Quantum Null Energy Condition Research Articles

Irreversibility, QNEC, and defects

We first present an analysis of infinitesimal null deformations for the entanglement entropy, which leads to a major simplification of the proof of the C, F and A-theorems in quantum field theory. Next, we study the quantum null energy condition (QNEC) on the light-cone for a CFT. Finally, we combine these tools in order to establish the irreversibility of renormalization group flows on planar d-dimensional defects, embedded in D-dimensional conformal field theories. This proof completes and unifies all known defect irreversibility theorems for defect dimensions d ≤ 4. The F-theorem on defects (d = 3) is a new result using information-theoretic methods. For d ≥ 4 we also establish the monotonicity of the relative entropy coefficient proportional to Rd−4. The geometric construction connects the proof of irreversibility with and without defects through the QNEC inequality in the bulk, and makes contact with the proof of strong subadditivity of holographic entropy taking into account quantum corrections.

Erasure Tolerant Quantum Memory and the Quantum Null Energy Condition in Holographic Systems.

Investigating principles for storage of quantum information at finite temperature with minimal need for active error correction is an active area of research. We bear upon this question in two-dimensional holographic conformal field theories via the quantum null energy condition that we have shown earlier to implement the restrictions imposed by quantum thermodynamics on such many-body systems. We study an explicit encoding of a logical qubit into two similar chirally propagating excitations of finite von Neumann entropy on a finite temperature background whose erasure can be implemented by an appropriate inhomogeneous and instantaneous energy-momentum inflow from an infinite energy memoryless bath due to which the system transits to a thermal state. Holographically, these fast erasure processes can be depicted by generalized AdS-Vaidya geometries described previously in which no assumption of specific form of bulk matter is needed. We show that the quantum null energy condition gives analytic results for the minimal finite temperature needed for the deletion which is larger than the initial background temperature in consistency with Landauer's principle. In particular, we find a simple expression for the minimum final temperature needed for the erasure of a large number of encoding qubits. We also find that if the encoding qubits are localized over an interval shorter than a specific localization length, then the fast erasure process is impossible, and furthermore this localization length is the largest for an optimal amount of encoding qubits determined by the central charge. We estimate the optimal encoding qubits for realistic protection against fast erasure. We discuss possible generalizations of our study for novel constructions of fault-tolerant quantum gates operating at finite temperature.

Entropy variations and light ray operators from replica defects

We study the defect operator product expansion (OPE) of displacement operators in free and interacting conformal field theories using replica methods. We show that as n approaches 1 a contact term can emerge when the OPE contains defect operators of twist d − 2. For interacting theories and general states we give evidence that the only possibility is from the defect operator that becomes the stress tensor in the n → 1 limit. This implies that the quantum null energy condition (QNEC) is always saturated for CFTs with a twist gap. As a check, we show independently that in a large class of near vacuum states, the second variation of the entanglement entropy is given by a simple correlation function of averaged null energy operators as studied by Hofman and Maldacena. This suggests that sub-leading terms in the defect OPE are controlled by a defect version of the spin-3 non-local light ray operator and we speculate about the possible origin of such a defect operator. For free theories this contribution condenses to a contact term that leads to violations of QNEC saturation.

Possible signature of entanglement in null-energy-condition-violating inflation

In recent years, the classical version of the null energy condition (NEC) has been enhanced into its quantum generalization, the quantum null energy condition (QNEC). The right-hand side of the QNEC inequality is non-zero; it is multiplied by a factor of ћ and recovers the standard NEC in the classical limit. Moreover, the second derivative of entanglement entropy also plays a role in determining the right-hand side. In this work, we study several examples of theories of inflation that violate the standard NEC, yet still, possibly obey the QNEC. This suggests that the deviation of the standard NEC arises due to the existence of entanglement between regions inside and outside of the cosmological horizon. Possible connection to quantum fluctuations of de Sitter spacetime is also studied.

Quantum Thermodynamics of Holographic Quenches and Bounds on the Growth of Entanglement from the Quantum Null Energy Condition

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.

Approximate Recovery and Relative Entropy I: General von Neumann Subalgebras

We prove the existence of a universal recovery channel that approximately recovers states on a von Neumann subalgebra when the change in relative entropy, with respect to a fixed reference state, is small. Our result is a generalization of previous results that applied to type-I von Neumann algebras by Junge at al. [arXiv:1509.07127]. We broadly follow their proof strategy but consider here arbitrary von Neumann algebras, where qualitatively new issues arise. Our results hinge on the construction of certain analytic vectors and computations/estimations of their Araki–Masuda L_p norms. We comment on applications to the quantum null energy condition.

Relative Entropy of Coherent States on General CCR Algebras

For a subalgebra of a generic CCR algebra, we consider the relative entropy between a general (not necessarily pure) quasifree state and a coherent excitationthereof. We give a unified formula for this entropy in terms of single-particle modular data. Further, we investigate changes of the relative entropy along subalgebras arising from an increasing family of symplectic subspaces; here convexity of the entropy (as usually considered for the Quantum Null Energy Condition) is replaced with lower estimates for the second derivative, composed of “bulk terms” and “boundary terms”. Our main assumption is that the subspaces are in differential modular position, a regularity condition that generalizes the usual notion of half-sided modular inclusions. We illustrate our results in relevant examples, including thermal states for the conformal U(1)-current.

