IA Scholar Query: Topological duality for distributive lattices, and applications.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 01 Dec 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Simulating hyperbolic space on a circuit board
https://scholar.archive.org/work/l3szjt56tnclffwqvd7eazmkwa
The Laplace operator encodes the behavior of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. A key input for the Laplace equation is the curvature of space. Here we discuss and experimentally demonstrate that the spectral ordering of Laplacian eigenstates for hyperbolic (negatively curved) and flat two-dimensional spaces has a universally different structure. We use a lattice regularization of hyperbolic space in an electric-circuit network to measure the eigenstates of a 'hyperbolic drum', and in a time-resolved experiment we verify signal propagation along the curved geodesics. Our experiments showcase both a versatile platform to emulate hyperbolic lattices in tabletop experiments, and a set of methods to verify the effective hyperbolic metric in this and other platforms. The presented techniques can be utilized to explore novel aspects of both classical and quantum dynamics in negatively curved spaces, and to realise the emerging models of topological hyperbolic matter.Patrick M Lenggenhager, Alexander Stegmaier, Lavi K Upreti, Tobias Hofmann, Tobias Helbig, Achim Vollhardt, Martin Greiter, Ching Hua Lee, Stefan Imhof, Hauke Brand, Tobias Kießling, Igor Boettcher, Titus Neupert, Ronny Thomale, Tomáš Bzdušekwork_l3szjt56tnclffwqvd7eazmkwaThu, 01 Dec 2022 00:00:00 GMTTopological Matter and Fractional Entangled Geometry
https://scholar.archive.org/work/ijlpsyes2jbe7nryynyciyaaea
Here, we review our progress on a geometrical approach of quantum physics and topological crystals starting from nature, electrodynamics of planets and linking with Dirac magnetic monopoles and gauge fields. The Bloch sphere of a quantum spin-1/2 particle can also acquire an integer topological charge in the presence of a radial magnetic field. We show that the global topological properties are revealed from the poles of the surface allowing a correspondence between smooth fields, metric and quantum distance. The information is transported from each pole to the equatorial plane on a thin Dirac string. We develop the theory, "the quantum topometry" in space and time, and present applications on transport from a Newtonian approach, on a quantized photo-electric effect from circular dichroism of light towards topological band structures of crystals. The occurrence of robust edge modes related to the topological lattice models are revealed analytically when deforming the sphere or ellipse onto a cylinder. The topological properties of the quantum Hall effect, the quantum anomalous Hall effect and the quantum spin Hall effect on the honeycomb lattice can be measured locally in the Brillouin zone from the light-matter coupling. The formalism allows us to include interaction effects from the momentum space. Interactions may also result in fractional entangled geometry within the curved space. We develop a relation between entangled wavefunction in quantum mechanics, coherent superposition of geometries, a way to one-half topological numbers and Majorana fermions. We show realizations in topological matter. We present a relation between axion electrodynamics, topological insulators on a surface of a cube and the two-spheres' model via the meron.Karyn Le Hurwork_ijlpsyes2jbe7nryynyciyaaeaTue, 29 Nov 2022 00:00:00 GMTAdS/CFT Correspondence with a 3D Black Hole Simulator
https://scholar.archive.org/work/y4qhz32d7fgcdonbwtt43yrq2a
The AdS/CFT correspondence has been insightful for high-energy and condensed matter physics alike. An application of this correspondence is the duality between the entanglement entropy of Anti-de Sitter (AdS) black holes and lower-dimensional conformal field theories (CFT). To explicitly demonstrate this correspondence we simulate the effect a 3D black hole geometry has on Dirac fields by employing a square lattice of fermions with inhomogeneous tunnelling couplings. Simulating a 3D BTZ black hole horizon, we numerically obtain an area law behaviour that is in agreement with the corresponding 2D CFT with a central charge that depends on the cosmological constant of the AdS space. A systematic numerical investigation of various 3D black hole profiles suggests that all 3D black holes give an entropic behaviour that can be represented by the same CFT.Aydin Deger, Jiannis K. Pachoswork_y4qhz32d7fgcdonbwtt43yrq2aMon, 28 Nov 2022 00:00:00 GMTTurbulence as Clebsch Confinement
https://scholar.archive.org/work/qrlmjshh65cfddfvw4x3lbhb44
