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arxiv: 2512.22997 · v3 · pith:FHHY26MInew · submitted 2025-12-28 · ✦ hep-th · math-ph· math.MP· math.PR· quant-ph

Generalised Entanglement Entropies from Unit-Invariant Singular Value Decomposition

Pawel Caputa , Abhigyan Saha , Piotr Su{\l}kowski This is my paper

Pith reviewed 2026-05-21 17:18 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPmath.PRquant-ph
keywords entanglement entropysingular value decompositionnon-Hermitian operatorsbiorthogonal quantum mechanicsChern-Simons theoryrandom matricesscale invariance
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The pith

Generalized entanglement entropies derived from unit-invariant singular value decomposition remain unchanged under rescalings of non-Hermitian or rectangular operators.

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

The paper introduces versions of entanglement entropy that stay fixed when the underlying operators undergo certain scale transformations or normalizations. These versions rest on a modified singular value decomposition that enforces invariance under left, right, or simultaneous diagonal rescalings. The constructions apply directly to operators that are non-Hermitian or rectangular, situations in which standard von Neumann entropy can shift with arbitrary choices of basis weights or dimensions. The authors test the resulting spectra in biorthogonal quantum mechanics, where scale behavior is central, and extend the checks to simple bipartite states, random-matrix ensembles tied to quantum chaos, and Chern-Simons theory. In each setting the new entropies produce stable, physically interpretable values that do not depend on the chosen normalization.

Core claim

By replacing the ordinary singular value decomposition with its unit-invariant counterparts, one obtains entanglement entropies whose spectra are unchanged under left-unitary, right-unitary, or bi-unitary diagonal transformations. These entropies are therefore well-defined for non-Hermitian matrices and for rectangular operators connecting spaces of unequal dimension or unequal metric weighting. The invariance removes the dependence on arbitrary rescalings that otherwise affects the standard von Neumann entropy, yielding consistent entropic measures across the tested physical frameworks.

What carries the argument

The unit-invariant singular value decomposition, a modification of the standard SVD that remains unchanged under left, right, or bi-diagonal scale transformations and supplies the singular values used to define the generalized entropies.

If this is right

  • In biorthogonal quantum mechanics the entropies remain consistent when states or operators are rescaled.
  • Random-matrix ensembles relevant to quantum chaos and holography produce normalization-independent entropic spectra.
  • Chern-Simons theory yields stable entanglement measures under the same invariance conditions.
  • The construction extends naturally to any rectangular operator whose input and output spaces carry different dimensions or metric weights.

Where Pith is reading between the lines

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

  • The same invariance property may remove normalization ambiguities that appear in entropy calculations for open or dissipative quantum systems.
  • Direct comparison with laboratory realizations of non-Hermitian Hamiltonians could test whether the UISVD spectra match observed information measures.
  • The approach may link to other scale-invariant information quantities already studied in holographic models.

Load-bearing premise

The unit-invariant modifications of the singular value decomposition produce entropies that possess genuine physical meaning and practical advantages over ordinary von Neumann entropy when the operators are non-Hermitian or rectangular.

What would settle it

An explicit calculation on a simple non-Hermitian rectangular matrix that shows the proposed entropy value changing after a left or right diagonal rescaling, or a concrete biorthogonal model in which the new entropy contradicts an independently computed physical observable.

read the original abstract

We introduce generalisations of von Neumann entanglement entropy that are invariant with respect to certain scale transformations. These constructions are based on the Unit-Invariant Singular Value Decomposition (UISVD) in its left-, right-, and bi-invariant incarnations, which are variations of the standard Singular Value Decomposition (SVD) that remain invariant under the corresponding class of diagonal transformations. These measures are naturally defined for non-Hermitian or rectangular operators and remain useful when the input and output spaces possess different dimensions or metric weights. We apply the UISVD entropy and discuss its advantages in the physically interesting framework of Biorthogonal Quantum Mechanics, whose important aspect is indeed the behaviour under scale transformations. Further, we illustrate features of UISVD-based entropies in other well-known setups, from simple quantum mechanical bipartite states to random matrices relevant to quantum chaos and holography, and in the context of Chern-Simons theory. In all cases, the UISVD yields stable, physically meaningful entropic spectra that are invariant under rescalings and normalisations.

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.

Referee Report

2 major / 1 minor

Summary. The paper introduces generalisations of von Neumann entanglement entropy based on Unit-Invariant Singular Value Decomposition (UISVD) in left-, right-, and bi-invariant forms. These constructions modify standard SVD to ensure invariance under corresponding classes of diagonal rescalings and are applied to non-Hermitian or rectangular operators. The authors claim that the resulting entropies are stable and physically meaningful when used in biorthogonal quantum mechanics, random-matrix models of quantum chaos and holography, and Chern-Simons theory, remaining useful when input and output spaces have different dimensions or metric weights.

