Pith. sign in

REVIEW 4 major objections 5 minor 70 references

Symmetry-enforced minimal entanglement and correlation in quantum spin chains

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper derives the exact symmetry-enforced minimum of Rényi entanglement for integer-spin chains with unbroken SO(3) and translation symmetry.

desk verdict Solid MPS derivation of symmetry-enforced entropy bounds, but the type-II branch only reaches the bound as an infimum — the abstract overstates 'minimal'. read the letter →

arxiv 2412.20765 v2 pith:IYQFTV5Q submitted 2024-12-30 cond-mat.str-el cond-mat.quant-gasmath-phmath.MPquant-ph

Kangle Li , Liujun Zou This is my paper · ORCID
classification cond-mat.str-elcond-mat.quant-gasmath-phmath.MPquant-ph
keywords matrixproductstatesSO(3)symmetryRényientanglemententropysymmetry-enforcedspinchainscorrelationlengthClebsch-Gordancoefficientsarealaw
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 asks how little entanglement a quantum spin chain can have when SO(3) spin-rotation symmetry and lattice translation symmetry are both unbroken. Working in the matrix product state (MPS) formalism, it proves exact lower bounds on the Rényi-α entanglement entropy of a long block for every α>0 and every integer spin J, and it constructs explicit states that saturate those bounds. The result matters because it shows that symmetry alone, independent of any Hamiltonian, forces a quantitative floor on entanglement, and that this floor grows with J. The paper also proves that no such symmetric state can have exactly zero correlation length, and that a minimally entangled state need not be the state with the shortest correlation length.

What carries the argument

The central object is the transfer matrix $T[A]=\sum_i (A^i)^*\otimes A^i$ of a uniform matrix product state, together with its dominant left and right eigenvectors $v_{l,r}$. For an injective uMPS the spectra of the reduced density matrices are $\mathrm{eig}(\rho_{\mathrm{t.c.}})=\mathrm{eig}((v_lv_r)^{\otimes 2})$ and $\mathrm{eig}(\rho_{\mathrm{o.c.}})=\mathrm{eig}(v_lv_r)$, so entanglement is read directly from the dominant eigenvectors. The SO(3) symmetry enters through the Wigner-Eckart structure of the tensor, with Clebsch-Gordan coefficients fixing how bond irreps couple to the physical spin $J$, and a key proposition shows that the dominant eigenvectors are block-diagonal singlets in each bond irrep sector. The argument splits into extendable tensors, whose bond sectors all have spin at least $J/2$, and generic tensors containing smaller sectors; the latter are forced into a valence-bond-like relation between low and high spin sectors, which produces the second branch of the bound. The explicit saturating states are the type-I state with a single bond sector of spin $J/2$ and the type-II state with bond sectors $0$ and $J$ plus a small $\varepsilon$.

What would settle it

A direct check is to run variational MPS optimization with increasing bond dimension for an SO(3)- and translation-symmetric integer-spin chain in a gapped symmetric phase and compare the converged large-block one-cut Rényi-α entropy with $F_\alpha(J)$; finding a converged value below $F_\alpha(J)$, or exhibiting any symmetric state with smaller entropy, would falsify the uMPS-achievability assumption or the bound as a universal statement. Within the paper's MPS setting, a search for any SO(3)-symmetric injective tensor whose one-cut entropy violates the bound would settle the theorem directly.

Watch

Extended reading notes

Core claim

Within the set of translation-invariant matrix product states (uMPS) with unbroken SO(3) and translation symmetry on a spin-$J$ chain ($J$ a positive integer), the paper establishes that for every $\alpha>0$ the two-cut Rényi-$\alpha$ entropy satisfies $S_\alpha(\rho_{\mathrm{t.c.}}) \ge 2F_\alpha(J)$ and the one-cut entropy satisfies $S_\alpha(\rho_{\mathrm{o.c.}}) \ge F_\alpha(J)$, with $F_\alpha(J)=\min\{\ln(J+1), -\frac{1}{\alpha-1}\ln\!\left(\frac{1+(2J+1)^{1-\alpha}}{2^\alpha}\right)\}$, where $\alpha=1$ is read as the von Neumann limit. Both bounds are tight: they are approached by explicit type-I states, built from a single bond-sector of spin $J/2$ and equivalent to AKLT-like states, and by type-II states, which contain valence-bond-like $(0,J)$ and $(J,0)$ blocks together with a small injection parameter $\varepsilon$. For $\alpha\to1$ these become $S(\rho_{\mathrm{t.c.}})\ge\min\{2\ln(J+1),\ln(4(2J+1))\}$ and $S(\rho_{\mathrm{o.c.}})\ge\min\{\ln(J+1),\ln(2\sqrt{2J+1})\}$. The paper further proves that a symmetric uMPS cannot have zero correlation length, because the transfer matrix of any such tensor is traceless by the Clebsch-Gordan properties, and it gives numerical examples of symmetric states with correlation length shorter than that of the AKLT state.

