Left-Right Relative Entropy

Mostafa Ghasemi

arxiv: 2411.09406 · v4 · pith:CSGNYT4Anew · submitted 2024-11-14 · ✦ hep-th · math-ph· math.MP· quant-ph

Left-Right Relative Entropy

Mostafa Ghasemi This is my paper

Pith reviewed 2026-05-23 17:50 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPquant-ph

keywords left-right relative entropyboundary statesconformal field theoryrelative entanglement sectorsmodular S-matrixKullback-Leibler divergenceNIM-representationst Hooft anomalies

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The pith

Left-right relative entropy vanishes between certain orthogonal boundary states in 2D CFTs, defining new equivalence classes called relative entanglement sectors.

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

The paper introduces left-right relative entropy as a distinguishability measure for boundary states in two-dimensional conformal field theories by tracing over left- or right-moving modes. The quantity reduces to a Kullback-Leibler divergence fixed entirely by the modular S-matrix and boundary data. In models such as the Ising, tricritical Ising, and su(2)_k WZW theories, this entropy is zero for some pairs of reduced states even when the full boundary states are orthogonal. This leads to relative entanglement sectors as equivalence classes of states that cannot be distinguished by the measure. The sectors transform according to NIM-representations of global symmetries and display structures tied to Z2 't Hooft anomalies.

Core claim

The left-right relative entropy between certain reduced boundary states vanishes even though the corresponding global boundary states are orthogonal. This observation motivates the introduction of relative entanglement sectors, defined as equivalence classes of boundary states that are indistinguishable with respect to left-right relative entropy. These sectors transform as NIM-representations of global symmetries and exhibit level-dependent structures that mirror Z2 't Hooft anomalies.

What carries the argument

left-right relative entropy, obtained by tracing over left- or right-moving modes of regularized boundary states on a circle and reducing to a Kullback-Leibler divergence fixed by the modular S-matrix and boundary data

If this is right

Exact expressions for the left-right relative entropy exist in diagonal CFTs in terms of modular data.
Left-right Renyi relative entropies and quantum fidelity follow from the same formalism.
Relative entanglement sectors transform as NIM-representations of global symmetries.
The sectors exhibit level-dependent structures that mirror Z2 't Hooft anomalies.
The construction links quantum information measures on boundary states to anomaly constraints in CFTs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The sectors could serve as a diagnostic for anomaly-protected phases when boundary states are realized in lattice or condensed-matter systems.
The framework may extend to nondiagonal CFTs or higher-dimensional analogs where modular data is replaced by other fusion rules.
Comparing left-right relative entropy to other boundary entanglement measures could reveal which information quantities are sensitive to the same anomaly structures.

Load-bearing premise

Tracing over left- or right-moving modes of regularized boundary states on a circle produces a valid probability distribution determined entirely by the modular S-matrix and boundary data.

What would settle it

An explicit computation in one of the studied models showing nonzero left-right relative entropy between two states placed in the same relative entanglement sector, or a mismatch between the derived probability distribution and the modular S-matrix entries.

read the original abstract

The concept of distinguishability lies at the heart of quantum information theory. We introduce \textit{left-right relative entropy} as a quantitative measure of distinguishability within the space of boundary states in two-dimensional conformal field theories (CFTs). By tracing over either the left- or right-moving modes, we derive a universal formula for arbitrary regularized boundary states defined on a circle. Remarkably, the resulting quantity reduces to a Kullback--Leibler divergence, where the associated probability distribution is determined entirely by the modular $\mathcal{S}$-matrix and the boundary data. For diagonal CFTs, we obtain exact expressions for the left-right relative entropy in terms of modular data, and extend the framework to define left-right R\'enyi relative entropies and quantum fidelity. Applying this formalism to the Ising model, tricritical Ising model, and $\widehat{su}(2)_k$ WZW model, we uncover a striking phenomenon: the left-right relative entropy between certain reduced boundary states vanishes even though the corresponding global boundary states are orthogonal. This observation motivates the introduction of \textit{relative entanglement sectors}, defined as equivalence classes of boundary states that are indistinguishable with respect to left-right relative entropy. These sectors transform as NIM-representations of global symmetries and exhibit level-dependent structures that mirror $\mathbb{Z}_2$ 't Hooft anomalies. Our findings establish an unexpected bridge between quantum information measures, boundary conformal symmetry, and quantum anomaly constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines left-right relative entropy from tracing boundary states in 2d CFTs, reduces it to KL divergence via the S-matrix, and uses vanishing cases to define relative entanglement sectors that track anomalies.

read the letter

The one thing to know is that this paper introduces left-right relative entropy as a distinguishability measure on reduced boundary states and shows it can vanish for orthogonal global states, which then defines new equivalence classes called relative entanglement sectors. These sectors are claimed to transform as NIM representations and mirror Z2 't Hooft anomalies in the examples given. That is the core new object and the main observation driving the rest of the work. The reduction to a Kullback-Leibler divergence whose probabilities come from the modular S-matrix and boundary coefficients is presented as the technical step that makes everything calculable. For diagonal CFTs the paper gives exact expressions in terms of modular data, and it works out the Ising, tricritical Ising, and su(2)_k cases explicitly. Those calculations are the concrete content that lets the vanishing and the level-dependent structures appear. The connection between a quantum-information quantity and anomaly constraints is the part that could interest people working on boundary states. The soft spot is the tracing step itself. The construction assumes that tracing left- or right-movers on the regularized circle states produces a normalized probability distribution fixed solely by the S-matrix and the boundary data, with no extra cutoff dependence. If that holds without state-dependent corrections, the KL interpretation and the sector classification follow directly. The abstract states it as universal, but the validity of the probability distribution is load-bearing for the vanishing results and the anomaly mirroring. The examples are presented as consequences of the formula, so the paper must verify the distribution property at least in those models. This work is aimed at readers who already use modular data and boundary CFT techniques and want to see an information-theoretic angle on anomaly constraints. Someone looking for new calculable quantities from the S-matrix would find the explicit formulas useful. It is coherent enough on its own terms to deserve a serious referee who can check the derivation of the probability distribution and the NIM-representation claim against the modular data. I would send it for review.

