arXiv subjects

Gilad Lifschytz

Publications and source records attributed to Gilad Lifschytz.

At least 19 recordsLinked to original sources

arXiv2026

Entanglement groups

We propose to define entanglement in terms of local unitary transformations acting on some parts of a system that can be undone by local unitary transformations acting on other parts. This leads to a characterization of entanglement in terms of groups. We refer to these as entanglement groups, and we refer to this notion as $g$-entanglement. We discuss the physical meaning of entanglement groups and contrast $g$-entanglement with other, more conventional definitions of entanglement. For pure states, entanglement groups are constructed as certain quotients of the stabilizer group and its subgroups. For mixed states, entanglement groups can be constructed from stabilizers of the purification. We analyze the structure of entanglement groups, show that they have properties which correspond to monogamy of entanglement, and explore the restrictions placed by separability. We show that $g$-entanglement underlies several well-known quantum tasks.

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arXiv2025

Entanglement groups for mixed states

We extend an operational characterization of entanglement in terms of stabilizer groups from pure states to mixed states. For a density matrix $ρ_{AB}$, a stabilizer is a factorized unitary matrix $u_A \otimes u_B$ that, under conjugation, leaves $ρ_{AB}$ invariant. The entanglement group is a quotient of the stabilizer group, in which one-party stabilizers are considered trivial. This definition relates the entanglement of a density matrix to the entanglement of its purification. We give general properties of entanglement groups for mixed states, then discuss special properties for separable states. For a separable state, the entanglement group may be non-trivial. However it can only arise from multi-party entanglement with the purifying system.

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arXiv2022

Dressing bulk spinor fields in AdS${}_3$

We continue the program of bulk reconstruction for fermionic fields. We reconstruct, from the CFT, the Dirac fermion field in $AdS_{3}$ coupled to a Chern-Simon gauge field. We show that the three conditions; solving the equation of motion, satisfying expected transformation under modular flow and a simple charge distribution at infinity are all compatible and all produce the same bulk operator. We also compute the bulk-boundary tree level three point function from the CFT construction.

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arXiv2022

Reconstructing bulk equation of motion using CFT modular Hamiltonians

In the framework of bulk reconstruction, we elucidate the relationship between the action of CFT modular Hamiltonians on bulk operators, the possible equation of motion for the bulk operators, and the charge distribution at infinity corresponding to such bulk fields. In particular for scalar fields interacting with gravity or with gauge fields, we show how CFT considerations of the action of the modular Hamiltonian constrain the possible bulk equation of motion to be consistent with bulk gauge invariance and diffeomorphism invariance. In fact we show that requiring that the action of the modular Hamiltonian on a dressed bulk scalar operator be compatible with some unknown simple equation of motion, fixes, under reasonable assumptions, both the equation of motion and the action of the modular Hamiltonian once the first order $\frac{1}{N}$ terms are known.

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arXiv2021

Light-ray moments as endpoint contributions to modular Hamiltonians

We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator $J^{(n)}$ of modular weight $n$ over a spacelike surface passing through $x = 0$. For $\vert n \vert \geq 2$ the modular Hamiltonian associated with a division of space at $x = 0$ picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at $x = 0$. The endpoint contribution is a sum of light-ray moments of the perturbing operator $J^{(n)}$ and its descendants. For perturbations on null planes only moments of $J^{(n)}$ itself contribute.

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arXiv2020

Dressing bulk fields in AdS${}_3$

We study a set of CFT operators suitable for reconstructing a charged bulk scalar field $ϕ$ in AdS${}_3$ (dual to an operator ${\cal O}$ of dimension $Δ$ in the CFT) in the presence of a conserved spin-$n$ current in the CFT. One has to sum a tower of smeared non-primary scalars $\partial_{+}^{m} J^{(m)}$, where $J^{(m)}$ are primaries with twist $Δ$ and spin $m$ built from ${\cal O}$ and the current. The coefficients of these operators can be fixed by demanding that bulk correlators are well-defined: with a simple ansatz this requirement allows us to calculate bulk correlators directly from the CFT. They are built from specific polynomials of the kinematic invariants up to a freedom to make field redefinitions. To order $1/N$ this procedure captures the dressing of the bulk scalar field by a radial generalized Wilson line.

