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Tensor renormalization group approach to entanglement entropy

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Pith's one-line read A tensor renormalization group method computes entanglement entropy for single intervals of arbitrary size, giving central charge c=0.49997(8) for the critical Ising model.

desk verdict A genuinely new binary-trimming construction for TRG entanglement entropy with a convincing Ising central-charge test, though the paper never checks positivity of the effective density matrix. read the letter →

arxiv 2509.02185 v1 pith:PI3UHS5R submitted 2025-09-02 hep-lat

Takahiro Hayazaki , Daisuke Kadoh , Shinji Takeda , Gota Tanaka This is my paper · ORCID
classification hep-lat
keywords tensorrenormalizationgroupentanglemententropyreduceddensitymatrixHOTRGcentralchargeIsingmodelvonNeumannnetwork
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 proposes a tensor renormalization group (TRG) method to compute the entanglement entropy of a one-dimensional quantum system for a single-interval subsystem of any size, not just the half-space cases studied before. The key move is to represent the reduced density matrix as a two-dimensional tensor network, coarse-grain it with the higher-order TRG algorithm, and then trim away isometry factors that do not affect the von Neumann entropy, leaving a much smaller matrix. The structure of the leftover network is set by the binary expansion of the interval length, so the final matrix has size set by the Hamming weight of that length. Tested at the critical point of the two-dimensional Ising model, the method reproduces the conformal-field-theory formula and gives central charge c=0.49997(8) at bond dimension D=96, consistent with the exact c=1/2. If it works generally, it offers a way to compute entanglement entropy in theories where Monte Carlo methods face sign problems.

What carries the argument

The central object is the trimmed tensor network ρ~A, obtained from the HOTRG coarse-grained representation of the reduced density matrix by eliminating isometry pairs UU† whose contraction is the identity. It carries the argument because it shrinks a D^ℓ × D^ℓ matrix to D^h × D^h with h the Hamming weight of the binary expansion of ℓ, while preserving the von Neumann entropy. The load-bearing identity is Tr(UρU† log(UρU†)) = Tr(ρ log ρ), valid when U†U=I, which licenses the trimming.

What would settle it

Take a small lattice, such as L=16 or 32, where the reduced density matrix can be formed exactly before coarse-graining, run the HOTRG plus trimming procedure, and inspect the final matrix. If any eigenvalue of the trimmed matrix is negative, or if its von Neumann entropy differs from the exact value by more than the truncation error, the method's central claim fails.

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Extended reading notes

Core claim

The paper claims that for a one-dimensional quantum system, the reduced density matrix of a single-interval subsystem can be coarse-grained with HOTRG and then reduced to a much smaller 'trimmed' tensor network whose von Neumann entropy equals the original entanglement entropy up to truncation error. The arrangement of the leftover isometries is controlled by the binary expansion of the interval length ℓ, and the final matrix has dimension set by the Hamming weight of ℓ. Testing this on the two-dimensional Ising model at criticality, the method reproduces the CFT scaling formula and yields c=0.49997(8) at D=96, matching c=1/2.

Load-bearing premise

The method treats the approximate reduced density matrix as a genuine quantum state; if its eigenvalues are not all non-negative, the logarithm in the entropy is not defined.

Editorial extensions

If this is right

  • Computing entanglement entropy no longer requires the replica trick and the n→1 extrapolation; the entropy is evaluated directly from the approximate reduced density matrix.
  • Because TRG is free from sign problems, the method is a candidate for entanglement-entropy computations in theories where Monte Carlo methods struggle.
  • For intervals whose length has a small Hamming weight, such as powers of two, the extra cost is modest, keeping large-system simulations feasible.
  • The same construction extends to higher-dimensional quantum systems for hyperrectangular subsystems, with (d−1)-dimensional isometries replacing the scalar ones.
  • The central charge extracted at criticality gives a direct numerical check of universal CFT predictions in lattice models.

