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PPS UMR 7126 – Laboratoire
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Amar Hadzihasanovic

String diagrams and the algebra of entanglement

The GHZ and W states are two entangled quantum states of three qubits, that are inequivalent in the sense that one cannot be turned into the other by local (single-qubit) operations; this is reflected in their different communicational properties and use in cryptographic protocols. A few years ago, Coecke and Kissinger showed that one can associate, to the two states, two Frobenius algebras in the category of Hilbert spaces - a type of algebra with a well-understood string diagram representation, which could hopefully provide a bridge between algebraic, computational and topological aspects of quantum entanglement. We present a complete graphical axiomatisation of the relations between the GHZ and W states/algebras: the ZW calculus. This calculus refines the pre-existing ZX calculus, while keeping its most desirable characteristics, such as the undirectedness of diagrams; comes with an explicit normalisation procedure; provides an original decomposition of the category of qubits, with a prominent "fermionic" fragment; and hints at a topological explanation of its components and axioms.