Load-bearing premise

The paper's Working Assumption 2 says that the minimal values of the entropy limits, and also of the correlation length, are achieved by a translation-invariant matrix product state; if that fails, the MPS-derived bounds would not be the true global minima over all symmetric states.

Editorial extensions

If this is right

  • If the bounds are correct, every gapped symmetric ground state of an integer-spin chain representable by an MPS must have one-cut von Neumann entropy at least $\min\{\ln(J+1),\ln(2\sqrt{2J+1})\}$, and two-cut entropy twice that amount.
  • For $J=1$, the AKLT state saturates the bound for all $\alpha>0$, making it the minimal-entanglement state among symmetric spin-1 uMPS.
  • No SO(3)- and translation-symmetric integer-spin uMPS can be a renormalization-group fixed point: its correlation length is always positive, with a spectral lower bound $|\lambda_2|\ge \frac{1}{D^2-1}$ for bond dimension $D$.
  • A minimally entangled state does not have to minimize the correlation length; spin-1 states with bond sectors $\frac12\oplus\frac32$ or $(\frac12)^2\oplus\frac32$ have numerically smaller second-largest transfer-matrix eigenvalues than the AKLT state.
  • Since the bound grows like $\ln J$ for large $J$, symmetric integer-spin chains must become more entangled as $J$ grows, consistent with semiclassical expectations and with Lieb-Schultz-Mattis-type reasoning applied to the half-integer case.

Reading between the lines

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

  • If the paper's Working Assumption 2 is correct, the same lower bounds should hold for all symmetric states, not only uMPS; a natural test is to perform variational MPS optimization with increasing bond dimension for $J=2,3$ and check whether the numerically converged one-cut entropies approach $F_\alpha(J)$.
  • The traceless-transfer-matrix argument is not specific to SO(3): for any non-Abelian symmetry group whose symmetric tensors are traceless by representation theory, the same style of argument would give a positive lower bound on the correlation length.
  • The type-II saturation branch suggests a general mechanism: the symmetry-preserving entropy minimum can be approached as the limit of a family of states that spontaneously break translation symmetry at exactly $\varepsilon=0$, so the minimal symmetric entanglement may often sit at a symmetry-breaking threshold.
  • The observation that the AKLT-like states are not the shortest-correlation states points to a practical search strategy: add off-diagonal spin blocks to AKLT-like tensors and perturbatively track the second-largest eigenvalue, which may locate the true minimal-correlation state.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies translation-invariant matrix product states (uMPS) for integer-spin chains with unbroken SO(3) and translation symmetry, and derives lower bounds on the Rényi-α entanglement entropies of a long segment (one-cut and two-cut) as well as a no-zero-correlation-length theorem. The main results are closed-form bounds: for the two-cut entropy the bound is min{2 ln(J+1), -2/(α-1) ln((1+(2J+1)^{1-α})/2^α)}, with the one-cut bound being half of the second term or ln(J+1), respectively. The proofs use the structural theory of symmetric uMPS, the Wigner-Eckart theorem, and properties of Clebsch-Gordan coefficients. The authors construct two families of states, type-I (single J/2 bond sector, e.g., AKLT-type) and type-II (0⊕J bond sectors with an ε perturbation), and show that these states approach the bounds. They also prove that an SO(3)-symmetric uMPS cannot have vanishing correlation length and give a corollary lower bound on the second-largest transfer-matrix eigenvalue. The extension of the entanglement bounds from uMPS to arbitrary symmetric states is explicitly identified in Section V as two working assumptions, one about the existence of Rényi-entropy limits and one about the achievability of the minima by translation-invariant MPS.

Significance. If the results are read as infimum statements within the uMPS class, the paper provides a clean, analytic answer to a natural question: exact symmetry-enforced lower bounds on Rényi entropies for integer spin chains, with explicit families that approach the bounds. The spin-1 AKLT state is correctly identified as exactly saturating the bound for all α, and the no-zero-correlation-length proof via a trace argument on Clebsch-Gordan blocks is elegant. The appendices give substantial supporting detail, including a treatment of non-injective uMPS and the structure of dominant eigenvectors. The main caveat is that the type-II family does not attain the claimed minimum in the unbroken-translation regime, so the headline statement needs to be restated as an infimum; this is a genuine interpretive and technical point that affects the abstract and the statement of the main theorems.

