Generalised Entanglement Entropies from Unit-Invariant Singular Value Decomposition

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.

53 pages, 3 tables, 7 figures

References in corpus (35)

• Holographic Derivation of Entanglement Entropy from AdS/CFT
• Entanglement Entropy and Quantum Field Theory
• String-net condensation: A physical mechanism for topological phases
• Building up spacetime with quantum entanglement
• Chaos in quantum channels
• Biorthogonal Quantum Mechanics
• Holographic Entanglement of Purification
• Entanglement spectrum and entropy in topological non-Hermitian systems and non-unitary conformal field theories
• Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1+1d
• Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in Weyl semimetals
• Holographic Pseudo Entropy
• Onsager Relations in Coupled Electric, Thermoelectric and Spin Transport: The Ten-Fold Way
• Revisiting the radio interferometer measurement equation. II. Calibration and direction-dependent effects
• Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
• Pseudo Entropy in dS/CFT and Time-like Entanglement Entropy
• Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models
• Timelike entanglement entropy
• Multi-terminal Thermoelectric Transport in a Magnetic Field: Bounds on Onsager Coefficients and Efficiency
• Hydrodynamic transport in strongly coupled disordered quantum field theories
• Holography in de Sitter Space via Chern-Simons Gauge Theory
• Aspects of Pseudo Entropy in Field Theories
• CFT duals of three-dimensional de Sitter gravity
• Multi-Boundary Entanglement in Chern-Simons Theory and Link Invariants
• Rényi entropies and negative central charges in non-Hermitian quantum systems
• Entanglement Entropy and the Colored Jones Polynomial
• Signature of quantum chaos in operator entanglement in 2d CFTs
• Topological pseudo entropy
• SVD Entanglement Entropy
• PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields
• Temporal Entanglement Entropy as a probe of Renormalization Group Flow
• Entropic trade-off relations for quantum operations
• Entanglement Hamiltonian and effective temperature of non-Hermitian quantum spin ladders
• Entangelment Entropy on Generalized Brillouin Zone
• Musings on SVD and pseudo entanglement entropies
• Complexity for link complement States in Chern Simons Theory