1,451 research outputs found
Teleportation, Braid Group and Temperley--Lieb Algebra
We explore algebraic and topological structures underlying the quantum
teleportation phenomena by applying the braid group and Temperley--Lieb
algebra. We realize the braid teleportation configuration, teleportation
swapping and virtual braid representation in the standard description of the
teleportation. We devise diagrammatic rules for quantum circuits involving
maximally entangled states and apply them to three sorts of descriptions of the
teleportation: the transfer operator, quantum measurements and characteristic
equations, and further propose the Temperley--Lieb algebra under local unitary
transformations to be a mathematical structure underlying the teleportation. We
compare our diagrammatical approach with two known recipes to the quantum
information flow: the teleportation topology and strongly compact closed
category, in order to explain our diagrammatic rules to be a natural
diagrammatic language for the teleportation.Comment: 33 pages, 19 figures, latex. The present article is a short version
of the preprint, quant-ph/0601050, which includes details of calculation,
more topics such as topological diagrammatical operations and entanglement
swapping, and calls the Temperley--Lieb category for the collection of all
the Temperley--Lieb algebra with physical operations like local unitary
transformation
Environment and classical channels in categorical quantum mechanics
We present a both simple and comprehensive graphical calculus for quantum
computing. In particular, we axiomatize the notion of an environment, which
together with the earlier introduced axiomatic notion of classical structure
enables us to define classical channels, quantum measurements and classical
control. If we moreover adjoin the earlier introduced axiomatic notion of
complementarity, we obtain sufficient structural power for constructive
representation and correctness derivation of typical quantum informatic
protocols.Comment: 26 pages, many pics; this third version has substantially more
explanations than previous ones; Journal reference is of short 14 page
version; Proceedings of the 19th EACSL Annual Conference on Computer Science
Logic (CSL), Lecture Notes in Computer Science 6247, Springer-Verlag (2010
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
Atemporal diagrams for quantum circuits
A system of diagrams is introduced that allows the representation of various
elements of a quantum circuit, including measurements, in a form which makes no
reference to time (hence ``atemporal''). It can be used to relate quantum
dynamical properties to those of entangled states (map-state duality), and
suggests useful analogies, such as the inverse of an entangled ket. Diagrams
clarify the role of channel kets, transition operators, dynamical operators
(matrices), and Kraus rank for noisy quantum channels. Positive (semidefinite)
operators are represented by diagrams with a symmetry that aids in
understanding their connection with completely positive maps. The diagrams are
used to analyze standard teleportation and dense coding, and for a careful
study of unambiguous (conclusive) teleportation. A simple diagrammatic argument
shows that a Kraus rank of 3 is impossible for a one-qubit channel modeled
using a one-qubit environment in a mixed state.Comment: Minor changes in references. Latex 32 pages, 13 figures in text using
PSTrick
Strong Complementarity and Non-locality in Categorical Quantum Mechanics
Categorical quantum mechanics studies quantum theory in the framework of
dagger-compact closed categories.
Using this framework, we establish a tight relationship between two key
quantum theoretical notions: non-locality and complementarity. In particular,
we establish a direct connection between Mermin-type non-locality scenarios,
which we generalise to an arbitrary number of parties, using systems of
arbitrary dimension, and performing arbitrary measurements, and a new stronger
notion of complementarity which we introduce here.
Our derivation of the fact that strong complementarity is a necessary
condition for a Mermin scenario provides a crisp operational interpretation for
strong complementarity. We also provide a complete classification of strongly
complementary observables for quantum theory, something which has not yet been
achieved for ordinary complementarity.
Since our main results are expressed in the (diagrammatic) language of
dagger-compact categories, they can be applied outside of quantum theory, in
any setting which supports the purely algebraic notion of strongly
complementary observables. We have therefore introduced a method for discussing
non-locality in a wide variety of models in addition to quantum theory.
The diagrammatic calculus substantially simplifies (and sometimes even
trivialises) many of the derivations, and provides new insights. In particular,
the diagrammatic computation of correlations clearly shows how local
measurements interact to yield a global overall effect. In other words, we
depict non-locality.Comment: 15 pages (incl. 5 appendix). To appear: LiCS 201
Depicting qudit quantum mechanics and mutually unbiased qudit theories
We generalize the ZX calculus to quantum systems of dimension higher than
two. The resulting calculus is sound and universal for quantum mechanics. We
define the notion of a mutually unbiased qudit theory and study two particular
instances of these theories in detail: qudit stabilizer quantum mechanics and
Spekkens-Schreiber toy theory for dits. The calculus allows us to analyze the
structure of qudit stabilizer quantum mechanics and provides a geometrical
picture of qudit stabilizer theory using D-toruses, which generalizes the Bloch
sphere picture for qubit stabilizer quantum mechanics. We also use our
framework to describe generalizations of Spekkens toy theory to higher
dimensional systems. This gives a novel proof that qudit stabilizer quantum
mechanics and Spekkens-Schreiber toy theory for dits are operationally
equivalent in three dimensions. The qudit pictorial calculus is a useful tool
to study quantum foundations, understand the relationship between qubit and
qudit quantum mechanics, and provide a novel, high level description of quantum
information protocols.Comment: In Proceedings QPL 2014, arXiv:1412.810
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