Von Neumann Entropy in QFT

Abstract

In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones O ⊂ O of the spacetime, where the closure of O is contained in O. Given a QFT net A of local von Neumann algebras A(O), we consider the von Neumann entropy S A (O, O) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras A(O) ⊂ A( O) (split property). We show that this canonical entanglement entropy S A (O, O ) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). To this end, we first study the notion of von Neumann entropy of a closed real linear subspace of a complex Hilbert space, that we then estimate for the local free fermion subspaces. We further consider the lower entanglement entropy S A (O, O), the infimum of the vacuum von Neumann entropy of F , where F here runs over all the intermediate, discrete type I von Neumann algebras. We prove that S A (O, O) is finite for the local chiral conformal net generated by finitely many commuting U (1)-currents.Von Neumann entropy is the basic concept in quantum information and extends the classical Shannon's information entropy notion to the non commutative setting.As is well known, a state ω on a matrix algebra M is given by a density matrix ρ, namely ω(T ) = Tr(ρT ), T ∈ M . The von Neumann entropy of ω is given byEntanglement entropy is a measure of quantum information by the degree of quantum entanglement of a quantum state.Let's consider a bipartite, finite-dimensional quantum system M = A ⊗ B, where A and B are matrix algebras. Given a pure state ω on the matrix algebra M , let ρ A and ρ B the density matrices associated with the restrictions of ω to A and B respectively on A and B. The entanglement entropy of ω is defined as(1)The above definition directly extends to the case M = k A k ⊗ B k is a direct sum, where A k , B k are matrix algebras and the restrictions ω| A k ⊗B k are pure (not normalised). It also extends to the infinite-dimensional case where A k , B k are type I factors; in this last case, however, the entanglement entropy may be infinite. Entanglement is certainly one of the main feature of quantum physics and there is a famous, long standing debate on its interpretation (EPR paradox, Bell's inequalities, etc.). An overview of the matter lies beyond the purpose of this introduction.The role of entanglement in Quantum Field Theory is more recent and increasingly important; it represents a piece of the quantum information framework in this subject. It appears in relation with several primary research topics in theoretical physics as area theorems, c-theorems, quantum null energy inequality, etc.

TT¯ , the entanglement wedge cross section, and the breakdown of the split property

“…This is why the reflected entropy is so useful for us in understanding if/how the split property holds in a given theory for a given split distance, s. For more comprehensive discussion on this point, we direct the reader to Refs. [54,75,77]. The split property has been shown to follow from the nuclearity condition which we describe now.…”

Reflected entropy, symmetries and free fermions

“…In [10] this was proven to be finite for free fermions in d = 2, and this is expected to be the case for most QFT models -see also [11][12][13].…”

“…In this expression J AB is the Tomita-Takesaki conjugation corresponding to the algebra AB and the state, and A ∨ B is the algebra generated by the two algebras A and B. This therefore defines a canonical von Neumann entropy [10], R(A, B) ≡ S(N AB ) .…”

Reflected entropy for free scalars

“…So far, it has not been rigorously proven that reflected entropy should be finite in general, 3 although it is believed to be so at least for most QFTs -see [9] and also [34][35][36]. This was proven to be the case for free fermions in (1 + 1) dimensions in [9] and confirmed later in [5], where we explicitly evaluated it for that theory as a function of the conformal cross ratio. The calculations in [31] also yield finite answers.…”

“…Also, J AB is the Tomita-Takesaki modular conjugation operator associated to the algebra of AB and the corresponding state. The von Neumann entropy associated to this type-I factor defines the reflected entropy [9] R(A, B) ≡ S(N AB ) .…”