arXiv subjects

Jonathan Sorce

Publications and source records attributed to Jonathan Sorce.

29 records · Page 2Linked to original sources

arXiv2022

Modular Commutators in Conformal Field Theory

The modular commutator is a recently discovered multipartite entanglement measure that quantifies the chirality of the underlying many-body quantum state. In this Letter, we derive a universal expression for the modular commutator in conformal field theories in $1+1$ dimensions and discuss its salient features. We show that the modular commutator depends only on the chiral central charge and the conformal cross ratio. We test this formula for a gapped $(2+1)$-dimensional system with a chiral edge, i.e., the quantum Hall state, and observe excellent agreement with numerical simulations. Furthermore, we propose a geometric dual for the modular commutator in certain preferred states of the AdS/CFT correspondence. For these states, we argue that the modular commutator can be obtained from a set of crossing angles between intersecting Ryu-Takayanagi surfaces.

cond-mat.str-el↗

arXiv2022

A canonical Hamiltonian for open quantum systems

If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniquely specified by requiring the dissipator's jump operators to be traceless, extending a uniqueness result known previously in the special case of Markovian master equations. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system's bare Hamiltonian.

quant-ph↗

arXiv2022

Approximate Petz recovery from the geometry of density operators

We derive a new bound on the effectiveness of the Petz map as a universal recovery channel in approximate quantum error correction using the second sandwiched Rényi relative entropy $\tilde{D}_{2}$. For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order-$ε$ accuracy whenever the data processing inequality for $\tilde{D}_{2}$ is saturated up to terms of order $ε^2$ times the inverse Hilbert space dimension. Conceptually, our result is obtained by extending arXiv:2011.03473, in which we studied exact saturation of the data processing inequality using differential geometry, to the case of approximate saturation. Important roles are played by (i) the fact that the exponential of the second sandwiched Rényi relative entropy is quadratic in its first argument, and (ii) the observation that the second sandwiched Rényi relative entropy satisfies the data processing inequality even when its first argument is a non-positive Hermitian operator.

quant-ph↗

arXiv2022

Geometric conditions for saturating the data processing inequality

The data processing inequality (DPI) is a scalar inequality satisfied by distinguishability measures on density matrices. For some distinguishability measures, saturation of the scalar DPI implies an operator equation relating the arguments of the measure. These results are typically derived using functional analytic techniques. In a complementary approach, we use geometric techniques to derive a formula that gives an operator equation from DPI saturation for any distinguishability measure; moreover, for a broad class of distinguishability measures, the derived operator equation is sufficient to imply saturation as well. Our operator equation coincides with known results for the sandwiched Rényi relative entropies, and gives new results for $α$-$z$ Rényi relative entropies and a family of of quantum $f$-divergences, which we compute explicitly.

quant-ph↗

arXiv2021

The Markov gap for geometric reflected entropy

The reflected entropy $S_R(A:B)$ of a density matrix $ρ_{AB}$ is a bipartite correlation measure lower-bounded by the quantum mutual information $I(A:B)$. In holographic states satisfying the quantum extremal surface formula, where the reflected entropy is related to the area of the entanglement wedge cross-section, there is often an order-$N^2$ gap between $S_R$ and $I$. We provide an information-theoretic interpretation of this gap by observing that $S_R - I$ is related to the fidelity of a particular Markov recovery problem that is impossible in any state whose entanglement wedge cross-section has a nonempty boundary; for this reason, we call the quantity $S_R - I$ the Markov gap. We then prove that for time-symmetric states in pure AdS$_3$ gravity, the Markov gap is universally lower bounded by $\log(2) \ell_{\text{AdS}}/2 G_N$ times the number of endpoints of the cross-section. We provide evidence that this lower bound continues to hold in the presence of bulk matter, and comment on how it might generalize above three bulk dimensions. Finally, we explore the Markov recovery problem controlling $S_R - I$ using fixed area states. This analysis involves deriving a formula for the quantum fidelity -- in fact, for all the sandwiched Rényi relative entropies -- between fixed area states with one versus two fixed areas, which may be of independent interest. We discuss, throughout the paper, connections to the general theory of multipartite entanglement in holography.

hep-th↗

arXiv2020

Holographic scattering requires a connected entanglement wedge

In AdS/CFT, there can exist local 2-to-2 bulk scattering processes even when local scattering is not possible on the boundary; these have previously been studied in connection with boundary correlation functions. We show that boundary regions associated with these scattering configurations must have $O(1/G_N)$ mutual information, and hence a connected entanglement wedge. One of us previously argued for this statement from the boundary theory using operational tools in quantum information theory. We improve that argument to make it robust to small errors and provide a proof in the bulk using focusing arguments in general relativity. We also provide a direct link to entanglement wedge reconstruction by showing that the bulk scattering region must lie inside the connected entanglement wedge. Our construction implies the existence of nonlocal quantum computation protocols that are exponentially more efficient than the optimal protocols currently known.