Significance. If the UISVD entropies can be shown to correspond to physically redundant rescalings while preserving an interpretation as measures of correlation or information, the construction would offer a practical extension of entanglement entropy to non-Hermitian and rectangular settings that appear in open quantum systems and certain holographic models. The invariance properties address a concrete technical issue in biorthogonal frameworks, and the applications to random matrices and Chern-Simons suggest potential utility for chaos and topological field theory contexts.

major comments (2)
  1. [biorthogonal QM discussion] The manuscript demonstrates mathematical invariance of the UISVD singular values under left/right diagonal rescalings but does not derive why these rescalings represent physically redundant degrees of freedom in the target models. In the biorthogonal quantum mechanics application, the necessity of the invariance for physical equivalence is asserted rather than obtained from the underlying dynamics or inner-product structure.
  2. [applications sections] No explicit derivations, error analysis, or quantitative comparisons against independent physical observables or known limits (e.g., Hermitian reduction or large-N limits) are supplied for the claimed stability and physical meaningfulness of the entropic spectra. The central claim that UISVD yields advantages over standard von Neumann entropy therefore rests on unshown technical details.
minor comments (1)
  1. [UISVD definition] The notation distinguishing the left-, right-, and bi-invariant UISVD variants should be introduced with explicit matrix definitions at the beginning of the technical construction section to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major points below and will incorporate revisions to provide explicit derivations of the physical redundancy in biorthogonal quantum mechanics and to supply quantitative comparisons and error analysis in the applications sections.

read point-by-point responses

  1. Referee: [biorthogonal QM discussion] The manuscript demonstrates mathematical invariance of the UISVD singular values under left/right diagonal rescalings but does not derive why these rescalings represent physically redundant degrees of freedom in the target models. In the biorthogonal quantum mechanics application, the necessity of the invariance for physical equivalence is asserted rather than obtained from the underlying dynamics or inner-product structure.

    Authors: We thank the referee for highlighting this distinction. The manuscript establishes the mathematical invariance under left- and right-diagonal rescalings via the UISVD construction. In biorthogonal quantum mechanics the left and right eigenvectors are defined only up to independent rescalings that preserve the biorthogonal relation with respect to the non-Hermitian inner product; these overall scales do not affect any physical observables or the dynamics generated by the non-Hermitian Hamiltonian. The UISVD entropy is therefore invariant precisely under these redundant choices. We will revise the relevant section to derive this redundancy explicitly from the inner-product structure and the underlying non-Hermitian Schrödinger equation, rather than asserting it. revision: yes

  2. Referee: [applications sections] No explicit derivations, error analysis, or quantitative comparisons against independent physical observables or known limits (e.g., Hermitian reduction or large-N limits) are supplied for the claimed stability and physical meaningfulness of the entropic spectra. The central claim that UISVD yields advantages over standard von Neumann entropy therefore rests on unshown technical details.

    Authors: We agree that additional explicit support is needed. In the revised manuscript we will add (i) a direct derivation showing that UISVD reduces to ordinary SVD (and hence to standard von Neumann entropy) when the operator is Hermitian, (ii) numerical comparisons in the random-matrix ensembles that quantify stability under rescaling and convergence in the large-N limit to known results from quantum-chaos literature, and (iii) an error analysis together with comparison to independent topological invariants in the Chern-Simons application. These additions will make the claimed advantages over standard von Neumann entropy concrete and verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the UISVD-based entropy definitions

full rationale

The paper defines the Unit-Invariant Singular Value Decomposition (UISVD) in its left-, right-, and bi-invariant forms as a direct mathematical modification of standard SVD that enforces invariance under specified diagonal rescalings. Generalized entanglement entropies are then constructed explicitly from the resulting singular values. These definitions are applied to concrete examples in biorthogonal quantum mechanics, random matrices, and Chern-Simons theory to illustrate stability and invariance properties. No load-bearing step reduces by construction to a fitted parameter renamed as a prediction, a self-citation chain, or an ansatz smuggled through prior work; the invariance follows immediately from the algebraic construction of UISVD itself, and physical interpretations are presented as consequences of the definitions rather than presupposed inputs. The derivation chain is therefore self-contained and definitional.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the mathematical definition of UISVD variants and the assumption that they yield physically meaningful entropies; no free parameters or new entities with independent evidence are mentioned.

axioms (1)
  • standard math Standard singular value decomposition properties extend to the unit-invariant variants under diagonal transformations.
    The paper modifies SVD for invariance but relies on its core decomposition properties.
invented entities (1)
  • Unit-Invariant Singular Value Decomposition (UISVD) in left-, right-, and bi-invariant forms no independent evidence
    purpose: To construct scale-invariant entanglement entropies for non-Hermitian and rectangular operators.
    Newly introduced variations of SVD presented as the basis for the generalized entropies.

pith-pipeline@v0.9.0 · 5727 in / 1278 out tokens · 59853 ms · 2026-05-21T17:18:43.680531+00:00 · methodology

discussion (0)

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