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arXiv2020

Endpoint contributions to excited-state modular Hamiltonians

We compute modular Hamiltonians for excited states obtained by perturbing the vacuum with a unitary operator. We use operator methods and work to first order in the strength of the perturbation. For the most part we divide space in half and focus on perturbations generated by integrating a local operator $J$ over a null plane. Local operators with weight $n \geq 2$ under vacuum modular flow produce an additional endpoint contribution to the modular Hamiltonian. Intuitively this is because operators with weight $n \geq 2$ can move degrees of freedom from a region to its complement. The endpoint contribution is an integral of $J$ over a null plane. We show this in detail for stress tensor perturbations in two dimensions, where the result can be verified by a conformal transformation, and for scalar perturbations in a CFT. This lets us conjecture a general form for the endpoint contribution that applies to any field theory divided into half-spaces.

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arXiv2019

Bulk reconstruction for spinor fields in AdS/CFT

We develop the representation of free spinor fields in the bulk of Lorentzian anti-de Sitter space in terms of smeared operators in the dual conformal field theory. To do this we expand the bulk field in a complete set of normalizable modes, work out the extrapolate dictionary for spinor fields, and show that the bulk field can be reconstructed from its near-boundary behavior. In some cases chirality and reality conditions can be imposed in the bulk. We study the action of the CFT modular Hamiltonian on bulk fermions to show that they transform with the expected spinor Lie derivative, and we calculate bulk--boundary two-point functions starting from CFT correlators.

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arXiv2019

Emergence of spacetime from the algebra of total modular Hamiltonians

We study the action of the CFT total modular Hamiltonian on the CFT representation of bulk fields with spin. In the vacuum of the CFT the total modular Hamiltonian acts as a bulk Lie derivative, reducing on the RT surface to a boost perpendicular to the RT surface. This enables us to reconstruct bulk fields with spin from the CFT. On fields with gauge redundancies the total modular Hamiltonian acts as a bulk Lie derivative together with a compensating bulk gauge (or diffeomorphism) transformation to restore the original gauge. We consider the Lie algebra generated by the total modular Hamiltonians of all spherical CFT subregions and define weakly-maximal Lie subalgebras as proper subalgebras containing a maximal set of total modular Hamiltonians. In a CFT state with a bulk dual, we show that the bulk spacetime parametrizes the space of these weakly-maximal Lie subalgebras. Each such weakly-maximal Lie subalgebra induces Lorentz transformations at a particular point in the bulk manifold. The bulk metric dual to a pure CFT state is invariant at each point under this transformation. This condition fixes the metric up to a conformal factor that can be computed from knowledge of the equation parametrizing extremal surfaces. This gives a holographic notion of the invariance of a pure CFT state under CFT modular flow.

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arXiv2018

Does boundary quantum mechanics imply quantum mechanics in the bulk?

Perturbative bulk reconstruction in AdS/CFT starts by representing a free bulk field $ϕ^{(0)}$ as a smeared operator in the CFT. A series of $1/N$ corrections must be added to $ϕ^{(0)}$ to represent an interacting bulk field $ϕ$. These corrections have been determined in the literature from several points of view. Here we develop a new perspective. We show that correlation functions involving $ϕ^{(0)}$ suffer from ambiguities due to analytic continuation. As a result $ϕ^{(0)}$ fails to be a well-defined linear operator in the CFT. This means bulk reconstruction can be understood as a procedure for building up well-defined operators in the CFT which thereby singles out the interacting field $ϕ$. We further propose that the difficulty with defining $ϕ^{(0)}$ as a linear operator can be re-interpreted as a breakdown of associativity. Presumably $ϕ^{(0)}$ can only be corrected to become an associative operator in perturbation theory. This suggests that quantum mechanics in the bulk is only valid in perturbation theory around a semiclassical bulk geometry.

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arXiv2017

Local bulk physics from intersecting modular Hamiltonians

We show that bulk quantities localized on a minimal surface homologous to a boundary region correspond in the CFT to operators that commute with the modular Hamiltonian associated with the boundary region. If two such minimal surfaces intersect at a point in the bulk then CFT operators which commute with both extended modular Hamiltonians must be localized at the intersection point. We use this to construct local bulk operators purely from CFT considerations, without knowing the bulk metric, using intersecting modular Hamiltonians. For conformal field theories at zero and finite temperature the appropriate modular Hamiltonians are known explicitly and we recover known expressions for local bulk observables.

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arXiv2017

Striped anyonic fluids

The choice of statistics for a quantum particle is almost always a discrete one: either bosonic or fermionic. Anyons are the exceptional case for which the statistics can take a range of intermediate values. Holography provides an opportunity to address the question of how the behavior of interacting anyons depends on the choice of statistics. In this paper, we analyze the spectrum of a strongly coupled, gapless fluid of anyons described holographically by the D3-D7' model with alternative boundary conditions. We investigate how these alternative boundary conditions impact the instability of the gapless homogeneous phase toward the formation of spatial order. In addition, we also show that for a particular, limiting choice of the alternative boundary conditions, this holographic system can be interpreted as describing strongly coupled (2+1)-dimensional QED. In this case, the instability leads to a spontaneous, spatially modulated magnetic field.

