Entanglement entropy by tensor renormalization group approach

Daisuke Kadoh; Gota Tanaka; Shinji Takeda; Takahiro Hayazaki

REVIEW 2 major objections 6 minor 18 references

Entanglement entropy by tensor renormalization group approach

T0 review · 2 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A tensor-network algorithm extracts central charge $c = 0.49997(8)$ from entanglement entropy.

desk verdict The binary-expansion construction is a genuine extension of TRG to arbitrary subsystem sizes, and c = 0.49997(8) is an impressive benchmark, but the paper defers the key tensor definitions to an unpublished work and only tests h <= 2, so the arbitrary-size claim is not yet fully demonstrated. read the letter →

arxiv 2502.02030 v2 pith:4Y47TOHO submitted 2025-02-04 hep-lat

Takahiro Hayazaki , Daisuke Kadoh , Shinji Takeda , Gota Tanaka This is my paper · ORCID

classification hep-lat

keywords tensorrenormalizationgroupentanglemententropyquantumIsingmodelcentralchargeHOTRGreduceddensitymatrixlatticefieldtheoryconformal

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 proposes a tensor renormalization group algorithm that computes the entanglement entropy of a single connected segment of arbitrary length in a one-dimensional quantum system. The reduced density matrix of a Gibbs state is written as a $(1+1)$-dimensional tensor network, and the higher-order tensor renormalization group (HOTRG) coarse-grains it while trimming isometries that do not affect the entropy. The authors test the method on the critical one-dimensional quantum Ising model and extract the central charge $c = 0.49997(8)$ at bond dimension $D = 96$, matching the theoretical value $c = 1/2$. If correct, this is the first general TRG construction for subsystem-size-dependent entanglement entropy, avoiding the replica trick and the $s\to 1$ extrapolation used in Monte Carlo computations.

What carries the argument

The central object is the tensor network representation of the reduced density matrix $\rho_A$ of a Gibbs state, built from the transfer matrix of the $(1+1)$-dimensional classical Ising model obtained by splitting the imaginary-time evolution into small steps. Coarse-graining proceeds with the higher-order tensor renormalization group (HOTRG): pairs of tensors are contracted into a new tensor $T' = U^\dagger M U$ using an isometry $U$ built from the $D$ largest eigenvectors of $M^\dagger M$. Because $U$ satisfies $U^\dagger U = I$ but not $U U^\dagger = I$, isometries attached to the boundary of the subsystem can be removed without changing the entanglement entropy $S_A = -\operatorname{Tr}(\rho_A \log \rho_A)$. The generalized algorithm decomposes the trimmed network into a core matrix $C$ and a boundary factor $B$, whose shape is fixed by the binary digits of the subsystem size $l$ (and $l-1$); contracting $C$ and $B$ gives $\rho_A$ at cost $O(D^3 h)$, where $h$ is the Hamming weight of $l$.

What would settle it

Recompute the effective central charge $c(l)$ at larger bond dimensions $D > 96$ and larger system sizes; if the deviation at small $l$ from a constant effective central charge does not shrink toward $c = 1/2$ as $D$ grows, the choice of the fitting range $7\le l\le 768$ is responsible for the agreement, and the extracted central charge would shift outside the reported error when the window is varied.

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Extended reading notes

Core claim

The central claim is that entanglement entropy as a function of subsystem size can be obtained directly from tensor coarse-graining, without the replica trick. The paper constructs the reduced density matrix $\rho_A$ of a Gibbs state as a tensor network, applies HOTRG to it, and shows that the isometries attached to the boundary of the subsystem can be removed using $U^\dagger U = I$, leaving a trimmed network composed of a core matrix and a boundary factor. The core matrix is a coarse-grained reduced density matrix of the half-space subsystem, while the boundary factor is assembled from isometries according to the binary representation of the subsystem size $l$ and of $l-1$; contracting the two gives $\rho_A$ with cost $O(D^3 h)$, where $h$ is the Hamming weight of $l$. For the critical transverse-field Ising model on a 1024-site lattice with a large time direction, the computed entanglement entropy follows the finite-size conformal scaling formula $S_A(l,L) = (c/3)\log(L\sin(\pi l/L)) + k$, and fitting yields $c = 0.49997(8)$ for $D = 96$, in agreement with $c = 1/2$. The paper takes this benchmark as evidence that the method is valid for arbitrary subsystem sizes and suitable for zero-temperature quantum systems.