hep-th↗

arXiv2020

Large-$d$ phase transitions in holographic mutual information

In the AdS/CFT correspondence, the entanglement entropy of subregions in the boundary CFT is conjectured to be dual to the area of a bulk extremal surface at leading order in $G_N$ in the holographic limit. Under this dictionary, distantly separated regions in the CFT vacuum state have zero mutual information at leading order, and only attain nonzero mutual information at this order when they lie close enough to develop significant classical and quantum correlations. Previously, the separation at which this phase transition occurs for equal-size ball-shaped regions centered at antipodal points on the boundary was known analytically only in $3$ spacetime dimensions. Inspired by recent explorations of general relativity at large-$d$, we compute the separation at which the phase transition occurs analytically in the limit of infinitely many spacetime dimensions, and find that distant regions cannot develop large correlations without collectively occupying the entire volume of the boundary theory. We interpret this result as illustrating the spatial decoupling of holographic correlations in the large-$d$ limit, and provide intuition for this phenomenon using results from quantum information literature. We also compute the phase transition separation numerically for a range of bulk spacetime dimensions from $4$ to $21$, where analytic results are intractable but numerical results provide insight into the dimension-dependence of holographic correlations. For bulk dimensions above $5$, our exact numerical results are well approximated analytically by working to next-to-leading order in the large-$d$ expansion.

hep-th↗

arXiv2019

Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT

We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-$N$ CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For 'tree' tensor networks (i.e., those that are constructed by discretizing spacetime with non-intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the 'holographic entanglement of purification' conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.

hep-th↗

arXiv2019

Holographic entanglement entropy is cutoff-covariant

In the AdS/CFT correspondence, it is often convenient to regulate infinite quantities in asymptotically anti-de Sitter spacetimes by introducing a sharp cutoff in a radial coordinate. This procedure is a priori coordinate-dependent, and may not be well-motivated in full, covariant general relativity; however, the fact that physically meaningful quantities such as the entanglement entropy can be obtained by such a regulation procedure suggests some underlying covariance. In this paper, we provide a careful treatment of the radial cutoff procedure for computing holographic entanglement entropy in asymptotically anti-de Sitter spacetimes. We prove two results that are frequently assumed in the literature, but that have not been carefully addressed: (i) that the choice of a "globally minimal surface" among several extremal candidates is independent of the choice of regulator, and (ii) that finite CFT quantities such as the mutual information which involve "divergence-cancelling" sums of holographic entanglement entropies are well-defined. Our results imply that the "globally minimal surface" prescription for computing holographic entanglement entropy is well-posed from the perspective of general relativity, and thus support the belief that this is the correct prescription for identifying the entanglement wedge of a boundary subregion in AdS/CFT. We also comment on the geometric source of state-dependent divergences in the holographic entanglement entropy, and identify precisely the regime of validity of the "vacuum subtraction" protocol for regulating infinite entanglement entropies in arbitrary states by comparing them to the entanglement entropies of identical regions in the vacuum. Our proofs make use of novel techniques for the covariant analysis of extremal surfaces, which are explained in detail and may find use more broadly in the study of holographic entanglement entropy.

hep-th↗

arXiv2019

Holographic Tensor Networks in Full AdS/CFT

We present a general procedure for constructing tensor networks for geometric states in the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. Given a state in a large-$N$ CFT with a static, semiclassical gravitational dual, our procedure produces a tensor network for the boundary state whose internal geometry matches (a discretization of) the bulk spacetime geometry. By invoking the "holographic entanglement of purification" conjecture, our construction can be made to capture the structure of the bulk spacetime at sub-AdS scales.

hep-th↗

arXiv2017

Gedanken Experiments to Destroy a Black Hole II: Kerr-Newman Black Holes Cannot be Over-Charged or Over-Spun

We consider gedanken experiments to destroy an extremal or nearly extremal Kerr-Newman black hole by causing it to absorb matter with sufficient charge and/or angular momentum as compared with energy that it cannot remain a black hole. It was previously shown by one of us that such gedanken experiments cannot succeed for test particle matter entering an extremal Kerr-Newman black hole. We generalize this result here to arbitrary matter entering an extremal Kerr-Newman black hole, provided only that the non-electromagnetic contribution to the stress-energy tensor of the matter satisfies the null energy condition. We then analyze the gedanken experiments proposed by Hubeny and others to over-charge and/or over-spin an initially slightly non-extremal Kerr-Newman black hole. Analysis of such gedanken experiments requires that we calculate all effects on the final mass of the black hole that are second-order in the charge and angular momentum carried into the black hole, including all self-force effects. We obtain a general formula for the full second order correction to mass, $δ^2 M$, which allows us to prove that no gedanken experiments of the generalized Hubeny type can ever succeed in over-charging and/or over-spinning a Kerr-Newman black hole, provided only that the non-electromagnetic stress-energy tensor satisfies the null energy condition. Our analysis is based upon Lagrangian methods, and our formula for the second-order correction to mass is obtained by generalizing the canonical energy analysis of Hollands and Wald to the Einstein-Maxwell case. Remarkably, we obtain our formula for $δ^2 M$ without having to explicitly compute self-force or finite size effects. Indeed, in an appendix, we show explicitly that our formula incorporates both the self-force and finite size effects for the special case of a charged body slowly lowered into an uncharged black hole.

gr-qc↗