A Construction of Dynamical Entropy on CAR Algebras

Kyouhei Ohmura; Noboru Watanabe

REVIEW 3 major objections 4 minor 17 references

A Construction of Dynamical Entropy on CAR Algebras

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper defines a dynamical entropy on CAR algebras and shows it vanishes on a two-state spin model.

desk verdict A clear but flawed attempt to transplant AOW entropy to CAR algebras: the Markov-chain state collapses to the maximally mixed state, so the entropy they define is not the von Neumann entropy of that chain. read the letter →

arxiv 1908.01955 v1 pith:NGOMJ3EN submitted 2019-08-06 math-ph math.MP

Kyouhei Ohmura , Noboru Watanabe This is my paper · ORCID

classification math-phmath.MP

keywords dynamicalentropyCARalgebrafermionsystemsAOWquantumMarkovchainUmegakiconditionalexpectationoperatorpartitionspinsystem

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

The paper sets out to build a dynamical entropy for CAR algebras, the C*-algebras describing fermionic spin systems. It adapts the Accardi-Ohya-Watanabe (AOW) entropy, originally formulated through quantum Markov chains on matrix algebras, to the CAR setting by introducing an Umegaki conditional expectation from $A_{[0,n]}$ to $A_0$. The resulting object, Definition 9, is the supremum over operator partitions $\gamma$ of the limsup of the Shannon entropy of the trajectory probabilities $P_{i_1\cdots i_n}=\operatorname{Tr}_0\,\theta^n(\gamma_{i_n})\cdots\theta(\gamma_{i_1})\rho\,\theta(\gamma_{i_1})^*\cdots\theta^n(\gamma_{i_n})^*$. If the construction is sound, it supplies a per-step information rate for fermionic time evolutions, and the authors test it on a $2\times2$ spin model where it returns zero for every initial state.

What carries the argument

The load-bearing object is the trajectory probability $P_{i_1\cdots i_n}=\operatorname{Tr}_0\,\theta^n(\gamma_{i_n})\cdots\theta(\gamma_{i_1})\rho\,\theta(\gamma_{i_1})^*\cdots\theta^n(\gamma_{i_n})^*$, a time-ordered correlation kernel on the CAR algebra. It is built from an operator partition $\gamma$ of the identity, a $*$-automorphism $\theta$ of $A_0$, and the stationary density $\rho$; the conditional expectation $E_{\gamma,\theta}$ in equation (47) propagates this weight from one site to the next while replacing each intermediate site by the normalized even part $A_{k,+}=a_ka_k^*+a_k^*a_k=1$. The machinery turns the Markov-chain density $\rho_n$ into a classical probability distribution, so the Shannon entropy of that distribution, divided by $n$ and subjected to a limsup, is the dynamical entropy.

What would settle it

Compute $h_{\varphi_0}(\theta)$ for a CAR automorphism that is not a single-site unitary rotation and is expected to produce positive dynamical entropy, such as a Bogoliubov transformation or a one-step lattice shift on an infinite CAR algebra; if formula (52) returns zero because the intermediate even parts are forced to be the identity, then the construction is not measuring the full dynamics.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that the correct dynamical entropy for a quadruple $(A,\varphi_0,\gamma,\theta)$ is $h_{\varphi_0}(\theta):=\sup_\gamma\{-\limsup_{n\to\infty}\frac{1}{n}\sum_{i_1,\cdots,i_n}P_{i_1\cdots i_n}\log P_{i_1\cdots i_n}\}$, and that this is well defined because $\sum P_{i_1\cdots i_n}=1$. The density on $A_{[1,n]}$ produced by iterating the conditional expectation factorizes into an even part $(1/2^n)(A_{n,+}\otimes\cdots\otimes A_{1,+})$ times the boundary weight built from $P_{i_1\cdots i_n}$, so the entropy reduces to the Shannon entropy of that classical trajectory distribution. The authors further claim that for the $2\times2$ matrix algebra with partition $\gamma_1=a_0^*a_0$, $\gamma_2=a_0a_0^*$ and automorphism $\theta(a)=UaU^*$ with $U=e^{ia_0^*a_0}$, the only nonzero trajectory probabilities are $P_{1\cdots1}=\lambda$ and $P_{2\cdots2}=1-\lambda$, making the entropy zero for every $\lambda$.