Load-bearing premise

The reported central charge depends on excluding small subsystem sizes from the fit, on the assumption that the visible non-constant behavior of the effective central charge at small $l$ is a lattice artifact; if that non-constant behavior is a genuine physical effect, the fitted value $0.49997(8)$ would be biased and the quoted error would not capture the bias.

Editorial extensions

If this is right

For the critical 1D quantum Ising model, the extracted central charge converges toward $c = 1/2$ as the bond dimension grows: $0.4998(2)$ at $D = 64$, $0.4999(1)$ at $D = 80$, and $0.49997(8)$ at $D = 96$.
The same algorithm applies to any subsystem size $l$, not only half the system, with the post-coarse-graining contraction cost scaling as $O(D^3 h)$ where $h$ is the Hamming weight of $l$.
Entanglement entropy can be obtained without the replica trick, so no extrapolation to Rényi index $s\to 1$ is needed.
At a time direction 16 times the spatial length, the entropy is effectively the ground-state value, so zero-temperature entanglement entropy is accessible in this framework.
The authors expect the method to extend to higher-dimensional theories and to studies of phase transitions using entanglement entropy as a probe.

Reading between the lines

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

A natural next test is to apply the same trimmed-network construction to models where Monte Carlo has a sign problem; if the tensor network can be formed, the method would yield entanglement entropy where the replica-based Monte Carlo route fails.
The reported error bar is computed with the fit parameter $k$ held fixed and the fitting window chosen after seeing $c(l)$, so the $0.49997(8)$ figure should be interpreted as conditional on that window; a window-independent estimate would be a stronger check.
The boundary-factor decomposition is specific to a one-dimensional subsystem whose boundary is two points; extending the idea to two spatial dimensions would require a new rule for trimming the network along a one-dimensional boundary, and the binary-decomposition structure does not carry over directly.
If the method survives those extensions, the holographic relation between entanglement entropy and geometry noted in the outlook could become a concrete lattice-level test rather than a formal correspondence.

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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

The core idea is genuinely new: prior TRG entanglement entropy work handles half-space bipartitions, and the binary-expansion construction of the boundary factor B and core matrix C for arbitrary subsystem size looks like a real extension. The derivation of the reduced density matrix tensor network from Suzuki-Trotter is standard but competently done, and the trimming argument dropping boundary isometries via U^dagger U = I is mathematically sound. The numerical benchmark is also solid as far as it goes: the convergence in alpha is checked properly, the observed S_A(l) = S_A(L-l) symmetry is a good sanity check, and the extracted central charge c = 0.49997(8) at D=96 matches theory almost too well. The table showing D-dependence moving monotonically toward 0.5 is credible. The soft spot is the one the stress test flags, and it is real. Section 2.1 says 'See [17] for details on definitions' for the transfer matrix T, and reference [17] is listed as 'Work in progress.' That means the precise form of the rank-4 tensor, including how the physical indices and the Trotter decomposition map into it, cannot be checked. The boundary-factor construction in Figs. 4-7 is described, but the box structures depend on b_k and the general contraction pattern is only illustrated for specific cases. The numerical section further restricts to Hamming weight h <= 2, i.e. l = 2^m + q with q a single power of two. So the claimed O(D^3 h) contraction and the h > 2 cases that exercise the full binary logic are never tested. If the deferred definitions contain a subtle sign error or the contraction pattern differs for h > 2, the central methodological claim would fail even though the benchmark is clean. The benchmark alone cannot close that gap because it only exercises the special cases. The fit-range selection is a second, milder issue. The effective central charge c(l) visibly deviates for small l, and the fit range 7 <= l <= 768 is chosen after seeing that deviation. The quoted error is a max-deviation with k fixed, so it does not include the systematic uncertainty from the cutoff choice. In a one-benchmark proceedings paper this is acceptable, but it keeps the significance moderate. Overall, the paper is a promising proceedings contribution, not a complete method paper. The authors are clearly thinking seriously and the algorithm as sketched is coherent. The missing details and the untested h > 2 cases are not flaws in what is presented; they are gaps in what can be independently verified. A serious referee should engage with this, because the idea is worth pursuing and the benchmark is encouraging. For my own work I would not cite it yet, pending the full paper with the tensor definitions and h > 2 tests. Recommendation: send to peer review, but require the algorithm specification and ideally one h > 2 test before acceptance.

