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

Artur O. Lopes; Jader E. Brasil; Josue Knorst

REVIEW 4 major objections 5 minor 29 references

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

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that, for a fixed measure, generic bounded continuous channel maps L satisfy the purification condition, so quantum channels of the form φ_L(ρ)=∫L(v)ρL(v)† dμ(v) have well-defined Lyapunov exponents whenever ∫|L|²log|L|…

desk verdict A nice explicit example and a plausible genericity question, but both main theorems have real gaps: Lemma 9.14's covering argument is invalid, and Theorem 7.3 silently drops the stochasticity assumption that makes the Lyapunov exponents well-defined. read the letter →

arxiv 1908.08942 v3 pith:4DUDRAMA submitted 2019-08-22 math.DS math-phmath.MPmath.PRquant-ph

Jader E. Brasil , Josue Knorst , Artur O. Lopes This is my paper · ORCID

classification math.DSmath-phmath.MPmath.PRquant-ph MSC 37H1537A3081P45

keywords quantumchannelsLyapunovexponentspurificationconditiongenericpropertieschannelentropyMarkovchainsrandommatrixproductsPhi-Ergproperty

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies quantum channels of the form $\phi_L(\rho)=\int L(v)\rho L(v)^\dagger\,d\mu(v)$, where $L$ is a measurable matrix-valued function and $\mu$ a measure on matrices. Its central claim is that for a fixed $\mu$, most choices of $L$—in the Baire-generic sense over bounded continuous functions with the uniform topology—are good enough to make the associated stochastic processes behave ergodically: irreducibility, the $\Phi$-Erg property, and the purification condition all hold at once. Under those hypotheses and a mild integrability condition $\int |L(v)|^2\log|L(v)|\,d\mu(v)<\infty$, the paper establishes that the Lyapunov exponents $\infty>\gamma_1\ge\gamma_2\ge\dots\ge\gamma_k\ge-\infty$ of the multiplicative process $W_n=L(v_n)\cdots L(v_1)$ are well defined. The paper then computes the top exponent for a channel built from a stationary Markov chain and obtains $\gamma_1=-\tfrac12 h$, where $h$ is the Markov entropy, with $\gamma_2=-\infty$. The point of the paper is to connect an information-theoretic entropy defined through a Ruelle-operator formalism to the genuinely dynamical quantity of a Lyapunov exponent, and to extend Lyapunov theory from the constant case $L=\mathrm{Id}$ to generic variable channels.

What carries the argument

The central object is the random product $W_n(\omega)=L(\omega_n)\cdots L(\omega_1)$ and its wedge powers $\bigwedge^p W_n$; the Lyapunov exponents are the almost-sure growth rates of the singular values, obtained through the identity $|\bigwedge^p W_n|=a_1(W_n)\cdots a_p(W_n)$. The argument runs on three structural conditions: irreducibility, which gives a Perron-Frobenius theorem for positive maps; the $\Phi$-Erg property, which guarantees a unique minimal invariant subspace and hence uniqueness of the invariant probability for the Markov kernel; and the purification condition, which forces the tail variable $Y_\infty=\lim W_n^*W_n/\operatorname{tr}(W_n^*W_n)$ to be a rank-one projection almost surely. The purification condition yields exponential decay of the expected second exterior power, $\int |\bigwedge^2(L(v_n)\cdots L(v_1))|\,d\mu^{\otimes n}\le C\beta^n$ with $\beta<1$, and this decay drives both the spectral gap $\gamma_2-\gamma_1<0$ and the Baire-category proof that purification is generic.

What would settle it

Take the two-state Markov example with transition matrix $P=\begin{pmatrix}p_{11}&p_{12}\\p_{21}&p_{22}\end{pmatrix}$ and invariant vector $\pi$, and compute numerically the finite-time growth rate $\frac1n\log\|W_n(\omega)^*W_n(\omega)\|^{1/2}$ for long $n$. If this does not converge to $-\frac12(\pi_1 h_1+\pi_2 h_2)$, where $h_j=-\sum_i p_{ij}\log p_{ij}$, the entropy formula fails; separately, constructing a bounded continuous $L$ that is purification-generic but fails $\phi_L^*(\mathrm{Id})=\mathrm{Id}$ would show that the stochastic hypothesis cannot be dropped from Theorem 7.3.

