UISVD Entropy and Invariant SVD Measures

• UISVD entropy is a family of gauge-invariant entanglement measures that use modified SVD methods to remain unaffected by diagonal rescalings and differing dimensions.
• It employs Left, Right, and Bi-invariant variants to derive invariant singular values, forming the basis for Rényi and von Neumann-type entropies.
• This approach robustly quantifies quantum correlations in non-Hermitian systems and rectangular matrices, advancing diagnostics in both quantum and information theory.

UISVD entropy comprises a family of scale-invariant entanglement measures derived by applying variations of Singular Value Decomposition (SVD) that are invariant under diagonal transformations, i.e., changes of units or local metric weights. These generalisations are well-defined for non-Hermitian or rectangular operators, enabling robust quantification of quantum correlations in systems where input and output spaces may differ in dimensionality or scaling. UISVD entropy measures remain unaffected by local rescalings that reflect arbitrary basis normalizations, establishing them as genuine, gauge-invariant diagnostics for entanglement and correlation in both physical and information-theoretic contexts (Caputa et al., 28 Dec 2025).

1. Fundamental Definition and UISVD Variants

Entanglement entropy traditionally relies on the singular values $\{\sigma_i\}$ produced by SVD of the coefficient matrix $A$ in the Schmidt decomposition of a pure bipartite state: $$|\psi\rangle = \sum_i \sigma_i |i\rangle_A \otimes |i\rangle_B,$$ where $\mathcal{H}_A$ and $\mathcal{H}_B$ are Hilbert spaces. The standard von Neumann entropy is invariant under unitaries but not under local diagonal (unit) rescalings of the form

$A \longmapsto D_A\,A\,D_B,$

for nonsingular diagonal matrices $D_A$, $D_B$ (Caputa et al., 28 Dec 2025).

Unit-Invariant Singular Value Decomposition (UISVD) addresses this by balancing the matrix $A$ so the singular values are invariant under specified diagonal rescalings:

• Left-Invariant UISVD (LUI-SVD): Uses diagonal $D^L$ (rows) so that $A^L = D^L A$, with singular values $\sigma^L(A)$. Invariant under $A \mapsto D A U$ for diagonal $D$, unitary $U$.
• Right-Invariant UISVD (RUI-SVD): Uses diagonal $D^R$ (columns) so that $A^R = A D^R$, with singular values $\sigma^R(A)$. Invariant under $A \mapsto U A D'$ for unitary $U$, diagonal $D'$.
• Bi-Invariant UISVD (BUI-SVD): Balances both sides: $A^B = D^{B_L} A D^{B_R}$, with geometric mean normalization per row and column. Singular values $\sigma^B(A)$ invariant under $A \mapsto D_1 A D_2$, diagonal $D_{1,2}$ (Caputa et al., 28 Dec 2025).

This structure produces singular values uniquely determined by the equivalence class under the relevant transformation group.

2. UISVD Entropy Measures: Formal Definitions

UISVD singular values enable construction of a family of Rényi and von Neumann-type entropies. For a matrix $A$, let $I\in\{L,R,B\}$ denote the invariance type and $\{\sigma^I_i\}$ the corresponding singular values. Define normalized weights: $$\hat{\sigma}^I_i = \frac{\sigma^I_i}{\sqrt{\sum_j (\sigma^I_j)^2}},$$ so that $\sum_i (\hat{\sigma}^I_i)^2 = 1$.

The order-$\alpha$ UISVD entropy is: $S_{\mathrm{UISVD}^{(\alpha)}^I}(A) = \frac{1}{1-\alpha} \ln\left( \sum_i \left( \hat{\sigma}^I_i \right)^{2\alpha} \right)$

Special cases:

• As $\alpha \to 1$ (von Neumann limit): $S_{\mathrm{UISVD}^{(1)}^I}(A) = -\sum_{i} (\hat{\sigma}^I_i)^2 \ln \left( (\hat{\sigma}^I_i)^2 \right)$
• As $\alpha \to \infty$ (min-entropy): $S_{\mathrm{UISVD}^{(\infty)}^I}(A) = -\ln \left( \max_i (\hat{\sigma}^I_i)^2 \right)$

All UISVD entropies are scale-invariant: $S^I(A) = S^I(D_1 A D_2)$ for appropriate $I$ and diagonal $D_1$, $D_2$ (Caputa et al., 28 Dec 2025).