Referee Report

2 major / 6 minor

Summary. The manuscript proposes a tensor renormalization group (TRG) method, based on the higher-order TRG (HOTRG) algorithm, to compute the entanglement entropy of a subsystem of arbitrary size in a one-dimensional quantum system. The reduced density matrix of a Gibbs state is represented as a (1+1)-dimensional tensor network, and the subsystem entropy is obtained from a 'trimmed' network in which boundary isometries are dropped using the isometry property U^dagger U = I. A binary decomposition of the subsystem size l is used to build a core matrix C and boundary factor B. The method is applied to the critical quantum Ising chain with L = 1024 and time extent alpha = 16, for subsystem sizes with Hamming weight h <= 2. The extracted central charge is c = 0.49997(8) for D = 96, in agreement with the theoretical value c = 1/2.

Significance. If the proposed algorithm is fully specified and validated, it would fill a genuine gap in the TRG toolbox: it would be the first TRG method to compute entanglement entropy for subsystems of arbitrary size, not just half of the system, with a claimed cost of O(D^3 h). The numerical benchmark is clean in the tested regime: the alpha -> infinity limit is checked in Fig. 8, the bond-dimension dependence in Table 1 is benign, and the extracted central charge is very close to the CFT prediction. However, the manuscript is not self-contained: the key algorithm definitions are deferred to a 'forthcoming' paper, and all numerical tests are restricted to Hamming weight h <= 2. These two limitations are load-bearing for the central claim and must be addressed before the result can be accepted as a general method.

major comments (2)

[Section 2.2 (after Eq. (7), Figs. 4-7)] The generalized algorithm for arbitrary subsystem size l is not fully specified in this manuscript. The definitions of the core matrix C and boundary factor B are delegated to reference [17], which is listed as 'Work in progress, (forthcoming)'. The rules for filling the blank boxes in Fig. 5 and the contraction patterns of Fig. 7 are presented pictorially, but the text does not give enough detail for an independent implementation. Since the central claim is a general TRG method for arbitrary l, this missing specification is load-bearing. The numerical tests in Section 3 cover only Hamming weight h <= 2 (l = 2^m + q with q = 0, 2^0, ..., 2^{m-1}); cases with h > 2, which would exercise the full boundary-factor construction and the claimed O(D^3 h) cost, are not presented. The authors should include the complete definitions (e.g., in an appendix) or make the forthcoming reference available before publication.
[Section 3, Eq. (10) and Fig. 10] The fit range 7 <= l <= 768 is selected after observing that the effective central charge c(l) in Fig. 10 visibly deviates from the expected constant behavior for small l. This post hoc choice, together with the error estimate based on the maximum deviation with k fixed, does not account for the systematic uncertainty associated with the lower cutoff. The quoted central charge c = 0.49997(8) therefore carries an error bar that is likely underestimated. The authors should document the sensitivity of c to the fit range (for example, by repeating the fit with several lower cutoffs) or provide an independent criterion for selecting the cutoff.

minor comments (6)

[Section 2.1, Eq. (4)] The sentence 'To keep K and K' finite, we need to take the limit beta -> infinity' is imprecise; the standard Suzuki-Trotter limit is N -> infinity with tau = beta/N fixed, which also sends beta to infinity. Please clarify.
[Section 2.2, trimming argument] The entropy invariance statement S_A = -tr(U rho' U^dagger log(U rho' U^dagger)) = -tr(rho' log rho') is correct, but the text would be clearer if it explicitly stated that the nonzero spectrum is preserved under the isometric embedding U and that zero eigenvalues do not contribute to the von Neumann entropy.
[Section 3, Fig. 8] The caption and text should clarify the normalization in Fig. 8: it says 'All results are normalized by dividing them by the data at alpha = 1024', but it is not explicitly stated that the plotted quantity is S_A(alpha)/S_A(alpha=1024).
[Section 3, Fig. 9 vs. Fig. 10] The statement that 'The theoretical form accurately describes the data in the whole region' is in apparent tension with the later exclusion of small-l data; please reconcile these statements, for instance by specifying that the deviation is visible only in the derivative quantity c(l).
[Section 1, Introduction] The claim that 'no general TRG algorithm for subsystems of arbitrary size is yet known' should be substantiated with a brief discussion of how the prior TRG entanglement studies [12-15] are limited (for example, to half-space subsystems).
[Section 3, Eq. (9)] Please specify the range of m and q for which c(l) is computed, and note explicitly that l' is the next data point in the sequence for fixed q.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the algorithm is validated against an external theoretical value (c = 1/2), and parameter extraction is explicitly a fit.