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Extended reading notes

Core claim

The paper's discovery is twofold. First, Proposition 9.1 states that for any measure $\mu$ supported on more than one point, the set of bounded continuous functions $L$ for which the pair $(L,\mu)$ satisfies the purification condition is generic in $B(M_k)$: it contains a dense $G_\delta$ set obtained by intersecting open dense sets over countably many projections. Combined with the earlier genericity of irreducibility and of the $\Phi$-Erg property, this means that the hypotheses of Theorem 7.2—irreducibility, $\Phi$-Erg, purification, and $\int |L(v)|^2\log|L(v)|\,d\mu(v)<\infty$—are satisfied by generic $L$. Theorem 7.3 then asserts that for generic $L$ the Lyapunov exponents are well defined, that $\gamma_2-\gamma_1<0$ (a spectral gap), and that the radial limit $\lim_{n\to\infty}\frac1n(\log|W_n(x)|-\log\|W_n\|)=0$ holds almost surely. The Markov-chain example, with $L=I$ and $\mu=\sum_{ij}\delta_{V_{ij}}$, $V_{ij}=\sqrt{p_{ij}}\,|i\rangle\langle j|$ for a column-stochastic matrix $P=(p_{ij})$, gives the exact formula $\gamma_1=\frac12\sum_{i,j}\pi_j p_{ij}\log p_{ij}=-\frac12 h$, where $h$ is the Shannon entropy of the stationary Markov measure; because $W_n^*W_n$ is always rank one, $\gamma_2=-\infty$.

Load-bearing premise

The proof assumes the channel is stochastic, so that $\phi_L^*(\mathrm{Id})=\mathrm{Id}$ and the expression $P(O_n)=\int_{O_n}\operatorname{tr}(W_n\rho W_n^*)\,d\mu^{\otimes n}$ is a genuine probability, but the generic set of maps $L$ is constructed over all bounded continuous functions without showing it remains generic among the stochastic channels.

Editorial extensions

If this is right

If the paper is right, then for a fixed measure $\mu$, the three hypotheses of irreducibility, $\Phi$-Erg, and purification hold simultaneously for a Baire-generic set of bounded continuous maps $L$, so the ergodic temporal-mean theorem applies generically.
Generic channels of this form have a well-defined full Lyapunov spectrum with a spectral gap whenever the top exponent is finite, and in the Markov example the spectrum degenerates with $\gamma_2=-\infty$.
The entropy of the Markov-chain channel is a dynamical observable: $h=-2\gamma_1$, so the Ruelle-operator entropy defined in the previous work can be recovered from the growth rate of random products.
The purification genericity result means the Lyapunov theory is not confined to the constant case $L=\mathrm{Id}$ but applies to a residual set of variable channel maps.
Because the Markov example's second singular value is exactly zero, the method also predicts that channels built from Dirac measures supported on rank-one Kraus operators produce degenerate spectra beyond the top exponent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The exact relation $\gamma_1=-\tfrac12 h$ suggests testing a broader conjecture: for Gibbs channels in the same thermodynamic formalism, the largest Lyapunov exponent may equal $-\tfrac12$ times the channel entropy, with the factor $1/2$ reflecting the Hilbert-Schmidt norm; finite-dimensional iterated-function-system examples would settle whether this is structural or special to the Markov constru
The paper constructs the generic purification set inside all bounded continuous functions but does not intersect it with the stochastic normalization $\phi_L^*(\mathrm{Id})=\mathrm{Id}$; a natural extension is to check whether purification remains generic in the uniform topology on stochastic channels, since the probability interpretation of the Lyapunov process needs that constraint.
The exponential decay of $\int|\bigwedge^2 W_n|$ under purification may control the speed of convergence of quantum trajectories to the unique invariant measure of the Markov kernel, giving quantitative mixing rates for generic channels.
A testable extension is to perturb the Markov example by adding a second measure component while keeping $L$ variable and bounded, and to see whether the identity $\gamma_1=-h/2$ survives or acquires correction terms from the non-rank-one part.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. The Markov-chain example that gives γ1 = -h/2 is a concrete, instructive computation, and the question of whether the purification condition is generic is worth asking. But the paper's two main claims — generic purification and generic Lyapunov exponents — are not proven as they stand, and the second is actually false as stated.

