Extension of information geometry to non-statistical systems some examples

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14345

Résumé

Our goal is to extend information geometry to situations where statistical modeling is not obvious. The setting is that of modeling experimental data. Quite often the data are not of a statistical nature. Sometimes also the model is not a statistical manifold. An example of the former is the description of the Bose gas in the grand canonical ensemble. An example of the latter is the modeling of quantum systems with density matrices. Conditional expectations in the quantum context are reviewed. The border problem is discussed: through conditioning the model point shifts to the border of the differentiable manifold.

Extension of information geometry to non-statistical systems some examples

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Our goal is to extend information geometry to situations where statistical modeling is not obvious. The setting is that of modeling experimental data. Quite often the data are not of a statistical nature. Sometimes also the model is not a statistical manifold. An example of the former is the description of the Bose gas in the grand canonical ensemble. An example of the latter is the modeling of quantum systems with density matrices. Conditional expectations in the quantum context are reviewed. The border problem is discussed: through conditioning the model point shifts to the border of the differentiable manifold.

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Extension of information geometry to non-statistical systems: some examples Jan Naudts and Ben Anthonis Universiteit Antwerpen Saclay, October 2015 J. Naudts and B. Anthonis, Extension of Information Geometry to Non-statistical Systems: Some Examples, in: Geometric Science of Information, GSI 2015 LNCS proceedings, F. Nielsen and F. Barbaresco eds., (Springer, 2015), ISBN 978-3-319-25039-7, p. 427–434 1/ 16 Outline I An abstract setting II The free Bose gas in the grand canonical ensemble III Quantum measurements IV Weak measurements 2/ 16 I An abstract setting The geometry of a statistical manifold M can be studied by means of the Kullback-Leibler divergence DKL(p, q) = i p(i) ln p(i) q(i) . (p and q are probability distributions) It is called a relative entropy in Statistical Physics. The Fisher information at a point pθ of M is given by Ik,l(θ) = Eθ ∂ ∂θk ln pθ ∂ ∂θl ln pθ . 3/ 16 The Fisher information can be obtained from the KL divergence by Ik,l (θ) = ∂2 ∂θk ∂θl D(p||pθ) p=pθ . In this expression it is not needed that p belongs to M. A straightforward calculation shows this. Definition The extended Fisher information of a pdf p (p not necessarily in M) is Ik,l (p) = ∂2 ∂θk ∂θl D(p||pθ) θ: p∈Fθ . Fθ denotes the ’fiber’ of all p best fitted by pθ. 4/ 16 Interestingly, the meaning of Ik,l (θ) becomes clearer when considering p not in M. p is the empirical measure, the result of an experiment. Given p, look for pθ which minimizes D(p||pθ). The divergence D(p||pθ) is a measure for the distance between an arbitrary p and a point pθ of the statistical manifold M. The inverse of the Fisher information matrix now indicates how well the model point pθ is determined by the experimental data p. 5/ 16 Proposition Ik,l (p) is covariant. Proposition If pθ belongs to the exponential family then Ik,l(p) is constant on the fibre Fθ. Proof follows from a Pythagorean relation. ◮ J. Naudts and B. Anthonis, The exponential family in abstract information theory, in: Geometric Science of Information, GSI 2013 LNCS proceedings, F. Nielsen and F. Barbaresco eds., (Springer, 2013), p. 265–272. ◮ B. Anthonis, Extension of information geometry for modelling non-statistical systems, PhD Thesis December 2014, arXiv:1501.00853. These results hold in an abstract setting. 6/ 16 Abstract setting It is not needed that p and pθ are probability distributions. Space X of possible experimental outcomes, Space M manifold of model points, D(x||m) : X × M → [0, +∞] divergence function. 