A group-theoretic approach to quantum entanglement

Abstract

We develop a unified framework connecting stabilizer-group methods, modular Hamiltonian, and entanglement measures for the characterization of multipartite quantum correlations. Building on the stabilizer group formalism, we review how bipartite and genuine tripartite entanglement can be identified through quotient constructions of local unitary stabilizer groups, and provide an intuitive set-theoretic interpretation of these structures. For a tripartite system, we introduce the modular mutual Hamiltonian, an operator-valued analog of mutual information, and prove the exact relation between its expectation value and the quantum mutual information. We show that reduced density matrices and modular Hamiltonian are invariant under stabilizer symmetries, and we establish that projected stabilizers commute with the modular mutual Hamiltonian. This leads to a modular characterization of factorizable states and demonstrates the rigidity of the stabilizer structure under modular flow. We further analyze the action of factorizable and non-factorizable unitary transformations, deriving perturbative corrections to entanglement. Motivated by the group-theoretic decomposition of multipartite correlations, we propose a residual quantity based on Renyi entropy and logarithmic negativity as a candidate measure of genuine tripartite entanglement. This measure vanishes for biseparable pure states and captures GHZ-type correlations. Together, these results provide a unified algebraic and information-theoretic description of multipartite entanglement, revealing deep connections between stabilizer symmetries, modular dynamics, and genuine quantum correlations, with applications to quantum information theory, quantum error correction, and quantum field theory.