3,847 research outputs found
The algebra of entanglement and the geometry of composition
String diagrams turn algebraic equations into topological moves that have
recurring shapes, involving the sliding of one diagram past another. We
individuate, at the root of this fact, the dual nature of polygraphs as
presentations of higher algebraic theories, and as combinatorial descriptions
of "directed spaces". Operations of polygraphs modelled on operations of
topological spaces are used as the foundation of a compositional universal
algebra, where sliding moves arise from tensor products of polygraphs. We
reconstruct several higher algebraic theories in this framework.
In this regard, the standard formalism of polygraphs has some technical
problems. We propose a notion of regular polygraph, barring cell boundaries
that are not homeomorphic to a disk of the appropriate dimension. We define a
category of non-degenerate shapes, and show how to calculate their tensor
products. Then, we introduce a notion of weak unit to recover weakly degenerate
boundaries in low dimensions, and prove that the existence of weak units is
equivalent to a representability property.
We then turn to applications of diagrammatic algebra to quantum theory. We
re-evaluate the category of Hilbert spaces from the perspective of categorical
universal algebra, which leads to a bicategorical refinement. Then, we focus on
the axiomatics of fragments of quantum theory, and present the ZW calculus, the
first complete diagrammatic axiomatisation of the theory of qubits.
The ZW calculus has several advantages over ZX calculi, including a
computationally meaningful normal form, and a fragment whose diagrams can be
read as setups of fermionic oscillators. Moreover, its generators reflect an
operational classification of entangled states of 3 qubits. We conclude with
generalisations of the ZW calculus to higher-dimensional systems, including the
definition of a universal set of generators in each dimension.Comment: v2: changes to end of Chapter 3. v1: 214 pages, many figures;
University of Oxford doctoral thesi
Classical Tensors and Quantum Entanglement II: Mixed States
Invariant operator-valued tensor fields on Lie groups are considered. These
define classical tensor fields on Lie groups by evaluating them on a quantum
state. This particular construction, applied on the local unitary group
U(n)xU(n), may establish a method for the identification of entanglement
monotone candidates by deriving invariant functions from tensors being by
construction invariant under local unitary transformations. In particular, for
n=2, we recover the purity and a concurrence related function (Wootters 1998)
as a sum of inner products of symmetric and anti-symmetric parts of the
considered tensor fields. Moreover, we identify a distinguished entanglement
monotone candidate by using a non-linear realization of the Lie algebra of
SU(2)xSU(2). The functional dependence between the latter quantity and the
concurrence is illustrated for a subclass of mixed states parametrized by two
variables.Comment: 23 pages, 4 figure
Balanced Tripartite Entanglement, the Alternating Group A4 and the Lie Algebra
We discuss three important classes of three-qubit entangled states and their
encoding into quantum gates, finite groups and Lie algebras. States of the GHZ
and W-type correspond to pure tripartite and bipartite entanglement,
respectively. We introduce another generic class B of three-qubit states, that
have balanced entanglement over two and three parties. We show how to realize
the largest cristallographic group in terms of three-qubit gates (with
real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group
dipoles and the enactment of Weyl/Coxeter group by entangling gates},
Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar "condensation" of
into the four-letter alternating group , obtained from a chain of
maximal subgroups. Group is realized from two B-type generators and found
to correspond to the Lie algebra . Possible
applications of our findings to particle physics and the structure of genetic
code are also mentioned.Comment: 14 page
- …