What the paper does well: the example is the strongest part. Modulo an indexing typo in W_n^*W_n, the calculation is essentially correct: the nonzero singular value is sqrt of the product of transition probabilities, and the ergodic limit gives γ1 = -h/2. That is a nice connection between channel entropy and a dynamical exponent, and it is new relative to the prior work the authors build on. The idea of adapting the Benoist–Fraas–Pautrat–Pellegrini framework to a variable L is also natural and worth pursuing.

The soft spots are not minor. Lemma 9.14 is supposed to reduce the uncountable intersection over all projections to a countable dense set. The proof says that for each π in a countable dense K2 there is a ball B(π,ε(π)) on which a fixed L satisfies the purification condition for nearby projections, and then claims the union of these balls covers P2. That covering step does not follow from density: the radii ε(π) may shrink too fast, leaving gaps. This is a genuine hole in the proof of the main genericity theorem.

The stochasticity problem is even more serious. Section 7 starts by assuming L is a Φ-Erg stochastic map, which is exactly what makes P(O_n)=∫ tr(W_n ρ W_n^*) dμ^n a probability. Theorem 7.3 then drops that assumption and just says “L is generic for μ”. But the genericity results of Section 9 are Baire statements in B(M_k), all bounded continuous L, and the stochastic condition is not generic. For k=1 and μ=(δ_0+δ_1)/2, the stochastic set is closed and nowhere dense. So a Baire-generic L will not be stochastic, P is not a probability, the martingale Proposition 6.5 and the “Pν-a.s.” statements lose their meaning, and the ergodic theorem from [10] does not apply. This is not a small fix; the statement of Theorem 7.3 is false as written. Restricting to stochastic L would require reworking the purification genericity on that subset, which the paper does not do.

The example itself satisfies stochasticity, so its conclusion is safe. But the two headline genericity results need major revision. This is a paper for specialists in the ergodic theory of quantum channels, and it deserves a serious referee — not a desk reject — but the referee should expect to send it back with substantial demands.

Referee Report

4 major / 5 minor

Summary. The paper studies quantum channels of the form φ_L(ρ)=∫ L(v)ρL(v)† dμ(v) for a fixed measure μ on M_k and a bounded continuous map L. It claims three main results: (i) the purification condition is generic in L for fixed μ; (ii) for generic L, under an integrability condition, the Lyapunov exponents γ_1≥...≥γ_k are well defined; and (iii) in a Markov-chain example, the top Lyapunov exponent equals −h/2, where h is the entropy of the associated Markov measure, and the second exponent is −∞. The paper adapts the framework of [10] to variable L and relies heavily on the authors' earlier work [12] for the Φ-Erg property and entropy definitions.

Significance. If the advertised results were correct, the paper would be a useful extension of the Lyapunov theory of quantum trajectories from the constant-identity case of [10] to a generic set of variable L, and the Markov-chain example would provide a concrete bridge between channel entropy and dynamical Lyapunov exponents. The explicit computation of γ_1=−h/2 is a valuable test case and is essentially correct modulo an indexing issue in the displayed formula for W_n^*W_n. However, the central genericity-to-Lyapunov implication is not established: Theorem 7.3 as stated is false, and the proof of the generic purification property rests on an invalid covering argument. These are load-bearing issues, not presentation problems.

major comments (4)