major comments (4)
  1. [Section III.D, Eq. (20)-(21); Theorem 2; Abstract] For parameter regions where the type-II branch is the smaller one (e.g., J≥7 at α=1), the stated lower bound is not attained by any state in S^TI_J. Eq. (21) gives S(ρ_t.c.(II)) = ln(4(2J+1)) + O(ε^2), and from Eq. (57) the leading correction is positive: the two-cut entropy is ln(4(2J+1)) + (ε^2/2) ln(2J+1) + O(ε^4). The ε→0 limit is the 2-periodic valence-bond tensor, which is excluded by the theorem's unbroken-translation assumption and has two-cut entropy ln(2(2J+1)), not ln(4(2J+1)). Consequently the abstract's 'minimal Rényi entropy is ...' and the Introduction's 'states saturating the lower bounds' are literally false unless 'minimal' is read as 'infimum'. The abstract, Theorems 2 and 3, and Section III.D should be restated in terms of an infimum that is approached by type-II states.
  2. [Section V, Working Assumption 2] Working Assumption 2 states that the minimal values of the limits can be achieved by a state described by a translation-invariant MPS. This is incompatible with the non-attainment described above: whenever the type-II branch is smaller, no translation-invariant MPS with unbroken translation symmetry achieves the infimum, so the exact minimum over all symmetric states, if it exists, is not certified by the type-I/type-II construction. The authors should either reformulate the working assumption and the main theorems in terms of infima and approaching sequences, or supply a separate existence argument for an attained minimum.
  3. [Section III.D, injectivity of type-II tensors] The injectivity proof for A^{J,m}(ε) after Eqs. (52)-(56) only examines eigenvectors of the assumed block-diagonal form X_a and then invokes continuity. To conclude that T[A_J(ε)] is injective for 0<ε≪1, the authors need to show that no other eigenvalue of T[A_J(ε)] can enter the peripheral spectrum. This can likely be repaired by combining the known peripheral spectrum {±1} at ε=0 with continuity of the whole spectrum, but that argument is not supplied and is load-bearing for the claim that type-II states are valid injective uMPS.
  4. [Appendix J, Eq. (167) and Lemma 7] The inequality labelled Eq. (167) is key for the α>1 case because it selects the valence-bond (j=0) configuration as the extremal one, but the final step is only justified by the sentence 'the last line is obtained by the monotonicity of the second last line with respect to j'. A derivation or a precise reference for this monotonicity should be provided. The analogous pairing inequalities in Eqs. (161) and (164) would also benefit from a one-line justification of the maximum over j.
minor comments (5)
  1. [Section III.E, Eq. (58)] The phrase 'can approached' should read 'can be approached'; more importantly, the theorem statements should systematically use 'infimum' or 'greatest lower bound' when type-II states are involved, rather than saying the bound is 'tight' or 'saturated' without qualification.
  2. [Section III.D, Eq. (20)] In the display of Eq. (20), the lower-right block ε B^{J,m}_{J,J} should specify the dimension of the zero block in the upper-left entry; the block form is clear from the text but the equation alone is slightly ambiguous about whether the upper-left block is 1×1 or empty.
  3. [Section III.E, cross-reference] The parenthetical reference 'also see Eq. (156) in Appendix III E' appears to be a broken cross-reference; it should point to the relevant equation in Appendix J.
  4. [Section IV.C] The numerical searches for the second-largest eigenvalue report values such as 0.1061 and 0.07624 as found by dual annealing, but there is no discussion of how the global minimum is certified or how many restarts were used. A brief statement about the reliability of these numerical minima would be helpful, since the paper explicitly does not claim a proof of the minimal correlation length.
  5. [Section II.B, Eq. (13)] The connected correlation function formula assumes a single non-degenerate second-largest eigenvalue with trivial Jordan structure; the text notes this in the following paragraph, but the notation λ_2 for the set of subleading eigenvalues is used loosely. This is acceptable but could be flagged in the equation itself.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: bounds derived from MPS theory, Wigner-Eckart/CG coefficients; self-citations are not load-bearing.

full rationale

The derivation is self-contained. Theorems 2 and 3 are proven within the stated uMPS setting using standard MPS canonical forms (Refs [30,31,36,40]), Perron-Frobenius theory of transfer matrices (Refs [39,40]), Wigner-Eckart/CG structure (Refs [8,44,46]), convexity/concavity properties of entropy functionals, and explicit constructions (type-I, type-II states). No parameter is fitted to data and no quantity called a prediction is defined in terms of the quantity it purportedly predicts. The self-citations (Refs [4,12]) appear only in the introduction as context or motivation and are not needed for the proofs of Theorems 2-4. The paper explicitly labels its two Working Assumptions in Section V (existence of the entropy limits and MPS achievability of the minima), so any reliance on these is an honest assumption rather than a disguised circular step. The type-II branch saturates the bound only as eps -> 0, and the paper itself records that the eps=0 limit is a non-injective, translation-breaking state excluded from the theorem; this affects whether 'minimum' should read 'infimum' and is a sharpness/correctness concern, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central derivation rests on standard MPS theory and CG coefficient identities. No free parameters are fitted to produce the bounds. Two working assumptions, both flagged by the authors in Sec. V, are needed to lift the uMPS results to all symmetric states.

