Topology of Quantum Gaussian States and Operations

TL;DR

This paper explores the topological properties of Gaussian states and operations in quantum systems, revealing new classifications and relationships that extend beyond Hamiltonian-based frameworks, with implications for quantum information and dynamical phases.

Contribution

It introduces a Hamiltonian-independent, operational framework for classifying topological Gaussian states and operations, uncovering nontrivial bosonic operations and complex fermionic relations.

Findings

Bosonic Gaussian states are all trivial.

Existence of nontrivial bosonic Gaussian operations.

Some fermionic states can be disentangled by certain operations.

Abstract

As is well-known in the context of topological insulators and superconductors, short-range-correlated fermionic pure Gaussian states with fundamental symmetries are systematically classified by the periodic table. We revisit this topic from a quantum-information-inspired operational perspective without referring to any Hamiltonians, and apply the formalism to bosonic Gaussian states as well as (both fermionic and bosonic) locality-preserving unitary Gaussian operations. We find that while bosonic Gaussian states are all trivial, there exist nontrivial bosonic Gaussian operations that cannot be continuously deformed into the identity under the locality and symmetry constraint. Moreover, we unveil unexpectedly complicated relations between fermionic Gaussian states and operations, pointing especially out that some of the former can be disentangled by the latter under the same symmetry constraint, while some cannot. In turn, we find that some topological operations are genuinely dynamical, in the sense that they cannot create any topological states from a trivial one, yet they are not connected to the identity. The notions of disentanglability and genuinely dynamical topology can be unified in a general picture of unitary-to-state homomorphism, and apply equally to interacting topological phases and quantum cellular automata.