696 research outputs found
Kochen-Specker theorem as a precondition for secure quantum key distribution
We show that (1) the violation of the Ekert 91 inequality is a sufficient
condition for certification of the Kochen-Specker (KS) theorem, and (2) the
violation of the Bennett-Brassard-Mermin 92 (BBM) inequality is, also, a
sufficient condition for certification of the KS theorem. Therefore the success
in each QKD protocol reveals the nonclassical feature of quantum theory, in the
sense that the KS realism is violated. Further, it turned out that the Ekert
inequality and the BBM inequality are depictured by distillable entanglement
witness inequalities. Here, we connect the success in these two key
distribution processes into the no-hidden-variables theorem and into witness on
distillable entanglement. We also discuss the explicit difference between the
KS realism and Bell's local realism in the Hilbert space formalism of quantum
theory.Comment: 4 pages, To appear in Phys. Rev.
Entropic Test of Quantum Contextuality
We study the contextuality of a three-level quantum system using classical
conditional entropy of measurement outcomes. First, we analytically construct
the minimal configuration of measurements required to reveal contextuality.
Next, an entropic contextual inequality is formulated, analogous to the
entropic Bell inequalities derived by Braunstein and Caves in [Phys. Rev. Lett.
{\bf 61}, 662 (1988)], that must be satisfied by all non-contextual theories.
We find optimal measurements for violation of this inequality. The approach is
easily extendable to higher dimensional quantum systems and more measurements.
Our theoretical findings can be verified in the laboratory with current
technology.Comment: 4 pages, 4 figure
Comment on ``All quantum observables in a hidden-variable model must commute simultaneously"
Malley discussed {[Phys. Rev. A {\bf 69}, 022118 (2004)]} that all quantum
observables in a hidden-variable model for quantum events must commute
simultaneously. In this comment, we discuss that Malley's theorem is indeed
valid for the hidden-variable theoretical assumptions, which were introduced by
Kochen and Specker. However, we give an example that the local hidden-variable
(LHV) model for quantum events preserves noncommutativity of quantum
observables. It turns out that Malley's theorem is not related with the LHV
model for quantum events, in general.Comment: 3 page
Negativity and contextuality are equivalent notions of nonclassicality
Two notions of nonclassicality that have been investigated intensively are:
(i) negativity, that is, the need to posit negative values when representing
quantum states by quasiprobability distributions such as the Wigner
representation, and (ii) contextuality, that is, the impossibility of a
noncontextual hidden variable model of quantum theory (also known as the
Bell-Kochen-Specker theorem). Although both of these notions were meant to
characterize the conditions under which a classical explanation cannot be
provided, we demonstrate that they prove inadequate to the task and we argue
for a particular way of generalizing and revising them. With the refined
version of each in hand, it becomes apparent that they are in fact one and the
same. We also demonstrate the impossibility of noncontextuality or
nonnegativity in quantum theory with a novel proof that is symmetric in its
treatment of measurements and preparations.Comment: 5 pages, published version (modulo some supplementary material
Reply to Comments of Bassi, Ghirardi, and Tumulka on the Free Will Theorem
We show that the authors in the title have erred in claiming that our axiom
FIN is false by conflating it with Bell locality. We also argue that the
predictions of quantum mechanics, and in particular EPR, are fully Lorentz
invariant, whereas the Free Will Theorem shows that theories with a mechanism
of reduction, such as GRW, cannot be made fully invariant.Comment: We sharpen our theorem by replacing axiom FIN by a weaker axiom MIN
to answer the above authors' objection
A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
This paper is the second in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper, we study in depth the topos representation of the
propositional language, PL(S), for the case of quantum theory. In doing so, we
make a direct link with, and clarify, the earlier work on applying topos theory
to quantum physics. The key step is a process we term `daseinisation' by which
a projection operator is mapped to a sub-object of the spectral presheaf--the
topos quantum analogue of a classical state space. In the second part of the
paper we change gear with the introduction of the more sophisticated local
language L(S). From this point forward, throughout the rest of the series of
papers, our attention will be devoted almost entirely to this language. In the
present paper, we use L(S) to study `truth objects' in the topos. These are
objects in the topos that play the role of states: a necessary development as
the spectral presheaf has no global elements, and hence there are no
microstates in the sense of classical physics. Truth objects therefore play a
crucial role in our formalism.Comment: 34 pages, no figure
Relaxed Bell inequalities and Kochen-Specker theorems
The combination of various physically plausible properties, such as no
signaling, determinism, and experimental free will, is known to be incompatible
with quantum correlations. Hence, these properties must be individually or
jointly relaxed in any model of such correlations. The necessary degrees of
relaxation are quantified here, via natural distance and information-theoretic
measures. This allows quantitative comparisons between different models in
terms of the resources, such as the number of bits, of randomness,
communication, and/or correlation, that they require. For example, measurement
dependence is a relatively strong resource for modeling singlet state
correlations, with only 1/15 of one bit of correlation required between
measurement settings and the underlying variable. It is shown how various
'relaxed' Bell inequalities may be obtained, which precisely specify the
complementary degrees of relaxation required to model any given violation of a
standard Bell inequality. The robustness of a class of Kochen-Specker theorems,
to relaxation of measurement independence, is also investigated. It is shown
that a theorem of Mermin remains valid unless measurement independence is
relaxed by 1/3. The Conway-Kochen 'free will' theorem and a result of Hardy are
less robust, failing if measurement independence is relaxed by only 6.5% and
4.5%, respectively. An appendix shows the existence of an outcome independent
model is equivalent to the existence of a deterministic model.Comment: 19 pages (including 3 appendices); v3: minor clarifications, to
appear in PR
An entropic approach to local realism and noncontextuality
For any Bell locality scenario (or Kochen-Specker noncontextuality scenario),
the joint Shannon entropies of local (or noncontextual) models define a convex
cone for which the non-trivial facets are tight entropic Bell (or
contextuality) inequalities. In this paper we explore this entropic approach
and derive tight entropic inequalities for various scenarios. One advantage of
entropic inequalities is that they easily adapt to situations like bilocality
scenarios, which have additional independence requirements that are non-linear
on the level of probabilities, but linear on the level of entropies. Another
advantage is that, despite the nonlinearity, taking detection inefficiencies
into account turns out to be very simple. When joint measurements are conducted
by a single detector only, the detector efficiency for witnessing quantum
contextuality can be arbitrarily low.Comment: 12 pages, 8 figures, minor mistakes correcte
Is there contextuality for a single qubit?
It was presented by Cabello and Nakamura [A. Cabello, Phys. Rev. Lett. 90,
190401 (2003)], that the Kochen-Specker theorem applies to two dimensions if
one uses Positive Operator-Valued Measures. We show that contextuality in their
models is not of the Kochen-Specker type. It is rather the result of not
keeping track of the whole system on which the measurement is performed. This
is connected to the fact that there is no one-to-one correspondence between
POVM elements and projectors on the extended Hilbert space and the same POVM
element has to originate from two different projectors when used in Cabello's
and Nakamura's models. Moreover, we propose a hidden-variable formulation of
the above models.Comment: 4 pages, 1 figure, comments welcom
The Free Will Theorem
On the basis of three physical axioms, we prove that if the choice of a
particular type of spin 1 experiment is not a function of the information
accessible to the experimenters, then its outcome is equally not a function of
the information accessible to the particles. We show that this result is
robust, and deduce that neither hidden variable theories nor mechanisms of the
GRW type for wave function collapse can be made relativistic. We also establish
the consistency of our axioms and discuss the philosophical implications.Comment: 31 pages, 6figure
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