assumptions (7)
  • standard math Every uMPS tensor can be transformed to a direct sum of irreducible tensors (standard form).
    Used in proof of Theorem 1 and Lemma 6; see Appendix B and Refs. [36,56].
  • standard math Wigner-Eckart theorem: symmetric tensor components factor into CG coefficients times degeneracy parameters.
    Eq. (16) in Sec. II C; used throughout to structure SO(3)-symmetric tensors.
  • standard math Quantum Perron-Frobenius theorem: transfer matrix of a normalized uMPS has a real dominant eigenvalue 1 and dominant eigenvectors that can be chosen positive semi-definite.
    Sec. II B, Refs. [39,40]; underpins the spectral analysis.
  • domain assumption Fact 1 from Ref. [8]: single-bond-spin MPS tensors are injective and have strictly positive fixed points.
    Used to establish type-I states saturate bounds; Sec. III D.
  • standard math Strong subadditivity of von Neumann entropy (for alpha=1) and convexity/concavity properties of Renyi entropy functionals.
    Used in Appendix A and Appendix J to prove convergence and minimization; Refs. [49,50,51].
  • ad hoc to paper Working assumption 1: the limits lim_N lim_L S_alpha exist for all alpha for area-law symmetric states (proved only for alpha=1).
    Explicitly stated in Sec. V; needed to extend the MPS bounds to general states.
  • ad hoc to paper Working assumption 2: the minimal entropies and minimal correlation length are achieved by translation invariant MPS states.
    Explicitly stated in Sec. V; the proofs establish the bounds within uMPS only.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Symmetry-enforced minimal entanglement and correlation in quantum spin chains." pith.science (2026). https://pith.science/paper/IYQFTV5Q

@misc{pith2026241220765,
  author       = {Pith},
  title        = {Pith review of: Symmetry-enforced minimal entanglement and correlation in quantum spin chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IYQFTV5Q}},
  note         = {Machine review of arXiv:2412.20765}
}
abstract

The interplay between symmetry, entanglement and correlation is an interesting and important topic in quantum many-body physics. Within the framework of matrix product states, in this paper we study the minimal entanglement and correlation enforced by the $SO(3)$ spin rotation symmetry and lattice translation symmetry in a quantum spin-$J$ chain, with $J$ a positive integer. When neither symmetry is spontaneously broken, for a sufficiently long segment in a sufficiently large closed chain, we find that the minimal R\'enyi-$\alpha$ entropy compatible with these symmetries is $\min\{ -\frac{2}{\alpha-1}\ln(\frac{1}{2^\alpha}({1+\frac{1}{(2J+1)^{\alpha-1}}})), 2\ln(J+1) \}$, for any $\alpha\in\mathbb{R}^+$. In an infinitely long open chain with such symmetries, for any $\alpha\in\mathbb{R}^+$ the minimal R\'enyi-$\alpha$ entropy of half of the system is $\min\{ -\frac{1}{\alpha-1}\ln(\frac{1}{2^\alpha}({1+\frac{1}{(2J+1)^{\alpha-1}}})), \ln(J+1) \}$. When $\alpha\rightarrow 1$, these lower bounds give the symmetry-enforced minimal von Neumann entropies in these setups. Moreover, we show that no state in a quantum spin-$J$ chain with these symmetries can have a vanishing correlation length. Interestingly, the states with the minimal entanglement may not be a state with the minimal correlation length.

Figures

Figures reproduced from arXiv: 2412.20765 by the authors.

Figure 1
Figure 1. FIG. 1. In a chain with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Cartoon of a quantum spin chain with [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) The schematic of transfer matrix and left/right [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Structure of a symmetric tensor [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: , we show whether the type-I or type-II state has a smaller R´enyi entropy as a function of J and α. In [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Both [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

70 extracted references · 44 canonical work pages

  1. [1]

    So far we can only prove this statement for the special case where α = 1 (see Appendix A), but we are unable to prove or disprove it for other values of α

    For a state in an integer spin-J chain with SO(3) and translation symmetries, if Sα(ρt.c.(N, L)) and Sα(ρo.c.(N, L)) are bounded when we first take L → ∞ and then take N → ∞, then the limits lim N →∞ limL→∞ Sα(ρt.c.(N, L)) and limN →∞ limL→∞ Sα(ρo.c.(N, L)) exist, for any α ∈ R+. So far we can only prove this statement for the special case where α = 1 (se...

    work page

  2. [2]

    Also, the minimal correlation length can also be achieved by such a state

    Assuming that the above limits exist, the mini- mal values of these limits can be achieved by a state described by a translation invariant MPS. Also, the minimal correlation length can also be achieved by such a state. Below we enumerate some interesting open questions for future studies

    work page

  3. [3]

    Within the framework of uMPS, one can show that the limits lim N →∞ limL→∞ Sα(ρt.c.(N, L)) and lim N →∞ limL→∞ Sα(ρo.c.(N, L)) exist. However, as far as we know, it has not been shown in full generality that these limits exist for all translation symmetric states where Sα(ρt.c.(N, L)) and Sα(ρt.c.(N, L)) are bounded when we first take L → ∞ and then take ...

    work page

  4. [4]

    The uMPS considered in this paper can only represent states with a zero momen- tum. Can one prove that states with a 17 nonzero momentum must have larger val- ues of lim N →∞ limL→∞ Sα(ρt.c.(N, L)) and limN →∞ limL→∞ Sα(ρo.c.(N, L)) than the ones found in Theorems 2 and 3, when these limits exist? Also, even within the set of states with zero momentum, ca...

    work page

  5. [5]

