38 research outputs found
Phase groups and the origin of non-locality for qubits
We describe a general framework in which we can precisely compare the
structures of quantum-like theories which may initially be formulated in quite
different mathematical terms. We then use this framework to compare two
theories: quantum mechanics restricted to qubit stabiliser states and
operations, and Spekkens's toy theory. We discover that viewed within our
framework these theories are very similar, but differ in one key aspect - a
four element group we term the phase group which emerges naturally within our
framework. In the case of the stabiliser theory this group is Z4 while for
Spekkens's toy theory the group is Z2 x Z2. We further show that the structure
of this group is intimately involved in a key physical difference between the
theories: whether or not they can be modelled by a local hidden variable
theory. This is done by establishing a connection between the phase group, and
an abstract notion of GHZ state correlations. We go on to formulate precisely
how the stabiliser theory and toy theory are `similar' by defining a notion of
`mutually unbiased qubit theory', noting that all such theories have four
element phase groups. Since Z4 and Z2 x Z2 are the only such groups we conclude
that the GHZ correlations in this type of theory can only take two forms,
exactly those appearing in the stabiliser theory and in Spekkens's toy theory.
The results point at a classification of local/non-local behaviours by finite
Abelian groups, extending beyond qubits to finitary theories whose observables
are all mutually unbiased.Comment: 24 pages, many picture
The GHZ/W-calculus contains rational arithmetic
Graphical calculi for representing interacting quantum systems serve a number
of purposes: compositionally, intuitive graphical reasoning, and a logical
underpinning for automation. The power of these calculi stems from the fact
that they embody generalized symmetries of the structure of quantum operations,
which, for example, stretch well beyond the Choi-Jamiolkowski isomorphism. One
such calculus takes the GHZ and W states as its basic generators. Here we show
that this language allows one to encode standard rational calculus, with the
GHZ state as multiplication, the W state as addition, the Pauli X gate as
multiplicative inversion, and the Pauli Z gate as additive inversion.Comment: In Proceedings HPC 2010, arXiv:1103.226
Geometric Quantization and Epistemically Restricted Theories: The Continuous Case
It is possible to reproduce the quantum features of quantum states, starting
from a classical statistical theory and then limiting the amount of knowledge
that an agent can have about an individual system [5, 18].These are so called
epistemic restrictions. Such restrictions have been recently formulated in
terms of the symplectic geometry of the corresponding classical theory [19].
The purpose of this note is to describe, using this symplectic framework, how
to obtain a C*-algebraic formulation for the epistemically restricted theories.
In the case of continuous variables, following the groupoid quantization recipe
of E. Hawkins, we obtain a twisted group C*-algebra which is the usual Moyal
quantization of a Poisson vector space [12].Comment: In Proceedings QPL 2016, arXiv:1701.00242. 10 page
A Diagrammatic Axiomatisation for Qubit Entanglement
Diagrammatic techniques for reasoning about monoidal categories provide an
intuitive understanding of the symmetries and connections of interacting
computational processes. In the context of categorical quantum mechanics,
Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used
as the building blocks of a new graphical calculus, aimed at a diagrammatic
classification of multipartite qubit entanglement that would highlight the
communicational properties of quantum states, and their potential uses in
cryptographic schemes.
In this paper, we present a full graphical axiomatisation of the relations
between GHZ and W: the ZW calculus. This refines a version of the preexisting
ZX calculus, while keeping its most desirable characteristics: undirectedness,
a large degree of symmetry, and an algebraic underpinning. We prove that the ZW
calculus is complete for the category of free abelian groups on a power of two
generators - "qubits with integer coefficients" - and provide an explicit
normalisation procedure.Comment: 12 page
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
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 Functorial Construction of Quantum Subtheories
We apply the geometric quantization procedure via symplectic groupoids
proposed by E. Hawkins to the setting of epistemically restricted toy theories
formalized by Spekkens. In the continuous degrees of freedom, this produces the
algebraic structure of quadrature quantum subtheories. In the odd-prime finite
degrees of freedom, we obtain a functor from the Frobenius algebra in
\textbf{Rel} of the toy theories to the Frobenius algebra of stabilizer quantum
mechanics.Comment: 19 page
Quantum cube: A toy model of a qubit
Account of a system may depend on available methods of gaining information.
We discuss a simple discrete system whose description is affected by a specific
model of measurement and transformations. It is shown that the limited means of
investigating the system make the epistemic account of the model
indistinguishable from a constrained version of a qubit corresponding to the
convex hull of eigenstates of Pauli operators, Clifford transformations and
Pauli observables.Comment: 5 pages, 4 figures; Final version as published in Physics Letters