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arXiv:hep-th/0405111 · Physics Letters B

A finite entanglement entropy and the c-theorem

Horacio Casini, Marina Huerta

Published Sep 14, 2004• 420 citations
arXivHTMLPDF

Abstract

The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a mixed density matrix with nonzero entropy. This is usually called entanglement entropy, and it is known to be divergent in quantum field theory (QFT). However, it is possible to define a finite quantity F(A,B) for two given different subsets A and B which measures the degree of entanglement between their respective degrees of freedom. We show that the function F(A,B) is severely constrained by the Poincaré symmetry and the mathematical properties of the entropy. In particular, for one component sets in two-dimensional conformal field theories its general form is completely determined. Moreover, it allows to prove an alternative entropic version of the c-theorem for (1+1)-dimensional QFT. We propose this well-defined quantity as the meaningfull entanglement entropy and comment on possible applications in QFT and the black hole evaporation problem.

DOI
10.1016/j.physletb.2004.08.072

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The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a mixed density matrix with nonzero entropy. This is usually called entanglement entropy, and it is known to be divergent in quantum field theory (QFT). However, it is possible to define a finite quantity F(A,B) for two given different subsets A and B which measures the degree of entanglement between their respective degrees of freedom. We show that the function F(A,B) is severely constrained by the Poincaré symmetry and the mathematical properties of the entropy. In particular, for one component sets in two-dimensional conformal field theories its general form is completely determined. Moreover, it allows to prove an alternative entropic version of the c-theorem for (1+1)-dimensional QFT. We propose this well-defined quantity as the meaningfull entanglement entropy and comment on possible applications in QFT and the black hole evaporation problem.
We prove an entropic ccc-theorem in two dimensions by constructing a ccc-function from the mutual information between spacelike separated wedges, defined through the relative entropy. The proof follows Casini and Huerta's, with strong subadditivity of entanglement entropy replaced by increasing differences of the mutual information, which we prove directly in the continuum from the split property and standard properties of relative entropy. The benefit of relative entropy is that every quantity we use is finite and no cutoff ever needs to be introduced. The assumptions of the theorem are only the standard axioms of algebraic quantum field theory, the split property with finite mutual information across a gap (both implied by modular ppp-nuclearity), and a finite ultraviolet bound on the resulting ccc-functions. We also note that, formally, the ccc-function is proportional to the minimal averaged null energy of a state that reproduces the vacuum correlations between the two wedges.
The aim of this work is to compute the entanglement entropy of real and virtual particles by rewriting the generating functional of ${\ensuremath{\phi}}^{4}$ theory as a mean value between states and observables defined through the correlation functions. Then the von Neumann definition of entropy can be applied to these quantum states and in particular, for the partial traces taken over the internal or external degrees of freedom. This procedure can be done for each order in the perturbation expansion showing that the entanglement entropy for real and virtual particles behaves as ln(m0)\mathrm{ln}({m}_{0})ln(m0​). In particular, entanglement entropy is computed at first order for the correlation function of two external points showing that mutual information is identical to the external entropy and that conditional entropies are negative for all the domain of m0{m}_{0}m0​. In turn, from the definition of the quantum states, it is possible to obtain general relations between total traces between different quantum states of a ${\ensuremath{\phi}}^{r}$ theory. Finally, discussion about the possibility of taking partial traces over external degrees of freedom is considered, which implies the introduction of some observables that measure space-time points where an interaction occurs.
The degrees of freedom of any interacting quantum field theory are entangled in momentum space. Thus, in the vacuum state, the infrared degrees of freedom are described by a density matrix with an entanglement entropy. We derive a relation between this density matrix and the Wilsonian effective action obtained by integrating out degrees of freedom with spatial momentum above some scale. We argue that the entanglement entropy of and mutual information between subsets of field theoretic degrees of freedom at different momentum scales are natural observables in quantum field theory and demonstrate how to compute these in perturbation theory. The results may be understood heuristically based on the scale dependence of the coupling strength and number of degrees of freedom. We measure the rate at which entanglement between degrees of freedom declines as their scales separate and suggest that this decay is related to the property of decoupling in quantum field theory.
A bstract The entanglement entropy of spacetime regions A in odd-dimensional conformal field theories (CFTs) contains a universal constant term, (−1)d−12 F(A) {\left(-1\right)}^{\frac{d-1}{2}}\ F(A) (−1)2d−1​ F(A) − 1 d − 1 2 F A . This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of A . In the case of general three-dimensional CFTs, F ( A ) is positive definite and bounded below by the round disk result, F ( A ) ≥ F 0 ≡ F ( ∂A = 𝕊 1 ). Additionally, strong evidence has been provided that for every region A , F ( A ) /F 0 is maximized, within the space of CFT 3 ’s, by the free scalar field result. In this paper we show that while F ( A ) remains a local minimum around F 0 ≡ F ( ∂A = 𝕊 3 ) for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT 5 ’s. We argue that an analogous conjecture regarding the extremization of F ( A ) /F 0 for general regions within the space of theories fails in d = 5. We instead analyze the viability of the weaker bound, F ϵ /F 0 ≤ F ϵ /F 0 | free scalar , ∀CFT 5 for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function C T and the Euclidean partition function on the sphere, namely, C T /F 0 ≤ C T /F 0 | free scalar ≈ 0 . 314, which we show to hold for all known CFT 5 ’s. We also comment on possible extensions of this result to higher dimensions.
Black holes in general relativity carry an entropy whose value is given by the Bekenstein-Hawking formula, but whose statistical origin remains obscure. Such horizons also possess an entanglement entropy, which has a clear statistical meaning but no a priori relation to the dynamics of gravity. For free minimally-coupled scalar and spinor fields, these two quantities are intimately related: the entanglement entropy is the one-loop correction to the black hole entropy due to renormalization of Newton's constant. For gauge fields, the entanglement entropy and the one-loop correction to the black hole entropy differ. This dissertation addresses two issues concerning the entanglement entropy of gauge fields, and its relation black hole entropy. First, for abelian gauge fields Kabat identified a negative divergent contribution to the black hole entropy that is not part of the entanglement entropy, known as a ``contact term''. We show that the contact term can be attributed to an ambiguous expression for the gauge field's contribution to the Wald entropy. Moreover, in two-dimensional de Sitter space, the contact term arises from an incorrect treatment of zero modes and is therefore unphysical. In a manifestly gauge-invariant reduced phase space quantization of two-dimensional gauge theory, the gauge field contribution to the entropy is positive, finite, and equal to the entanglement entropy. This suggests that the contact term in more than two dimensions may also be unphysical. Second, we consider lattice gauge theory and point out that the Hilbert space corresponding to a region of space includes edge states that transform nontrivially under gauge transformations. By decomposing these edge states in irreducible representations of the gauge group, the entanglement entropy of an arbitrary state is shown to be a sum of a bulk entropy and a boundary entropy associated to the edge states. This entropy formula agrees with the two-dimensional results from the reduced phase space quantization. These results are applied to several examples, including the ground state in the strong coupling expansion of Kogut and Susskind, and the entropy of the edge states is found to be the dominant contribution to the entanglement entropy.