    In this paper, we have shown that a state de- scribed by SO(3) symmetric uMPS cannot have a zero correlation length. But what is the precise value of the minimal correlation length? Does it indeed diverge when J → ∞, as expected by combining the semiclassical analysis and Lieb- Schultz-Mattis theorem?

    work page

  6. [6]

    How can the considerations in this paper be generalized to multi-partite entanglement, other non-Abelian symmetry groups and/or higher di- mensions? For the case of other non-Abelian symmetries, note that our Proposition 1 still holds, and one needs to use the properties of the representations of these groups to find the mini- mal entanglement, as done in...

    work page

  7. [7]

    irreducible

    Can the minimal entanglement found in this work be used for some tasks relevant to quantum in- formation and quantum computation? Acknowledgments We thank Hoi Chun Po, Yimu Bao, Zhehao Dai, Rui- hua Fan, and Frank Pollmann for useful discussions. KL acknowledges the support by the National Nat- ural Science Foundation of China/Hong Kong RGC Collaborative ...

    work page

  8. [8]

    If c1 = c2, the largest eigenvalues and corresponding eigenvectors are determined by the largest eigenvalues and the corresponding eigenvectors of both P i(Ai 1)∗ ⊗ Ai 1 and P i(Ai 2)∗ ⊗ Ai 2. To unpack the meaning of this lemma, note that the transfer matrix T [A] can be grouped into a block diag- onal form of the following 4 blocks: c2 1 X i (Ai 1)∗ ⊗ A...

    work page

Show all 70 references
  1. [9]

    Now that X is an isometry, we have D2 ⩾ D1

    Denote the bond di- mension of A1 (A2) by D1 (D2). Now that X is an isometry, we have D2 ⩾ D1. As we assume, Ai 2 is irreducible, so it does not have any proper invariant subspace, which means that D1 = D2, otherwise XX † will be a projector onto a proper invariant subspace of Ai

  2. [10]

    This implies that |λ| = 1 only if Ai 1 is equivalent to Ai

    Therefore X is a unitary matrix. This implies that |λ| = 1 only if Ai 1 is equivalent to Ai

  3. [11]

    There- fore, |λ| = 1 if and only if Ai 1 and Ai 2 are equivalent

    On the other hand, if Ai 1 and Ai 2 are equivalent, then one can trans- form them to the same tensor without affecting the physical state, and in this case clearly |λ| = 1. There- fore, |λ| = 1 if and only if Ai 1 and Ai 2 are equivalent. In the statement of Lemma 2, if c1 > c...

  4. [12]

    For α > 1, H(α) is a convex function in ρ, i.e., H(P i tiρi) ⩽ P i tiH(ρi) for 0 ⩽ ti ⩽ 1, P i ti = 1, and ρi some density matrices (see Theorem 3.27 in Ref. [51])

  5. [13]

    Before moving to the proof, we need to understand the relation between one-cut R´ enyi entropy and two- cut R´ enyi entropy

    Hα( 1 n Ln i=1 ρi) = 1 nα P i Hα(ρi) These properties will be used in the following discus- sions. Before moving to the proof, we need to understand the relation between one-cut R´ enyi entropy and two- cut R´ enyi entropy. If the tensor is injective, then eig(ρo.c.) = eig(vlv...

  6. [14]

    (152)Hα(ρt.c.) can be expressed as Hα(ρt.c.) = 1 pα pX k=1 Hα(ρk)2

    Denote ρk+1 = A† kρkAk for 1 ⩽ k < p Hα(ρo.c.) = 1 pα pX k=1 Hα(˜ρk) (160) Using the inequality Hα(˜ρk) + Hα(˜ρk+1) ⩽ max j {Hα(ρj) + Hα(ρS−j)} = 1 + 1 (2J + 1)α−1 , (161) we find Hα(ρo.c.) ⩽ 1 2α 1 + 1 (2J + 1)α−1 (162) For two-cut entropy, from Eq. (152)Hα(ρt.c.) can be expr...

  7. [15]

    Gapped quantum liq- uids and topological order, stochastic local transfor- mations and emergence of unitarity,

    Bei Zeng and Xiao-Gang Wen, “Gapped quantum liq- uids and topological order, stochastic local transfor- mations and emergence of unitarity,” Phys. Rev. B 91, 125121 (2015), arXiv:1406.5090 [cond-mat.str-el]

    2015 arXiv

  8. [16]

    An introduction to spontaneous symmetry breaking,

    Aron J. Beekman, Louk Rademaker, and Jasper van Wezel, “An introduction to spontaneous symmetry breaking,” SciPost Phys. Lect. Notes , 11 (2019)

    2019

  9. [17]

    Hal Tasaki, Physics and Mathematics of Quantum Many-Body Systems (Springer Cham, 2020)

    2020

  10. [18]

    Entanglement-enabled symmetry-breaking orders,

    Cheng-Ju Lin and Liujun Zou, “Entanglement-enabled symmetry-breaking orders,” SciPost Phys. Core 7, 010 (2024)

    2024

  11. [19]

    Two soluble models of an antiferromagnetic chain,

    Elliott Lieb, Theodore Schultz, and Daniel Mattis, “Two soluble models of an antiferromagnetic chain,” Annals of Physics 16, 407–466 (1961)