Load-bearing premise

The construction stands on the assumption that at every intermediate time step the site observable can be replaced by its normalized even part $A_{k,+}=1$, so the bulk of the Markov chain is maximally mixed and the dynamics is read only through the boundary trajectory probabilities $P_{i_1\cdots i_n}$; if bulk correlations matter for information, this entropy will miss them.

Editorial extensions

If this is right

The CAR-algebra analogue of AOW entropy is now available, so fermionic lattice observables can be assigned a per-step information rate rather than only matrix-algebra observables.
For the occupation partition of a single spin, the entropy is exactly zero for every initial state $\lambda$, which the paper reads as consistency with the known vanishing of entropy for unitary automorphisms of finite-dimensional algebras.
Because the intermediate bulk factors in equation (49) are all $A_{k,+}=1$, the entropy depends on the dynamics only through the boundary distribution $P_{i_1\cdots i_n}$; any two automorphisms producing the same trajectory probabilities have equal entropy.

Reading between the lines

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

Because every intermediate site is forced to the even identity $A_{k,+}=1$, the construction reads the time evolution as a classical random walk on partition labels; I would expect $h_{\varphi_0}(\theta)$ to vanish for any finite-dimensional unitary automorphism, not only the $2\times2$ example.
The paper itself leaves lattice translations and Bogoliubov automorphisms for future work; a concrete extension would be to replace the single-site conditional expectation by a genuinely nonlocal family $\{E_n\}$ acting on $A_{[0,n+1]}$, and then check whether the resulting entropy is positive for free-fermion transport.
A testable consequence of this boundary-only construction is that computing (52) for a shift automorphism by labelling sites will still return zero, because the intermediate even factors project out bulk correlations; that would indicate the entropy is measuring boundary information rather than full fermionic dynamics.

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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

You should know this paper before citing it: the claimed construction of dynamical entropy on CAR algebras has a hole in the derivation. The authors define a Markov-chain density operator ρ_n in Eq. (50), but because each A_{k,+} is the identity (from the CAR anticommutator), ρ_n is simply 2^{-n} times the identity on the n-site system—independent of θ, γ, and ρ. Its von Neumann entropy is n log 2, not the -∑P log P of Eq. (52). The identification made in the AOW paper (Eq. (8)) relied on ρ_n being diagonal with eigenvalues P_i; that structure is absent here. So Definition 9 is not the AOW entropy of the Markov chain. It is a Shannon entropy of the trajectory distribution, which is a legitimate object, but the paper does not frame it that way.

What the paper does well: it is clearly written, gives a careful review of CAR algebras and Markov chains, and the 2×2 example is computed correctly. The fact that the entropy vanishes for the unitary automorphism is consistent with known results for finite-dimensional unitary dynamics. The definition is explicit and could in principle be studied on its own.

The soft spots are proportionate to this central flaw. Beyond the collapse of ρ_n, the construction only handles automorphisms of a single-site algebra, not the full CAR algebra; the authors concede this in the conclusion. There is also no comparison with other quantum dynamical entropies (e.g., Connes–Narnhofer–Thirring), which would help position the result. The example is too trivial to validate anything—it gives zero because the trajectory distribution is constant, not because of any deep property.

Who is this for? Specialists in noncommutative dynamical entropy or fermionic quantum information who want to see a concrete attempt to extend AOW entropy to CAR algebras. They will get some review material and a cautionary example of how conditional expectations can degenerate.

My recommendation: this deserves a serious referee, not a desk reject, because the topic is legitimate and the flaw is fixable by reframing the definition as a trajectory entropy. But as it stands, the central justification is incorrect. I would send it back for major revision with a clear request to either fix the Markov-chain derivation or explicitly redefine the quantity and drop the AOW-entropy claim.

Referee Report

3 major / 4 minor

Summary. The paper claims to construct a dynamical entropy for CAR algebras by adapting the Accardi-Ohya-Watanabe (AOW) construction. It first recalls the AOW entropy for von Neumann algebras, then reviews CAR algebras and the Markov-state framework of Accardi-Fidaleo-Mukhamedov. In Section 5 it introduces a conditional expectation E_{gamma,theta} of the form (47), derives a state rho_n on the n-time-step algebra in Eq. (50), and defines in Definition 9 the entropy h_{phi_0}(theta) as the supremum over partitions gamma of the limsup of (1/n) times -sum P_{i_1...i_n} log P_{i_1...i_n}, where P is given by Eq. (51). Section 6 computes this quantity for a 2x2 matrix algebra with a unitary automorphism, obtaining the value 0, and cites consistency with the known zero entropy of unitary automorphisms.

