Joint product numerical range and geometry of reduced density matrices

Abstract

The reduced density matrices of a many-body quantum system form a convex set, whose threedimensional projection Θ is convex in R 3 . The boundary ∂Θ of Θ may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces: symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti's theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range Π of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that ruled surface emerge naturally when taking a convex hull of Π. We show that, a ruled surface on ∂Θ sitting in Π has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of Θ, with two boundary pieces of symmetry breaking origin separated by two gapless lines.

Geometrical characterization of reduced density matrices reveals quantum phase transitions in many-body systems

“…itself, the authors proposed to characterize geometrically the two-particle reduced density matrices (2-RDMs) of ( ) , N 0 which would reflect the sudden change of the ground state [2,3].…”

“…Characterizing QPTs, which normally needs complicated theoretical calculations, becomes a fundamental problem to further study quantum matters. Here a group of physicists proposed to connect the geometrical properties of reduced density matrices (RDMs) of the physical system with its quantum phase transitions [2,3]. For a many-body system (with N particles) described by a Hamiltonian H(λ) containing some set of parameters λ, the ground state ( ) N 0 may change suddenly while the parameter λ changes smoothly, leading to a QPT.…”

“…For a many-body system (with N particles) described by a Hamiltonian H(λ) containing some set of parameters λ, the ground state ( ) N 0 may change suddenly while the parameter λ changes smoothly, leading to a QPT. To make QPTs more visible, instead of analyzing ( )itself, the authors proposed to characterize geometrically the two-particle reduced density matrices (2-RDMs) of ( ) , N 0 which would reflect the sudden change of the ground state [2,3].Concretely, the RDMs of many-body quantum states form a convex set. The boundary of low dimensional projections of this convex set may exhibit nontrivial geometry such as ruled surfaces.…”

“…Characterizing QPTs, which normally needs complicated theoretical calculations, becomes a fundamental problem to further study quantum matters. Here a group of physicists proposed to connect the geometrical properties of reduced density matrices (RDMs) of the physical system with its quantum phase transitions [2,3].…”

“…For example, the three-dimensional projection Θ is convex in . 3 Ruled surfaces emerges from the boundary ∂Θ of Θ. Shown in Figure 1 are the ruled surfaces for the Ising model in the large N limit.…”