    1961

  12. [20]

    Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Sys- tems on a Periodic Lattice,

    Masaki Oshikawa, “Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Sys- tems on a Periodic Lattice,” Phys. Rev. Lett.84, 1535– 1538 (2000), arXiv:cond-mat/9911137 [cond-mat.str- el]

    2000 arXiv

  13. [21]

    Lieb-Schultz-Mattis in higher dimen- sions,

    M. B. Hastings, “Lieb-Schultz-Mattis in higher dimen- sions,” Phys. Rev. B 69, 104431 (2004), arXiv:cond- mat/0305505 [cond-mat.str-el]

    2004

  14. [22]

    Matrix product states: Symmetries and two- body Hamiltonians,

    M. Sanz, M. M. Wolf, D. P´ erez-Garc ´ ıa, and J. I. Cirac, “Matrix product states: Symmetries and two- body Hamiltonians,” Phys. Rev. A 79, 042308 (2009), arXiv:0901.2223 [cond-mat.str-el]

    2009 arXiv

  15. [23]

    Classification of gapped symmetric phases in one- dimensional spin systems,

    Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen, “Classification of gapped symmetric phases in one- dimensional spin systems,” Phys. Rev. B 83, 035107 (2011), arXiv:1008.3745 [cond-mat.str-el]

    2011 arXiv

  16. [24]

    Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge,

    Dominic V. Else and Chetan Nayak, “Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge,” Phys. Rev. B 90, 235137 (2014), arXiv:1409.5436 [cond- mat.str-el]

    2014 arXiv

  17. [25]

    Nonzero Momentum Re- quires Long-Range Entanglement,

    Lei Gioia and Chong Wang, “Nonzero Momentum Re- quires Long-Range Entanglement,” Physical Review X 12, 031007 (2022), arXiv:2112.06946 [cond-mat.str-el]

    2022 arXiv

  18. [26]

    Lieb-Schultz-Mattis theorems and generaliza- tions in long-range interacting systems,

    Ruizhi Liu, Jinmin Yi, Shiyu Zhou, and Liujun Zou, “Lieb-Schultz-Mattis theorems and generaliza- tions in long-range interacting systems,” arXiv e-prints , arXiv:2405.14929 (2024), arXiv:2405.14929 [cond- mat.str-el]

    2024

  19. [27]

    Mixed-state quantum anomaly and multipartite en- tanglement,

    Leonardo A. Lessa, Meng Cheng, and Chong Wang, “Mixed-state quantum anomaly and multipartite en- tanglement,” arXiv e-prints , arXiv:2401.17357 (2024), arXiv:2401.17357 [cond-mat.str-el]

    2024 arXiv

  20. [28]

    Sym- metry enforced entanglement in maximally mixed states,

    Amin Moharramipour, Leonardo A. Lessa, Chong Wang, Timothy H. Hsieh, and Subhayan Sahu, “Sym- metry enforced entanglement in maximally mixed states,” arXiv e-prints , arXiv:2406.08542 (2024), arXiv:2406.08542 [quant-ph]

    2024 arXiv

  21. [29]

    Highly-entangled stationary states from strong symmetries,

    Yahui Li, Frank Pollmann, Nicholas Read, and Pablo Sala, “Highly-entangled stationary states from strong symmetries,” arXiv e-prints , arXiv:2406.08567 (2024), arXiv:2406.08567 [quant-ph]

    2024 arXiv

  22. [30]

    Colloquium: Area laws for the entanglement entropy,

    J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Area laws for the entanglement entropy,” Reviews of Modern Physics 82, 277–306 (2010), arXiv:0808.3773 [quant-ph]

    2010 arXiv

  23. [31]

    Area Laws in Quan- tum Systems: Mutual Information and Correlations,

    Michael M. Wolf, Frank Verstraete, Matthew B. Hast- ings, and J. Ignacio Cirac, “Area Laws in Quan- tum Systems: Mutual Information and Correlations,” Phys. Rev. Lett. 100, 070502 (2008), arXiv:0704.3906 [quant-ph]

    2008 arXiv

  24. [32]

    Daniel C Mattis, The theory of magnetism I: Statics and Dynamics (Springer Science & Business Media, 1981)

    1981

  25. [33]

    121 (Oxford University Press, 2003)

    Thierry Giamarchi, Quantum physics in one dimen- sion, Vol. 121 (Oxford University Press, 2003)

    2003

  26. [34]

    Assa Auerbach, Interacting electrons and quantum magnetism (Springer Science & Business Media, 2012)

    2012

  27. [35]

    An area law for one-dimensional quantum systems,

    M. B. Hastings, “An area law for one-dimensional quantum systems,” Journal of Statistical Mechan- ics: Theory and Experiment 2007, 08024 (2007), arXiv:0705.2024 [quant-ph]

    2007 arXiv

  28. [36]

