Faigle, U.

Schönhuth, A. (Alexander)

English

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Observation and Evolution ofFinite-dimensional Markov SystemsUlrich Faigle1 and Alexander Scho¨nhuth21University of Cologne2CWI Amsterdam1 Markov systemsA system S is an entity that can be in one of several states. Let S be the set of states of S. Ann-dimensional Markov representation is an injective map ρ : S → Q onto an affine hyperplaneQ of an n-dimensional Hilbert space H over R. We denote the inner product in H by 〈x|y〉and assumeQ = {x ∈ H | τ(x) = 1},where τ : H → R is a linear functional. Given the representation ρ, we identify S with Q andspeak of Q as the collection of (Markov) states of S.An n-dimensional Markov system S admits a standard representation σ : S → Q into theeuclidean coordinate space Rn with inner product〈x|y〉 = xT y =n∑i=1xiyi for all xT = (x1, . . . , yn), yT = (y1, . . . , yn) ∈ Rn.and, with 1T := (1, 1, . . . , 1), the affine hyperplaneQ = {x ∈ Rn | τ(x) = 1Tx = x1 + . . .+ xn = 1}.However, also other representations are of interest to the mathematical modeler:1.1 Quantum Markov systemsMotivated by the classical model of m-dimensional quantum systems, consider the (complex)Hilbert space Cm×m of complex (m×m)-matrices with inner product〈C|D〉 = tr(D∗C),where D∗ is the conjugate transpose of D and tr(A) denotes the trace of a matrix A. Recallthat a matrix C is self-adjoint (or hermitian) if C = C∗ and let H denote the collection of allself-adjoint (m×m)-matrices C. It is not difficult to see that H forms a real(!) Hilbert spaceof dimension n = m2. Letting I denote the identity matrix of Cm×m, we call the members ofthe hyperplaneD = {D ∈ H | tr(D) = 〈D|I〉 = 1}Markov density matrices and refer to a system with states corresponding to Markov densitymatrices a Markov quantum system.1.2 Quantum activity systems and quantum bitsWhile classical computation is based on boolean bits, quantum computation (see, e.g., [8])models activities by quantum bits (”qbits”), where one qbit has the formq = α|0〉+ β|1〉 with α, β ∈ C s.t. |α|2 + |β|2 = 1.The qbit q has has the interpretation that |0〉 is observed with probability |α|2 ≥ 0 and |1〉with probability |β|2 = 1− |α|2 ≥ 0.An n-dimensional quantum activity system is the n-fold tensor product A = A1 ⊗ · · · ⊗ Anof 1-dimensional quantum activity systems Ai. An n-dimensional quantum activity state (”n-qbit”) is therefore of the formq =∑k∈{0,1}nαk|k〉 with αk ∈ C and∑k |α|2 = 1 (1)and corresponds to the parameter vector v = (αk|k ∈ {0, 1}n) ∈ C2n with (squared) norm‖v‖2 = v∗v = |α1|2 + . . .+ |αn|2 = 1.Note that an n-qbit q in the form (1) cannot directly be interpreted a Markov state instandard form. The associated matrix Q = vv∗ is self-adjoint with tracetr(vv∗) = v∗v = |α1|2 + . . .+ |αn|2 = 1and hence a Markov density (in fact, a classical quantum density).1.3 Pseudo-boolean functions and cooperative gamesA real-valued set function v : 2N → R is a pseudo-boolean function (see [6]). Identifying thesubsets K ⊆ N with their associated boolean states |k〉, a pseudo-boolean function v can beviewed as a formal linear combinationv =∑k∈{0,1}nαk|k〉with the coefficients αk = v(K).From a game theoretic point of view, the pair Γ = (N, v) is a cooperative game with charac-teristic function v. The parameter v(K) is thought to reflect the ”value” of the coalitionK ⊆ Nin a given economic context. It is reasonable to assume that the game Γ is scaling-invariant.So we might equally well study the normalized game (N, v˜), wherev˜ ={0 if v ≡ 0v/‖v‖2 if ‖v‖2 = ∑K⊆N v(K)2 6= 0and think of a non-trivial cooperative game as a qbit with real coefficients.Remark 1.1. The Hadamard transformation H of a a 1-qbit is the linear transformation|k1 . . . kn〉 7→ H|k1〉 ⊗ · · · ⊗H|kn〉 (k1 . . . kn ∈ {0, 1}2). (2)The Hadamard coefficients αˆk of v correspond to the Banzhaf indices (see [2]), well-known insocial choice theory. (See, e.g., [7] for more applications of the Hadamard transformation tosocial choice problems and [5] for more on interaction indices).2 Observables and measurementsReturning to the general Markov state model with the n-dimensional Hilbert space H andQ = {v ∈ H | τ(v) = 1} relative to the system S, let us fix a particular basis B ⊆ Q.Remark 2.1. We think of B as the set of representatives of the ”ground states” of S.We call a function X : B → {0, 1} an information function. So X models a ”property”ground states b ∈ B may or may not have. Extending X