Loop Groups and QNEC

We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG. For the corresponding solitonic states, we prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound. As an intermediate result, we show that a Positive Energy Representation of a loop group LG can be extended to a PER of H^{s}(S^1,G) for s>3/2, where G is any compact, simple and simply connected Lie group. We also show the existence of the exponential map of the semidirect product LG rtimes R, with R a one-parameter subgroup of mathrm{Diff}_+(S^1), and we compute the adjoint action of H^{s+1}(S^1,G) on the stress energy tensor.

QNEC2 in deformed holographic CFTs

We use the quantum null energy condition in strongly coupled two-dimensional field theories (QNEC2) as diagnostic tool to study a variety of phase structures, including crossover, second and first order phase transitions. We find a universal QNEC2 constraint for first order phase transitions with kinked entanglement entropy and discuss in general the relation between the QNEC2-inequality and monotonicity of the Casini-Huerta c-function. We then focus on a specific example, the holographic dual of which is modelled by three-dimensional Einstein gravity plus a massive scalar field with one free parameter in the self-interaction potential. We study translation invariant stationary states dual to domain walls and black branes. Depending on the value of the free parameter we find crossover, second and first order phase transitions between such states, and the c-function either flows to zero or to a finite value in the infrared. We present evidence that evaluating QNEC2 for ground state solutions allows to predict the existence of phase transitions at finite temperature.

A R\xe9nyi quantum null energy condition: proof for free field theories

The Quantum Null Energy Condition (QNEC) is a lower bound on the stress-energy tensor in quantum field theory that has been proved quite generally. It can equivalently be phrased as a positivity condition on the second null shape derivative of the relative entropy Srel(ρ||σ) of an arbitrary state ρ with respect to the vacuum σ. The relative entropy has a natural one-parameter family generalization, the Sandwiched Rényi divergence Sn(ρ||σ), which also measures the distinguishability of two states for arbitrary n ∈ [1/2, ∞). A Rényi QNEC, a positivity condition on the second null shape derivative of Sn(ρ||σ), was conjectured in previous work. In this work, we study the Rényi QNEC for free and superrenormalizable field theories in spacetime dimension d > 2 using the technique of null quantization. In the above setting, we prove the Rényi QNEC in the case n > 1 for arbitrary states. We also provide counterexamples to the Rényi QNEC for n < 1.

Recovering the QNEC from the ANEC

We study the relative entropy in QFT comparing the vacuum state to a special family of purifications determined by an input state and constructed using relative modular flow. We use this to prove a conjecture by Wall that relates the shape derivative of relative entropy to a variational expression over the averaged null energy (ANE) of possible purifications. This variational expression can be used to easily prove the quantum null energy condition (QNEC). We formulate Wall’s conjecture as a theorem pertaining to operator algebras satisfying the properties of a half-sided modular inclusion, with the additional assumption that the input state has finite averaged null energy. We also give a new derivation of the strong superadditivity property of relative entropy in this context. We speculate about possible connections to the recent methods used to strengthen monotonicity of relative entropy with recovery maps.

Proof of the quantum null energy condition for free fermionic field theories

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 ${S}_{\mathrm{out}}$ 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 $p$ with area $A$. 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.

Quantum null energy condition, entanglement wedge nesting, and quantum focusing

We study the consequences of Entanglement Wedge Nesting for CFTs with holographic duals. The CFT is formulated on an arbitrary curved background, and we include the effects of curvature-squared couplings in the bulk. In this setup we find necessary and sufficient conditions for Entanglement Wedge Nesting to imply the Quantum Null Energy Condition in $d\leq 5$, extending its earlier holographic proofs. We also show that the Quantum Focusing Conjecture yields the Quantum Null Energy Condition as its nongravitational limit under these same conditions.

Local Quantum Energy Conditions in Non-Lorentz-Invariant Quantum Field Theories.

We provide the first example of local quantum energy conditions in quantum field theories that are not Lorentz invariant. We focus on field theories in two dimensions with infinite-dimensional symmetries, like the ones governed by the Bondi-van der Burg-Metzner-Sachs group that appear in the context of flat space holography. Reminiscent of holographic results on the quantum null energy condition, we prove that our new energy conditions saturate for states in the field theory that are dual to vacuum solutions of three-dimensional Einstein gravity with a vanishing cosmological constant.