Significance. If the construction were sound, it would provide a fermionic analogue of AOW dynamical entropy and would offer a tool for quantifying information in quantum spin systems. The paper is transparent, uses no fitted parameters, and the 2x2 computation is explicit and easy to follow; the attempted comparison with the known result for unitary automorphisms is a reasonable external consistency check. However, the central identification of -sum P log P with the entropy of the Markov-chain state fails because the state rho_n in Eq. (50) is maximally mixed and independent of the dynamics. The proposed quantity is therefore a classical Shannon entropy of the trajectory distribution P, not the AOW entropy of a quantum Markov chain on a CAR algebra. The claimed extension is not established, and the defect is not a local presentation issue but an error in the main construction.

major comments (3)

[Section 5, Eq. (50) and Definition 9] In Eq. (50), rho_n = (1/2^n) sum_{i_1,...,i_n} P_{i_1...i_n} A_{n,+} otimes ... otimes A_{1,+}. By the CAR relation (29), A_{k,+} = a_k a_k^* + a_k^* a_k = 1 for every k, so rho_n = (1/2^n) I on A_[1,n]. In particular, rho_n is maximally mixed and independent of theta, gamma, and rho. Its von Neumann entropy is n log 2. In the AOW derivation, Eq. (8) equals -sum P log P because rho_n is diagonal with eigenvalues P; here that eigenvalue structure is absent. Therefore Definition 9 is not the von Neumann entropy of the Markov-chain state and is not the AOW entropy of this chain. The central claim of the paper is unsupported.
[Section 6, Lemma 1 and Eq. (63)] The model computation illustrates the failure. For n = 1, Eq. (50) gives rho_1 = (1/2) I, so -Tr rho_1 log rho_1 = log 2, whereas the expression in Definition 9 gives H(lambda) = -lambda log lambda - (1-lambda) log(1-lambda); the two coincide only at lambda = 1/2. The reported value 0 in Eq. (63) follows from dividing the trajectory entropy by n, and the same value would be obtained for any dynamics with only two non-zero trajectory probabilities. Thus the computation does not test the entropy of the Markov-chain state, and Remark 5's consistency check with unitary automorphisms does not validate Definition 9.
[Section 5, Eq. (47)] The map E_{gamma,theta} defined in Eq. (47) is not shown to be an Umegaki conditional expectation from A_[0,n] to A_0. To be a conditional expectation it must satisfy E(1_{A_[0,n]}) = 1_{A_0}, E(a) = a for a in A_0, complete positivity, and the module property; none of these is verified in the text. The partial trace uses the normalized trace implicitly, and the role of Theta_n is not explained. Since the derivation of the state in Eq. (50) relies entirely on this map, this is a load-bearing gap in the construction.

minor comments (4)

[Throughout] There are numerous typographical errors, including 'oprator' (p. 7), 'Therfore' and 'developped' (p. 11), and 'Acca rdi' in the abstract; a careful proofreading pass is needed.
[Sections 2 and 5] The same symbol E_{gamma,theta} is used for the AOW transition expectation in Eq. (2) and for the CAR conditional expectation in Eq. (47), although these are mathematically different objects; distinct notation would avoid confusion.
[Section 7] The Conclusion states that lattice translations and Bogoliubov automorphisms require a reformulation of the entropy; this is a significant limitation and should be reflected in the abstract and introduction, not only in the concluding remarks.
[Section 6, Remark 5] The reference to [13] for the statement that unitary automorphisms have zero complexity would be more useful with a precise theorem number or page, since the property depends on the class of algebras and the definition of entropy used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entropy in Definition 9 is a direct construction with an external benchmark; the A_{k,+}=1 issue is a correctness gap, not a circular reduction.