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arXiv2016

Locality, bulk equations of motion and the conformal bootstrap

We develop an approach to construct local bulk operators in a CFT to order $1/N^2$. Since 4-point functions are not fixed by conformal invariance we use the OPE to categorize possible forms for a bulk operator. Using previous results on 3-point functions we construct a local bulk operator in each OPE channel. We then impose the condition that the bulk operators constructed in different channels agree, and hence give rise to a well-defined bulk operator. We refer to this condition as the "bulk bootstrap." We argue and explicitly show in some examples that the bulk bootstrap leads to some of the same results as the regular conformal bootstrap. In fact the bulk bootstrap provides an easier way to determine some CFT data, since it does not require knowing the form of the conformal blocks. This analysis clarifies previous results on the relation between bulk locality and the bootstrap for theories with a $1/N$ expansion, and it identifies a simple and direct way in which OPE coefficients and anomalous dimensions determine the bulk equations of motion to order $1/N^2$.

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arXiv2016

Asymmetric interiors for small black holes

We develop the representation of infalling observers and bulk fields in the CFT as a way to understand the black hole interior in AdS. We first discuss properties of CFT states which are dual to black holes. We then show that in the presence of a Killing horizon bulk fields can be decomposed into pieces we call ingoing and outgoing. The ingoing field admits a simple operator representation in the CFT, even inside a small black hole at late times, which leads to a simple CFT description of infalling geodesics. This means classical infalling observers will experience the classical geometry in the interior. The outgoing piece of the field is more subtle. In an eternal two-sided geometry it can be represented as an operator on the left CFT. In a stable one-sided geometry it can be described using entanglement via the PR construction. But in an evaporating black hole trans-horizon entanglement breaks down at the Page time, which means that for old black holes the PR construction fails and the outgoing field does not see local geometry. This picture of the interior allows the CFT to reconcile unitary Hawking evaporation with the classical experience of infalling observers.

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arXiv2015

Bulk equations of motion from CFT correlators

To O(1/N) we derive, purely from CFT data, the bulk equations of motion for interacting scalar fields and for scalars coupled to gauge fields and gravity. We first uplift CFT operators to mimic local AdS fields by imposing bulk microcausality. This requires adding an infinite tower of smeared higher-dimension double-trace operators to the CFT definition of a bulk field, with coefficients that we explicitly compute. By summing the contribution of the higher-dimension operators we derive the equations of motion satisfied by these uplifted CFT operators and show that we precisely recover the expected bulk equations of motion. We exhibit the freedom in the CFT construction which corresponds to bulk field redefinitions.

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arXiv2014

Finite N and the failure of bulk locality: Black holes in AdS/CFT

We consider bulk quantum fields in AdS/CFT in the background of an eternal black hole. We show that for black holes with finite entropy, correlation functions of semiclassical bulk operators close to the horizon deviate from their semiclassical value and are ill-defined inside the horizon. This is due to the large-time behavior of correlators in a unitary CFT, and means the region near and inside the horizon receives corrections. We give a prescription for modifying the definition of a bulk field in a black hole background, such that one can still define operators that mimic the inside of the horizon, but at the price of violating microcausality. For supergravity fields we find that commutators at spacelike separation generically ~ exp(-S/2). Similar results hold for stable black holes that form in collapse. The general lesson may be that a small amount of non-locality, even over arbitrarily large spacelike distances, is an essential aspect of non-perturbative quantum gravity.

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arXiv2014

Flowing holographic anyonic superfluid

We investigate the flow of a strongly coupled anyonic superfluid based on the holographic D3-D7' probe brane model. By analyzing the spectrum of fluctuations, we find the critical superfluid velocity, as a function of the temperature, at which the flow stops being dissipationless when flowing past a barrier. We find that at a larger velocity the flow becomes unstable even in the absence of a barrier.

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arXiv2014

Decoding the hologram: Scalar fields interacting with gravity

We construct smeared CFT operators which represent a scalar field in AdS interacting with gravity. The guiding principle is micro-causality: scalar fields should commute with themselves at spacelike separation. To O(1/N) we show that a correct and convenient criterion for constructing the appropriate CFT operators is to demand micro-causality in a three-point function with a boundary Weyl tensor and another boundary scalar. The resulting bulk observables transform in the correct way under AdS isometries and commute with boundary scalar operators at spacelike separation, even in the presence of metric perturbations.

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