full rationale

The paper's central claim is a new TRG algorithm for subsystem entanglement entropy; its validity is tested by comparing the fitted central charge c = 0.49997(8) with the independent theoretical value c = 1/2. This is a genuine external benchmark rather than a circular reduction: the theoretical form (8) is used for fitting, but the central charge itself is a free parameter determined from the numerical data. The choice of fit range 7 <= l <= 768 is motivated by the visibly non-constant effective central charge at small l, which is a standard model-selection step rather than an input that forces the final result. The error estimate reuses the fitted k, but this only affects the reported uncertainty, not the central value. The only self-citation to note is [17], 'Work in progress, (forthcoming)', used for tensor definitions; this is a reproducibility gap but does not make the derivation circular, since the numerical agreement with c = 1/2 is independent of that deferred definition. No equation reduces by construction to a fitted input, and no prediction is renamed as a fit. The paper honestly presents the central charge as obtained by fitting, not as a parameter-free prediction.

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

The central claim rests on the standard Suzuki-Trotter mapping, the Calabrese-Cardy formula used for fitting, and the algorithmic construction of the boundary factor. The only fitted quantities are the central charge c and constant k in Eq. (8); additionally, the time extent alpha=16 and the fit range 7<=l<=768 are chosen by hand. No new physical entities are introduced.

free parameters (4)

central charge c = 0.49997(8)
Extracted by fitting the computed S_A(l) to the CFT formula (8) with a least-squares fit.
constant k in Eq. (8) = not reported
Second parameter in the fit to the CFT entanglement entropy formula.
time extent alpha (=N/L) = 16
Chosen by hand as effectively infinite after observing convergence for alpha>=16 in Fig. 8.
fit range for central charge extraction = 7 <= l <= 768
Chosen post hoc after inspecting c(l) in Fig. 10 and excluding data with visible l-dependence.

assumptions (4)

standard math Suzuki-Trotter decomposition (Eq. 3) converges to the Gibbs state e^{-beta H}.
Standard discretization used to map the quantum chain to a (1+1)-dimensional classical Ising model.
domain assumption The CFT formula S_A(l,L) = (c/3) log(L/pi sin(pi l/L)) + k (Eq. 8) describes the entanglement entropy of the critical Ising chain on a cylinder.
Taken from Calabrese-Cardy; used as the fitting form to extract c. Assumes the finite-size system is described by a c=1/2 CFT.
domain assumption The HOTRG isometries U satisfy U^dagger U = I exactly, so dropping boundary isometries preserves the entanglement spectrum of the truncated network.
Used in Section 2.2 to justify trimming isometries from the network. True for orthonormal columns even when truncation makes U U^dagger != I.
ad hoc to paper The binary decomposition construction of the core matrix C and boundary factor B correctly reproduces the reduced density matrix for arbitrary subsystem size l.
Algorithmic construction introduced in this paper; its correctness is supported only by the numerical benchmark in Section 3, not by an independent proof.

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Cite this review

Pith. "Pith review of Entanglement entropy by tensor renormalization group approach." pith.science (2026). https://pith.science/paper/4Y47TOHO

@misc{pith2026250202030,
  author       = {Pith},
  title        = {Pith review of: Entanglement entropy by tensor renormalization group approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4Y47TOHO}},
  note         = {Machine review of arXiv:2502.02030}
}

abstract

We report on tensor renormalization group calculations of entanglement entropy in one-dimensional quantum systems. The reduced density matrix of a Gibbs state can be represented as a $1 + 1$-dimensional tensor network, which is analogous to the tensor network representation of the partition function. The HOTRG method is used to approximate the reduced density matrix for arbitrary subsystem sizes, from which we obtain the entanglement entropy. We test our method in the quantum Ising model and obtain the entanglement entropy of the ground state by taking the size of time direction to infinity. The central charge $c$ is obtained as $c = 0.49997(8)$ for a bond dimension $D=96$, which agrees with the theoretical value $c=1/2$ within the error.