3. Key Mathematical Properties

UISVD entropy enjoys several essential mathematical features:

• Scale Invariance: Entropy is unchanged under all diagonal rescalings of the input operator matching the selected invariance ($L$, $R$, or $B$).
• Adjoint/Transpose Robustness: The singular-value spectrum and therefore the entropy is invariant under Hermitian conjugation, especially for bi-invariant UISVD: $\sigma^B(A) = \sigma^B(A^\dagger)$.
• Convexity and Majorisation: For all $\alpha$, $S^I_{(\alpha)}$ is a nonincreasing function of $\alpha$, concave in the probability vector $(\hat{\sigma}^I_i)^2$, and satisfies the standard Rényi-entropy inequalities.

These properties render UISVD entropy robust and meaningful in settings where standard SVD-based entropies are ill-defined—such as non-Hermitian systems, metric-weighted spaces, or rectangular matrices.

4. Illustrative Examples and Applications

UISVD entropy is applicable across quantum physics, random matrix theory, and topological field theory:

• Simple Bipartite States: Under local rescalings on one subsystem, ordinary singular values alter while $\sigma^L(A)$ (or $\sigma^R(A)$) remain invariant. Bi-invariant entropy $S^B$ is fully insensitive to diagonal gauge changes on both subsystems.
• Biorthogonal Quantum Mechanics: Operators’ “reduced transition matrices” transform as $\tau_A \mapsto D^{-1}\tau_A D$. Using bi-invariant UISVD singular values, the resulting entropy

$$S^B(\tau_A) = -\sum_k (\hat{\sigma}_k^B)^2 \ln \left( (\hat{\sigma}_k^B)^2 \right)$$

is real, nonnegative, bounded, and invariant under all biorthonormal rescalings.

• Random Matrix Ensembles: For Ginibre (full-rank Gaussian) matrices, empirically $\sigma^L$ and $\sigma^R$ distributions converge to the quarter-circle law. BUI singular values yield a stretched quarter-circle form, with entropy $S^I$ exhibiting sub-Gaussian fluctuations in matrix size.
• Chern–Simons Link-Complement States: Link state manipulations correspond to diagonal actions; ordinary SVD entropies change, but LUI, RUI, and BUI entropies remain invariant under component sums or Dehn twist framings.

5. Physical and Information-Theoretic Significance

UISVD-derived entropies measure quantum and information-theoretic correlations free from artificial dependencies on basis scaling or units—purging gauge redundancies. They remain robust for open or non-Hermitian systems, in contrast to standard von Neumann entropy, which may be ill-defined when operators have non-positive spectra.

For tensor networks, multi-terminal transport, and gauge-fixing in computational models, arbitrary gauge freedoms reduce to diagonal transitivities; UISVD entropy thus provides concise, theoretically grounded, and gauge-invariant diagnostics of entanglement.

By generalising SVD to Left, Right, or Bi-Unit invariance and constructing entropic measures on these singular values, UISVD entropy fundamentally extends the theory of information quantification into domains with natural scale freedoms (Caputa et al., 28 Dec 2025).

6. Connection to Entropy-UID and Information Density

Recent work on balancing entropy and uniform information density in text generation models has shown that imposing joint constraints on global entropy and local surprisal yields smoother information flow and reduces local spikes while maintaining fluency and coherence (Shou, 20 Feb 2025). A plausible implication is that UISVD entropy could be further refined using joint regularisation schemes that penalise rapid changes in singular directions or spectrum variance over time intervals, thus promoting uniform latent subspace transitions.

Specifically, one may combine a UISVD-based entropy objective

$$H_{\mathrm{SVD}} = -\sum_i \tilde{\sigma}_i \log \tilde{\sigma}_i$$

with a penalty on abrupt spectral changes, enforcing smooth latent compression and information flow regularity. This analogy extends UISVD entropy’s utility from quantum frameworks to broader autoregressive or sequence modeling contexts (Shou, 20 Feb 2025).