We argue that in the strong turbulence phase, as opposed to the weak one, the Clebsch variables compactify to the sphere S_2 and are not observable as wave excitations. Various topologically nontrivial configurations of this confined Clebsch field are responsible for vortex sheets. Stability equations (CVS) for closed vortex surfaces (bubbles of Clebsch field) are derived and investigated. The exact non-compact solution for the stable vortex sheet family is presented. Compact solutions are proven not to exist by De Lellis and Brué. Asymptotic conservation of anomalous dissipation on stable vortex surfaces in the turbulent limit is discovered. We derive an exact formula for this anomalous dissipation as a surface integral of the square of velocity gap times the square root of minus local normal strain. Topologically stable time-dependent solutions, which we call Kelvinons, are introduced. They have a conserved velocity circulation around static loop; this makes them responsible for asymptotic PDF tails of velocity circulation, perfectly matching numerical simulations. The loop equation for circulation PDF as functional of the loop shape is derived and studied. This equation is exactly equivalent to the Schrödinger equation in loop space, with viscosity ν playing the role of Planck's constant. This equivalence opens the way for direct numerical simulation of turbulence on quantum computers. Kelvinons are fixed points of the loop equation at turbulent limit ν→ 0. Area law and the asymptotic scaling law for mean circulation at a large area are derived. The representation of the solution of the loop equation in terms of a singular stochastic equation for momentum loop trajectory is presented.Alexander Migdalwork_qrlmjshh65cfddfvw4x3lbhb44Mon, 28 Nov 2022 00:00:00 GMTA fundamental non-classical logic
https://scholar.archive.org/work/nbt4qkxni5ee3pnktefqhjhgfe
We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation. Finally, we discuss ways of extending these representations to lattices with a conditional or implication operation.Wesley H. Hollidaywork_nbt4qkxni5ee3pnktefqhjhgfeMon, 28 Nov 2022 00:00:00 GMTUnified Physics and Cosmology: the Theory of Everything
https://scholar.archive.org/work/z6bkz4quandsdnkmakzcjliu3m
Quantum Mechanics and General Relativity appears to be incompatible because we are using the wrong model of the universe. Nature does not use two separate rule-books, but uses two different viewpoints. To see how the conflicting demands of Quantum Mechanics and General Relativity can be easily satisfied, we need the true model of our universe. Ever since Hubble's law was discovered, scientists speculated that the analogy of an expanding balloon best described the shape of our universe. This view was rejected based on wrong assumptions and replaced by the presently accepted model of a flat and infinite universe, which is wrong! We are confident that we have measured the universe to be (3d) flat using two different methods. Unfortunately, neither method is capable of measuring the extrinsic curvature of a 3d hyper-surface: 1) We cannot measure the curvature of a 3 dimensional (hyper) surface using summation of angles in a triangle. That works for a 2 dimensional surface curving in the 3rd dimension. But for our case, we need the sum of solid angles (i.e. we need a tetrahedron, and not a triangle). The 'sum of angles of the triangle' checkup which we had applied to CMB (Cosmic Microwave Background) spots is bound to show that our universe is (3d) flat! 2) We cannot measure the curvature of our universe using the critical mass-energy density method of General Relativity (General Relativity can measure intrinsic curvature, but not the extrinsic curvature). That proves that the universe may not be 3d flat. But how can we be sure that it is curved? Here is another piece of clue which finally nails it: Our universe does have has a Center (although the Center does not lie anywhere in our 3d space). This can be easily proved: The Center of Mass equation is a powerful equation. In the vastness of our cosmos, we can consider each galaxy (or maybe a galaxy cluster) as a point mass. Even as the number of galaxies tend to infinity, we are still left with a single point center of mass. Simply invoking infinity i [...]Subhajit Waughwork_z6bkz4quandsdnkmakzcjliu3mMon, 28 Nov 2022 00:00:00 GMTReconnectads
https://scholar.archive.org/work/pxkol4hgdzattpwoikxseadwxm
We introduce a new operad-like structure that we call a reconnectad; the "input" of an element of a reconnectad is a finite simple graph, rather than a finite set, and "compositions" of elements are performed according to the notion of the reconnected complement of a subgraph. The prototypical example of a reconnectad is given by the collection of toric varieties of graph associahedra of Carr and Devadoss, with the structure operations given by inclusions of orbits closures. We develop the general theory of reconnectads, and use it to study the "wonderful reconnectad" assembled from homology groups of complex toric varieties of graph associahedra.Vladimir Dotsenko, Adam Keilthy, Denis Lyskovwork_pxkol4hgdzattpwoikxseadwxmMon, 28 Nov 2022 00:00:00 GMTBi-intermediate logics of trees and co-trees
https://scholar.archive.org/work/nf2zi4giizbnfg5wlmv5sp54tm
