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arXiv:1909.03144 · Journal of High Energy Physics

Rényi mutual information inequalities from Rindler positivity

David Blanco, Leandro Lanosa, Mauricio Leston, Guillem Pérez-Nadal

Published Dec 1, 2019• 4 citations
arXivHTMLPDF

Abstract

Abstract Rindler positivity is a property that holds in any relativistic Quantum Field Theory and implies an infinite set of inequalities involving the exponential of the Rényi mutual information I n (A i , $$ \overline{A} $$ A ¯ j ) between A i and $$ \overline{A} $$ A ¯ j , where A i is a spacelike region in the right Rindler wedge and $$ \overline{A} $$ A ¯ j is the wedge reflection of A j . We explore these inequalities in order to get local inequalities for I n (A, $$ \overline{A} $$ A ¯ ) as a function of the distance between A and its mirror region $$ \overline{A} $$ A ¯ . We show that the assumption, based on the cluster property of the vacuum, that I n goes to zero when the distance goes to infinity, implies the more stringent and simple condition that F n ≡ e (n−1)I n should be a completely monotonic function of the distance, meaning that all the even (odd) derivatives are non-negative (non-positive). In the case of a CFT, we show that conformal invariance implies stronger conditions, including a sort of monotonicity of the Rényi mutual information for pairs of balls. An application of these inequalities to obtain constraints for the OPE coefficients of the 4-point function of certain twist operators is also discussed.

