23 research outputs found
Semiclassical asymptotics and entropy
We study the entanglement of quantum states associated with submanifolds of
Kaehler manifolds. As a motivating example, we discuss the semiclassical
asymptotics of entanglement entropy of pure states on the two dimensional
sphere with the standard metric.Comment: To appear in Proceedings of QTS-12 (Prague, 2023
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Convex and Algebraic Geometry
The subjects of convex and algebraic geometry meet primarily in the theory of toric varieties. Toric geometry is the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Recent developments in toric geometry that were discussed during the workshop include applications to mirror symmetry, motivic integration and hypergeometric systems of PDE’s, as well as deformations of (unions of) toric varieties and relations to tropical geometry
Eigenvalue Distributions of Reduced Density Matrices
Given a random quantum state of multiple distinguishable or indistinguishable
particles, we provide an effective method, rooted in symplectic geometry, to
compute the joint probability distribution of the eigenvalues of its one-body
reduced density matrices. As a corollary, by taking the distribution's support,
which is a convex moment polytope, we recover a complete solution to the
one-body quantum marginal problem. We obtain the probability distribution by
reducing to the corresponding distribution of diagonal entries (i.e., to the
quantitative version of a classical marginal problem), which is then determined
algorithmically. This reduction applies more generally to symplectic geometry,
relating invariant measures for the coadjoint action of a compact Lie group to
their projections onto a Cartan subalgebra, and can also be quantized to
provide an efficient algorithm for computing bounded height Kronecker and
plethysm coefficients.Comment: 51 pages, 7 figure
Graph Representation of Topological Stabilizer States
Topological quantum states, especially these in topological stabilizer quantum error correction codes, are currently the focus of intense activity because of their potential for fault-tolerant operations. While every stabilizer state maps to a graph state under local Clifford operations, the graphs associated with topological stabilizer codes remain unknown. In this thesis, I show that the toric code graph is composed of only two kinds of subgraphs: star graphs and half graphs. The topological order of the toric code is identified with the existence of multiple star graphs, which reveals a nice connection between repetition codes and the toric code. The graph structure readily yields a log-depth and a constant-depth (including ancillae) circuit for state preparation. Next, I derive the necessary and sufficient conditions for a family of graph states to be in TQO-1, a class of quantum error correction code states whose code distance scales macroscopically with the number of physical qubits. Using these criteria, I consider a number of specific graph families, including the star and complete graphs, and the line graphs of complete and completely bipartite graphs, and discuss which are topologically ordered and how to construct the codewords. The formalism is then employed to construct several codes with macroscopic distance, including a three-dimensional topological code generated by local stabilizers that also has a macroscopic number of encoded logical qubits. Last, the connection between the characterization of topological order using graph theory and the hierarchy of topological order is analyzed
Quantum Correlations in the Minimal Scenario
In the minimal scenario of quantum correlations, two parties can choose from
two observables with two possible outcomes each. Probabilities are specified by
four marginals and four correlations. The resulting four-dimensional convex
body of correlations, denoted , is fundamental for quantum
information theory. It is here studied through the lens of convex algebraic
geometry. We review and systematize what is known and add many details,
visualizations, and complete proofs. A new result is that is
isomorphic to its polar dual. The boundary of consists of
three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds
of exposed extreme points. These share all basic properties with the usual
maximally CHSH-violating correlations. These patches are separated by cubic
surfaces of non-exposed extreme points. We provide a trigonometric
parametrization of all extreme points, along with their exposing Tsirelson
inequalities and quantum models. All non-classical extreme points (exposed or
not) are self-testing, i.e., realized by an essentially unique quantum model.
Two principles, which are specific to the minimal scenario, allow a quick and
complete overview: The first is the pushout transformation, the application of
the sine function to each coordinate. This transforms the classical polytope
exactly into the correlation body , also identifying the boundary
structures. The second principle, self-duality, reveals the polar dual, i.e.,
the set of all Tsirelson inequalities satisfied by all quantum correlations.
The convex body includes the classical correlations, a cross
polytope, and is contained in the no-signaling body, a 4-cube. These polytopes
are dual to each other, and the linear transformation realizing this duality
also identifies with its dual.Comment: We also discuss the sets of correlations achieved with fixed Hilbert
space dimension, fixed state or fixed observable
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
The minimal canonical form of a tensor network
Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable - in contrast to the well-known undecidability for PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization