179 research outputs found
On the 'Reality' of Observable Properties
This note contains some initial work on attempting to bring recent
developments in the foundations of quantum mechanics concerning the nature of
the wavefunction within the scope of more logical and structural methods. A
first step involves generalising and reformulating a criterion for the reality
of the wavefunction proposed by Harrigan & Spekkens, which was central to the
PBR theorem. The resulting criterion has several advantages, including the
avoidance of certain technical difficulties relating to sets of measure zero.
By considering the 'reality' not of the wavefunction but of the observable
properties of any ontological physical theory a novel characterisation of
non-locality and contextuality is found.
Secondly, a careful analysis of preparation independence, one of the key
assumptions of the PBR theorem, leads to an analogy with Bell locality, and
thence to a proposal to weaken it to an assumption of
`no-preparation-signalling' in analogy with no-signalling. This amounts to
introducing non-local correlations in the joint ontic state, which is, at
least, consistent with the Bell and Kochen-Specker theorems. The question of
whether the PBR result can be strengthened to hold under this relaxed
assumption is therefore posed.Comment: 8 pages, re-written with new section
Hardy's Non-locality Paradox and Possibilistic Conditions for Non-locality
Hardy's non-locality paradox is a proof without inequalities showing that
certain non-local correlations violate local realism. It is `possibilistic' in
the sense that one only distinguishes between possible outcomes (positive
probability) and impossible outcomes (zero probability). Here we show that
Hardy's paradox is quite universal: in any (2,2,l) or (2,k,2) Bell scenario,
the occurence of Hardy's paradox is a necessary and sufficient condition for
possibilistic non-locality. In particular, it subsumes all ladder paradoxes.
This universality of Hardy's paradox is not true more generally: we find a new
`proof without inequalities' in the (2,3,3) scenario that can witness
non-locality even for correlations that do not display the Hardy paradox. We
discuss the ramifications of our results for the computational complexity of
recognising possibilistic non-locality
Reflections on the PBR Theorem: Reality Criteria & Preparation Independence
This paper contains initial work on attempting to bring recent developments
in the foundations of quantum mechanics concerning the nature of the
wavefunction within the scope of more logical and structural methods. A first
step involves dualising a criterion for the reality of the wavefunction
proposed by Harrigan & Spekkens, which was central to the Pusey-Barrett-Rudolph
theorem. The resulting criterion has several advantages, including the
avoidance of certain technical difficulties relating to sets of measure zero.
By considering the 'reality' not of the wavefunction but of the observable
properties of any ontological physical theory a new characterisation of
non-locality and contextuality is found. Secondly, a careful analysis of
preparation independence, one of the key assumptions of the PBR theorem, leads
to a precise analogy with the kind of locality prohibited by Bell's theorem.
Motivated by this, we propose a weakening of the assumption to something
analogous to no-signalling. This amounts to allowing global or non-local
correlations in the joint ontic state, which nevertheless do not allow for
superluminal signalling. This is, at least, consistent with the Bell and
Kochen-Specker theorems. We find a counter-example to the PBR argument, which
violates preparation independence, but does satisfy this physically motivated
assumption. The question of whether the PBR result can be strengthened to hold
under the relaxed assumption is therefore posed.Comment: In Proceedings QPL 2014, arXiv:1412.810
A comonadic view of simulation and quantum resources
We study simulation and quantum resources in the setting of the
sheaf-theoretic approach to contextuality and non-locality. Resources are
viewed behaviourally, as empirical models. In earlier work, a notion of
morphism for these empirical models was proposed and studied. We generalize and
simplify the earlier approach, by starting with a very simple notion of
morphism, and then extending it to a more useful one by passing to a co-Kleisli
category with respect to a comonad of measurement protocols. We show that these
morphisms capture notions of simulation between empirical models obtained via
`free' operations in a resource theory of contextuality, including the type of
classical control used in measurement-based quantum computation schemes.Comment: To appear in Proceedings of LiCS 201
Tsirelson's bound and Landauer's principle in a single-system game
We introduce a simple single-system game inspired by the
Clauser-Horne-Shimony-Holt (CHSH) game. For qubit systems subjected to unitary
gates and projective measurements, we prove that any strategy in our game can
be mapped to a strategy in the CHSH game, which implies that Tsirelson's bound
also holds in our setting. More generally, we show that the optimal success
probability depends on the reversible or irreversible character of the gates,
the quantum or classical nature of the system and the system dimension. We
