scholarly journals The ZX&-calculus: A complete graphical calculus for classical circuits using spiders

2021 ◽  
Vol 340 ◽  
pp. 60-90
Author(s):  
Cole Comfort
Keyword(s):  
Graphical Calculus

scholarly journals Graphical calculus for Gaussian pure states

2011 ◽  
Vol 83 (4) ◽  
Author(s):  
Nicolas C. Menicucci ◽  
Steven T. Flammia ◽  
Peter van Loock
Keyword(s):  
Graphical Calculus ◽  
Pure States

scholarly journals A diagrammatic approach to link invariants of finite degree

2004 ◽  
Vol 94 (2) ◽  
pp. 295 ◽  
Author(s):  
Olof-Petter Östlund
Keyword(s):  
Finite Degree ◽  
Explicit Formulas ◽  
Connected Sum ◽  
Knot Invariants ◽  
Graphical Calculus ◽  
Link Invariants ◽  
Low Degree ◽  
Coalgebra Structure ◽  
Arrow Diagrams

In [5] M. Polyak and O. Viro developed a graphical calculus of diagrammatic formulas for Vassiliev link invariants, and presented several explicit formulas for low degree invariants. M. Goussarov [2] proved that this arrow diagram calculus provides formulas for all Vassiliev knot invariants. The original note [5] contained no proofs, and it also contained some minor inaccuracies. This paper fills the gap in literature by presenting the material of [5] with all proofs and details, in a self-contained form. Furthermore, a compatible coalgebra structure, related to the connected sum of knots, is introduced on the algebra of based arrow diagrams with one circle.

scholarly journals LOW ENTROPY OUTPUT STATES FOR PRODUCTS OF RANDOM UNITARY CHANNELS

2013 ◽  
Vol 02 (01) ◽  
pp. 1250018 ◽  
Author(s):  
BENOÎT COLLINS ◽  
MOTOHISA FUKUDA ◽  
ION NECHITA
Keyword(s):  
Free Probability ◽  
Bell State ◽  
Quantum Channels ◽  
Unitary Operators ◽  
Graphical Calculus ◽  
Additivity Problems ◽  
Communication Capacity ◽  
The Difference ◽  
Low Entropy ◽  
Math Phys

In this paper, we study the behavior of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings [Superadditivity of communication capacity using entangled inputs, Nat. Phys.5 (2009) 255–257] and Hayden–Winter [Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys.284(1) (2008) 263–280] to the additivity problems. In particular, we study in depth the difference of behavior between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in [B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Comm. Math. Phys.297(2) (2010) 345–370].

scholarly journals Lorentzian spin foam amplitudes: graphical calculus and asymptotics

2010 ◽  
Vol 27 (16) ◽  
pp. 165009 ◽  
Author(s):  
John W Barrett ◽  
R J Dowdall ◽  
Winston J Fairbairn ◽  
Frank Hellmann ◽  
Roberto Pereira
Keyword(s):  
Graphical Calculus ◽  
Spin Foam

scholarly journals FEYNMAN DIAGRAMS VIA GRAPHICAL CALCULUS

2002 ◽  
Vol 11 (07) ◽  
pp. 1095-1131 ◽  
Author(s):  
DOMENICO FIORENZA ◽  
RICCARDO MURRI
Keyword(s):  
Quantum Gravity ◽  
Asymptotic Expansions ◽  
Feynman Diagrams ◽  
Ribbon Graphs ◽  
Graphical Calculus

We use Reshetikhin-Turaev graphical calculus to define Feynman diagrams and prove that asymptotic expansions of Gaussian integrals can be written as a sum over a suitable family of graphs. We discuss how different kinds of interactions give rise to different families of graphs. In particular, we show how symmetric and cyclic interactions lead to "ordinary" and "ribbon" graphs respectively. As an example, the 't Hooft-Kontsevich model for 2D quantum gravity is treated in some detail.

scholarly journals FROM PLANAR GRAPHS TO EMBEDDED GRAPHS - A NEW APPROACH TO KAUFFMAN AND VOGEL'S POLYNOMIAL

2000 ◽  
Vol 09 (08) ◽  
pp. 975-986 ◽  
Author(s):  
RUI PEDRO CARPENTIER
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Direct Proof ◽  
Connected Components ◽  
Embedded Graphs ◽  
New Approach ◽  
Graphical Calculus ◽  
Embedded Graph

In [4] Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph G a polynomial, denoted [G], in three variables, A, B and a, satisfying the skein relations: [Formula: see text] and is defined in terms of a state-sum and the Dubrovnik polynomial for links. Using the graphical calculus of [4] it is shown that the polynomial of a planar graph can be calculated recursively from that of planar graphs with less vertices, which also allows the polynomial of an embedded graph to be calculated without resorting to links. The same approach is used to give a direct proof of uniqueness of the (normalized) polynomial restricted to planar graphs. In the case B=A-1 and a=A, it is proved that for a planar graph G we have [G]=2c-1(-A-A-1)v, where c is the number of connected components of G and v is the number of vertices of G. As a corollary, a necessary, but not sufficient, condition is obtained for an embedded graph to be ambient isotopic to a planar graph. In an appendix it is shown that, given a polynomial for planar graphs satisfying the graphical calculus, and imposing the first skein relation above, the polynomial extends to a rigid vertex regular isotopy invariant for embedded graphs, satisfying the remaining skein relations. Thus, when existence of the planar polynomial is guaranteed, this provides a direct way, not depending on results for the Dubrovnik polynomial, to show consistency of the polynomial for embedded graphs.

scholarly journals Dagger linear logic for categorical quantum mechanics

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Robin Cockett ◽  
Cole Comfort ◽  
Priyaa Srinivasan
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Physics ◽  
Linear Logic ◽  
Infinite Dimensional ◽  
Graphical Calculus ◽  
Finite Dimensional ◽  
Categorical Quantum Mechanics ◽  
Definition Of ◽  
Categorical Semantics

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.

Categories for Quantum Theory

2019 ◽  
Author(s):  
Chris Heunen ◽  
Jamie Vicary
Keyword(s):  
Quantum Theory ◽  
Category Theory ◽  
Hopf Algebras ◽  
Quantum Teleportation ◽  
Graphical Calculus ◽  
Frobenius Structures ◽  
Nonstandard Models ◽  
Background Material ◽  
Quantum Phenomena ◽  
High Level

Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to understand many high-level quantum phenomena. Here, we lay the foundations for this categorical quantum mechanics, with an emphasis on the graphical calculus that makes computation intuitive. We describe superposition and entanglement using biproducts and dual objects, and show how quantum teleportation can be studied abstractly using these structures. We investigate monoids, Frobenius structures and Hopf algebras, showing how they can be used to model classical information and complementary observables. We describe the CP construction, a categorical tool to describe probabilistic quantum systems. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Previous knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text, we point out links with many other areas, such as representation theory, topology, quantum algebra, knot theory and probability theory, and present nonstandard models including sets and relations. All results are stated rigorously and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.

Sign in / Sign up

Export Citation Format

Share Document