Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Protocols involving Multiparticle Entanglement and their Representations in the zx-calculus.

Anne Hillebrand

Published Jan 1, 2011• 8 citations
OpenAlex

Abstract

Quantum entanglement, described by Einstein as “spooky action at a distance”, is a key resource in many quantum protocols, like quantum teleportation and quantum cryptography. Yet entanglement makes protocols presented in Dirac notation difficult to follow and check. This is why Coecke nad Duncan have introduced a diagrammatic language for multi-qubit systems, called the red/green calculus or the zx-calculus [23]. This diagrammatic notation is both intuitive and formally rigorous. It is a simple, graphical, high level language that emphasises the composition of systems and naturally captures the essentials of quantum mechanics. One crucial feature that will be exploited here is the encoding of complementary observables and corresponding phase shifts. Reasoning is done by rewriting diagrams, i.e. locally replacing some part of a diagram. Diagrams are defined by their topology only; the number of inputs and outputs and the way they are connected. This exemplifies the ‘flow’ of information. For protocols involving multipartite entangled states, such as the GreenbergerHorne-Zeilinger and W -state, it will be shown that the zx-calculus provides a relatively easy and more intuitive presentation. Moreover, in this representation it is easier to check that protocols are correct. Protocols that will be discussed in detail are quantum teleportation, quantum cryptography, leader election, superdense coding and quantum direct communication with multipartite entangled states.

Related papers

Superdense Coding with GHZ and Quantum Key Distribution with W in the ZX-calculus

Anne Hillebrand

Quantum entanglement is a key resource in many quantum protocols, such as quantum teleportation and quantum cryptography. Yet entanglement makes protocols presented in Dirac notation difficult to verify. This is why Coecke and Duncan have introduced a diagrammatic language for quantum protocols, called the ZX-calculus. This diagrammatic notation is both intuitive and formally rigorous. It is a simple, graphical, high level language that emphasises the composition of systems and naturally captures the essentials of quantum mechanics. In the author's MSc thesis it has been shown for over 25 quantum protocols that the ZX-calculus provides a relatively easy and more intuitive presentation. Moreover, the author embarked on the task to apply categorical quantum mechanics on quantum security; earlier works did not touch anything but Bennett and Brassard's quantum key distribution protocol, BB84. Superdense coding with the Greenberger-Horne-Zeilinger state and quantum key distribution with the W-state are presented in the ZX-calculus in this paper.

arXiv:1210.0650•Electronic Proceedings in Theoretical Computer Science•Oct 1, 2012•11 citations

A Graphical Approach to Measurement-BasedQuantum Computing

Ross Duncan

Quantum computations are easily represented in the graphical notation known as the zx-calculus, also known as the red–green calculus. We demonstrate its use in reasoning about measurement-based quantum computing, where the graphical syntax directly captures the structure of the entangled states used to represent computations, and show that the notion of information flow within the entangled states gives rise to rewriting strategies for proving the correctness of quantum programs.

arXiv:1203.6242•Oxford University Press eBooks•Feb 21, 2013•27 citations

ZX-Calculus and Extended Hypergraph Rewriting Systems I: A Multiway Approach to Categorical Quantum Information Theory

Jonathan Gorard, Manojna Namuduri, Xerxes D. Arsiwalla

Categorical quantum mechanics and the Wolfram model offer distinct but complementary approaches to studying the relationship between diagrammatic rewriting systems over combinatorial structures and the foundations of physics; the objective of the present article is to begin elucidating the formal correspondence between the two methodologies in the context of the ZX-calculus formalism of Coecke and Duncan for reasoning diagrammatically about linear maps between qubits. After briefly summarizing the relevant formalisms, and presenting a categorical formulation of the Wolfram model in terms of adhesive categories and double-pushout rewriting systems, we illustrate how the diagrammatic rewritings of the ZX-calculus can be embedded and realized within the broader context of Wolfram model multiway systems, and illustrate some of the capabilities of the software framework (ZXMultiwaySystem) that we have developed specifically for this purpose. Finally, we present a proof (along with an explicitly computed example) based on the methods of Dixon and Kissinger that the multiway evolution graphs and branchial graphs of the Wolfram model are naturally endowed with a monoidal structure based on rulial composition that is, furthermore, compatible with the monoidal product of ZX-diagrams.