analyse the bounds obtained in light of Landauer's principle, showing the
entropic costs of the erasure associated with the game. This shows a connection
between the reversibility in fundamental operations embodied by Landauer's
principle and Tsirelson's bound, that arises from the restricted physics of a
unitarily-evolving single-qubit system.Comment: 7 pages, 5 figures, typos correcte
Consequences and applications of the completeness of Hardy's nonlocality
Logical nonlocality is completely characterized by Hardy's "paradox" in
(2,2,l) and (2,k,2) scenarios. We consider a variety of consequences and
applications of this fact. (i) Polynomial algorithms may be given for deciding
logical nonlocality in these scenarios. (ii) Bell states are the only entangled
two-qubit states which are not logically nonlocal under projective
measurements. (iii) It is possible to witness Hardy nonlocality with certainty
in a simple tripartite quantum system. (iv) Noncommutativity of observables is
necessary and sufficient for enabling logical nonlocality.Comment: corrected typos and includes some extra comments and explanations
contained in published versio
Reality of the quantum state: Towards a stronger ψ-ontology theorem
The Pusey-Barrett-Rudolph (PBR) no-go theorem provides an argument for the
reality of the quantum state by ruling out {\psi}-epistemic ontological
theories, in which the quantum state is of a statistical nature. It applies
under an assumption of preparation independence, the validity of which has been
subject to debate. We propose two plausible and less restrictive alternatives:
a weaker notion allowing for classical correlations, and an even weaker,
physically motivated notion of independence, which merely prohibits the
possibility of superluminal causal influences in the preparation process. The
latter is a minimal requirement for enabling a reasonable treatment of
subsystems in any theory. It is demonstrated by means of an explicit
{\psi}-epistemic ontological model that the argument of PBR becomes invalid
under the alternative notions of independence. As an intermediate step, we
recover a result which is valid in the presence of classical correlations.
Finally, we obtain a theorem which holds under the minimal requirement,
approximating the result of PBR. For this, we consider experiments involving
randomly sampled preparations and derive bounds on the degree of {\psi}
epistemicity that is consistent with the quantum-mechanical predictions. The
approximation is exact in the limit as the sample space of preparations becomes
infinite.Comment: rewordings and typos edited to agree with published versio
Quantum Advantage in Information Retrieval
Random access codes have provided many examples of quantum advantage in
communication, but concern only one kind of information retrieval task. We
introduce a related task - the Torpedo Game - and show that it admits greater
quantum advantage than the comparable random access code. Perfect quantum
strategies involving prepare-and-measure protocols with experimentally
accessible three-level systems emerge via analysis in terms of the discrete
Wigner function. The example is leveraged to an operational advantage in a
pacifist version of the strategy game Battleship. We pinpoint a characteristic
of quantum systems that enables quantum advantage in any bounded-memory
information retrieval task. While preparation contextuality has previously been
linked to advantages in random access coding we focus here on a different
characteristic called sequential contextuality. It is shown not only to be
necessary and sufficient for quantum advantage, but also to quantify the degree
of advantage. Our perfect qutrit strategy for the Torpedo Game entails the
strongest type of inconsistency with non-contextual hidden variables, revealing
logical paradoxes with respect to those assumptions.Comment: 15 pages, 11 figures; new presentation, additional figures and
reference
The Cohomology of Non-Locality and Contextuality
In a previous paper with Adam Brandenburger, we used sheaf theory to analyze
the structure of non-locality and contextuality. Moreover, on the basis of this
formulation, we showed that the phenomena of non-locality and contextuality can
be characterized precisely in terms of obstructions to the existence of global
sections.
Our aim in the present work is to build on these results, and to use the
powerful tools of sheaf cohomology to study the structure of non-locality and
contextuality. We use the Cech cohomology on an abelian presheaf derived from
the support of a probabilistic model, viewed as a compatible family of
distributions, in order to define a cohomological obstruction for the family as
a certain cohomology class. This class vanishes if the family has a global
section. Thus the non-vanishing of the obstruction provides a sufficient (but
not necessary) condition for the model to be contextual.
We show that for a number of salient examples, including PR boxes, GHZ
states, the Peres-Mermin magic square, and the 18-vector configuration due to
Cabello et al. giving a proof of the Kochen-Specker theorem in four dimensions,
the obstruction does not vanish, thus yielding cohomological witnesses for
contextuality.Comment: In Proceedings QPL 2011, arXiv:1210.029
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