A bi-Heyting algebra validates the Gödel-Dummett axiom (p→ q)∨ (q→ p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension 𝖻𝗂-𝖫𝖢 of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice Λ(𝖻𝗂-𝖫𝖢) of extensions of 𝖻𝗂-𝖫𝖢. We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of 𝖻𝗂-𝖫𝖢. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of 𝖻𝗂-𝖫𝖢. We introduce a sequence of co-trees, called the finite combs, and show that a logic in 𝖻𝗂-𝖫𝖢 is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest non-locally tabular extension of 𝖻𝗂-𝖫𝖢 and consequently, a unique pre-locally tabular extension of 𝖻𝗂-𝖫𝖢. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.N. Bezhanishvili, M. Martins, T. Moraschiniwork_nf2zi4giizbnfg5wlmv5sp54tmSun, 27 Nov 2022 00:00:00 GMTA variational principle in the parametric geometry of numbers
https://scholar.archive.org/work/2aouitjuz5ahtmagbkatb52puy
We extend the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establish a new connection to the metric theory via a variational principle that computes fractal dimensions of a variety of sets of number-theoretic interest. The proof relies on two novel ingredients: a variant of Schmidt's game capable of computing the Hausdorff and packing dimensions of any set, and the notion of templates, which generalize Roy's rigid systems. In particular, we compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of Dani's correspondence principle, the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Other applications include quantitative strengthenings of theorems due to Cheung and Moshchevitin, which originally resolved conjectures due to Starkov and Schmidt respectively; as well as dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing partial results due to Baker, Bugeaud, Cheung, Chevallier, Dodson, Laurent and Rynne.Tushar Das, Lior Fishman, David Simmons, Mariusz Urbańskiwork_2aouitjuz5ahtmagbkatb52puySat, 26 Nov 2022 00:00:00 GMTWeighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics
https://scholar.archive.org/work/yd2yjv4auff3hegja4ji5ffsse
We show that a compact weighted extremal Kahler manifold (as defined by the third named author) has coercive weighted Mabuchi energy with respect to a maximal complex torus in the reduced group of complex automorphisms. This provides a vast extension and a unification of a number of results concerning Kahler metrics satisfying special curvature conditions, including constant scalar curvature Kahler metrics, extremal Kahler metrics, Kahler-Ricci solitons and their weighted extensions. Our result implies the strict positivity of the weighted Donaldson-Futaki invariant of any non-product equivariant smooth K\"ahler test configuration with reduced central fibre, a property also known as weighted K-polystability on such test configurations. For a class of fibre-bundles, we use our result in conjunction with the recent results of Chen-Cheng, He, and Han-Li in order to characterize the existence of extremal Kahler metrics and Calabi-Yau cones associated to the total space, in terms of the coercivity of the weighted Mabuchi energy of the fibre. In particular, this yields an existence result for Sasaki-Einstein metrics on Fano toric fibrations, extending the results of Futaki-Ono-Wang in the toric Fano case, and of Mabuchi-Nakagawa in the case of Fano projective line bundles.Vestislav Apostolov, Simon Jubert, Abdellah Lahdiliwork_yd2yjv4auff3hegja4ji5ffsseSat, 26 Nov 2022 00:00:00 GMTDecomposition, Trivially-Acting Symmetries, and Topological Operators
https://scholar.archive.org/work/ww4lewt5izepdfsmytj4ou4uty
Trivially-acting symmetries in two-dimensional conformal field theory include twist fields of dimension zero which are local topological operators. We investigate the consequences of regarding these operators as part of the global symmetry of the theory. That is, we regard such a symmetry as a mix of topological defect lines (TDLs) and topological point operators (TPOs). TDLs related by a trivially-acting symmetry can join at a TPO to form non-trivial two-way junctions. Upon gauging, the local operators at those junctions can become vacua in a disjoint union of theories. Examining the behavior of the TPOs under gauging therefore allows us to refine decomposition by tracking the trivially-acting symmetries of each universe. Mixed anomalies between the TDLs and TPOs provide discrete torsion-like phases for the partition functions of these orbifolds, modifying the resulting decomposition. This framework also readily allows for the consideration of trivially-acting non-invertible symmetries.Daniel Robbins, Eric Sharpe, Thomas Vandermeulenwork_ww4lewt5izepdfsmytj4ou4utyFri, 25 Nov 2022 00:00:00 GMTA dichotomy theory for height functions
https://scholar.archive.org/work/je55i7cxxzb43ar7oh4hsaktmu
Height functions are random functions on a given graph, in our case integer-valued functions on the two-dimensional square lattice. We consider gradient potentials which (informally) lie between the discrete Gaussian and solid-on-solid model (inclusive). The phase transition in this model, known as the roughening transition, Berezinskii-Kosterlitz-Thouless transition, or localisation-delocalisation transition, was established rigorously in the 1981 breakthrough work of Fröhlich and Spencer. It was not until 2005 that Sheffield derived continuity of the phase transition. First, we establish sharpness, in the sense that covariances decay exponentially in the localised phase. Second, we show that the model is delocalised at criticality, in the sense that the set of potentials inducing localisation is open in a natural topology. Third, we prove that the pointwise variance of the height function is at least clog n in the delocalised regime, where n is the distance to the boundary, and where c>0 denotes a universal constant. This implies that the effective temperature of any potential cannot lie in the interval (0,c) (whenever it is well-defined), and jumps from 0 to at least c at the critical point. We call this range of forbidden values the effective temperature gap.Piet Lammerswork_je55i7cxxzb43ar7oh4hsaktmuFri, 25 Nov 2022 00:00:00 GMTA Representation-Theoretic Approach to qq-Characters
https://scholar.archive.org/work/6xejk5wplbcczcf2dfza6omcsm
We raise the question of whether (a slightly generalized notion of) qq-characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal gl 1 algebra, geometric engineering of adjoint matter produces an explicit vertex operator RR which computes certain qq-characters, namely Hirzebruch χ y -genera, completely analogously to how the R-matrix R computes q-characters. We give a geometric proof of the independence of preferred direction for the refined vertex in this and more general non-toric settings.Henry Liu, University of Oxford, UKwork_6xejk5wplbcczcf2dfza6omcsmThu, 24 Nov 2022 00:00:00 GMTQuantum cohomology of Grassmannian and unitary Dyson Brownian motion
https://scholar.archive.org/work/keq65xf4gne3hmxkfvc665q5ji
We study a class of commuting Markov kernels whose simplest element describes the movement of k particles on a discrete circle of size n conditioned to not intersect each other. Such Markov kernels are related to the quantum cohomology ring of Grassmannian, which is an algebraic object counting analytic maps from ℙ^1(ℂ) to the Grassmannian space of k-dimensional vector subspaces of ℂ^n with prescribed constraints at some points of ℙ^1(ℂ). We obtain a Berry-Esseen theorem and a local limit theorem for an arbitrary product of approximately n^2 Markov kernels belonging to the above class, when k is fixed. As a byproduct of those results, we derive asymptotic formulas for the quantum cohomology ring of the Grassmannian in terms of the heat kernel on SU (k).Jérémie Guilhot, Cédric Lecouvey, Pierre Tarragowork_keq65xf4gne3hmxkfvc665q5jiWed, 23 Nov 2022 00:00:00 GMTForeword
https://scholar.archive.org/work/gjz4xbea7ffsvd2yqyyus23zqi
George Zoupanoswork_gjz4xbea7ffsvd2yqyyus23zqiWed, 23 Nov 2022 00:00:00 GMTSpatial mixing and the random-cluster dynamics on lattices
https://scholar.archive.org/work/kz4iblu3inerrnrehjfcfz7vdi
An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high-temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random-cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this paper, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random-cluster dynamics in finite subsets of ℤ^d for general d≥ 2. These imply a host of new results, including optimal O(Nlog N) mixing for the random cluster dynamics on torii and boxes on N vertices in ℤ^d at all high temperatures and at sufficiently low temperatures, and for large values of q quasi-polynomial (or quasi-linear when d=2) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst-case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on ℤ^d for general d.Reza Gheissari, Alistair Sinclairwork_kz4iblu3inerrnrehjfcfz7vdiWed, 23 Nov 2022 00:00:00 GMTImaging Nanoscale Spin Textures with Scanning Diamond Quantum Microscopy
https://scholar.archive.org/work/cp7slo5flrdn5pb6a6z7bhob5y
Magnetic materials play a key role in the advancement of modern technology, from data storage mediums and giant magnetoresistance read heads for information processing, to medical applications like drug delivery and magnetic resonance imaging. Their research and development have progressed towards ever-reducing spatial and magnetic footprints, largely motivated by promising novel properties emerging at reduced dimensions and the miniaturization of technology. Sustained research progress will therefore require the timely development of scalable probing techniques with enhanced sensitivities. State-of-the-art material research requirements of non-perturbative measurements, with resolutions of at least a hundred nanometers, and sensitivities down to single magnetic atomic layers, are becoming increasingly inaccessible by conventional spatial imaging techniques. In this respect, quantum sensing via nitrogen-vacancy centres in diamond promises to address contemporary and prospective imaging needs. Employed on a scanning probe platform, the scanning diamond quantum microscope is capable of nanoscale imaging with an unprecedented magnetic sensitivity down to the nano-Tesla regime and minimal magnetic back-action on the target. Scanning diamond quantum microscopy