[Section 7, Theorem 7.3] Theorem 7.3 drops the stochasticity assumption that the rest of Section 7 uses to define P as a probability. Earlier in the section, P(O_n)=∫_{O_n} tr(W_n ρ W_n†) dμ^n is a probability only because φ_L^*(Id)=∫ L(v)†L(v) dμ(v)=Id. For a Baire-generic L in B(M_k) this identity fails; for example, for k=1 and μ=(δ_0+δ_1)/2, the stochastic set S={L:∫|L|^2 dμ=1} is closed and nowhere dense, so a generic L is outside S. Consequently the martingale Proposition 6.5, all 'Pν-a.s.' statements, and the invocation of Theorem 7.2 have no valid probability measure behind them, and Theorem 7.3 is false as stated. The authors must either add the stochasticity assumption to the theorem and prove the needed genericity statements inside the stochastic subset, or substantially revise the claim.
[Section 9, Lemma 9.14] The proof of Lemma 9.14 asserts that the balls B(π,ε(π)) centered at a countable dense set K2⊂P2 cover P2. This does not follow from density: in a complete metric space, a union of variable-radius balls around a dense set may fail to cover the whole space, since the radii ε(π) can shrink too quickly. The open set {π: L∈Pur_π} can be a proper open dense set containing K2, as in the standard example of R∖{√2} containing Q. Since Lemma 9.14 is the direct justification of Proposition 9.1, the proof of generic purification is incomplete. A different argument, such as uniform control of ε(π) or a different Baire-category scheme, is needed.
[Section 9, Proposition 9.6; Section 7] Even if generic purification in B(M_k) were proved, the perturbation used in Proposition 9.6, L_ε=L+εQ^*DQ, does not preserve the stochastic condition ∫L†L dμ=Id. The denseness results of Section 9 are therefore statements about the full space B(M_k), not about the stochastic subset needed for the probability interpretation in Section 7. No argument is given that the residual set of Φ-Erg and purifying L can be intersected with the stochastic set to produce a residual subset there; indeed the stochastic set is typically meager. Thus the advertised conclusion 'generically on L ... Lyapunov exponents are well defined' is not supported by the provided genericity proofs.
[Section 7, Theorem 7.2] The proof of Theorem 7.2 is not supplied; the text states only that the proof from [10] 'works here in our setting'. Since this theorem is the probabilistic heart of the Lyapunov-exponent result and the current setting includes non-constant, not necessarily normalized L, the paper should identify which steps of [10] use stochasticity and how they adapt. As written, the dependence on [10] is too coarse to verify the hypotheses, especially in view of the stochasticity issue raised above.

minor comments (5)

[Section 8] In the displayed formula for W_n(ω)^*W_n(ω), the Kronecker delta should be δ_{j_{k+1} i_k} rather than δ_{i_{k+1} j_k}, and the final rank-one projector should be |j_1⟩⟨j_1| with the ordering W_n=L(ω_n)...L(ω_1). This does not affect the final value γ_1=−h/2, but the indexing should be corrected.
[Section 6, Proposition 6.8] The notation 'Ech(k|∧2 W_n|/tr(W_n^*W_n))' uses Ech without definition; presumably it denotes expectation with respect to the probability P_ch, but this should be stated explicitly.
[Section 5] The phrase 'By abuse of language we consider V_i: Ω→M_k as a random variable V_i(ω)=ω_i' is confusing because V_i was not introduced; the coordinates of ω=(ω_1,ω_2,...) are already matrices, so W_n is simply the product L(ω_n)...L(ω_1). Please align the notation.
[Section 7, Theorem 7.3(a)] The clause 'whenever γ_1=−∞' in item (a) is unclear: if γ_1=−∞, the difference γ_2−γ_1 is not defined, so the statement that γ_2−γ_1<0 needs separate interpretation or a limiting formulation.
[Abstract and Section 2] There are minor typographical issues, e.g., 'dm(v)' in the abstract should be 'dμ(v)', and 'funtion' in Definition 2.2 should be 'function'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Lyapunov argument is adapted from external [10] and the generic purification proof is self-contained; self-citations to [12] supply prior genericity and entropy results but are not repackaged conclusions.

