6 means that h ( ρ ) = h $^{′}$( ρ ), since smooth functions whose derivatives agree at a dense set of points are identical. Eq. 13 is the maximum. Furthermore, this is a local max- imum (with respect to all variations), because the gradi- ent of h ( ρ ) is orthogonal to unitary rotations and there- fore locally orthogonal to off-diagonal variations. This is also a global maximum, because log L $^{′}$( ρ ) is a convex function. Thus, Eq. 13 maximizes the hedged likelihood. Now, consider some other hedging function h $^{′}$( ρ ). If h $^{′}$( ρ ) is not unitarily invariant, then there exists some point ρ such that, in the neighborhood of ρ , the gradi- ent of h $^{′}$( ρ ) is not orthogonal to unitary rotations. Sup- pose that the measured basis {| k 〉} is the diagonal basis of ρ , and the measured frequencies n$_{k}$ are such that ˆ ρ$_{H}$ (given by Eq. 13) is in the neighborhood of ρ . Then, at the point ˆ ρ$_{H}$ , the gradient of L ( ρ ) is orthogonal to off-diagonal variations, but the gradient of h $^{′}$( ρ ) is not. This means that the gradient of the hedged likelihood does not vanish, and thus its maximum cannot coincide with Eq. 13. ∗ Electronic address: robin@blumekohout.com [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Press, 2000). [2] Z. Hradil, Phys. Rev. A 55 , R1561 (1997). [3] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Physical Review A 64 , 052312 (2001). [4] Z. Hradil, J. Reh´ acek, J. Fiurasek, and M. Jezcaronek, in Quantum state estimation , edited by M. G. A. Paris and J. Rehacek (Berlin, Germany : Springer-Verlag, 2004, 2004), vol. 649 of Lecture Notes In Physics , pp. 59–112. [5] H. Haeffner, W. Haensel, C. F. Roos, J. Benhelm, D. C. al kar, M. Chwalla, T. Koerber, U. D. Rapol, M. Riebe, P. O. Schmidt, et al., Nature 438 , 643 (2005). [6] R. Blume-Kohout (2006), quant-ph/0611080 . [7] G. J. Lidstone, Transactions of the Faculty of Actuaries 8 , 182 (1920). [8] E. Ristad, Arxiv preprint cmp-lg/9508012 (1995). [9] T. H. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991). [10] Q. Xie and A. Barron, IEEE Transactions on Information Theory 46 , 431 (2000). [11] K. Vogel and H. Risken, Phys. Rev. A 40 , 2847 (1989). [12] B. Clarke and A. Barron, Journal of Statistical Planning and Inference (1994). [13] R. Krichevskiy, IEEE Transactions on Information The- ory 44 , 296 (1998). [14] D. Braess, J. Forster, T. Sauer, and H. Simon, Lecture Notes in Computer Science pp. 380–394 (2002). [15] H. Jeffreys, Theory of Probability (Oxford University Press, 1998). If h $^{′}$( ρ ) is unitarily invariant, then for every measured basis, the maximization can safely be restricted to diago- nal matrices ρ = ∑ k p$_{k}$ | k 〉〈 k | , and it reduces to a classical problem, maximizing To reproduce the “add β ” rule, the gradient of log L ′ must vanish – for all { n$_{k}$ } – at p$_{H}$ = { n$_{k}$ + β N $^{}}$. This implies and since this condition is automatically satisfied by Eq. 1, at every point p$_{H}$ = { n$_{k}$ + β N } for any { n$_{k}$ } . These points are dense in the simplex on which h $^{′}$( p ) is defined. This log L $^{′}$( p ) = log h $^{′}$( p ) + log L ( p ) . ∇ log h $^{′}$| p$_{H}$ = −∇ log L| $_{p$_{H}$}$, ∇ log h $^{′}$| p$_{H}$ = ∇ log h | p$_{H}$