full rationale

The paper defines a new functional on CAR algebras in Definition 9, Eq. (52), as a supremum over partitions of a Shannon entropy rate of the trajectory probabilities P_{i1...in} introduced in Eq. (51). This is a self-contained construction rather than a fitted prediction: no parameter is tuned to a target data set, and the formula is not derived by assuming the conclusion. The authors' own works [14,17] appear only in Remark 2 as comments on extensions of AOW entropy and are not load-bearing for the new definition. The consistency check in Section 6 and Remark 5 invokes Neshveyev-Størmer [13], which is an external standard reference, so the validation does not create a self-citation loop. There is an internal mathematical gap unrelated to circularity: from Eq. (29) each A_{k,+}=a_k a_k^*+a_k^*a_k equals the identity, so the state rho_n in Eq. (50) is (1/2^n) times the identity on the tensor product; its von Neumann entropy is n log 2, whereas Eq. (52) uses -sum P log P. That inconsistency undermines the claim that Eq. (52) is the AOW entropy of the constructed Markov chain, but it is not a reduction of the output to an input by construction, a fitted parameter renamed as a prediction, or a load-bearing self-citation. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities; it relies on standard CAR algebra results, prior Markov chain constructions, and a new ad hoc conditional expectation.

assumptions (5)

standard math CAR canonical anticommutation relations {a_i, a_j*}=δ_ij 1, {a_i,a_j}=0
The entire construction is built on the CAR algebra generated by these relations, Section 3.
domain assumption The Markov chain construction on CAR algebras from Accardi-Fidaleo-Mukhamedov [2] is valid
The paper uses the existence and properties of Markov states/chains on CAR algebras from [2] without proof, Section 4.
domain assumption The AOW entropy construction on matrix algebras from Accardi-Ohya-Watanabe [6] is valid
The paper adapts the AOW entropy definition (Section 2) and relies on its properties as motivation.
ad hoc to paper The specific form of the Umegaki conditional expectation in (47) is a valid conditional expectation from A_[0,n] to A_0
This map is introduced in this paper without a proof that it is a well-defined, completely positive, identity-preserving conditional expectation; its action is only computed in a special case.
ad hoc to paper The entropy rate is defined via limsup and sup over partitions; no proof that the sup is finite or that the limit exists
Definition 9 assumes the supremum over γ is meaningful and the limsup exists; no proof or example with nonzero entropy is given.

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Cite this review

Pith. "Pith review of A Construction of Dynamical Entropy on CAR Algebras." pith.science (2026). https://pith.science/paper/NGOMJ3EN

@misc{pith2026190801955,
  author       = {Pith},
  title        = {Pith review of: A Construction of Dynamical Entropy on CAR Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGOMJ3EN}},
  note         = {Machine review of arXiv:1908.01955}
}

abstract

The dynamical entropy on von Neumann algebras defined by Accardi, Ohya and Watanabe (AOW entropy) is a natural noncommutative extension of the classical dynamical entropy. On the other hand, quantum spin lattice systems currently used in quantum computing and communication processes are mathematically described by $C^*$-algebras called CAR algebras. Therefore, in order to obtain the average amount of quantum information and to calculate the uncertainty of the dynamics of quantum spin systems, it is necessary to define dynamical entropy on CAR algebras. In this paper, we formulate dynamical entropy on CAR algebras based on the construction of the AOW entropy. Moreover, we compute the introduced entropy for a $2 \times 2$ matrix algebra case which has relation to the quantum spin system.

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Accardi: Noncommutative Markov chains, in School of M athematical Physics, Camerino, 268-295 (1974)

L. Accardi: Noncommutative Markov chains, in School of M athematical Physics, Camerino, 268-295 (1974)

work page 1974
[2]

Accardi, F

L. Accardi, F. Fidaleo, F. Mukhamedov : Markov states and chains on the CAR algebra, Inf. Dymen. Anal. Quantum Probab., Rel. Top. 10, 165-184 (2007)

work page 2007
[3]

Accardi, A

L. Accardi, A. Frigerio, J. Lewis : Quantum stochastic pr ocesses. Publ. RIMS Kyoto Univ. 18, 97-133 (1982)

work page 1982
[4]

Accardi, N

L. Accardi, N. Obata : Foundations of Quantum Probabilit y Theory. Makino Pub. Co., Tokyo (2003)

work page 2003
[5]

Accardi, M

L. Accardi, M. Ohya, N. Watanabe : Note on quantum dynamic al entropies, Report on Math. Phys., 38, 457-469 (1996)

work page 1996
[6]

Accardi, M

L. Accardi, M. Ohya, N. Watanabe : Dynamical entropy thro ugh quantum Markov chains, Open Sys. & Information Dyn. 4, 71-87 (1997)

work page 1997
[7]