2 examples: ◮ Bose gas in the grand canonical ensemble ◮ Quantum measurements 7/ 16 II The free Bose gas in the grand canonical ensemble For instance, a gas of photons. ◮ The energy levels are denoted ε1, ε2, · · · . ◮ Each level εj can contain an arbitrary number nj of photons. ◮ The total number N = j nj of photons is usually not known in advance. ◮ A measurement yields the sequence of numbers n1, nj , · · · . ◮ The standard trick is to define probabilities pj = nj /N. ◮ Working with the empirical measure on the space of all finite sequences of integers is better, but is technically involved. ◮ Why not take directly X = {n1, n2, · · · , 0, 0, 0, · · · : ni ∈ N}? 8/ 16 The manifold M is a two-parameter family of probability distributions pβ,µ(n) = 1 Z(β, µ) exp(−β j ǫj nj + βµ j nj ), β > 0 and µ < ǫj for all j, normalization given by Z(β, µ) = j 1 1 − exp(−β(ǫj − µ)) . Choose the divergence function D(n||β, µ) = ln Z(β, µ) − j nj (−βǫj + βµ). 9/ 16 Minimization of β, µ → D(n||β, µ) reproduces standard textbook results for β and µ. The metric tensor g(β, µ) can be calculated. The extended Fisher information matrix is given by Ik,l (n) = g(β, µ) for all n ∈ Fβ,µ. It indicates how well β and µ are determined by the data set n. In the PhD thesis covariant derivatives ∇β and ∇µ are introduced. A method is described to calculate connection coefficients ωc ab for which ∇a∂b = ωc ab∂c. 10/ 16 III Quantum measurements The state of a quantum system is a wave function ψ or a density matrix ρ ◮ ψ is a normalized element of a Hilbert space ◮ ρ is a non-negative trace-class operator with trace 1 ◮ in the finite case ρ is an Hermitean matrix with eigenvalues pi ≥ 0, i pi = 1. ◮ ψ determines the density matrix |ψ ψ| which is an orthogonal projection onto Cψ. A quantum experiment measuring ψ reveals the diagonal part of the matrix |ψ ψ| in a basis dictated by the experimental setup. 11/ 16 Example Let ψ = 1√ 2 1 1 ∈ C2 . Then |ψ ψ| = 1 2 1 1 1 1 . The result of a measurement in the basis 1 0 , 0 1 is 1 2 1 0 0 1 . Axiom The result of a quantum measurement is always a conditional expectation. The condition is that the resulting density matrix must be diagonal in the basis of the measurement. 12/ 16 A definition of a quantum conditional expectation is found in D. Petz, Quantum Information Theory and Quantum Statistics (Springer, 2008) The theory of quantum conditional expectations ◮ originated with the work of Accardi and Cecchini; ◮ relies on Tomita-Takesaki theory. The map |ψ ψ| → diag |ψ ψ| is a quantum conditional expectation in the sense of Petz. 13/ 16 The quantum analogue of the Kullback-Leibler divergence is given by D(σ||ρ) = Tr σ ln σ − Tr σ ln ρ. It satisfies D(σ||ρ) ≥ 0, with equality if and only if σ = ρ. Choose a model manifold M consisting of diagonal matrices. Let σc = diag σ. Then the Pythagorean relation holds (special case of Theorem 9.3 of Petz) D(σ||ρ) = D(σ||σc ) + D(σc||ρ) for all σ, for all ρ ∈ M. Hence, minimizing ρ ∈ M → D(σ||ρ) may be replaced by minimizing ρ ∈ M → D(σc||ρ). 14/ 16 IV Weak measurements ◮ Quantum measurements concern tiny effects ◮ The measurements usually destroy the state of the measured system ◮ This is known as the collapse of the wave function ◮ Recent experiments overcome this difficulty ⇒ weak measurements Theoretical support comes from Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351–1354 (1988) 15/ 16 Our interpretation What happens if the model manifold M does not consist of matrices diagonal in the basis of the experiment? ◮ Assume ρ > 0 for all ρ ∈ M. ◮ Project M onto Mc = {ρc : ρ ∈ M}. ◮ Minimize ρ → D(σc ||ρc). ◮ Then also ρ → D(σ||ρc) is minimized. If now Mc lies in the border of {ρ : ρ > 0} then the value of D(σ||ρc ) can become large and very sensitive to the experimental outcome σ. We conjecture that this amplification effect is exploited in weak measurements. 16/ 16