    An area law and sub-exponential algorithm for 1D systems,

    Itai Arad, Alexei Kitaev, Zeph Landau, and Umesh Vazirani, “An area law and sub-exponential algorithm for 1D systems,” arXiv e-prints , arXiv:1301.1162 (2013), arXiv:1301.1162 [quant-ph]

    2013 arXiv

  29. [37]

    An area law for entanglement from exponential de- cay of correlations,

    Fernando G. S. L. Brand˜ ao and Micha l Horodecki, “An area law for entanglement from exponential de- cay of correlations,” Nature Physics 9, 721–726 (2013), arXiv:1309.3789 [quant-ph]

    2013 arXiv

  30. [38]

    Realistic Area-Law Bound on Entan- glement from Exponentially Decaying Correlations,

    Jaeyoon Cho, “Realistic Area-Law Bound on Entan- glement from Exponentially Decaying Correlations,” Physical Review X 8, 031009 (2018), arXiv:1706.09379 [quant-ph]

    2018 arXiv

  31. [39]

    Area law of noncritical ground states in 1D long-range interacting systems,

    Tomotaka Kuwahara and Keiji Saito, “Area law of noncritical ground states in 1D long-range interacting systems,” Nature Communications 11, 4478 (2020), arXiv:1908.11547 [quant-ph]

    2020 arXiv

  32. [40]

    Geometric and renormalized entropy in conformal field theory,

    Christoph Holzhey, Finn Larsen, and Frank Wilczek, “Geometric and renormalized entropy in conformal field theory,” Nuclear Physics B 424, 443–467 (1994), arXiv:hep-th/9403108 [hep-th]

    1994 arXiv

  33. [41]

    En- tanglement in Quantum Critical Phenomena,

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, “En- tanglement in Quantum Critical Phenomena,” Phys. Rev. Lett. 90, 227902 (2003), arXiv:quant-ph/0211074 [quant-ph]

    2003 arXiv

  34. [42]

    Entanglement entropy and quantum field theory,

    Pasquale Calabrese and John Cardy, “Entanglement entropy and quantum field theory,” Journal of Statis- tical Mechanics: Theory and Experiment 2004, 06002 (2004), arXiv:hep-th/0405152 [hep-th]

    2004 arXiv

  35. [43]

    Entangle- ment entropy and conformal field theory,

    Pasquale Calabrese and John Cardy, “Entangle- ment entropy and conformal field theory,” Journal of Physics A Mathematical General 42, 504005 (2009), arXiv:0905.4013 [cond-mat.stat-mech]

    2009 arXiv

  36. [44]

    Finitely correlated states on quantum spin chains,

    M. Fannes, B. Nachtergaele, and R. F. Werner, “Finitely correlated states on quantum spin chains,” Communications in Mathematical Physics 144, 443– 490 (1992)

    1992

  37. [45]

    Matrix product state representations,

    D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, “Matrix product state representations,” Quan- tum Info. Comput. 7, 401–430 (2007), arXiv:quant- ph/0608197 [quant-ph]

    2007

  38. [46]

    String Order and Symmetries in Quantum Spin Lattices,

    D. P´ erez-Garc ´ ıa, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac, “String Order and Symmetries in Quantum Spin Lattices,” Phys. Rev. Lett.100, 167202 (2008), arXiv:0802.0447 [cond-mat.str-el]

    2008 arXiv

  39. [47]

    Matrix 33 product states, projected entangled pair states, and variational renormalization group methods for quan- tum spin systems,

    F. Verstraete, V. Murg, and J. I. Cirac, “Matrix 33 product states, projected entangled pair states, and variational renormalization group methods for quan- tum spin systems,” Advances in Physics 57, 143–224 (2008), arXiv:0907.2796 [quant-ph]

    2008 arXiv

  40. [48]

    A practical introduction to tensor net- works: Matrix product states and projected entangled pair states,

    Rom´ an Or´ us, “A practical introduction to tensor net- works: Matrix product states and projected entangled pair states,” Annals of Physics 349, 117–158 (2014), arXiv:1306.2164 [cond-mat.str-el]

    2014 arXiv

  41. [49]

    The density-matrix renormaliza- tion group in the age of matrix product states,

    Ulrich Schollw¨ ock, “The density-matrix renormaliza- tion group in the age of matrix product states,” An- nals of Physics 326, 96–192 (2011), arXiv:1008.3477 [cond-mat.str-el]

    2011 arXiv

  42. [50]

    Matrix product states and projected entangled pair states: Concepts, symme- tries, theorems,

    J. Ignacio Cirac, David P´ erez-Garc ´ ıa, Norbert Schuch, and Frank Verstraete, “Matrix product states and projected entangled pair states: Concepts, symme- tries, theorems,” Rev. Mod. Phys. 93, 045003 (2021), arXiv:2011.12127

    2021 arXiv

  43. [51]

    Rigorous results on valence-bond ground states in antiferromagnets,

    Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki, “Rigorous results on valence-bond ground states in antiferromagnets,” Phys. Rev. Lett. 59, 799– 802 (1987)

    1987

  44. [52]

