256 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
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
A complete characterisation of All-versus-Nothing arguments for stabiliser states
An important class of contextuality arguments in quantum foundations are the
All-versus-Nothing (AvN) proofs, generalising a construction originally due to
Mermin. We present a general formulation of All-versus-Nothing arguments, and a
complete characterisation of all such arguments which arise from stabiliser
states. We show that every AvN argument for an n-qubit stabiliser state can be
reduced to an AvN proof for a three-qubit state which is local
Clifford-equivalent to the tripartite GHZ state. This is achieved through a
combinatorial characterisation of AvN arguments, the AvN triple Theorem, whose
proof makes use of the theory of graph states. This result enables the
development of a computational method to generate all the AvN arguments in
on n-qubit stabiliser states. We also present new insights into
the stabiliser formalism and its connections with logic.Comment: 18 pages, 6 figure
Chain of Hardy-type local reality constraints for qubits
Non-locality without inequality is an elegant argument introduced by L. Hardy
for two qubit systems, and later generalised to qubits, to establish
contradiction of quantum theory with local realism. Interestingly, for
this argument is actually a corollary of Bell-type inequalities, viz. the
CH-Hardy inequality involving Bell correlations, but for greater than 2 it
involves -particle probabilities more general than Bell-correlations. In
this paper, we first derive a chain of completely new local realistic
inequalities involving joint probabilities for qubits, and then, associated
to each such inequality, we provide a new Hardy-type local reality constraint
without inequalities. Quantum mechanical maximal violations of the chain of
inequalities and of the associated constraints are also studied by deriving
appropriate Cirel'son type theorems. These results involving joint
probabilities more general than Bell correlations are expected to provide a new
systematic tool to investigate entanglement.Comment: 10 pages, Late
Categorical Quantum Dynamics
We use strong complementarity to introduce dynamics and symmetries within the
framework of CQM, which we also extend to infinite-dimensional separable
Hilbert spaces: these were long-missing features, which open the way to a
wealth of new applications. The coherent treatment presented in this work also
provides a variety of novel insights into the dynamics and symmetries of
quantum systems: examples include the extremely simple characterisation of
symmetry-observable duality, the connection of strong complementarity with the
Weyl Canonical Commutation Relations, the generalisations of Feynman's clock
construction, the existence of time observables and the emergence of quantum
clocks.
Furthermore, we show that strong complementarity is a key resource for
quantum algorithms and protocols. We provide the first fully diagrammatic,
theory-independent proof of correctness for the quantum algorithm solving the
Hidden Subgroup Problem, and show that strong complementarity is the feature
providing the quantum advantage. In quantum foundations, we use strong
complementarity to derive the exact conditions relating non-locality to the
structure of phase groups, within the context of Mermin-type non-locality
arguments. Our non-locality results find further application to quantum
cryptography, where we use them to define a quantum-classical secret sharing
scheme with provable device-independent security guarantees.
All in all, we argue that strong complementarity is a truly powerful and
versatile building block for quantum theory and its applications, and one that
should draw a lot more attention in the future.Comment: Thesis submitted for the degree of Doctor of Philosophy, Oxford
University, Michaelmas Term 2016 (273 pages
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