arXiv:2010.02752•Oct 5, 2020•10 citations

Introduction to ZX Calculus

Subhojit Halder

While the quantum circuit model remains the dominant framework for representing and executing quantum algorithms, it is not always the most optimal framework for representing the quantum circuit’s deeper structure. In a standard circuit diagram, computation is shown mainly as a time ordered sequence of gates acting on qubit wires. ZX calculus, by contrast, represents the same process using two families of generators, the Z spiders and the X spiders, together with wires and Hadamard structure, so that algebraic identities, entanglement structure, and rewrite based simplifications become visible directly in the diagram itself. ZX calculus is not another visual representation for the same quantum circuit we have been using, but a graphical calculus in which diagrams carry both computational meaning and transformation rules. This is also the reason for the name ZX ; the formalism is built around the interaction between Z -type and X -type observables, phases, and basis changes. By turning matrix manipulations into diagrammatic rewrites, ZX calculus reduces much of the redundancies and abstract nature usually associated with quantum computation and provides a more structural language for understanding, optimizing, and verifying quantum processes.

Apress eBooks•Jan 1, 2026•0 citations

A Diagrammatic Axiomatisation for Qubit Entanglement

Amar Hadzihasanovic

Diagrammatic techniques for reasoning about monoidal categories provide an intuitive understanding of the symmetries and connections of interacting computational processes. In the context of categorical quantum mechanics, Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used as the building blocks of a new graphical calculus, aimed at a diagrammatic classification of multipartite qubit entanglement that would highlight the communicational properties of quantum states, and their potential uses in cryptographic schemes. In this paper, we present a full graphical axiomatisation of the relations between GHZ and W: the ZW calculus. This refines a version of the preexisting ZX calculus, while keeping its most desirable characteristics: undirectedness, a large degree of symmetry, and an algebraic underpinning. We prove that the ZW calculus is complete for the category of free abelian groups on a power of two generators - "qubits with integer coefficients" - and provide an explicit normalisation procedure.

arXiv:1501.07082•Jan 28, 2015•0 citations

Quantum Information Flow under String-Diagram Rewriting

Yi-Yu Lin, Chen-Ye Li

We revisit the notion of ``quantum information flow'' introduced in Bob Coecke's early work and seek to give it an explicit string-diagrammatic formalization. Given a semantics preserving string-diagram rewriting sequence, we first distinguish apparent through-paths, which depend on the current graphical presentation, from genuine through paths, which can be compatibly inherited through successive rewrites to a terminal decoupled bare wire factor. We then formally define a ``Coecke flow line'' in terms of this compatible inheritance relation. In particular, we use the ZX calculus, i.e., the ZX string diagram rewriting system, to illustrate the resulting formalism. In the physical setting of quantum protocols, a Coecke flow can be interpreted as a constrained, quasi-local, line-like presentation of a target bare wire morphism factor within the protocol string diagram.

