Riemannian geometry of positive definite matrices: Matrix means and quantum Fisher information

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1 Riemannian geometry of positive definite matrices: Matrix means and quantum Fisher information Dénes Petz Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences POB 127, H-1364 Budapest, Hungary Home page: petz Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.1/25

2 Abstract Riemannian geometries are studied on the manifold of positive definite matrices. One motivation is a geometric view of matrix means which can be a midpoint of geodesics in some very special examples. The other motivation is a information-geometric approach to the state space of a quantum system. The state space consists of positive definite matrices of trace 1. The statistical distance should be monotone with respect to coarse-grainings and the Riemmanian metrics with this property are the quantum Fisher informations. Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.2/25

3 Main references T. Ando, C-K. Li and R. Mathias, Geometric means, Linear Algebra Appl. 385(2004), F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246(1980), M. Moakher, A differential geometric approach to the geometric mean of symmetric positive definite matrices, SIAM J. Matrix Anal. Appl. 26(2005), L. T. Skovgaard, A Riemannian geometry of the multivariate normal model, Scand. J. Statistics, ), Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.3/25

4 Means of positive numbers A function positive numbers if may be called a mean of (i) for every. (ii) for every. (iii) If, then. (iv) If and, then. (v) is continuous. (vi). The geometric mean, the arithmetic mean and the harmonic mean are the most known examples, sometimes they are called Pythagorean means. Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.4/25

5 Motivation Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.5/25

6 Generalization 1. Instead of arithmetic mean consider other means, for example geometric. 2. Consider means of positive definite matrices. 3. Choose a geometry such that the mean is the midpoint of the connecting geodesic curve. 4. From the mean of two matrices get the mean of three (or more) matrices by a symmetrization algorithm: Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.6/25

7 Information geometry A positive definite matrix might be considered as the variance of the multivariate normal distribution (with 0 mean): The simplest way to construct an information geometry is to start with an information potential function and to introduce the Riemannian metric by the Hessian of the potential which will be the Boltzmann entropy is a constant Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.7/25

8 Riemannian metric The real symmetric matrices can be identified with the Euclidean space of dimension and the positive definite matrices form an open set. Therefore the set of Gaussians has a simple and natural manifold structure. The tangent space at each foot point is the set of symmetric matrices. The Riemannian metric is defined as where gives and are tangents at. The differentiation easily Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.8/25

9 Geodesic is Theorem. The geodesic connecting Proof: Due to the property be a curve. Let, then we may assume that. This will be used for the perturbation such that. in the form of the curve We want to differentiate the length. Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.9/25 at with respect to

10 Differentiation Now we integrate by part the second term:, the first term wanishes here and the Since. is 0 for every perturbation derivative at Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.10/25

11 Geodesic distance On the other hand is the, we can conclude that does not depend on. The distance is and geodesic curve between or generally stands for the Hilbert Schmidt norm. where Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.11/25

12 Kubo-Ando matrix means when mean: is invertible. An important example is the geometric which is the midpoint of the geodesic. Extension of the geometric mean to three positive definite matrices can be done by the symmetrization procedure: Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.12/25

13 View of symmetrization B' C A' A C' B Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.13/25

14 Isometry On the space of self-adjoint matrices we have a natural flat geometry: The distance of and is and the geodesic connecting them is ( ). Let For a function be the manifold of all positive definite the mapping matrices. will be used to determine a Riemannian geometry in such a way that the mapping should be isometric. Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.14/25

15 Orthogonal decomposition If we use the affine parametrization, then the tangent space the foot point orthogonal decomposition consists of the self-adjoint matrices and has an at We denote the two subspaces by and, respectively. These subspaces become othogonal (by definition) also in the Rimannian geometry we introduce. Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.15/25

16 The induced geometry is, then the tangent of the curve If and we have we similarly have For from. These formulas determine the Riemannian metric on Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.16/25

17 Geodesic is and Due to the isometry property the geodesic between with the midpoint which is the quasi-arithmetic mean corresponding to the. function Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.17/25

18 Monotone metrics Let be the manifold of positive definite matrices of trace 1. Assume that for each a Riemannian metric (inner product) is given such that the condition holds for every positive trace preserving linear mapping and for every and. Then there is a matrix mean such that where is the right multiplication. is the left multiplication by and Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.18/25

19 Means of positive matrices The key result in the theory is that operator means are in a 1-to-1 correspondence with functions satisfying conditions: (i) (ii) (iii) if and if. (iv) (v) is operator monotone. is continuous. Operator monotone function: implies. The correspondence is given by when is invertible. Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.19/25

20 Quantum Fisher information ), then (with Diag 1: If, then and 2: If independently on the function. was already discussed. 3: The particular case Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.20/25

21 Kubo-Mori metric, can be used as Von Neumann entropy, information potential and This corresponds to the logarithmic mean and Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.21/25

22 My conjecture The scalar curvature is monotone under coarse graining. Another form of the conjecture is to consider the scalar curvature along curves of Gibbs states: Conjecture: the scalar curvature is monotone decreasing function of. For matrices can be proved by simple computation. For matrices numerical computation gives positive result. (Some partial results are in the PhD thesis of Attila Andai.) The conjecture has an interpretation as uncertainty relation. Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.22/25

23 My relevant works D. Petz, Monotone metrics on matrix spaces, Linear Algebra Appl. 244(1996), D. Petz, Covariance and Fisher information in quantum mechanics, J. Phys. A: Math. Gen. 35(2002), D. Petz and R. Temesi, Means of positive numbers and matrices. Dedicated to Professor Pál Rózsa on the occasion of his 80th birthday, SIAM J. Matrix Anal. Appl. 27(2006), F. Hia and D. Petz, Quasi-arithmetic means of matrices, in preparation. Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.23/25

24 Another conjecture The symmetrization algorithm is convergent for any Kubo-Mori matrix means. Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.24/25

25 A relevant book Non-Euckidean Geometry and its Applications, Debrecen August 22, 2008 p.25/25