Reading between the lines

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

  • A natural next test is to apply the method to a theory without a known CFT prediction, where the extracted central charge could be cross-checked against other methods; the paper validates only the Ising case.
  • Because the trimming relies on the algebraic identity for the von Neumann entropy, the method implicitly assumes the approximate reduced density matrix is positive semidefinite; checking the eigenvalue spectrum after truncation would make the method's range of validity concrete.
  • For very long intervals with many 1 bits in their binary expansion, the cost exponent grows with Hamming weight, so choosing subsystem sizes whose binary form is sparse could make larger or higher-dimensional computations practical.
  • The binary-box structure suggests similar trimming could produce inexpensive estimators for Rényi entropies or mutual information by replacing the logarithm with other functions of the reduced density matrix.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a tensor renormalization group (TRG) method for computing the entanglement entropy of a single-interval subsystem in a one-dimensional quantum system. The reduced density matrix is represented as a (1+1)-dimensional tensor network, and the authors develop a HOTRG-based coarse-graining procedure in which the isometries that do not affect the entropy are trimmed away, leaving a smaller matrix whose von Neumann entropy approximates the desired entanglement entropy. The method is designed for arbitrary interval size, not just half-space, with the final matrix size controlled by the Hamming weight of the binary representation of the interval length. The method is tested on the two-dimensional classical Ising model at criticality, and the extracted central charge c=0.49997(8) at D=96 agrees with the CFT prediction c=0.5, with consistent results for D=64,80 and for fixed ratios x=ℓ/L.

Significance. If the method is sound, it offers a potentially valuable sign-problem-free route to entanglement entropy without the replica trick, extending previous TRG-based work that was largely limited to half-space partitions. The binary-trimming construction is conceptually elegant and the numerical agreement with c=0.5 across several bond dimensions and subsystem ratios is encouraging. The paper also gives a clear computational-cost estimate in terms of the Hamming weight. However, the central claim depends on the positivity of the object whose von Neumann entropy is evaluated, and this is not established; the numerical validation also relies on fit ranges and error estimates that are not fully standard. These issues need to be addressed before the method can be considered reliably demonstrated.

major comments (2)
  1. [§2.2 and §3.1] The derivation uses the identity Tr(U ρ̃ U† log(U ρ̃ U†)) = Tr(ρ̃ log ρ̃) to discard isometries, and evaluates S_A as −Tr(ρ̃_A log ρ̃_A). This is valid only when ρ̃_A is Hermitian positive semidefinite. The paper never verifies positivity of the initial "density matrix" (the quotes in §3.1 are telling) or its preservation under HOTRG truncation and the trimming step. For a closed network the classical Ising tensor (3.1) gives positive Boltzmann weights, but the reduced density matrix with open indices, and especially the truncated/trimmed network, need not be PSD. If any negative eigenvalues appear, the von Neumann entropy is not well defined, the trace identity fails, and the numerical agreement with c=0.5 could be accidental. The authors should report the spectrum of ρ̃_A for the parameters used in Table 1, or prove that positivity is preserved by their coarse-graining/trimming. This i
  2. [§3.2] The error estimate for the central charge is obtained by fixing k1 to its fitted central value and taking the maximal deviation of solutions of Eq. (3.2) for individual ℓ. This ignores the covariance between c and k1 and the uncertainty in the choice of fit ranges (7≤ℓ≤768 and the plateau 16≤xL≤128 are selected from the data). The reported errors such as 0.49997(8) are therefore not a reliable measure of the method's accuracy. A standard fitting procedure with full covariance, or a bootstrap over the fit range, should be reported. This is important because the central validation claim is the agreement with c=0.5.
minor comments (5)
  1. [§2.1] Typo: "subsysmtem" should be "subsystem".
  2. [Table 1] Typo: "T able 1" should be "Table 1".
  3. [Fig. 5] The labels and the binary-path markings in Fig. 5 are hard to read at normal print size; please use larger fonts and clearer arrows.
  4. [Eq. (3.3)] In the definition of the effective central charge, it would help to state explicitly that L is fixed and that the formula uses S_A(L,ℓ) and S_A(L,ℓ′). The notation is clear from context but should be spelled out.
  5. [§3.1] The paper calls the object in Fig. 1b a "density matrix" in quotation marks. This is confusing because the numerical test uses the isotropic classical Ising model. The authors should either justify the terminology or use a neutral term such as "tensor network state" and clarify the relation to a genuine quantum density matrix.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EE algorithm follows from tensor-network topology and isometry properties, and the central-charge comparison is an external benchmark.

