114 research outputs found
Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone
We investigate the completely positive semidefinite cone ,
a new matrix cone consisting of all matrices that admit a Gram
representation by positive semidefinite matrices (of any size). In particular
we study relationships between this cone and the completely positive and doubly
nonnegative cones, and between its dual cone and trace positive non-commutative
polynomials.
We use this new cone to model quantum analogues of the classical independence
and chromatic graph parameters and , which are roughly
obtained by allowing variables to be positive semidefinite matrices instead of
scalars in the programs defining the classical parameters. We can
formulate these quantum parameters as conic linear programs over the cone
. Using this conic approach we can recover the bounds in
terms of the theta number and define further approximations by exploiting the
link to trace positive polynomials.Comment: Fixed some typo
How to make unforgeable money in generalised probabilistic theories
We discuss the possibility of creating money that is physically impossible to
counterfeit. Of course, "physically impossible" is dependent on the theory that
is a faithful description of nature. Currently there are several proposals for
quantum money which have their security based on the validity of quantum
mechanics. In this work, we examine Wiesner's money scheme in the framework of
generalised probabilistic theories. This framework is broad enough to allow for
essentially any potential theory of nature, provided that it admits an
operational description. We prove that under a quantifiable version of the
no-cloning theorem, one can create physical money which has an exponentially
small chance of being counterfeited. Our proof relies on cone programming, a
natural generalisation of semidefinite programming. Moreover, we discuss some
of the difficulties that arise when considering non-quantum theories.Comment: 27 pages, many diagrams. Comments welcom
The operational significance of the quantum resource theory of Buscemi nonlocality
Although entanglement is necessary for observing nonlocality in a Bell
experiment, there are entangled states which can never be used to demonstrate
nonlocal correlations. In a seminal paper [PRL 108, 200401 (2012)] F. Buscemi
extended the standard Bell experiment by allowing Alice and Bob to be asked
quantum, instead of classical, questions. This gives rise to a broader notion
of nonlocality, one which can be observed for every entangled state. In this
work we study a resource theory of this type of nonlocality referred to as
Buscemi nonlocality. We propose a geometric quantifier measuring the ability of
a given state and local measurements to produce Buscemi nonlocal correlations
and establish its operational significance. In particular, we show that any
distributed measurement which can demonstrate Buscemi nonlocal correlations
provides strictly better performance than any distributed measurement which
does not use entanglement in the task of distributed state discrimination. We
also show that the maximal amount of Buscemi nonlocality that can be generated
using a given state is precisely equal to its entanglement content. Finally, we
prove a quantitative relationship between: Buscemi nonlocality, the ability to
perform nonclassical teleportation, and entanglement. Using this relationship
we propose new discrimination tasks for which nonclassical teleportation and
entanglement lead to an advantage over their classical counterparts.Comment: 10+11 pages, 1 figure. Updated presentation and added clarifications
regarding the choice of free set. Comments welcome
On constructions of quantum-secure device-independent randomness expansion protocols
Device-independent randomness expansion protocols aim to expand a short uniformly random string into a much longer one whilst guaranteeing that their output is truly random. They are device-independent in the sense that this guarantee does not dependent on the specifics of an implementation. Rather, through the observation of nonlocal correlations we can conclude that the outputs generated are necessarily random. This thesis reports a general method for constructing these protocols and evaluating their security. Using this method, we then construct several explicit protocols and analyse their performance on noisy qubit systems. With a view towards near-future quantum technologies, we also investigate whether randomness expansion is possible using current nonlocality experiments. We find that, by combining the recent theoretical and experimental advances, it is indeed now possible to reliably and securely expand randomness
Quantum entanglement: insights via graph parameters and conic optimization
In this PhD thesis we study the effects of quantum entanglement, one of quantum mechanics most peculiar features, in nonlocal games and communication problems in zero-error information theory. A nonlocal game is a thought experiment in which two cooperating players, who are forbidden to communicate, want to perform a certain task.
Zero-error information theory is the mathematical field that studies communication problems where no error is tolerated. The unifying link among the various scenarios we consider is their combinatorial nature and in particular their reformulations as graph theoretical problems, mainly concerning the chromatic and stability numbers and some quantum generalizations thereof.
In this thesis we propose a novel approach to the study of these quantum graph parameters using the paradigm of conic optimization. For that, we introduce and study the completely positive semidefinite cone, a new matrix cone consisting of all symmetric matrices that admit a Gram representation by positive semidefinite matrices.
Furthermore, we investigate whether entanglement allows for better-than-classical communication schemes in some well-known problems from zero-error information theory. For example we study the channel coding problem, which asks a sender to transmit data reliably to a receiver in the presence of noise, as well as some of its generalizations
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