has been recently used to image a number of novel magnetic systems, however a large subset of materials, including various antiferromagnets and two-dimensional magnets, remains unexplored. Moreover, the technique is capable of secondary sensing modalities that are relatively underdeveloped and require systematic studies to understand their potentials and limitations. This thesis describes the work I have undertaken towards implementing a scanning diamond quantum microscope operational in cryogenic and ambient environments. I utilise this setup to develop imaging capabilities to study nanoscale non-trivial spin textures in topological (anti)ferromagnetic materials. The capabilities developed and physical insights revealed in this work establish the diamond quantu [...]Anthony Tan Kok Cheng, Apollo-University Of Cambridge Repository, Mete Ataturework_cp7slo5flrdn5pb6a6z7bhob5yTue, 22 Nov 2022 00:00:00 GMTAsymptotically isometric codes for holography
https://scholar.archive.org/work/h7dr3dsffze23actqdx3uz52cm
The holographic principle suggests that the low energy effective field theory of gravity, as used to describe perturbative quantum fields about some background has far too many states. It is then natural that any quantum error correcting code with such a quantum field theory as the code subspace is not isometric. We discuss how this framework can naturally arise in an algebraic QFT treatment of a family of CFT with a large-N limit described by the single trace sector. We show that an isometric code can be recovered in the N →∞ limit when acting on fixed states in the code Hilbert space. Asymptotically isometric codes come equipped with the notion of simple operators and nets of causal wedges. While the causal wedges are additive, they need not satisfy Haag duality, thus leading to the possibility of non-trivial entanglement wedge reconstructions. Codes with complementary recovery are defined as having extensions to Haag dual nets, where entanglement wedges are well defined for all causal boundary regions. We prove an asymptotic version of the information disturbance trade-off theorem and use this to show that boundary theory causality is maintained by net extensions. We give a characterization of the existence of an entanglement wedge extension via the asymptotic equality of bulk and boundary relative entropy or modular flow. While these codes are asymptotically exact, at fixed N they can have large errors on states that do not survive the large-N limit. This allows us to fix well known issues that arise when modeling gravity as an exact codes, while maintaining the nice features expected of gravity, including, among other things, the emergence of non-trivial von Neumann algebras of various types.Thomas Faulkner, Min Liwork_h7dr3dsffze23actqdx3uz52cmTue, 22 Nov 2022 00:00:00 GMTThe Role of Cytonemes and Diffusive Transport in the Establishment of Morphogen Gradients
https://scholar.archive.org/work/35klvfrqxbhshb7paq5gmcmcmq
Spatial distributions of morphogens provide positional information in developing systems, but how the distributions are established and maintained remains an open problem. Transport by diffusion has been the traditional mechanism, but recent experimental work has shown that cells can also communicate by filopodia-like structures called cytonemes that make direct cell-to-cell contacts. Here we investigate the roles each may play individually in a complex tissue and how they can jointly establish a reliable spatial distribution of a morphogen.Jay A. Stotsky, Hans G. Othmerwork_35klvfrqxbhshb7paq5gmcmcmqTue, 22 Nov 2022 00:00:00 GMTMarginal independence and an approximation to strong subadditivity
https://scholar.archive.org/work/krgsnqlmazg2hj5qc6lpbrhiiu
Given a multipartite quantum system, what are the possible ways to impose mutual independence among some of the parties, and the presence of correlations among others, such that there exists a quantum state which satisfies these demands? This question and the related notion of a pattern of marginal independence (PMI) were introduced in arXiv:1912.01041, and then argued in arXiv:2204.00075 to distill the essential information for the derivation of the holographic entropy cone. Here we continue the general analysis initiated in arXiv:1912.01041, focusing in particular on the implications of the necessary condition for the saturation of subadditivity. This condition, which we dub Klein's condition, will be interpreted as an approximation to strong subadditivity for PMIs. We show that for an arbitrary number of parties, the set of PMIs compatible with this condition forms a lattice, and we investigate several of its structural properties. In the discussion we highlight the role played by the meet-irreducible elements in the solution of the quantum marginal independence problem, and by the coatoms in the holographic context. To make the presentation self-contained, we review the key ingredients from lattice theory as needed.Temple He, Veronika E. Hubeny, Massimiliano Rotawork_krgsnqlmazg2hj5qc6lpbrhiiuMon, 21 Nov 2022 00:00:00 GMT