21 research outputs found
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
The ZX-calculus is incomplete for quantum mechanics
We prove that the ZX-calculus is incomplete for quantum mechanics. We suggest
the addition of a new 'color-swap' rule, of which currently no analytical
formulation is known and which we suspect may be necessary, but not sufficient
to make the ZX-calculus complete.Comment: In Proceedings QPL 2014, arXiv:1412.810
A first-order logic for string diagrams
Equational reasoning with string diagrams provides an intuitive means of
proving equations between morphisms in a symmetric monoidal category. This can
be extended to proofs of infinite families of equations using a simple
graphical syntax called !-box notation. While this does greatly increase the
proving power of string diagrams, previous attempts to go beyond equational
reasoning have been largely ad hoc, owing to the lack of a suitable logical
framework for diagrammatic proofs involving !-boxes. In this paper, we extend
equational reasoning with !-boxes to a fully-fledged first order logic called
with conjunction, implication, and universal quantification over !-boxes. This
logic, called !L, is then rich enough to properly formalise an induction
principle for !-boxes. We then build a standard model for !L and give an
example proof of a theorem for non-commutative bialgebras using !L, which is
unobtainable by equational reasoning alone.Comment: 15 pages + appendi
Quantum Correlations and Quantum Non-Locality: A Review and a Few New Ideas
In this paper we make an extensive description of quantum non-locality, one
of the most intriguing and fascinating facets of quantum mechanics. After a
general presentation of several studies on this subject, we consider if quantum
non-locality, and the friction it carries with special relativity, can
eventually find a "solution" by considering higher dimensional spaces.Comment: 1
Mermin Non-Locality in Abstract Process Theories
The study of non-locality is fundamental to the understanding of quantum
mechanics. The past 50 years have seen a number of non-locality proofs, but its
fundamental building blocks, and the exact role it plays in quantum protocols,
has remained elusive. In this paper, we focus on a particular flavour of
non-locality, generalising Mermin's argument on the GHZ state. Using strongly
complementary observables, we provide necessary and sufficient conditions for
Mermin non-locality in abstract process theories. We show that the existence of
more phases than classical points (aka eigenstates) is not sufficient, and that
the key to Mermin non-locality lies in the presence of certain algebraically
non-trivial phases. This allows us to show that fRel, a favourite toy model for
categorical quantum mechanics, is Mermin local. We show Mermin non-locality to
be the key resource ensuring the device-independent security of the HBB CQ
(N,N) family of Quantum Secret Sharing protocols. Finally, we challenge the
unspoken assumption that the measurements involved in Mermin-type scenarios
should be complementary (like the pair X,Y), opening the doors to a much wider
class of potential experimental setups than currently employed. In short, we
give conditions for Mermin non-locality tests on any number of systems, where
each party has an arbitrary number of measurement choices, where each
measurement has an arbitrary number of outcomes and further, that works in any
abstract process theory.Comment: In Proceedings QPL 2015, arXiv:1511.0118
Interacting Frobenius Algebras are Hopf
Theories featuring the interaction between a Frobenius algebra and a Hopf
algebra have recently appeared in several areas in computer science: concurrent
programming, control theory, and quantum computing, among others. Bonchi,
Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive
law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the
opposite approach, and show that interacting Frobenius algebras form Hopf
algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the
underlying object---the so-called phase group---and investigate the effects of
finite dimensionality of the underlying model. We recover the system of Bonchi
et al as a subtheory in the prime power dimensional case, but the more general
theory does not arise from a distributive law.Comment: 32 pages; submitte