Functions of Markov Processes and Algebraic Measures

Fannes, Mark; Nachtergaele, Bruno; Slegers, L.

doi:10.1142/s0129055x92000030

Cited by 19 publications

(37 citation statements)

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“…We state the following proposition from [1] Proposition II.1: Given the triplet (U, ρ, (E a ) a∈K ) there exists a unique translation invariant probability measure µ on …”

Section: A Manifestly Positive Algebraic Measuresmentioning

confidence: 99%

“…In [1] an equation for φ(dw) is derived in terms of a Markov operator T µ on C(W), the space of continuous functions on the simplex W. In addition the support of the measure is also characterized. For functions of Markov processes Blackwell [4] obtained a formula similar to equation (8) however there was no clear connection of the measure with the Markov operator and the support of the measure was also not explicitly characterized.…”

Section: Theorem Ii2 ([1])mentioning

confidence: 99%

“…These measures known as manifestly positive algebraic measures and their properties were first studied in [1]. A one to one correspondence was shown in [1] a between manifestly positive algebraic measures and hidden Markov process.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we follow the approach of [1], wherein a formula similar to (4) was derived. Moreover, in [1] the support of the Blackwell measure was explicitly characterized. We see that in the case of the particular noise model that we study in this paper the support of the measure φ(dw) simplifies and leads to an analytic solution of the entropy rate.…”

Section: Introductionmentioning

confidence: 99%

“…In section II-A we review some key results on algebraic measures from [1] that we will use in this paper. The description of the noise model and the support of the measure φ(dw) is computed in in section III.…”

Section: Introductionmentioning

confidence: 99%

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Entropy rate calculations of algebraic measures

Marchand¹,

Mulherkar

Nachtergaele

2012

2012 IEEE International Symposium on Information Theory Proceedings

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Abstract-Let K = {0, 1, ..., q − 1}. We use a special class of translation invariant measures on K Z called algebraic measures to study the entropy rate of a hidden Markov processes. Under some irreducibility assumptions of the Markov transition matrix we derive exact formulas for the entropy rate of a general q state hidden Markov process derived from a Markov source corrupted by a specific noise model. We obtain upper bounds on the error when using an approximation to the formulas and numerically compute the entropy rates of two and three state hidden Markov models.

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“…We state the following proposition from [1] Proposition II.1: Given the triplet (U, ρ, (E a ) a∈K ) there exists a unique translation invariant probability measure µ on …”

Section: A Manifestly Positive Algebraic Measuresmentioning

confidence: 99%

Section: Theorem Ii2 ([1])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Entropy rate calculations of algebraic measures

Marchand¹,

Mulherkar

Nachtergaele

2012

2012 IEEE International Symposium on Information Theory Proceedings

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Finitely correlated states on quantum spin chains

1992

Self Cite

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We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic "Neel ordered" states. The ergodic components have exponential decay of correlations. All states considered can be obtained as "local functions" of states of a special kind, so-called "purely generated states," which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.

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Entangled Markov chains

Accardi

Fidaleo

2005

Annali di Matematica

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Motivated by the problem of finding a satisfactory quantum generalization of the classical random walks, we construct a new class of quantum Markov chains which are at the same time purely generated and uniquely determined by a corresponding classical Markov chain. We argue that this construction yields as a corollary, a solution to the problem of constructing quantum analogues of classical random walks which are "entangled" in a sense specified in the paper. The formula giving the joint correlations of these quantum chains is obtained from the corresponding classical formula by replacing the usual matrix multiplication by Schur multiplication. The connection between Schur multiplication and entanglement is clarified by showing that these quantum chains are the limits of vector states whose amplitudes, in a given basis (e.g. the computational basis of quantum information), are complex square roots of the joint probabilities of the corresponding classical chains. In particular, when restricted to the projectors on this basis, the quantum chain reduces to the classical one. In this sense we speak of entangled lifting, to the quantum case, of a classical Markov chain. Since random walks are particular Markov chains, our general construction also gives a solution to the problem that motivated our study. In view of possible applications to quantum statistical mechanics too, we prove that the ergodic type of an entangled Markov chain with finite state space (thus excluding random walks) is completely determined by the corresponding ergodic type of the underlying classical chain

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Functions of Markov Processes and Algebraic Measures

Cited by 19 publications

References 0 publications

Entropy rate calculations of algebraic measures

Entropy rate calculations of algebraic measures

Finitely correlated states on quantum spin chains

Entangled Markov chains

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