38 research outputs found
Programming with Quantum Communication
This work develops a formal framework for specifying, implementing, and
analysing quantum communication protocols. We provide tools for developing
simple proofs and analysing programs which involve communication, both via
quantum channels and exhibiting the LOCC (local operations, classical
communication) paradigm
Programming Telepathy: Implementing Quantum Non-Locality Games
Quantum pseudo-telepathy is an intriguing phenomenon which results from the
application of quantum information theory to communication complexity. To
demonstrate this phenomenon researchers in the field of quantum communication
complexity devised a number of quantum non-locality games. The setting of these
games is as follows: the players are separated so that no communication between
them is possible and are given a certain computational task. When the players
have access to a quantum resource called entanglement, they can accomplish the
task: something that is impossible in a classical setting. To an observer who
is unfamiliar with the laws of quantum mechanics it seems that the players
employ some sort of telepathy; that is, they somehow exchange information
without sharing a communication channel. This paper provides a formal framework
for specifying, implementing, and analysing quantum non-locality games
De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure
Elaborating on our joint work with Abramsky in quant-ph/0402130 we further
unravel the linear structure of Hilbert spaces into several constituents. Some
prove to be very crucial for particular features of quantum theory while others
obstruct the passage to a formalism which is not saturated with physically
insignificant global phases.
First we show that the bulk of the required linear structure is purely
multiplicative, and arises from the strongly compact closed tensor which,
besides providing a variety of notions such as scalars, trace, unitarity,
self-adjointness and bipartite projectors, also provides Hilbert-Schmidt norm,
Hilbert-Schmidt inner-product, and in particular, the preparation-state
agreement axiom which enables the passage from a formalism of the vector space
kind to a rather projective one, as it was intended in the (in)famous Birkhoff
& von Neumann paper.
Next we consider additive types which distribute over the tensor, from which
measurements can be build, and the correctness proofs of the protocols
discussed in quant-ph/0402130 carry over to the resulting weaker setting. A
full probabilistic calculus is obtained when the trace is moreover linear and
satisfies the \em diagonal axiom, which brings us to a second main result,
characterization of the necessary and sufficient additive structure of a both
qualitatively and quantitatively effective categorical quantum formalism
without redundant global phases. Along the way we show that if in a category a
(additive) monoidal tensor distributes over a strongly compact closed tensor,
then this category is always enriched in commutative monoids.Comment: Essential simplification of the definitions of orthostructure and
ortho-Bornian structure: the key new insights is captured by the definitions
in terms of commutative diagrams on pages 13 and 14, which state that if in a
category a (additive) monoidal tensor distributes over a strongly compact
closed tensor, then this category is always enriched in commutative monoid
The Measurement Calculus
Measurement-based quantum computation has emerged from the physics community
as a new approach to quantum computation where the notion of measurement is the
main driving force of computation. This is in contrast with the more
traditional circuit model which is based on unitary operations. Among
measurement-based quantum computation methods, the recently introduced one-way
quantum computer stands out as fundamental.
We develop a rigorous mathematical model underlying the one-way quantum
computer and present a concrete syntax and operational semantics for programs,
which we call patterns, and an algebra of these patterns derived from a
denotational semantics. More importantly, we present a calculus for reasoning
locally and compositionally about these patterns.
We present a rewrite theory and prove a general standardization theorem which
allows all patterns to be put in a semantically equivalent standard form.
Standardization has far-reaching consequences: a new physical architecture
based on performing all the entanglement in the beginning, parallelization by
exposing the dependency structure of measurements and expressiveness theorems.
Furthermore we formalize several other measurement-based models:
Teleportation, Phase and Pauli models and present compositional embeddings of
them into and from the one-way model. This allows us to transfer all the theory
we develop for the one-way model to these models. This shows that the framework
we have developed has a general impact on measurement-based computation and is
not just particular to the one-way quantum computer.Comment: 46 pages, 2 figures, Replacement of quant-ph/0412135v1, the new
version also include formalization of several other measurement-based models:
Teleportation, Phase and Pauli models and present compositional embeddings of
them into and from the one-way model. To appear in Journal of AC
The dagger lambda calculus
We present a novel lambda calculus that casts the categorical approach to the
study of quantum protocols into the rich and well established tradition of type
theory. Our construction extends the linear typed lambda calculus with a linear
negation of "trivialised" De Morgan duality. Reduction is realised through
explicit substitution, based on a symmetric notion of binding of global scope,
with rules acting on the entire typing judgement instead of on a specific
subterm. Proofs of subject reduction, confluence, strong normalisation and
consistency are provided, and the language is shown to be an internal language
for dagger compact categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models
In the categorical approach to the foundations of quantum theory, one begins
with a symmetric monoidal category, the objects of which represent physical
systems, and the morphisms of which represent physical processes. Usually, this
category is taken to be at least compact closed, and more often, dagger
compact, enforcing a certain self-duality, whereby preparation processes
(roughly, states) are inter-convertible with processes of registration
(roughly, measurement outcomes). This is in contrast to the more concrete
"operational" approach, in which the states and measurement outcomes associated
with a physical system are represented in terms of what we here call a "convex
operational model": a certain dual pair of ordered linear spaces -- generally,
{\em not} isomorphic to one another. On the other hand, state spaces for which
there is such an isomorphism, which we term {\em weakly self-dual}, play an
important role in reconstructions of various quantum-information theoretic
protocols, including teleportation and ensemble steering. In this paper, we
characterize compact closure of symmetric monoidal categories of convex
operational models in two ways: as a statement about the existence of
teleportation protocols, and as the principle that every process allowed by
that theory can be realized as an instance of a remote evaluation protocol ---
hence, as a form of classical probabilistic conditioning. In a large class of
cases, which includes both the classical and quantum cases, the relevant
compact closed categories are degenerate, in the weak sense that every object
is its own dual. We characterize the dagger-compactness of such a category
(with respect to the natural adjoint) in terms of the existence, for each
system, of a {\em symmetric} bipartite state, the associated conditioning map
of which is an isomorphism
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