full rationale

The paper's derivation chain is not circular by construction. Theorem 7.2 is explicitly taken over from Benoist, Fraas, Pautrat and Pellegrini [10], an external reference, and the paper only adapts the proof to a general L after assuming irreducibility, Phi-Erg and purification. The genuinely new generic-purification claim (Propositions 6.11 and 9.1) is proved in Section 9 by a Baire-category argument: the purification set is expressed as an intersection over a countable dense set of projections of open dense sets Pur_n_pi, with no use of the Lyapunov conclusion or of a fitted parameter. The only load-bearing items attributed to the authors' own previous paper [12] are the generic Phi-Erg/irreducibility statement quoted in Proposition 6.10 and the properties of the Markov-chain example in Section 8; those are prior parameter-free results whose statement does not include the target Lyapunov exponents, so citing them is normal reliance on earlier work rather than a reduction of the new result to its inputs. No quantity is fitted and then renamed a prediction, and the entropy formula gamma1 = -h/2 is obtained by an explicit computation of limits of products of the example's matrices. A mathematical gap exists but is not circularity: Theorem 7.3 passes from 'L generic for mu' in B(M_k) to P_nu-a.s. statements, while the probability P(O_n) = integral tr(W_n rho W_n^dagger) d mu^n is normalized only when Phi*_L(Id) = Id; the residual set produced by Proposition 6.10 and Proposition 9.1 is not shown to intersect the stochastic set. This is a missing assumption in the theorem's statement, not an identity between input and output.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim uses the authors' previous paper [12] for Phi-Erg genericity, the entropy definition, and the Markov example's purification and irreducibility, and uses [10] for the entire Lyapunov-exponent machinery. These are legitimate references, but the current paper does not re-prove them and one step in the new proof (Lemma 9.14) is incomplete. No numerical parameters are fitted anywhere.

assumptions (4)

standard math The space B(M_k) of bounded continuous L with sup norm is complete, so Baire category applies.
Used implicitly whenever Baire category is invoked for the set of L; the paper works in B(M_k).
ad hoc to paper The proofs of [10] for L(v)=v extend verbatim to arbitrary measurable L.
Propositions 6.5, 6.6, 6.8 and Theorem 7.2 are asserted to have the same proofs; this is an unreproved transfer of a martingale and contraction argument to non-constant, possibly non-invertible L.
ad hoc to paper A generic L in B(M_k) can be taken stochastic or normalized for the Lyapunov theorem.
Section 7 needs phi_L stochastic so that P(O_n) is a probability; Theorem 7.3 does not state or prove this.
standard math For the Markov example, the ergodic theorem applies to the occupation counts X_{ij,n}.
Used to pass from sums of log p_{ij} weighted by X_{ij,n}/n to expectations under the stationary Markov measure.

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Pith. "Pith review of Lyapunov exponents for Quantum Channels: an entropy formula and generic properties." pith.science (2026). https://pith.science/paper/4DUDRAMA

@misc{pith2026190808942,
  author       = {Pith},
  title        = {Pith review of: Lyapunov exponents for Quantum Channels: an entropy formula and generic properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4DUDRAMA}},
  note         = {Machine review of arXiv:1908.08942}
}

abstract

We denote by $M_k$ the set of $k$ by $k$ matrices with complex entries. We consider quantum channels $\phi_L$ of the form: given a measurable function $L:M_k\to M_k$ and a measure $\mu$ on $M_k$ we define the linear operator $\phi_L:M_k \to M_k$, by the law $\rho \,\to\,\phi_L(\rho) = \int_{M_k} L(v) \rho L(v)^\dagger \, \dm(v).$ On a previous work the authors show that for a fixed measure $\mu$ it is generic on the function $L$ the $\Phi$-Erg property (also irreducibility). Here we will show that the purification property is also generic on $L$ for a fixed $\mu$. Given $L$ and $\mu$ 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 $\Phi$-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 on $L$, if $\int |L(v)|^2 \log |L(v)|\, \ d\mu(v)<\infty$, then the Lyapunov exponents $\infty > \gamma_1\geq \gamma_2\geq ...\geq \gamma_k\geq -\infty$ are well defined. On 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 $-\frac{1}{2} \,h$, where $h$ is the entropy of the associated Markov probability.