Referee Report

1 major / 0 minor

Summary. The paper introduces left-right relative entropy as a distinguishability measure for boundary states in 2d CFTs. Tracing over left- or right-moving modes of regularized boundary states on a circle yields a universal formula reducing to a Kullback-Leibler divergence whose probability distribution is fixed by the modular S-matrix and boundary data. Exact expressions are given for diagonal CFTs; applications to the Ising, tricritical Ising, and su(2)_k WZW models show vanishing left-right relative entropy between certain orthogonal global boundary states. This motivates relative entanglement sectors (equivalence classes under the entropy) that transform as NIM-representations of global symmetries and exhibit level-dependent structures mirroring Z2 't Hooft anomalies.

Significance. If the central derivations hold, the work supplies a new information-theoretic probe of boundary states and anomaly constraints in CFTs, potentially useful for classifying exotic phases. The reduction to an S-matrix-determined KL divergence and the exact diagonal-CFT formulas would be concrete strengths; the reported vanishing results in the three example models would then provide falsifiable predictions for the proposed sectors.

major comments (1)

[Abstract (paragraph on derivation of universal formula)] Abstract (paragraph on derivation of universal formula): the claim that tracing left- or right-movers on regularized circle boundary states produces a valid probability distribution determined entirely by the modular S-matrix and boundary coefficients is load-bearing. The manuscript must supply the explicit steps showing that the resulting weights are guaranteed positive and sum to one independently of cutoff details, cylinder mapping, or the specific Ishibashi decomposition; without this, the KL interpretation, the exact formulas, the vanishing entropy in the Ising/tricritical Ising/su(2)_k examples, and the NIM-representation structure for relative entanglement sectors all fail to follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for recognizing the potential of left-right relative entropy as a new probe of boundary states and anomalies. We address the single major comment below and will incorporate the requested clarifications.

read point-by-point responses

Referee: Abstract (paragraph on derivation of universal formula): the claim that tracing left- or right-movers on regularized circle boundary states produces a valid probability distribution determined entirely by the modular S-matrix and boundary coefficients is load-bearing. The manuscript must supply the explicit steps showing that the resulting weights are guaranteed positive and sum to one independently of cutoff details, cylinder mapping, or the specific Ishibashi decomposition; without this, the KL interpretation, the exact formulas, the vanishing entropy in the Ising/tricritical Ising/su(2)_k examples, and the NIM-representation structure for relative entanglement sectors all fail to follow.

Authors: We agree that an explicit, self-contained derivation of positivity and normalization is essential. While the manuscript sketches the reduction to the S-matrix-determined weights in Section 3 via the cylinder-to-circle mapping and the modular transformation of Ishibashi states, we acknowledge that the steps establishing cutoff independence and summation to unity are not written out in sufficient detail. In the revised version we will add a dedicated subsection (new Section 3.1) that proceeds as follows: (i) start from the regularized boundary state |B⟩_ε = ∑_i B_i exp(−ε Δ_i) |i⟩⟩; (ii) perform the partial trace over right-movers on the cylinder of length L, yielding a reduced density matrix whose eigenvalues are p_j = |∑_i B_i S_{ji}|^2 / Z, where Z is the cylinder partition function; (iii) invoke unitarity of S together with the Cardy consistency condition on the B_i to prove p_j ≥ 0; (iv) use the completeness relation ∑_j S_{ji} S^*_{jk} = δ_{ik} to show ∑_j p_j = 1 independently of ε and L in the limit ε → 0. The same algebra holds for the left-moving trace. These steps will be verified explicitly for the diagonal models used later in the paper. We believe this addition will fully secure the subsequent results without changing any conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external modular data

full rationale

The paper derives a universal formula for left-right relative entropy by tracing left- or right-movers on regularized boundary states, then states that the result reduces to a Kullback-Leibler divergence whose probability weights are fixed by the standard modular S-matrix and boundary coefficients (abstract and derivation paragraph). The S-matrix is an external CFT object, not defined or fitted inside the paper. Exact expressions for diagonal CFTs and applications to Ising/tricritical Ising/su(2)_k use known modular data and boundary coefficients as inputs; no parameter is fitted to a subset and then relabeled as a prediction, no self-citation chain justifies a uniqueness theorem, and no ansatz is smuggled via prior work by the same author. The introduction of relative entanglement sectors follows directly from the computed vanishing values rather than redefining the input. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on standard CFT modular data and the validity of the partial trace operation on boundary states; no free parameters are mentioned. The paper introduces two new entities without independent evidence outside the construction itself.

axioms (1)

domain assumption The modular S-matrix of a 2d CFT determines a probability distribution when combined with boundary data after tracing left or right modes
Invoked to obtain the universal formula reducing to KL divergence

invented entities (2)

left-right relative entropy no independent evidence
purpose: Quantitative measure of distinguishability for reduced boundary states
Newly defined quantity whose properties are derived in the paper
relative entanglement sectors no independent evidence
purpose: Equivalence classes of boundary states indistinguishable under the new entropy measure
Motivated by observed vanishing of the entropy for orthogonal states

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discussion (0)

Forward citations

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