Musings on SVD and pseudo entanglement entropies

TL;DR

The paper analyzes pseudo-entropy $S^{\phi|\psi}_{\mathrm{P}}$ and SVD entropy $S^{\phi|\psi}_{\mathrm{SVD}}$ as post-selection generalizations of entanglement entropy, introducing their excesses as potential distance-like measures between pre- and post-selected states. It applies these quantities to link complement states from Chern–Simons theory, deriving explicit formulae and large-$k$ asymptotics for diverse link families (U(1) and SU$(2)$ CS theories, torus and Borromean links), and demonstrates that the imaginary part of $S_{\mathrm{P}}$ can diagnose knot chirality in several cases. Across quantum-mechanical examples (SU$(2)$ and SU$(1,1)$ coherent states, GHZ and W states) and topological states, the work reveals conditions under which entropy excess behaves as a (pseudo-)metric and highlights when triangle inequalities are satisfied. The study uncovers universal structures in large-$k$ limits (e.g., zeta-function contributions) and suggests broader connections to quantum complexity and holographic notions, while outlining open questions about phase sensitivity and the operational meaning of the imaginary part. Overall, the paper provides a unified framework to quantify state differences via generalized entropies in both simple and topological quantum systems, with potential implications for knot theory and quantum information geometry.

Abstract

Pseudo-entropy and SVD entropy are generalizations of the entanglement entropy that involve post-selection. In this work we analyze their properties as measures on the spaces of quantum states and argue that their excess provides useful characterization of a difference between two (i.e. pre-selected and post-selected) states, which shares certain features and in certain cases can be identified as a metric. In particular, when applied to link complement states that are associated to topological links via Chern-Simons theory, these generalized entropies and their excess provide a novel quantification of a difference between corresponding links. We discuss the dependence of such entropy measures on the level of Chern-Simons theory and determine their asymptotic values for certain link states. We find that imaginary part of the pseudo-entropy is sensitive to, and can diagnose chirality of knots. We also consider properties of these entropy measures for simpler quantum mechanical systems, such as generalized SU(2) and SU(1,1) coherent states, and tripartite GHZ and W states.

Figures (20)

• Figure 1: Representative example of a generic torus link $\mathrm{T}(p,q)$ with $p=4$ and $q=12$.
• Figure 2: Examples of twist knots $\mathcal{K}_p$ for $p=0$ (unknot), 1 (trefoil) , $-1$ (figure-8), $2,-2,3,-3,4$ (in anti-clockwise order, starting from the unknot at the top).
• Figure 3: Links of the form $\mathcal{K}\#{2^{2}_{1}}$ for $\mathcal{K}=0_1$ (unknot)$, 3_1 \textrm{ (trefoil knot)}$ and $4_1$ (figure-8 knot) respectively.
• Figure 4: Entanglement entropy as a function of $j$ for $\theta=\pi/2$ (dots) vs asymptotic formula \ref{['AsymptotjEQ']} (solid blue curve).
• Figure 5: Pseudo-entropy and SVD entropy excess for $t_2=1$. Left: pseudo-entropy excess for $\phi_{12}=\pi/6$ (blue curve) and SVD excess (orange curve). Right: pseudo-entropy excess for $t_2=1$ and $t_1=0.4$.