linearly to all of H, X correspondsto an element x ∈ H such that〈x|b〉 = X(b) for all b ∈ B.Assume that S happens to be in the Markov state q =∑b∈B qbb and definepiq(r) =∑b∈B:X(b)=rqb (r = 0, 1).We call X (statistically) observable in the state q if piq(r) ≥ 0 holds for r = 0, 1.3 Evolution of Markov systemsA Markov (evolution) operator relative to the Markov system S, represented as the hyperplaneQ of the Hilbert space H is a linear transformation µ : H → H such that µ(q) ∈ Q holds forall q ∈ Q.A (generalized) Markov chain is a pair (µ, q(0)) where µ is a Markov operator and q a Markovstate. The pair (µ, q(0)) stands short for the Markov evolution of states in discrete time whenthe Markov system S is in state q(0) at time t = 0:q(t) = µ(q(t−1)) = µt(q(0)) for t = 1, 2, . . ..Examples of Markov chains relative to the standard representation are, of course, classicalMarkov chains, where µ is represented by a probability transition matrix.Other examples arise from the Schro¨dinger wave evolution in quantum activity systems.3.1 Evolution and measurementThe concept of a measurement can be naturally be put into context with evolution. We call afamily X = {µr | r ∈ R} of linear operators µr : H → H a Markov measurement with (finite)scale R iffµX :=∑r∈Rµa is a Markov operator. (3)In light of (3), we write (X, q) as a unifying notation for both a Markov measurement X andan associated Markov chain (µX , q) and refer to it as a Markov measurement chain. A Markovmeasurement chain is invariant if µX(q) = q.Now consider concatenating measurements (w := r1...rn)µw(q) := µrn(...(µr1(q))...)and observe that, by multinomial expansion, µtX =∑w∈Rt µw. We call a Markov measurementchain (X, q) (statistically) observable iffτ(µw(q)) ≥ 0 for all w ∈ R∗.3.2 Equivalence and minimality of Markov measurementsWe call two Markov measurement chainsX1 = ({µr : H1 → H1 | r ∈ R}, q1) and X2 = ({ρr : H2 → H2 | r ∈ R}, q2)where, possibly, dimH1 6= dimH2, equivalent iffτ1(µr¯(q1)) = τ2(µr¯(q2)) for all r¯ ∈ R∗ =∑t≥0Rt.We write(X1, q1) ∼ (X2, q2)in that case.We call a Markov measurement chain (X, q) on H minimal iff dimH is minimal among allMarkov measurement chains that are equivalent to (X, q). (See also [4] for details on how toperform equivalence tests efficiently.)3.3 Decomposition of Markov measurementsWe present the following new theorem:Theorem 3.1 (Decomposition of invariant Markov measurement chains). Let X = ({µr :H → H | r ∈ R}, q) be a minimal, observable, invariant Markov measurement chain. Letd := dim(EigµX (1). Then there are minimal, observable, invariant Markov measurement chainsXi := ({µ(i)r : Hi → Hi | r ∈ R}, qi) i = 1, ..., dsuch that(i) q = q1 + ...+ qd(ii) (X, qi) ∼ (Xi, qi)(iii) dim(EigµXi(1)) = 1.(iv) H ∼= H1 ⊗ ...⊗Hd.Remark 3.2. dim EigµX (1) ≥ 1, see [3].One may perceive this theorem as a building block for a unifying theory of classification for,for example, hidden Markov processes, quantum random walks and action-based cooperationsystems emerging from game theory [10].References[1] J.-P. Aubin: Cooperative fuzzy games. Math. of Operations Research 6 (1981), 1-13.[2] J.F. Banzhaf: Weighted voting does not work: A mathematical analysis. Rutgers LawReview 19 (1965), 317-347.[3] U. Faigle and A. Scho¨nhuth: Asymptotic mean stationarity of sources with finite evolutiondimension, IEEE Trans. Information Theory 53 (2007), 2342-2348.[4] U. Faigle and A. Scho¨nhuth: Efficient tests for equivalence of hidden Markov processes andquantum random walks, IEEE Trans. Information Theory, 57 (2011), 1746-1753.[5] M. Grabisch, J.-L. Marichal, R. Mesiar and E. Pap, Aggregation Functions. Encyclopediaof Mathematics and its Applications vol. 127, Cambridge, 2009.[6] P.L. Hammer and S. Rudeanu, Boolean Methods in Operations Research and RelatedAreas. Springer-Verlag, 1968.[7] G. Kalai: A Fourier-theoretic perspective on the Condorcet paradox and Arrow’s theorem.Advances in Applied Mathematics 29 (2002), 412-426.[8] A. Yu. Kitaev, A.H. Shen and M.N. Vyalyi, Classical and Quantum Computation, GraduateStudies in Mathematics vol. 47, American Mathematical Society, 2002.[9] A. Scho¨nhuth, Discrete-valued stochastic vector spaces. Doctoral dissertation (in German),Universita¨t zu Ko¨ln, 2006.[10] J. Voss, A System-theoretic Approach to Multi-Agent Models. Doctoral dissertation (inGerman), Universita¨t zu Ko¨ln, 2012.

Observation and evolution of finite-dimensional Markov systems

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