Entropy Distribution of Localised States

Abstract

We study the geometric distribution of the relative entropy of a charged localised state in Quantum Field Theory. With respect to translations, the second derivative of the vacuum relative entropy is zero out of the charge localisation support and positive in mean over the support of any single charge. For a spatial strip, the asymptotic mean entropy density is πE, with E the corresponding vacuum charge energy. In a conformal QFT, for a charge in a ball of radius r, the relative entropy is non linear, the asymptotic mean radial entropy density is πE and Bekenstein's bound is satisfied. We also study the null deformation case. We construct, operator algebraically, a positive selfadjoint operator that may be interpreted as the deformation generator, we thus get a rigorous form of the Averaged Null Energy Condition that holds in full generality. In the one dimensional conformal U (1)-current model, we give a complete and explicit description of the entropy distribution of a localised charged state in all points of the real line; in particular, the second derivative of the relative entropy is strictly positive in all points where the charge density is non zero, thus the Quantum Null Energy Condition holds here for these states and is not saturated in these points. * Supported by the ERC Advanced Grant 669240 QUEST "Quantum Algebraic Structures and Models", MIUR FARE R16X5RB55W QUEST-NET and GNAMPA-INdAM. where ϕ and ψ are restricted to A(W ), E loc = (ξ, K ρ,W ξ) is the charge local energy, with K ρ,W the Rindler modular Hamiltonian in presence of the charge ρ, andis half of the conditional entropy of ρ, independent of ψ (see Sect. 3.1.2); d(ρ) is the DHR statistical dimension [17], which is equal the square root of the Jones index of ρ [33,34]. We shall also have a corresponding formula for S(ψ||ϕ).Formula (1) is the key to study the dependence of S(ϕ||ψ) on W . In particular, if W t is the shifted wedge by a null translation by t ≥ 0, ρ 1 , . . . ρ n are charges localised in W and S(t) is the relative entropy between ϕ| A(Wt) and ψ| A(Wt) we have S ′′ (t) = 0 , 1 The convexity of S is not an intrinsic concept, as it depends on re-parameterizing t. However, in Section 3.2.2, we shall have a natural deformation parameter t, the half-sided modular translation parameter.

Relative entropy for coherent states from Araki formula

“…In the Lagrangian approach to QFT, this is the usual symmetry φ (x) → −φ (x) 12. This result has been found in the past using other methods.…”

“…which is much better behaved than the entropy (see for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14] for actual calculations). In fact, the relative entropy has an expression directly in the continuous theory for type III algebras in terms of the Araki formula [15].…”

Entanglement entropy and superselection sectors. Part I. Global symmetries

“…Several results about the entropy of charged states analogous to the ones in this section have previously appeared in the literature for specific models. See for example [53][54][55][56][57][58][59].…”

Entropy of coherent excitations

“…We do not attempt here to give an even schematic sketch of the past and actual research on the subject, we rather refer to [12], and reference therein, for a rigorous approach to the subject. Indeed, this paper is a continuation of the work done in [12], especially in Sect. 4, where a first and detailed analysis of the vacuum relative entropy of a localised state has been provided for the chiral, conformal net of von Neumann algebras associated with U (1)-current.…”

“…dx d is the space volume. The above formula can be expressed also as a space integration detecting boundary entropy contributions (Lemma 5.3) that are not visible in the one-dimensional chiral case [12].…”