    Valence bond ground states in isotropic quan- tum antiferromagnets,

    Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki, “Valence bond ground states in isotropic quan- tum antiferromagnets,” Communications in Mathe- matical Physics 115, 477–528 (1988)

    1988

  45. [53]

    Spectral properties of positive maps on c*-algebras,

    David E Evans and Raphael Høegh-Krohn, “Spectral properties of positive maps on c*-algebras,” Journal of the London Mathematical Society 2, 345–355 (1978)

    1978

  46. [54]

    Quantum channels and operations: Guide tour,

    Michael Wolf, “Quantum channels and operations: Guide tour,” (2012)

    2012

  47. [55]

    Equivalence of the variational matrix prod- uct method and the density matrix renormalization group applied to spin chains,

    J. Dukelsky, M. A. Mart ´ ın-Delgado, T. Nishino, and G. Sierra, “Equivalence of the variational matrix prod- uct method and the density matrix renormalization group applied to spin chains,” EPL (Europhysics Letters) 43, 457–462 (1998), arXiv:cond-mat/9710310 [cond-mat.str-el]

    1998 arXiv

  48. [56]

    The matrix product approach to quantum spin ladders,

    J. M. Rom´ an, G. Sierra, J. Dukelsky, and M. A. Mart ´ ın-Delgado, “The matrix product approach to quantum spin ladders,” Journal of Physics A Math- ematical General 31, 9729–9759 (1998), arXiv:cond- mat/9802150 [cond-mat.str-el]

    1998

  49. [57]

    The non-Abelian density matrix renormalization group algorithm,

    I. P. McCulloch and M. Gul´ acsi, “The non-Abelian density matrix renormalization group algorithm,” EPL (Europhysics Letters) 57, 852–858 (2002), arXiv:cond- mat/0012319 [cond-mat.str-el]

    2002

  50. [58]

    Tensor network decompositions in the presence of a global symmetry,

    Sukhwinder Singh, Robert N. C. Pfeifer, and Guifr´ e Vidal, “Tensor network decompositions in the presence of a global symmetry,” Phys. Rev. A82, 050301 (2010), arXiv:0907.2994 [cond-mat.str-el]

    2010 arXiv

  51. [59]

    Tensor network states and algorithms in the presence of a global SU(2) symmetry,

    Sukhwinder Singh and Guifre Vidal, “Tensor network states and algorithms in the presence of a global SU(2) symmetry,” Phys. Rev. B 86, 195114 (2012), arXiv:1208.3919 [cond-mat.str-el]

    2012 arXiv

  52. [60]

    Wigner–eckart theorem for an arbitrary group or lie algebra,

    Vishnu K Agrawala, “Wigner–eckart theorem for an arbitrary group or lie algebra,” Journal of Mathemat- ical Physics 21, 1562–1565 (1980)

    1980

  53. [61]

    Brian C Hall and Brian C Hall, Lie groups, Lie alge- bras, and representations (Springer, 2013)

    2013

  54. [62]

    42 (Springer, 1977)

    Jean-Pierre Serre et al., Linear representations of finite groups, Vol. 42 (Springer, 1977)

    1977

  55. [63]

    Michael A Nielsen and Isaac L Chuang, Quantum com- putation and quantum information (Cambridge univer- sity press, 2010)

    2010

  56. [64]

    Entanglement monotones,

    Guifre Vidal, “Entanglement monotones,” Journal of Modern Optics 47, 355–376 (2000), arXiv:quant- ph/9807077 [quant-ph]

    2000

  57. [65]

    Fumio Hiai and D´ enes Petz, Introduction to matrix analysis and applications (Springer Science & Business Media, 2014)

    2014

  58. [66]

    Dmitri ˘ ı Aleksandrovich Varshalovich, Anatol ¨ ı Nikolae- vitch Moskalev, and Valerij Kel’manov ˇ ıc Khersonskii, Quantum theory of angular momentum (World Scien- tific, 1988) p. 192

    1988

  59. [67]

    A Fundamen- tal Property of Quantum-Mechanical Entropy,

    Elliott H. Lieb and Mary Beth Ruskai, “A Fundamen- tal Property of Quantum-Mechanical Entropy,” Les rencontres physiciens-math´ ematiciens de Strasbourg - RCP25 19, 1–2 (1973), talk:3

    1973

  60. [68]

    Infinitely many information inequal- ities,

    Frantisek Matus, “Infinitely many information inequal- ities,” in 2007 IEEE International Symposium on In- formation Theory (IEEE, 2007) pp. 41–44

    2007

  61. [69]

    The structure of r´ enyi entropic inequalities,

    Noah Linden, Mil´ an Mosonyi, and Andreas Winter, “The structure of r´ enyi entropic inequalities,” Proceed- ings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, 20120737 (2013)

    2013

  62. [70]

    Matrix product density operators: Renormal- ization fixed points and boundary theories,

    J. I. Cirac, D. P´ erez-Garc ´ ıa, N. Schuch, and F. Ver- straete, “Matrix product density operators: Renormal- ization fixed points and boundary theories,” Annals of Physics 378, 100–149 (2017), arXiv:1606.00608 [quant- ph]

    2017 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.