The GHZ/W-calculus contains rational arithmetic

Merry, Alex; Roy, Shibdas

doi:10.4204/eptcs.52.4

Cited by 14 publications

(9 citation statements)

References 18 publications

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“…In this paper, we set out to improve and complete the axiomatisation started in [11], [12] of the relations between the GHZ and W 3-qubit quantum states.…”

Section: Discussionmentioning

confidence: 99%

“…In particular, it is provable in the ZW calculus that , and , form a strongly complementary pair in the sense of [22]. Moreover, the ZW calculus completes the axiomatisation of the GHZ/W calculus with additive inverses, as started in [11], and can be used to encode rational arithmetic as suggested there.…”

mentioning

confidence: 90%

“…In particular, a diagrammatic theory of Frobenius algebras is the basis of the ZX calculus [9], whose completeness for the important stabiliser fragment of quantum mechanics has recently been proven [10]. In [11], [12], a graphical axiomatisation of the relations between the GHZ and W algebras was started, with a similar calculus in mind; but this was not brought to completion, and only results about universality and classification were obtained.…”

Section: Introductionmentioning

confidence: 99%

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A Diagrammatic Axiomatisation for Qubit Entanglement

Hadzihasanovic

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract-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.

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“…In this paper, we set out to improve and complete the axiomatisation started in [11], [12] of the relations between the GHZ and W 3-qubit quantum states.…”

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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A Diagrammatic Axiomatisation for Qubit Entanglement

Hadzihasanovic

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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“…Other interesting instances of the special and the antispecial requirements have been considered in [11,19].…”

Section: Lemma 43mentioning

confidence: 99%

(Modular) Effect Algebras are Equivalent to (Frobenius) Antispecial Algebras

Pavlović

Seidel

2017

Electron. Proc. Theor. Comput. Sci.

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Effect algebras are one of the generalizations of Boolean algebras proposed in the quest for a quantum logic. Frobenius algebras are a tool of categorical quantum mechanics, used to present various families of observables in abstract, often nonstandard frameworks. Both effect algebras and Frobenius algebras capture their respective fragments of quantum mechanics by elegant and succinct axioms; and both come with their conceptual mysteries. A particularly elegant and mysterious constraint, imposed on Frobenius algebras to characterize a class of tripartite entangled states, is the antispecial law. A particularly contentious issue on the quantum logic side is the modularity law, proposed by von Neumann to mitigate the failure of distributivity of quantum logical connectives. We show that, if quantum logic and categorical quantum mechanics are formalized in the same framework, then the antispecial law of categorical quantum mechanics corresponds to the natural requirement of effect algebras that the units are each other's unique complements; and that the modularity law corresponds to the Frobenius condition. These correspondences lead to the equivalence announced in the title. Aligning the two formalisms, at the very least, sheds new light on the concepts that are more clearly displayed on one side than on the other (such as e.g. the orthogonality). Beyond that, it may also open up new approaches to deep and important problems of quantum mechanics (such as the classification of complementary observables).

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“…This calculus was also shown to have refined the graphical calculus of complementary observables [4], which was already previously shown to have many applications and admit automation. The GHZ/W calculus has now been shown in [6] to allow for faithfully encoding standard rational arithmetic as well.…”

Section: Introductionmentioning

confidence: 99%

Towards Normal Forms for GHZ∕W Calculus

Roy¹

2011

AIP Conference Proceedings

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Recently, a novel GHZ/W graphical calculus has been established to study and reason more intuitively about interacting quantum systems. The compositional structure of this calculus was shown to be well-equipped to sufficiently express arbitrary mutlipartite quantum states equivalent under stochastic local operations and classical communication (SLOCC). However, it is still not clear how to explicitly identify which graphical properties lead to what states. This can be achieved if we have well-behaved normal forms for arbitrary graphs within this calculus. This article lays down a first attempt at realizing such normal forms for a restricted class of such graphs, namely simple and regular graphs. These results should pave the way for the most general cases as part of future work.

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The GHZ/W-calculus contains rational arithmetic

Cited by 14 publications

References 18 publications

A Diagrammatic Axiomatisation for Qubit Entanglement

A Diagrammatic Axiomatisation for Qubit Entanglement

(Modular) Effect Algebras are Equivalent to (Frobenius) Antispecial Algebras

Towards Normal Forms for GHZ∕W Calculus

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