DOI
10.1007/jhep12(2019)078

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Abstract Rindler positivity is a property that holds in any relativistic Quantum Field Theory and implies an infinite set of inequalities involving the exponential of the Rényi mutual information I n (A i , A‾\overline{A}A A ¯ j ) between A i and A‾\overline{A}A A ¯ j , where A i is a spacelike region in the right Rindler wedge and A‾\overline{A}A A ¯ j is the wedge reflection of A j . We explore these inequalities in order to get local inequalities for I n (A, A‾\overline{A}A A ¯ ) as a function of the distance between A and its mirror region A‾\overline{A}A A ¯ . We show that the assumption, based on the cluster property of the vacuum, that I n goes to zero when the distance goes to infinity, implies the more stringent and simple condition that F n ≡ e (n−1)I n should be a completely monotonic function of the distance, meaning that all the even (odd) derivatives are non-negative (non-positive). In the case of a CFT, we show that conformal invariance implies stronger conditions, including a sort of monotonicity of the Rényi mutual information for pairs of balls. An application of these inequalities to obtain constraints for the OPE coefficients of the 4-point function of certain twist operators is also discussed.
A bstract In this paper, we explore the concept of pseudo Rényi entropy within the context of quantum field theories (QFTs). The transition matrix is constructed by applying operators situated in different regions to the vacuum state. Specifically, when the operators are positioned in the left and right Rindler wedges respectively, we discover that the logarithmic term of the pseudo Rényi entropy is necessarily real. In other cases, the result might be complex. We provide direct evaluations of specific examples within 2-dimensional conformal field theories (CFTs). Furthermore, we establish a connection between these findings and the pseudo-Hermitian condition. Our analysis reveals that the reality or complexity of the logarithmic term of pseudo Rényi entropy can be explained through this pseudo-Hermitian framework. Additionally, we investigate the divergent term of the pseudo Rényi entropy. Interestingly, we observe a universal divergent term in the second pseudo Rényi entropy within 2-dimensional CFTs. This universal term is solely dependent on the conformal dimension of the operator under consideration. For n -th pseudo Rényi entropy ( n ≥ 3), the divergent term is intricately related to the specific details of the underlying theory.
We study a proper definition of Rényi mutual information (RMI) in quantum field theory as defined via the Petz Rényi relative entropy. Unlike the standard definition, the RMI we compute is a genuine measure of correlations between subsystems, as evidenced by its non-negativity and monotonicity under local operations. Furthermore, the RMI is UV finite and well defined in the continuum limit. We develop a replica path integral approach for the RMI in quantum field theories and evaluate it explicitly in 1+1D conformal field theory using twist fields. We prove that it bounds connected correlation functions and check our results against exact numerics in the massless free fermion theory.
Abstract The mutual information I(A, B) of pairs of spatially separated regions satisfies, for any d-dimensional CFT, a set of structural physical properties such as positivity, monotonicity, clustering, or Poincaré invariance, among others. If one imposes the extra requirement that I(A, B) is extensive as a function of its arguments (so that the tripartite information vanishes for any set of regions, I3(A, B, C ) ≡ 0), a closed geometric formula involving integrals over ∂A and ∂B can be obtained. We explore whether this “Extensive Mutual Information” model (EMI), which in fact describes a free fermion in d = 2, may similarly correspond to an actual CFT in general dimensions. Using the long-distance behavior of IEMI(A, B) we show that, if it did, it would necessarily include a free fermion, but also that additional operators would have to be present in the model. Remarkably, we find that IEMI(A, B) for two arbitrarily boosted spheres in general d exactly matches the result for the free fermion current conformal block GΔ=(d−1),J=1d {G}_{\Delta =\left(d-1\right),J=1}^d GΔ=(d−1),J=1d​ G ∆ = d − 1 , J = 1 d . On the other hand, a detailed analysis of the subleading contribution in the long-distance regime rules out the possibility that the EMI formula represents the mutual information of any actual CFT or even any limit of CFTs. These results make manifest the incompleteness of the set of known constraints required to describe the space of allowed entropy functions in QFT.
The conditional quantum mutual information I(A; B|C) of a tripartite state ρABC is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A; B|C) = I(A; B|D) for a four-party pure state on systems ABCD. The conditional mutual information also underlies the squashed entanglement, an entanglement measure that satisfies all of the axioms desired for an entanglement measure. As such, it has been an open question to find Rényi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different α-Rényi generalizations Iα(A; B|C) of the conditional mutual information, some of which we can prove converge to the conditional mutual information in the limit α → 1. Furthermore, we prove that many of these generalizations satisfy non-negativity, duality, and monotonicity with respect to local operations on one of the systems A or B (with it being left as an open question to prove that monotonicity holds with respect to local operations on both systems). The quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Rényi conditional mutual informations defined here with respect to the Rényi parameter α. We prove that this conjecture is true in some special cases and when α is in a neighborhood of one.
A bstract We explore a large class of correlation measures called the α − z Rényi mutual informations (RMIs). Unlike the commonly used notion of RMI involving linear combinations of Rényi entropies, the α − z RMIs are positive semi-definite and monotonically decreasing under local quantum operations, making them sensible measures of total (quantum and classical) correlations. This follows from their descendance from Rényi relative entropies. In addition to upper bounding connected correlation functions between subsystems, we prove the much stronger statement that for certain values of α and z , the α − z RMIs also lower bound certain connected correlation functions. We develop an easily implementable replica trick which enables us to compute the α − z RMIs in a variety of many-body systems including conformal field theories, free fermions, random tensor networks, and holography.
Abstract We derive the property of strong superadditivity of mutual information arising from the Markov property of the vacuum state in a conformal field theory and strong subadditivity of entanglement entropy. We show this inequality encodes unitarity bounds for different types of fields. These unitarity bounds are precisely the ones that saturate for free fields. This has a natural explanation in terms of the possibility of localizing algebras on null surfaces. A particular continuity property of mutual information characterizes free fields from the entropic point of view. We derive a general formula for the leading long distance term of the mutual information for regions of arbitrary shape which involves the modular flow of these regions. We obtain the general form of this leading term for two spheres with arbitrary orientations in spacetime, and for primary fields of any tensor representation. For free fields we further obtain the explicit form of the leading term for arbitrary regions with boundaries on null cones.