8,739 research outputs found
Dynamics of quantum causal structures
It was recently suggested that causal structures are both dynamical, because
of general relativity, and indefinite, due to quantum theory. The process
matrix formalism furnishes a framework for quantum mechanics on indefinite
causal structures, where the order between operations of local laboratories is
not definite (e.g. one cannot say whether operation in laboratory A occurs
before or after operation in laboratory B). Here we develop a framework for
"dynamics of causal structures", i.e. for transformations of process matrices
into process matrices. We show that, under continuous and reversible
transformations, the causal order between operations is always preserved.
However, the causal order between a subset of operations can be changed under
continuous yet nonreversible transformations. An explicit example is that of
the quantum switch, where a party in the past affects the causal order of
operations of future parties, leading to a transition from a channel from A to
B, via superposition of causal orders, to a channel from B to A. We generalise
our framework to construct a hierarchy of quantum maps based on transformations
of process matrices and transformations thereof.Comment: 13+5 pages, 4 figures. Two appendices added. Published versio
Abstract Tensor Systems as Monoidal Categories
The primary contribution of this paper is to give a formal, categorical
treatment to Penrose's abstract tensor notation, in the context of traced
symmetric monoidal categories. To do so, we introduce a typed, sum-free version
of an abstract tensor system and demonstrate the construction of its associated
category. We then show that the associated category of the free abstract tensor
system is in fact the free traced symmetric monoidal category on a monoidal
signature. A notable consequence of this result is a simple proof for the
soundness and completeness of the diagrammatic language for traced symmetric
monoidal categories.Comment: Dedicated to Joachim Lambek on the occasion of his 90th birthda
A categorical semantics for causal structure
We present a categorical construction for modelling causal structures within
a general class of process theories that include the theory of classical
probabilistic processes as well as quantum theory. Unlike prior constructions
within categorical quantum mechanics, the objects of this theory encode
fine-grained causal relationships between subsystems and give a new method for
expressing and deriving consequences for a broad class of causal structures. We
show that this framework enables one to define families of processes which are
consistent with arbitrary acyclic causal orderings. In particular, one can
define one-way signalling (a.k.a. semi-causal) processes, non-signalling
processes, and quantum -combs. Furthermore, our framework is general enough
to accommodate recently-proposed generalisations of classical and quantum
theory where processes only need to have a fixed causal ordering locally, but
globally allow indefinite causal ordering.
To illustrate this point, we show that certain processes of this kind, such
as the quantum switch, the process matrices of Oreshkov, Costa, and Brukner,
and a classical three-party example due to Baumeler, Feix, and Wolf are all
instances of a certain family of processes we refer to as in
the appropriate category of higher-order causal processes. After defining these
families of causal structures within our framework, we give derivations of
their operational behaviour using simple, diagrammatic axioms.Comment: Extended version of a LICS 2017 paper with the same titl
The sl(n)-WZNW Fusion Ring: a combinatorial construction and a realisation as quotient of quantum cohomology
A simple, combinatorial construction of the sl(n)-WZNW fusion ring, also
known as Verlinde algebra, is given. As a byproduct of the construction one
obtains an isomorphism between the fusion ring and a particular quotient of the
small quantum cohomology ring of the Grassmannian Gr(k,k+n). We explain how our
approach naturally fits into known combinatorial descriptions of the quantum
cohomology ring, by establishing what one could call a
`Boson-Fermion-correspondence' between the two rings. We also present new
recursion formulae for the structure constants of both rings, the fusion
coefficients and the Gromov-Witten invariants.Comment: 61 pages, 2 eps figures; revised version accepted for publication in
Advances in Mathematics: some minor typos removed, rewording of the proof to
Corollary 6.9 and figure in Example 8.3 change
Quantum Picturalism
The quantum mechanical formalism doesn't support our intuition, nor does it
elucidate the key concepts that govern the behaviour of the entities that are
subject to the laws of quantum physics. The arrays of complex numbers are kin
to the arrays of 0s and 1s of the early days of computer programming practice.
In this review we present steps towards a diagrammatic `high-level' alternative
for the Hilbert space formalism, one which appeals to our intuition. It allows
for intuitive reasoning about interacting quantum systems, and trivialises many
otherwise involved and tedious computations. It clearly exposes limitations
such as the no-cloning theorem, and phenomena such as quantum teleportation. As
a logic, it supports `automation'. It allows for a wider variety of underlying
theories, and can be easily modified, having the potential to provide the
required step-stone towards a deeper conceptual understanding of quantum
theory, as well as its unification with other physical theories. Specific
applications discussed here are purely diagrammatic proofs of several quantum
computational schemes, as well as an analysis of the structural origin of
quantum non-locality. The underlying mathematical foundation of this high-level
diagrammatic formalism relies on so-called monoidal categories, a product of a
fairly recent development in mathematics. These monoidal categories do not only
provide a natural foundation for physical theories, but also for proof theory,
logic, programming languages, biology, cooking, ... The challenge is to
discover the necessary additional pieces of structure that allow us to predict
genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures,
some colo
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