction of the vec and vech operators [H. Neudecker, J. Amer. Statist. Assoc. 64 (1969) 953-963; H.V. Henderson, S.R. Searle, Canad. J. Statist. 7 (1979) 65-81; J.R. Magnus, H. Neudecker, SIAM J. Algebraic Discrete Methods 1 (1980) 422-449; J.R. Magnus, H. Neudecker, Econometric Theory 2 (1986) 157-190]. Only relatively basic properties of these operators are needed. Arguments that appeal to the existence of the singular value decomposition or to related decompositions are not needed; nor is it necessary to introduce matrix differentials. © 2000 Elsevier Science Inc.
Igor Devetak, Andreas Winter
ISIT 2003
Ronen Feldman, Martin Charles Golumbic
Ann. Math. Artif. Intell.
Richard M. Karp, Raymond E. Miller
Journal of Computer and System Sciences
Heng Cao, Haifeng Xi, et al.
WSC 2003