Generalised Entanglement Entropies from Unit-Invariant Singular Value Decomposition

TL;DR

This work addresses the problem that standard SVD-based entropies conflate genuine quantum correlations with artefacts of scale and unit choices. It introduces Unit-Invariant Singular Values (UISVD) in left, right, and bi variants (LUI, RUI, BUI) and defines corresponding entropies for states and transition matrices, ensuring covariance under diagonal scaling and unitaries. The authors derive explicit expressions, normalisations, and geometric interpretations, prove spectral convergence results for random matrices, and demonstrate invariances in diverse contexts, including Haar ensembles, Chern–Simons link states, and Biorthogonal Quantum Mechanics. TheUISVD framework yields stable, real, bounded entropic spectra that provide a robust, convention-free lens on quantum correlations, with potential applications in holography, topological field theories, and non-Hermitian quantum systems.

Abstract

We introduce generalisations of von Neumann entanglement entropy that are invariant with respect to certain scale transformations. These constructions are based on Unit-Invariant Singular Value Decomposition (UISVD) with its right-, left-, and bi-invariant incarnations, which itself are variations of the standard Singular Value Decomposition (SVD) that remain invariant under (appropriate set 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 behavior under scale transformations. Further, we illustrate features of UISVD-based entropies in other representative settings, 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.

Figures (9)

• Figure 1: Empirical densities for the invariants $\{\sigma_k^\mathrm{L}\}$, $\{\sigma_k^\mathrm{R}\}$, and $\{\sigma_k^{\mathrm{B}}\}$ with $n=200$. LUI and RUI follow the quarter-circle law of singular value distribution. BUI has dimension dependent stretched supports $\approx$[$0,2.67\sqrt{n}$] for GUE, and $\approx$[$0,3.77\sqrt{n}$] for GOE.
• Figure 2: Subsystem dimensions $d_\mathbb{A},d_\mathbb{B}=10$
• Figure 3: Subsystem dimensions $d_\mathbb{A},d_\mathbb{B}=40$
• Figure 5: Connected sum of left and right trefoil knots gives a "granny knot".
• Figure 6: Connected sum of the trefoil knot $3_1$ with $2\mathcal{N}^2_1$

Theorems & Definitions (38)

• Theorem 2.1: Quarter-circle law for LUI- and RUI-SVD
• proof
• Theorem 2.2: Stretched quarter-circle law for BUI-SVD
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem B.1: Vershynin2018HDP
• Theorem B.2: Marchenko-Pastur law BaiSilverstein2010