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Works this paper leans on

29 extracted references · 28 canonical work pages

[10]

Benoist, M

T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini. Invariant m easure for quantum trajectories. Prob. Theory and Related Fields , 174, no. 1–2, 307–334 (2019)

work page 2019
[12]

J. E. Brasil, J. Knorst and A. O. Lopes, Thermodynamic Formalis m for Quantum Channels: Entropy, Pressure, Gibbs Channels and generic properties, arXiv (2019)

work page 2019
[1]

Alicki and M Fannes, Quantum Dynamical Systems, Oxford Uni- versity Press (2001)

R. Alicki and M Fannes, Quantum Dynamical Systems, Oxford Uni- versity Press (2001)

work page 2001
[2]

Avila and J

A. Avila and J. Bochi, Trieste Lecture notes on Lyapunov expone nts Part 1, ICTP (2008) 22

work page 2008
[3]

Avila and J

A. Avila and J. Bochi. A formula with some applications to the theory of Lyapunov exponents. Israel J. Math., 131: 125-137, (2002)

work page 2002
[4]

Avila and M

A. Avila and M. Viana, Simplicity of Lyapunov spectra: a suﬃcient criterion, Port. Math. 64 (2007), no. 3, 311-376

work page 2007
[5]

Backes, M

L. Backes, M. Poletti and P. Varandas, Simplicity of Lyapunov sp ec- trum for linear cocycles over non-uniformly hyperbolic systems, to ap- pear in Ergodic Theory and Dynamical Systems

work page
[6]

Baraviera, C

A. Baraviera, C. Lardizabal, A. O. Lopes, and M. T. Cunha. A the r- modynamic formalism for density matrices in quantum information. Applied Mathematics Research Express , 2010 (1): 63–118, 2010

work page 2010

Show all 29 references

[7]

Baraviera, C

A. Baraviera, C. Lardizabal, A. O. Lopes, and M. T. Cunha, Quan - tum Stochastic Processes, Quantum Iterated Function Systems and Entropy, Sao Paulo Journal of Math. Sciences, Vol. 5, N 1, 51-84 (2011)

2011
[8]

Baraviera, C

A. Baraviera, C. Lardizabal, A. O. Lopes, and M. T. Cunha, A dy- namical point of view of Quantum Information: entropy and pressu re, Dynamics, Games and Science I, DYNA 2008, Edit. M. Peixoto, A. Pinto and D Rand, pp 81-122 Springer Verlag (2011)

2011
[9]

Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Springer Verlag

L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Springer Verlag
[11]

Bougerol and J

P. Bougerol and J. Lacroix. Products of Random Matrices with Ap- plications to Schr¨ odinger Operators, Birkh¨ auser Boston, Inc.,Boston (1985)

1985
[13]

Duarte and S

P. Duarte and S. Klein Continuity, positivity and simplicity of the Lyapunov exponents for-quasi periodic cocycles, Arxiv (2017)

2017
[14]

Duarte and S

P. Duarte and S. Klein, Continuity of the Lyapunov exponents of linear cocycles. Publicacoes Matematicas do IMPA, 31 o Coloquio Brasileiro de Matematica IMPA (2017). 23

2017
[15]

D. E. Evans and R. Høegh-Krohn. Spectral properties of pos itive maps on C*-algebras. Journal of the London Mathematical Society , 2(2):345– 355, 1978

1978
[16]

Lidar, Lecture Notes on the Theory of Open Quantum Syste ms, arXiv (2019)

D. Lidar, Lecture Notes on the Theory of Open Quantum Syste ms, arXiv (2019)

2019
[17]

Froyland, C

G. Froyland, C. Gonzalez-Tokman and A. Quas, Stochastic sta bility of Lyapunov exponents and Oseledets splittings for semi-invertible matrix cocycles. Comm. Pure Appl. Math. 68, no. 11, 2052-2081 (2015)