Quantum Markov Chains: A unification approach

L. Accardi, A. Souissi, S. Gheteb : Quantum Markov Chains : uniﬁcation approach, arXiv: 1811.00500, 1 Nov (2018)

work page Pith review arXiv 2018
[8]

Araki, H

H. Araki, H. Moriya : Equilibrium Statistical Mechanics of Fermion Lattice Systems, Rev. in Math. Phys., 15, 02, pp. 93-198 (2003)

work page 2003

Show all 17 references

[9]

Bilingsley : Ergodic Theory and Information, Wiley, N ew York (1965)

L. Bilingsley : Ergodic Theory and Information, Wiley, N ew York (1965)

1965
[10]

Bratteli, D

O. Bratteli, D. W. Robinson : Opertor Algebras and Quant um Statistical Mechanics II, Springer, New York (1987). 14

1987
[11]

A. N. Kolmogorov : Theory of transmission of informatio n. Am. Math. Soc. Transl. Ser. 2 33, 291-321 (1963)

1963
[12]

Moriya : Separability condition for composite syste ms of distinguishable fermions, arXiv : quant-ph/0405166v2, 5 Sep (2005)

H. Moriya : Separability condition for composite syste ms of distinguishable fermions, arXiv : quant-ph/0405166v2, 5 Sep (2005)

2005 arXiv
[13]

Neshveyev, E

S. Neshveyev, E. Størmer : Dynamical Entropy in Operato r Algebras, Springer, Berlin (2006)

2006
[14]

Ohmura, N

K. Ohmura, N. Watanabe : Quantum Dynamical Mutual Entro py Based on AOW Entropy, Open Syst. Inf. Dyn., 26, 2, pp. 1950009-1 - pp. 1950009-16 (2019)

2019
[15]

M. Ohya, D. Petz : Quantum Entropy and its Use, Springer, Berlin (1993)

1993
[16]

Umegaki : Conditional expectations in an operator al gebra, IV, (entropy and infor- mation), Kodai Mathematical Seminar Reports, 14, 59-85 (1962)

H. Umegaki : Conditional expectations in an operator al gebra, IV, (entropy and infor- mation), Kodai Mathematical Seminar Reports, 14, 59-85 (1962)

1962
[17]

Watanabe, M

N. Watanabe, M. Muto : Note on transmitted complexity fo r quantum dynamical sys- tems, Philosophical Transactions Royal Society A 375 , pp. 20160396-1-pp. 20160936-16 (2017) 15

2017

Pith tools

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

[1] [1]

Accardi: Noncommutative Markov chains, in School of M athematical Physics, Camerino, 268-295 (1974)

L. Accardi: Noncommutative Markov chains, in School of M athematical Physics, Camerino, 268-295 (1974)

work page 1974

[2] [2]

Accardi, F

L. Accardi, F. Fidaleo, F. Mukhamedov : Markov states and chains on the CAR algebra, Inf. Dymen. Anal. Quantum Probab., Rel. Top. 10, 165-184 (2007)

work page 2007

[3] [3]

Accardi, A

L. Accardi, A. Frigerio, J. Lewis : Quantum stochastic pr ocesses. Publ. RIMS Kyoto Univ. 18, 97-133 (1982)

work page 1982

[4] [4]

Accardi, N

L. Accardi, N. Obata : Foundations of Quantum Probabilit y Theory. Makino Pub. Co., Tokyo (2003)

work page 2003

[5] [5]

Accardi, M

L. Accardi, M. Ohya, N. Watanabe : Note on quantum dynamic al entropies, Report on Math. Phys., 38, 457-469 (1996)

work page 1996

[6] [6]

Accardi, M

L. Accardi, M. Ohya, N. Watanabe : Dynamical entropy thro ugh quantum Markov chains, Open Sys. & Information Dyn. 4, 71-87 (1997)

work page 1997

[7] [7]

Quantum Markov Chains: A unification approach

L. Accardi, A. Souissi, S. Gheteb : Quantum Markov Chains : uniﬁcation approach, arXiv: 1811.00500, 1 Nov (2018)

work page Pith review arXiv 2018

[8] [8]

Araki, H

H. Araki, H. Moriya : Equilibrium Statistical Mechanics of Fermion Lattice Systems, Rev. in Math. Phys., 15, 02, pp. 93-198 (2003)

work page 2003