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arXiv:2610.03576

Sharp null-energy lower bounds for Page-like entropy excursions in two-dimensional conformal field theory

Soufiane Karmoudi, Mohammed Daoud A

Published Oct 2, 2026• 0 citations
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Abstract

We derive sharp lower bounds on the energy cost of Page-like entanglement entropy excursions in two-dimensional conformal field theory. For states satisfying the primary quantum null energy condition (QNEC), the positive null energy $E_+$ obeys a sharp lower bound with two branches separated by the threshold $ΔS=(c/6)\ln 2$. We also derive a sharp lower bound on the net energy $E_{\rm net}$, with a coefficient larger by a factor of two than the value quoted by Abdolrahimi and Page [Phys. Rev. D 92, 083005 (2015)] and, in the QNEC-saturating sector, we obtain a sharp lower bound on the negative energy $E_-$. The constants are sharp as infima over smooth conformal vacua. We determine the entropy profiles that minimize $E_+$ for general QNEC states and the one that minimizes $E_-$ in the saturating sector. Finally, for an excursion containing an infinite sequence of small sub-excursions within a finite duration, we derive a necessary condition for the total positive energy to remain finite, which is stronger than requiring the entropy to have finite total variation.

DOI
10.48550/arxiv.2610.03576

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The quantum null energy condition (QNEC) is a lower bound on the expectation value of the null-null component of the energy-momentum tensor in terms of null variations of the entanglement entropy. A stronger version of the QNEC (the primary QNEC) is expected to hold in 1 + 1 dimensional conformal field theories (CFT). QNEC has been shown to impose nontrivial quantum thermodynamic restrictions on irreversible entropy production in quenches in 1 + 1 dimensional holographic CFTs. It is therefore natural to study if QNEC imposes similar bounds in other quench setups. In this paper we study QNEC in the Calabrese-Cardy global and local joining quenches using standard CFT techniques. In the global quench we show that the primary QNEC must hold at sufficiently early times and find that it imposes bounds on the four point correlators of twist fields in a boundary state. This is a constraint on the set of boundary states that satisfy the primary QNEC. Furthermore, we find that a violation of the primary QNEC implies a violation of the averaged null energy condition (ANEC) in a conformally transformed frame. In the local quench we find similar bounds on four point correlators from both the primary and the usual QNEC.
The Quantum Null Energy Condition (QNEC) relates energy to the second variation of entropy in relativistic quantum field theory. We use the QNEC inequality to bound entanglement entropy in quenches. At early times the entanglement entropy grows quadratically in time, and the QNEC provides an upper bound on the prefactor. We demonstrate that the bound is tight, by showing that it is saturated in certain quench protocols: boundary state quenches in conformal field theories in any dimensions. In higher than two dimensions we compute entanglement entropy using AdS/CFT. Our results are the first purely field theoretic applications of the QNEC.
The quantum null energy condition (QNEC) is a lower bound on the energy-momentum tensor in terms of the variation of the entanglement entropy of a subregion along a null direction. To gain insights into quantum thermodynamics of many-body systems, we study if the QNEC restricts irreversible entropy production in quenches driven by energy-momentum inflow from an infinite memoryless bath in two-dimensional holographic theories. We find that an increase in both entropy and temperature, as implied by the Clausius inequality of classical thermodynamics, is necessary but not sufficient to not violate QNEC in quenches leading to transitions between thermal states with momentums that are dual to Banados-Teitelboim-Zanelli geometries. For an arbitrary initial state, we can determine the lower and upper bounds on the increase of entropy (temperature) for a fixed increase in temperature (entropy). Our results provide explicit instances of quantum lower and upper bounds on irreversible entropy production whose existence has been established in literature. We also find monotonic behavior of the nonsaturation of the QNEC with time after a quench, and analytically determine their asymptotic values. Our study shows that the entanglement entropy of an interval of length l always thermalizes in time l/2 with an exponent 3/2. Furthermore, we determine the coefficient of initial quadratic growth of entanglement analytically for any l, and show that the slope of the asymptotic ballistic growth of entanglement for a semi-infinite interval is twice the difference of the entropy densities of the final and initial states. We determine explicit upper and lower bounds on these rates of growth of entanglement.
Energy conditions play an important role in constraining the dynamics of quantum field theories as well as gravitational theories. For example, in semi-classical gravity, the achronal averaged null energy condition (AANEC) can be used to prove that it is always slower to traverse through a wormhole than to travel around via its exterior. Such conditions prevent causality violations that would lead to paradoxes. Recent advances have been made in proving previously conjectured energy conditions directly in quantum field theory (QFT) as well as in uncovering new ones. This thesis will be an exploration of various energy inequalities in conformal field theories (CFTs) as well as semi-classical quantum gravity. At the core of this work lies the recently proved quantum null energy condition (QNEC). The QNEC bounds the null energy flowing past a point by a certain second shape derivative of entanglement entropy. We will demonstrate that the QNEC represents a deep connection between causality, energy and entanglement in quantum field theories. We explore this connection first in the context of holographic CFTs. Calculations in holographic theories will lead us to conjecture that the so-called diagonal QNEC is saturated in all interacting QFTs. We will then provide further, independent evidence that this conjecture holds for all CFTs with a twist gap by explicitly calculating shape derivatives of entanglement entropy using defect CFT techniques.
A bstract Utilizing quantum information theory, it has been shown that irreversible entropy production is bounded from both below and above in physical processes. Both these bounds are positive and generalize the Clausius inequality. Such bounds are, however, obtained from distance measures in the space of states, which are hard to define and compute in quantum field theories. We show that the quantum null energy condition (QNEC) can be utilized to obtain both lower and upper bounds on irreversible entropy production for quenches leading to transitions between thermal states carrying uniform momentum density in two dimensional holographic conformal field theories. We achieve this by refining earlier methods and developing an algebraic procedure for determining HRT surfaces in arbitrary Bañados-Vaidya geometries which are dual to quenches involving transitions between general quantum equilibrium states (e.g. thermal states) where the QNEC is saturated. We also discuss results for the growth and thermalization of entanglement entropy for arbitrary initial and final temperatures and momentum densities. The rate of quadratic growth of entanglement just after the quench depends only on the change in the energy density and is independent of the entangling length. For sufficiently large entangling lengths, the entanglement tsunami phenomenon can be established. Finally, we study recovery of the initial state from the evolving entanglement entropy and argue that the Renyi entropies should give us a refined understanding of scrambling of quantum information.
The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition, which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that $⟨{T}_{kk}{⟩}_{p}\ensuremath{\ge}{\mathrm{lim}}_{A\ensuremath{\rightarrow}0}(\frac{\ensuremath{\hbar}}{2\ensuremath{\pi}A}{S}_{\mathrm{out}}^{\ensuremath{'}\ensuremath{'}})$, where Sout{S}_{\mathrm{out}}Sout​ is the entanglement entropy restricted to one side of a codimension-2 surface $\mathrm{\ensuremath{\Sigma}}$, which is deformed in the null direction about a neighborhood of point ppp with area AAA. A proof of QNEC has been given before, which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and methods, we prove the QNEC for fermionic field theories.