Figures

Figures reproduced from arXiv: 2502.02030 by the authors.

**Figure 1.** Tensor network representation of the partition function 𝑍 and the reduced density matrix 𝜌𝐴. The horizontal axis represents the temporal direction, while the vertical axis represents the spatial direction. eigenvectors of 𝑀†𝑀 with 𝐷cut largest eigenvalues. Since 𝑈 is a submatrix of a unitary matrix, it no longer satisfies 𝑈𝑈† = 𝐼, although it still satisfies 𝑈 †𝑈 = 𝐼. The HOTRG procedure is shown in [PITH_FULL_IMAG… view at source ↗

**Figure 2.** The left figure in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Coarse-graining procedure for 𝜌𝐴. shown in Fig. 3d. We give an algorithm that generalizes this method to arbitrary subsystem size 𝑙. In the following discussion, we take 𝐿 = 2 𝑛 and 𝑁 = 𝛼·2 𝑛 (𝛼 ∈ N). It is convenient to use the binary representations of 𝑙 and 𝑙 − 1 for the generalized algorithm: 𝑙 = ∑︁𝑛−1 𝑖=0 2 𝑖 𝑎𝑖 (𝑎𝑖 = 0, 1), (7a) 𝑙 − 1 = ∑︁𝑛−1 𝑖=0 2 𝑖 𝑏𝑖 (𝑏𝑖 = 0, 1). (7b) The trimmed tensor 𝜌𝐴 is made of two el… view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: For example, in the case of 𝐿 = 16 and 𝑙 = 5, we have 𝑎3𝑎2𝑎1𝑎0 = 0101, 𝑏3𝑏2𝑏1𝑏0 = 0100 and 𝑟 = 0 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 4.** Figure 4: Core matrix 𝐶 and boundary factor 𝐵. (a) Case of 𝑏𝑘 = 0. (b) Case of 𝑏𝑘 = 1 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Structure of the box specified by 𝑏𝑘 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: Boundary factor for 𝐿 = 16 and 𝑙 = 5. (a) 𝑙 > 𝐿/2 (b) 𝑙 ≤ 𝐿/2 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 8.** Figure 8: and [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 10.** Figure 10: Central charge 𝑐(𝑙) computed by (9). 𝐷cut central charge 64 0.4998(2) 80 0.4999(1) 96 0.49997(8) [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [17]

Hayazaki, D

T. Hayazaki, D. Kadoh, S. Takeda and G. Tanaka,Work in progress, (forthcoming)

work page

[2] [1]

Calabrese and J.L

P. Calabrese and J.L. Cardy,Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152]. 8 Entanglement entropy by tensor renormalization group approach Gota Tanaka

arXiv 2004

[3] [2]

Calabrese and J

P. Calabrese and J. Cardy,Entanglement entropy and conformal field theory,J. Phys. A42 (2009) 504005 [0905.4013]

arXiv 2009

[4] [3]

Ryu and T

S. Ryu and T. Takayanagi,Aspects of Holographic Entanglement Entropy, JHEP 08(2006) 045 [hep-th/0605073]

arXiv 2006

[5] [4]

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A. Velytsky,Entanglement entropy in SU(N) gauge theory, PoS LATTICE2008(2008) 256 [0809.4502]

work page Pith review arXiv 2008

[6] [5]

Buividovich and M.I

P.V. Buividovich and M.I. Polikarpov,Numerical study of entanglement entropy in SU(2) lattice gauge theory,Nucl. Phys. B802 (2008) 458 [0802.4247]

arXiv 2008

[7] [6]

Nakagawa, A

Y. Nakagawa, A. Nakamura, S. Motoki and V.I. Zakharov,Entanglement entropy of SU(3) Yang-Mills theory,PoS LAT2009(2009) 188 [0911.2596]

arXiv 2009

[8] [7]

Nakagawa, A

Y. Nakagawa, A. Nakamura, S. Motoki and V.I. Zakharov,Quantum entanglement in SU(3) lattice Yang-Mills theory at zero and finite temperatures, PoS LATTICE2010(2010) 281 [1104.1011]

arXiv 2010