6,567 research outputs found
A Diagrammatic Axiomatisation for Qubit Entanglement
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.Comment: 12 page
On Khovanov's cobordism theory for su(3) knot homology
We reconsider the su(3) link homology theory defined by Khovanov in
math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With
some slight modifications, we describe the theory as a map from the planar
algebra of tangles to a planar algebra of (complexes of) `cobordisms with
seams' (actually, a `canopolis'), making it local in the sense of Bar-Natan's
local su(2) theory of math.GT/0410495.
We show that this `seamed cobordism canopolis' decategorifies to give
precisely what you'd both hope for and expect: Kuperberg's su(3) spider defined
in q-alg/9712003. We conjecture an answer to an even more interesting question
about the decategorification of the Karoubi envelope of our cobordism theory.
Finally, we describe how the theory is actually completely computable, and
give a detailed calculation of the su(3) homology of the (2,n) torus knots.Comment: 49 page
A complete graphical calculus for Spekkens' toy bit theory
While quantum theory cannot be described by a local hidden variable model, it
is nevertheless possible to construct such models that exhibit features
commonly associated with quantum mechanics. These models are also used to
explore the question of {\psi}-ontic versus {\psi}-epistemic theories for
quantum mechanics. Spekkens' toy theory is one such model. It arises from
classical probabilistic mechanics via a limit on the knowledge an observer may
have about the state of a system. The toy theory for the simplest possible
underlying system closely resembles stabilizer quantum mechanics, a fragment of
quantum theory which is efficiently classically simulable but also non-local.
Further analysis of the similarities and differences between those two theories
can thus yield new insights into what distinguishes quantum theory from
classical theories, and {\psi}-ontic from {\psi}-epistemic theories.
In this paper, we develop a graphical language for Spekkens' toy theory.
Graphical languages offer intuitive and rigorous formalisms for the analysis of
quantum mechanics and similar theories. To compare quantum mechanics and a toy
model, it is useful to have similar formalisms for both. We show that our
language fully describes Spekkens' toy theory and in particular, that it is
complete: meaning any equality that can be derived using other formalisms can
also be derived entirely graphically. Our language is inspired by a similar
graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens'
toy bit theory and stabilizer quantum mechanics can be analysed and compared
using analogous graphical formalisms.Comment: Major revisions for v2. 22+7 page
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