We study the question of how reliably one can distinguish two quantum field theories (QFTs). Each QFT defines a probability distribution on the space of fields. The relative entropy provides a notion of proximity between these distributions and quantifies the number of measurements required to distinguish between them. In the case of nearby conformal field theories, this reduces to the Zamolodchikov metric on the space of couplings. Our formulation quantifies the information lost under renormalization group flow from the UV to the IR and leads us to a quantification of fine-tuning. This formalism also leads us to a criterion for distinguishability of low energy effective field theories generated by the string theory landscape.We study the question of how reliably one can distinguish two quantum field theories (QFTs). Each QFT defines a probability distribution on the space of fields. The relative entropy provides a notion of proximity between these distributions and quantifies the number of measurements required to distinguish between them. In the case of nearby conformal field theories, this reduces to the Zamolodchikov metric on the space of couplings. Our formulation quantifies the information lost under renormalization group flow from the UV to the IR and leads us to a quantification of fine-tuning. This formalism also leads us to a criterion for distinguishability of low energy effective field theories generated by the string theory landscape.

arXiv:1410.6809Oct 24, 20141 citation

Defect Relative Entropy

Mostafa Ghasemi

Distinguishability is central to quantum information theory, but a quantitative measure for distinguishing topological defects--realizations of generalized symmetries in quantum field theory (QFT)--has been lacking. We introduce the notion of the \textit{defect relative entropy} to fill this gap for topological defects in two-dimensional conformal field theories (CFTs). For defects on a circle, we derive a universal formula that reduces defect relative entropy to a Kullback--Leibler divergence determined entirely by the modular $\mathcal{S}$ -matrix and defect coefficients. Thus, the algebraic data governing modular transformations also determines defect distinguishability. A striking consequence is that certain distinct defects can have vanishing relative entropy when restricted to one side, implying that an observer confined to that side cannot distinguish them. This gives rise to information-theoretic equivalence classes of defects, which we term \textit{defect relative sectors}. We further introduce the sandwiched defect Rényi relative entropy and defect fidelity, derive general formulas for these quantities. Explicit calculations in the Ising model, tricritical Ising model, and $\widehat{su}(2)_k$ WZW models illustrate our results.Distinguishability is central to quantum information theory, but a quantitative measure for distinguishing topological defects--realizations of generalized symmetries in quantum field theory (QFT)--has been lacking. We introduce the notion of the \textit{defect relative entropy} to fill this gap for topological defects in two-dimensional conformal field theories (CFTs). For defects on a circle, we derive a universal formula that reduces defect relative entropy to a Kullback--Leibler divergence determined entirely by the modular $\mathcal{S}$-matrix and defect coefficients. Thus, the algebraic data governing modular transformations also determines defect distinguishability. A striking consequence is that certain distinct defects can have vanishing relative entropy when restricted to one side, implying that an observer confined to that side cannot distinguish them. This gives rise to information-theoretic equivalence classes of defects, which we term \textit{defect relative sectors}. We further introduce the sandwiched defect Rényi relative entropy and defect fidelity, derive general formulas for these quantities. Explicit calculations in the Ising model, tricritical Ising model, and $\widehat{su}(2)_k$ WZW models illustrate our results.

hep-thCross-listedarXiv:2601.21875Jan 30, 20260 citations

Relative entropy and proximity of quantum field theories

Alexander Maloney, Jonathan J. Heckman, Vijay Balasubramanian

Jul 2, 20210 citations

The g-theorem and quantum information theory

Horacio Casini, Ignacio Salazar Landea, Gonzalo Torroba

We study boundary renormalization group flows between boundary conformal field theories in 1 + 1 dimensions using methods of quantum information theory. We define an entropic g-function for theories with impurities in terms of the relative entanglement entropy, and we prove that this g-function decreases along boundary renormalization group flows. This entropic g-theorem is valid at zero temperature, and is independent from the g-theorem based on the thermal partition function. We also discuss the mutual information in boundary RG flows, and how it encodes the correlations between the impurity and bulk degrees of freedom. Our results provide a quantum-information understanding of (boundary) RG flow as increase of distinguishability between the UV fixed point and the theory along the RG flow.We study boundary renormalization group flows between boundary conformal field theories in 1 + 1 dimensions using methods of quantum information theory. We define an entropic g-function for theories with impurities in terms of the relative entanglement entropy, and we prove that this g-function decreases along boundary renormalization group flows. This entropic g-theorem is valid at zero temperature, and is independent from the g-theorem based on the thermal partition function. We also discuss the mutual information in boundary RG flows, and how it encodes the correlations between the impurity and bulk degrees of freedom. Our results provide a quantum-information understanding of (boundary) RG flow as increase of distinguishability between the UV fixed point and the theory along the RG flow.