2015
[18]

Katok and B

A. Katok and B. Hasselblatt, Introduction to the Modern Theo ry of Dynamical Systems, Cambridge Press
[19]

A. O. Lopes, J. K. Mengue, J. Mohr and R. R. Souza, Entropy a nd Variational Principle for one-dimensional Lattice Systems with a gen - eral a-priori probability: positive and zero temperature, Erg. Th eory and Dyn Systems, 35 (6), 1925-1961 (2015)

2015
[20]

Ma˜ ne, Ergodic Theory and Diﬀerentiable Dynamics, SpringerVerlag

R. Ma˜ ne, Ergodic Theory and Diﬀerentiable Dynamics, SpringerVerlag
[21]

/suppress Lozi´ nski, K.˙Zyczkowski, and W

A. /suppress Lozi´ nski, K.˙Zyczkowski, and W. S/suppress lomczy´ nski. Quantum iterated function systems. Physical Review E , 68(4): 046110, 2003

2003
[22]

Parry and M

W. Parry and M. Pollicott. Zeta functions and the periodic orbit s truc- ture of hyperbolic dynamics, Ast´ erisqueVol 187-188 1990

1990
[23]

Petz, Quantum Information Theory and Quantum Statistics , Springer Verlag (2008)

D. Petz, Quantum Information Theory and Quantum Statistics , Springer Verlag (2008)

2008
[24]

Pollicott and M

M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, Springer Verlag
[25]

Schrader

R. Schrader. Perron-Frobenius theory for positive maps on trace ideals. arXiv preprint math-ph/0007020 , 2000

2000 arXiv
[26]

S/suppress lomczy´ nski.Dynamical entropy, Markov operators, and iterated function systems

W. S/suppress lomczy´ nski.Dynamical entropy, Markov operators, and iterated function systems . Wydaw. UJ, 2003

2003
[27]

Viana and K

M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambrig e Press (2016)

2016
[28]

Viana, Lectures on Lyapunov Exponents, Cambrige Press ( 2016)

M. Viana, Lectures on Lyapunov Exponents, Cambrige Press ( 2016)

2016
[29]

M. M. Wolf. Quantum channels & operations: Guided tour. Lecture notes available at http://www-m5. ma. tum. de/foswiki/pub M, 5, 2012. 24

2012

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

Benoist, M

T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini. Invariant m easure for quantum trajectories. Prob. Theory and Related Fields , 174, no. 1–2, 307–334 (2019)

work page 2019

[2] [12]

J. E. Brasil, J. Knorst and A. O. Lopes, Thermodynamic Formalis m for Quantum Channels: Entropy, Pressure, Gibbs Channels and generic properties, arXiv (2019)

work page 2019

[3] [1]

Alicki and M Fannes, Quantum Dynamical Systems, Oxford Uni- versity Press (2001)

R. Alicki and M Fannes, Quantum Dynamical Systems, Oxford Uni- versity Press (2001)

work page 2001

[4] [2]

Avila and J

A. Avila and J. Bochi, Trieste Lecture notes on Lyapunov expone nts Part 1, ICTP (2008) 22

work page 2008

[5] [3]

Avila and J

A. Avila and J. Bochi. A formula with some applications to the theory of Lyapunov exponents. Israel J. Math., 131: 125-137, (2002)

work page 2002

[6] [4]

Avila and M

A. Avila and M. Viana, Simplicity of Lyapunov spectra: a suﬃcient criterion, Port. Math. 64 (2007), no. 3, 311-376

work page 2007

[7] [5]

Backes, M

L. Backes, M. Poletti and P. Varandas, Simplicity of Lyapunov sp ec- trum for linear cocycles over non-uniformly hyperbolic systems, to ap- pear in Ergodic Theory and Dynamical Systems

work page

[8] [6]

Baraviera, C

A. Baraviera, C. Lardizabal, A. O. Lopes, and M. T. Cunha. A the r- modynamic formalism for density matrices in quantum information. Applied Mathematics Research Express , 2010 (1): 63–118, 2010

work page 2010