A Dynamical Point of View of Quantum Information: Entropy and Pressure

Baraviera, Alexandre; Lardizabal, Carlos F.; Lopes, Artur O.; Cunha, M. O. Terra

doi:10.1007/978-3-642-11456-4_7

Cited by 4 publications

(3 citation statements)

References 30 publications

(30 reference statements)

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“…This definition is a generalization of the analogous concept presented on the papers [6], [8] and [7].…”

Section: Entropymentioning

confidence: 99%

“…Example 8.5 in section 8 (related to Markov Chains) will show that our definition is a natural extension of the classical concept of entropy. We point out that the definition of entropy we will consider here is a generalization of the concept described on the papers [6], [8] and [7]. This particular way of defining entropy is inspired by results of [25] which consider iterated function systems.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Thermodynamic Formalism for Quantum Channels: Entropy, Pressure, Gibbs channels and generic properties

E.¹,

Josue²,

O.³

2019

Preprint

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Denote M k the set of complex k by k matrices. We will analyze here quantum channels φ L of the following kind: given a measurable function L : M k → M k and the measure µ on M k we define the linear operatorA recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where L was the identity.Under some mild assumptions on the quantum channel φ L we analyze the eigenvalue property for φ L and we define entropy for such channel. For a fixed µ (the a priori measure) and for a given an Hamiltonian H : M k → M k we present a variational principle of pressure (associated to such H) and relate it to the eigenvalue problem. We introduce the concept of Gibbs channel.We also show that for a fixed µ (with more than one point in the support) the set of L such that it is Φ-Erg (also irreducible) for µ is a generic set.We describe a related process X n , n ∈ N, taking values on the projective space P (C k ) and analyze the question of existence of invariant probabilities.We also consider an associated process ρ n , n ∈ N, with values on D k (D k is the set of density operators). Via the barycenter we associate the invariant probabilities mentioned above with the density operator fixed for φ L .

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“…This definition is a generalization of the analogous concept presented on the papers [6], [8] and [7].…”

Section: Entropymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Thermodynamic Formalism for Quantum Channels: Entropy, Pressure, Gibbs channels and generic properties

E.¹,

Josue²,

O.³

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…This shows that the concept that was introduced is natural. This definition of entropy is a generalization of the concept described on the papers [6], [8] and [7]. This particular way of defining entropy is in some sense inspired by results of [26] which considers iterated function systems.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov exponents for Quantum Channels: an entropy formula and generic properties

E.¹,

Josue²,

O.³

2019

Preprint

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We denote by M k the set of k by k matrices with complex entries. We consider quantum channels φ L of the form: given a measurable function L : M k → M k and a measure µ on M k we define the linear operator φ L :On a previous work the authors show that for a fixed measure µ it is generic on the function L the Φ-Erg property (also irreducibility). Here we will show that the purification property is also generic on L for a fixed µ.Given L and µ there are two related stochastic process: one takes values on the projective space P (C k ) and the other on matrices in M k . The Φ-Erg property and the purification condition are good hypothesis for the discrete time evolution given by the natural transition probability. In this way it will follow that generically onOn the previous work it was presented the concepts of entropy of a channel and of Gibbs channel; and also an example (associated to a stationary Markov chain) where this definition of entropy (for a quantum channel) matches the Kolmogorov-Shanon definition of entropy. We estimate here the larger Lyapunov exponent for the above mentioned example and we show that it is equal to − 1 2 h, where h is the entropy of the associated Markov probability.

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A Dynamical Point of View of Quantum Information: Entropy and Pressure

Cited by 4 publications

References 30 publications

Thermodynamic Formalism for Quantum Channels: Entropy, Pressure, Gibbs channels and generic properties

Thermodynamic Formalism for Quantum Channels: Entropy, Pressure, Gibbs channels and generic properties

Lyapunov exponents for Quantum Channels: an entropy formula and generic properties

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