quant-pharXiv:2608.09823•Aug 11, 2026•0 citations
Quantum entanglement, described by Einstein as “spooky action at a distance”, is a key resource in many quantum protocols, like quantum teleportation and quantum cryptography. Yet entanglement makes protocols presented in Dirac notation difficult to follow and check. This is why Coecke nad Duncan have introduced a diagrammatic language for multi-qubit systems, called the red/green calculus or the zx-calculus [23]. This diagrammatic notation is both intuitive and formally rigorous. It is a simple, graphical, high level language that emphasises the composition of systems and naturally captures the essentials of quantum mechanics. One crucial feature that will be exploited here is the encoding of complementary observables and corresponding phase shifts. Reasoning is done by rewriting diagrams, i.e. locally replacing some part of a diagram. Diagrams are defined by their topology only; the number of inputs and outputs and the way they are connected. This exemplifies the ‘flow’ of information. For protocols involving multipartite entangled states, such as the GreenbergerHorne-Zeilinger and W -state, it will be shown that the zx-calculus provides a relatively easy and more intuitive presentation. Moreover, in this representation it is easier to check that protocols are correct. Protocols that will be discussed in detail are quantum teleportation, quantum cryptography, leader election, superdense coding and quantum direct communication with multipartite entangled states.
Quantum entanglement is a key resource in many quantum protocols, such as quantum teleportation and quantum cryptography. Yet entanglement makes protocols presented in Dirac notation difficult to verify. This is why Coecke and Duncan have introduced a diagrammatic language for quantum protocols, called the ZX-calculus. This diagrammatic notation is both intuitive and formally rigorous. It is a simple, graphical, high level language that emphasises the composition of systems and naturally captures the essentials of quantum mechanics. In the author's MSc thesis it has been shown for over 25 quantum protocols that the ZX-calculus provides a relatively easy and more intuitive presentation. Moreover, the author embarked on the task to apply categorical quantum mechanics on quantum security; earlier works did not touch anything but Bennett and Brassard's quantum key distribution protocol, BB84. Superdense coding with the Greenberger-Horne-Zeilinger state and quantum key distribution with the W-state are presented in the ZX-calculus in this paper.
Categorical quantum mechanics and the Wolfram model offer distinct but complementary approaches to studying the relationship between diagrammatic rewriting systems over combinatorial structures and the foundations of physics; the objective of the present article is to begin elucidating the formal correspondence between the two methodologies in the context of the ZX-calculus formalism of Coecke and Duncan for reasoning diagrammatically about linear maps between qubits. After briefly summarizing the relevant formalisms, and presenting a categorical formulation of the Wolfram model in terms of adhesive categories and double-pushout rewriting systems, we illustrate how the diagrammatic rewritings of the ZX-calculus can be embedded and realized within the broader context of Wolfram model multiway systems, and illustrate some of the capabilities of the software framework (ZXMultiwaySystem) that we have developed specifically for this purpose. Finally, we present a proof (along with an explicitly computed example) based on the methods of Dixon and Kissinger that the multiway evolution graphs and branchial graphs of the Wolfram model are naturally endowed with a monoidal structure based on rulial composition that is, furthermore, compatible with the monoidal product of ZX-diagrams.
While the quantum circuit model remains the dominant framework for representing and executing quantum algorithms, it is not always the most optimal framework for representing the quantum circuit’s deeper structure. In a standard circuit diagram, computation is shown mainly as a time ordered sequence of gates acting on qubit wires. ZX calculus, by contrast, represents the same process using two families of generators, the Z spiders and the X spiders, together with wires and Hadamard structure, so that algebraic identities, entanglement structure, and rewrite based simplifications become visible directly in the diagram itself. ZX calculus is not another visual representation for the same quantum circuit we have been using, but a graphical calculus in which diagrams carry both computational meaning and transformation rules. This is also the reason for the name ZX ; the formalism is built around the interaction between Z -type and X -type observables, phases, and basis changes. By turning matrix manipulations into diagrammatic rewrites, ZX calculus reduces much of the redundancies and abstract nature usually associated with quantum computation and provides a more structural language for understanding, optimizing, and verifying quantum processes.
Diagrammatic techniques for reasoning about monoidal categories provide an intuitive understanding of the symmetries and connections of interacting computational processes. In the context of categorical quantum mechanics, Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used as the building blocks of a new graphical calculus, aimed at a diagrammatic classification of multipartite qubit entanglement that would highlight the communicational properties of quantum states, and their potential uses in cryptographic schemes. In this paper, we present a full graphical axiomatisation of the relations between GHZ and W: the ZW calculus. This refines a version of the preexisting ZX calculus, while keeping its most desirable characteristics: undirectedness, a large degree of symmetry, and an algebraic underpinning. We prove that the ZW calculus is complete for the category of free abelian groups on a power of two generators - "qubits with integer coefficients" - and provide an explicit normalisation procedure.
We revisit the notion of ``quantum information flow'' introduced in Bob Coecke's early work and seek to give it an explicit string-diagrammatic formalization. Given a semantics preserving string-diagram rewriting sequence, we first distinguish apparent through-paths, which depend on the current graphical presentation, from genuine through paths, which can be compatibly inherited through successive rewrites to a terminal decoupled bare wire factor. We then formally define a ``Coecke flow line'' in terms of this compatible inheritance relation. In particular, we use the ZX calculus, i.e., the ZX string diagram rewriting system, to illustrate the resulting formalism. In the physical setting of quantum protocols, a Coecke flow can be interpreted as a constrained, quasi-local, line-like presentation of a target bare wire morphism factor within the protocol string diagram.