full rationale

The paper's derivation chain is self-contained and does not reduce to its own inputs. The reduced density matrix is represented as a (d+1)-dimensional tensor network using the standard Suzuki-Trotter / path-integral construction (Sec. 2.1), and the proposed algorithm in Sec. 2.2 coarse-grains this network with HOTRG isometries U defined from M†M eigenvectors. The key trimming step uses the identity Tr(U ρ̃ U† log(U ρ̃ U†)) = Tr(ρ̃ log ρ̃) with U†U = I, which is a mathematical property of isometries, not an input of the target result. The entanglement entropy is then evaluated from the trimmed network ρ̃_A, and the numerical test compares the extracted central charge against the known CFT value c = 0.5 using the theoretical formula (3.2). Fitting c and k1 to the computed EE values and comparing with an external prediction is standard benchmarking, not a fitted-input-called-prediction circularity. The paper does cite work by its own authors ([19] Takeda, [23] Kadoh-Nakayama), but these citations are historical or optional-algorithm remarks, not load-bearing for the central claim. The main caveat—that the trimmed ρ̃_A is not explicitly shown to be positive semidefinite, so the von Neumann logarithm may be ill-defined—is a rigor/correctness concern, not a circularity of the derivation. Therefore no circular step is present.

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

The central algorithm itself introduces no fitted physical parameters; the listed free parameters are analysis choices in the numerical validation. The main axioms are the standard tensor-network representation of density matrices and the standard equivalence between the isotropic classical Ising model and the quantum Ising chain at criticality. No new physical entities are posited.

free parameters (4)
  • fit range lower bound ℓ_min = 7
    Chosen from effective central charge plot (Fig. 13) to exclude small-ℓ lattice artifacts; affects the extracted central charge.
  • fit range upper bound ℓ_max = 768
    Chosen from the same plot; affects the extracted central charge in the first analysis.
  • plateau fit range for xL = 16 to 128
    Chosen from effective central charge plot (Fig. 17); affects the extracted central charge in the fixed-x analysis.
  • temporal extent ratio α = 16
    Chosen from Fig. 10 where convergence is achieved; approximates the ground-state (zero-temperature) limit.
assumptions (4)
  • domain assumption The reduced density matrix of a d-dimensional quantum system admits a (d+1)-dimensional tensor network representation with locally connected homogeneous tensors.
    Stated in Section 2.1; relies on locality and translation invariance of the Hamiltonian.
  • domain assumption The isotropic classical Ising model tensor network at Tc describes the same entanglement scaling as the (1+1)-dimensional quantum Ising chain, so the CFT formula (3.2) applies.
    Used in Section 3.1, citing [29] and prior works [15,25]; a standard equivalence, stated but not derived in this paper.
  • standard math HOTRG isometries satisfy U†U=I by construction.
    Used in Section 2.2 to drop isometry pairs and to derive the trace identity.
  • standard math Tr(ρ log ρ) is invariant under isometric conjugation UρU† when U†U=I and ρ is positive semidefinite.
    Used in Section 2.2 for the trimming step.

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Pith. "Pith review of Tensor renormalization group approach to entanglement entropy." pith.science (2026). https://pith.science/paper/PI3UHS5R

@misc{pith2026250902185,
  author       = {Pith},
  title        = {Pith review of: Tensor renormalization group approach to entanglement entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PI3UHS5R}},
  note         = {Machine review of arXiv:2509.02185}
}
abstract

We propose a method to compute the entanglement entropy (EE) using the tensor renormalization group (TRG) method. The reduced density matrix of a $d$-dimensional quantum system is represented as a $(d+1)$-dimensional tensor network. We develop an explicit algorithm for $d=1$ that enables the calculation of EE for single-interval subsystems of arbitrary size. We test our method in two-dimensional tensor network of the Ising model. The central charge is obtained as $c=0.49997(8)$ for $D=96$, which agrees with the theoretical prediction within an error, demonstrating the accuracy and reliability of our proposed method.

Figures

Figures reproduced from arXiv: 2509.02185 by the authors.

