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arXiv:2411.09696 · Letters in Mathematical Physics

Petz–Rényi relative entropy in QFT from modular theory

Markus B. Fröb, Leonardo Sangaletti

Published Mar 17, 2025• 4 citations
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Abstract

Abstract We consider the generalization of the Araki–Uhlmann formula for relative entropy to Petz–Rényi relative entropy. We compute this entropy for a free scalar field in the Minkowski wedge between the vacuum and a coherent state, as well as for the free chiral current in a thermal state. In contrast to the relative entropy which in these cases only depends on the symplectic form and thus reduces to the classical entropy of a wave packet, the Petz–Rényi relative entropy also depends on the symmetric part of the two-point function and is thus genuinely quantum. We also consider the relation with standard subspaces, where we define the Rényi entropy of a vector and show that it admits an upper bound given by the entropy of the vector.

DOI
10.1007/s11005-025-01923-2

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Abstract We consider the generalization of the Araki–Uhlmann formula for relative entropy to Petz–Rényi relative entropy. We compute this entropy for a free scalar field in the Minkowski wedge between the vacuum and a coherent state, as well as for the free chiral current in a thermal state. In contrast to the relative entropy which in these cases only depends on the symplectic form and thus reduces to the classical entropy of a wave packet, the Petz–Rényi relative entropy also depends on the symmetric part of the two-point function and is thus genuinely quantum. We also consider the relation with standard subspaces, where we define the Rényi entropy of a vector and show that it admits an upper bound given by the entropy of the vector.
We numerically investigate the Araki-Uhlmann relative entropy in Quantum Field Theory, focusing on a free massive scalar field in 1+1-dimensional Minkowski spacetime. Using Tomita-Takesaki modular theory, we analyze the relative entropy between a coherent state and the vacuum state, with several types of test functions localized in the right Rindler wedge. Our results confirm that relative entropy decreases with increasing mass and grows with the size of the spacetime region, aligning with theoretical expectations.
Abstract Utilizing the Tomita–Takesaki modular theory, we derive a closed-form analytic expression for the Araki–Uhlmann relative entropy between a single-mode squeezed state and the vacuum state in a free relativistic massive scalar Quantum Field Theory within wedge regions of Minkowski spacetime. Similarly to the case of coherent states, this relative entropy is proportional to the smeared Pauli–Jordan distribution. Consequently, the Araki–Uhlmann entropy between a single-mode squeezed state and the vacuum satisfies all expected properties: it remains positive, increases with the size of the Minkowski region under consideration, and decreases as the mass parameter grows.
We numerically investigate the Araki-Uhlmann relative entropy in Quantum Field Theory, focusing on a free massive scalar field in 1 + 1 -dimensional Minkowski spacetime. Using Tomita-Takesaki modular theory, we analyze the relative entropy between a coherent state and the vacuum state, with several types of test functions localized in the right Rindler wedge. Our results confirm that relative entropy decreases with increasing mass and grows with the size of the spacetime region, aligning with theoretical expectations.
We make a rigorous computation of the relative entropy between the vacuum state and a coherent state for a free scalar in the framework of algebraic description of quantum field theory (AQFT). We study the case of the Rindler wedge. Previous calculations including path integral methods and computations from the lattice give a result for such relative entropy which involves integrals of expectation values of the energy-momentum stress tensor along the considered region. However, the stress tensor is in general nonunique. That means that if we start with some stress tensor, then we can ``improve'' it adding a conserved term without modifying the Poincar\'e charges. On the other hand, the presence of such an improving term affects the naive expectation for the relative entropy by a nonvanishing boundary contribution along the entangling surface. In other words, this means that there is an ambiguity in the usual formula for the relative entropy coming from the nonuniqueness of the stress tensor. The main motivation of this work is to solve this puzzle. We first show that all choices of stress tensor except the canonical one are not allowed by positivity and monotonicity of the relative entropy. Then we fully compute the relative entropy between the vacuum and a coherent state in the framework of AQFT using the Araki formula and the techniques of modular theory. After all, both results coincide and give the usual expression for the relative entropy calculated with the canonical stress tensor.
The quantum Rényi relative entropies play a prominent role in quantum information theory, finding applications in characterizing error exponents and strong converse exponents for quantum hypothesis testing and quantum communication theory. On a different thread, quantum Gaussian states have been intensely investigated theoretically, motivated by the fact that they are more readily accessible in the laboratory than are other, more exotic quantum states. In this paper, we derive formulas for the quantum Rényi relative entropies of quantum Gaussian states. We consider both the traditional (Petz) Rényi relative entropy as well as the more recent sandwiched Rényi relative entropy, finding formulas that are expressed solely in terms of the mean vectors and covariance matrices of the underlying quantum Gaussian states. Our development handles the hitherto elusive case for the Petz–Rényi relative entropy when the Rényi parameter is larger than one. Finally, we also derive a formula for the max-relative entropy of two quantum Gaussian states, and we discuss some applications of the formulas derived here.