A general proof of the quantum null energy condition

We prove a conjectured lower bound on 〈T__(x)〉ψ in any state ψ of a CFT on Minkowski space, dubbed the Quantum Null Energy Condition (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through x. Our proof involves a combination of the two independent methods that were used recently to prove the weaker Averaged Null Energy Condition (ANEC). In particular the properties of modular Hamiltonians under shape deformations for the state ψ play an important role, as do causality considerations. We study the two point function of a “probe” operator mathcal{O} in the state ψ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider causality on modular time for the modular evolved probe operators, which we constrain using Tomita-Takesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect Operator Product Expansion to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. We argue that the proof extends to more general relativistic QFT with an interacting UV fixed point and also prove a higher spin version of the QNEC. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the Ryu-Takayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of any QFT with an interacting UV fixed point.

A modular toolkit for bulk reconstruction

We introduce new tools for studying modular flow in AdS/CFT. These tools allow us to efficiently extract bulk information related to causality and locality. For example, we discuss the relation between analyticity in modular time and entanglement wedge nesting which can then be used to extract the location of the Ryu-Takayanagi (RT) surface directly from the boundary theory. Probing the RT surface close to the boundary our results reduce to the recent proof of the Quantum Null Energy Condition. We focus on heavy probe operators whose correlation functions are determined by spacelike geodesics. These geodesics interplay with the RT surface via a set of rules that we conjecture and give evidence for using the replica trick.

Quantum null energy condition and its (non)saturation in 2d CFTs

We consider the Quantum Null Energy Condition (QNEC) for holographic conformal field theories in two spacetime dimensions (CFT_22). We show that QNEC saturates for all states dual to vacuum solutions of AdS_33 Einstein gravity, including systems that are far from thermal equilibrium. If the Ryu-Takayanagi surface encounters bulk matter QNEC does not need to be saturated, whereby we give both analytical and numerical examples. In particular, for CFT_22 with a global quench dual to AdS_33-Vaidya geometries we find a curious half-saturation of QNEC for large entangling regions. We also address order one corrections from quantum backreactions of a scalar field in AdS_33 dual to a primary operator of dimension h in a large central charge expansion and explicitly compute both, the backreacted Ryu–Takayanagi surface part and the bulk entanglement contribution to EE and QNEC. At leading order for small entangling regions the contribution from bulk EE exactly cancels the contribution from the back-reacted Ryu-Takayanagi surface, but at higher orders in the size of the region the contributions are almost equal while QNEC is not saturated. For a half-space entangling region we find that QNEC is gapped by h/4h/4 in the large h expansion.

Entropy Distribution of Localised States

We study the geometric distribution of the relative entropy of a charged localised state in Quantum Field Theory. With respect to translations, the second derivative of the vacuum relative entropy is zero out of the charge localisation support and positive in mean over the support of any single charge. For a spatial strip, the asymptotic mean entropy density is $\pi E$, with $E$ the corresponding vacuum charge energy. In a conformal QFT, for a charge in a ball of radius $r$, the relative entropy is non linear, the asymptotic mean radial entropy density is $\pi E$ and Bekenstein's bound is satisfied. We also study the null deformation case. We construct, operator algebraically, a positive selfadjoint operator that may be interpreted as the deformation generator, we thus get a rigorous form of the Averaged Null Energy Condition that holds in full generality. In the one dimensional conformal $U(1)$-current model, we give a complete and explicit description of the entropy distribution of a localised charged state in all points of the real line; in particular, the second derivative of the relative entropy is strictly positive in all points where the charge density is non zero, thus the Quantum Null Energy Condition holds here for these states and is not saturated in these points.

Violation of the quantum null-energy condition in a holographic wormhole and infrared effects

We examine the quantum null energy condition (QNEC) for a $2+1$-dimensional conformal field theory (CFT) at strong coupling in the background of a wormhole spacetime by employing the AdS/CFT correspondence. First, we numerically construct a novel $3+1$-dimensional vacuum AdS black hole solution with non-trivial topology, which is dual to a wormhole geometry connecting two flat universes. Although the bulk null energy condition (NEC) is not violated, the NEC for the holographic stress-energy tensor is violated near the wormhole throat. Next, we investigate the entanglement entropy for a half-space anchored to the boundary wormhole throat. We propose a natural prescription for regularizing the IR divergent part of the entanglement entropy and show that the QNEC is violated at the throat. This is the first counterexample to the QNEC, indicating that IR effects are crucial.

Constraining quantum fields using modular theory

Tomita-Takesaki modular theory provides a set of algebraic tools in quantum field theory that is suitable for the study of the information-theoretic properties of states. For every open set in spacetime and choice of two states, the modular theory defines a positive operator known as the relative modular operator that decreases monotonically under restriction to subregions. We study the consequences of this operator monotonicity inequality for correlation functions in quantum field theory. We do so by constructing a one-parameter Rényi family of information-theoretic measures from the relative modular operator that inherit monotonicity by construction and reduce to correlation functions in special cases. In the case of finite quantum systems, this Rényi family is the sandwiched Rényi divergence and we obtain a simple proof of its monotonicity. Its monotonicity implies a class of constraints on correlation functions in quantum field theory, only a small set of which were known to us. We explore these inequalities for free fields and conformal field theory. We conjecture that the second null derivative of Rényi divergence is non-negative which is a generalization of the quantum null energy condition to the Rényi family.