Figure 1
Figure 1. Tensor network representation of e −∆βHˆ and e −βHˆ for d = 1. Contraction of indices in the vertical direction (the dotted lines) corresponds to the periodic boundary condition in spatial direction. Consider d = 1 and take a single interval as the subsystem A. For that case, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Tensor network representation of ρA in d = 1. It is important to note that these tensor networks ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. HOTRG algorithm for the two-dimensional tensor network. The tensor M is obtained by con￾tracting two tensor Ts inside dotted loop. The isometry matrix U is a set of eigenvectors of the matrix M†M corresponding to D largest eigenvalues. The entanglement entropy is evaluated by applying the HOTRG algorithm to the reduced density matrix ρA. Let T (0) be an initial tensor representing ρA. The algorithm is applied altern… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Coarse-graining procedure for ρA. To illustrate our method, we first consider the case of L = N = 8 and ℓ = 3 as shown in Fig. 4a, where six external lines remain open, and ρA is a D3 × D3 matrix. Figure 4b, 4c and 4d denote the approximations of ρA after one, two, and…
Figure 5
Figure 5. Figure 5: ρ ′ A and ˜ρA. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Inside of each dotted box Bk defined in Fig. 5c, the isometry matrices U (k) and U (k)† are contracted as illustrated in Fig. 6a if ak = 0, and as illustrated in Fig. 6b if ak = 1. In both cases, the line towards the top is contracted with the line towards the bottom e…
Figure 7
Figure 7. Figure 7: Trimmed network ˜ρA for n = 3, ℓ = 2 and ℓ = 7. 𝐶 𝐵!"# 𝐵!"$ 𝐵% ⋯ [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: Trimmed network ˜ρA. 𝑇 ! … 𝛼 [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 10
Figure 10. Figure 10: Entanglement entropy at T = Tc as a function of ℓ for several values of α = 1, 2, 4, . . . , 128. The bond dimension and the lattice size are fixed to D = 64 and L = 1024 respectively. All results are normalized by the values at α = 1024 [PITH_FULL_IMAGE:figures/full…
Figure 11
Figure 11. Figure 11: Entanglement entropy at T = Tc as a function of ℓ at D = 96 and α = 16. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1 10 100 Fitting by theoretical form Entanglement entropy Subsystem size [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: Zoom of [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]
Figure 13
Figure 13. Figure 13: Effective central charge c(ℓ) computed by (3.3) at T = Tc. As a second analysis, let us consider the case of fixed x ≡ ℓ/L. For fixed x, we have SA,x(L) ≡ SA(L, xL) = c 3 log L + k ′ 1 (x) , (3.5) with k ′ 1 (x) = k1 + c 3 log (sin xπ) . (3.6) The effective central ch…
Figure 14
Figure 14. Figure 14: Temperature dependence of entanglement entropy around the critical point at α = 16 and D = 96. The ratio x = ℓ/L = 1/2 is fixed and the total size is varied in the range L = 128 − 2048. The dotted grey line shows the location of the critical temperature Tc. 2.260 2.26…
Figure 15
Figure 15. Figure 15: Temperature dependence of effective central charge defined in (3.7) near critical point. The parameters T, α, D and x are the same as those in [PITH_FULL_IMAGE:figures/full_fig_p012_15.png]
Figure 16
Figure 16. Figure 16: L dependence of the entanglement entropy at T = Tc for various values of the ratio x = 1/2, 1/4, 1/8, and 1/16 at D = 96 and α = 16. 0.49 0.5 0.51 0.52 0.53 0.54 1 10 100 1000 x=1/2 x=1/4 x=1/8 x=1/16 Effective central charge Subsystem size xL [PITH_FULL_IMAGE:figure…
Figure 17
Figure 17. Figure 17: Effective central charge cx(L) in (3.7) at T = Tc as a function of L with x = 1/2, 1/4, 1/8 and 1/16 at D = 96 and α = 16. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]
Figure 18
Figure 18. Figure 18: Tensor Tijkℓ. 𝑇 𝑖 𝑗 𝑘 ℓ 𝑇 𝑚 𝑜 𝑛 [PITH_FULL_IMAGE:figures/full_fig_p014_18.png]

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