arXiv:1607.00390Journal of High Energy PhysicsOct 1, 2016137 citations

Left-Right Relative Entropy

Mostafa Ghasemi

The concept of distinguishability lies at the heart of quantum information theory. We introduce \textit{left-right relative entropy} as a quantitative measure of distinguishability within the space of boundary states in two-dimensional conformal field theories (CFTs). By tracing over either the left- or right-moving modes, we derive a universal formula for arbitrary regularized boundary states defined on a circle. Remarkably, the resulting quantity reduces to a Kullback--Leibler divergence, where the associated probability distribution is determined entirely by the modular $\mathcal{S}$ -matrix and the boundary data. For diagonal CFTs, we obtain exact expressions for the left-right relative entropy in terms of modular data, and extend the framework to define \textit{Sandwiched Left-Right Rényi relative entropies} and \textit{left-right fidelity}. Applying this formalism to the Ising model, tricritical Ising model, and $\widehat{su}(2)_k$ WZW model, we uncover a striking phenomenon: the left-right relative entropy between certain reduced boundary states vanishes even though the corresponding global boundary states are orthogonal. This observation motivates the introduction of \textit{relative entanglement sectors}, defined as equivalence classes of boundary states that are indistinguishable with respect to left-right relative entropy. These sectors transform as NIM-representations of global symmetries and exhibit level-dependent structures that mirror $\mathbb{Z}_2$ 't Hooft anomalies. Our findings establish an unexpected bridge between quantum information measures, boundary conformal symmetry, and quantum anomaly constraints.The concept of distinguishability lies at the heart of quantum information theory. We introduce \textit{left-right relative entropy} as a quantitative measure of distinguishability within the space of boundary states in two-dimensional conformal field theories (CFTs). By tracing over either the left- or right-moving modes, we derive a universal formula for arbitrary regularized boundary states defined on a circle. Remarkably, the resulting quantity reduces to a Kullback--Leibler divergence, where the associated probability distribution is determined entirely by the modular $\mathcal{S}$-matrix and the boundary data. For diagonal CFTs, we obtain exact expressions for the left-right relative entropy in terms of modular data, and extend the framework to define \textit{Sandwiched Left-Right Rényi relative entropies} and \textit{left-right fidelity}. Applying this formalism to the Ising model, tricritical Ising model, and $\widehat{su}(2)_k$ WZW model, we uncover a striking phenomenon: the left-right relative entropy between certain reduced boundary states vanishes even though the corresponding global boundary states are orthogonal. This observation motivates the introduction of \textit{relative entanglement sectors}, defined as equivalence classes of boundary states that are indistinguishable with respect to left-right relative entropy. These sectors transform as NIM-representations of global symmetries and exhibit level-dependent structures that mirror $\mathbb{Z}_2$ 't Hooft anomalies. Our findings establish an unexpected bridge between quantum information measures, boundary conformal symmetry, and quantum anomaly constraints.

arXiv:2411.09406Nov 14, 20240 citations

Quantum information measures and their applications in quantum field theory

David Blanco

In the last decades, it has been understood that a wide variety of phenomena in quantum field theory (QFT) can be characterised using quantum information measures, such as the entanglement entropy of a state and the relative entropy between quantum states in the same Hilbert space. In this thesis, we use these and other tools from quantum information theory to study several interesting problems in quantum field theory. The topics analysed range from the study of the Aharonov-Bohm effect in QFT using entanglement entropy, to the consistence of the Ryu-Takayanagi formula (proposed in the context of the AdS/CFT duality) using properties of relative entropy. We show that relative entropy can also be used to obtain new interesting quantum energy inequalities, that constrain the spatial distribution of negative energy density.In the last decades, it has been understood that a wide variety of phenomena in quantum field theory (QFT) can be characterised using quantum information measures, such as the entanglement entropy of a state and the relative entropy between quantum states in the same Hilbert space. In this thesis, we use these and other tools from quantum information theory to study several interesting problems in quantum field theory. The topics analysed range from the study of the Aharonov-Bohm effect in QFT using entanglement entropy, to the consistence of the Ryu-Takayanagi formula (proposed in the context of the AdS/CFT duality) using properties of relative entropy. We show that relative entropy can also be used to obtain new interesting quantum energy inequalities, that constrain the spatial distribution of negative energy density.

arXiv:1702.07384Feb 23, 20170 citations

\mathcal{S}

-matrix and defect coefficients. Thus, the algebraic data governing modular transformations also determines defect distinguishability. A striking consequence is that certain distinct defects can have vanishing relative entropy when restricted to one side, implying that an observer confined to that side cannot distinguish them. This gives rise to information-theoretic equivalence classes of defects, which we term \textit{defect relative sectors}. We further introduce the sandwiched defect Rényi relative entropy and defect fidelity, derive general formulas for these quantities. Explicit calculations in the Ising model, tricritical Ising model, and

\widehat{su}(2)_k

WZW models illustrate our results.

\mathcal{S}

-matrix and the boundary data. For diagonal CFTs, we obtain exact expressions for the left-right relative entropy in terms of modular data, and extend the framework to define \textit{Sandwiched Left-Right Rényi relative entropies} and \textit{left-right fidelity}. Applying this formalism to the Ising model, tricritical Ising model, and

\widehat{su}(2)_k

WZW model, we uncover a striking phenomenon: the left-right relative entropy between certain reduced boundary states vanishes even though the corresponding global boundary states are orthogonal. This observation motivates the introduction of \textit{relative entanglement sectors}, defined as equivalence classes of boundary states that are indistinguishable with respect to left-right relative entropy. These sectors transform as NIM-representations of global symmetries and exhibit level-dependent structures that mirror

\mathbb{Z}_2

't Hooft anomalies. Our findings establish an unexpected bridge between quantum information measures, boundary conformal symmetry, and quantum anomaly constraints.