2,603 research outputs found
The Inflation Technique for Causal Inference with Latent Variables
The problem of causal inference is to determine if a given probability
distribution on observed variables is compatible with some causal structure.
The difficult case is when the causal structure includes latent variables. We
here introduce the for tackling this problem. An
inflation of a causal structure is a new causal structure that can contain
multiple copies of each of the original variables, but where the ancestry of
each copy mirrors that of the original. To every distribution of the observed
variables that is compatible with the original causal structure, we assign a
family of marginal distributions on certain subsets of the copies that are
compatible with the inflated causal structure. It follows that compatibility
constraints for the inflation can be translated into compatibility constraints
for the original causal structure. Even if the constraints at the level of
inflation are weak, such as observable statistical independences implied by
disjoint causal ancestry, the translated constraints can be strong. We apply
this method to derive new inequalities whose violation by a distribution
witnesses that distribution's incompatibility with the causal structure (of
which Bell inequalities and Pearl's instrumental inequality are prominent
examples). We describe an algorithm for deriving all such inequalities for the
original causal structure that follow from ancestral independences in the
inflation. For three observed binary variables with pairwise common causes, it
yields inequalities that are stronger in at least some aspects than those
obtainable by existing methods. We also describe an algorithm that derives a
weaker set of inequalities but is more efficient. Finally, we discuss which
inflations are such that the inequalities one obtains from them remain valid
even for quantum (and post-quantum) generalizations of the notion of a causal
model.Comment: Minor final corrections, updated to match the published version as
closely as possibl
Which causal structures might support a quantum-classical gap?
A causal scenario is a graph that describes the cause and effect
relationships between all relevant variables in an experiment. A scenario is
deemed `not interesting' if there is no device-independent way to distinguish
the predictions of classical physics from any generalised probabilistic theory
(including quantum mechanics). Conversely, an interesting scenario is one in
which there exists a gap between the predictions of different operational
probabilistic theories, as occurs for example in Bell-type experiments. Henson,
Lal and Pusey (HLP) recently proposed a sufficient condition for a causal
scenario to not be interesting. In this paper we supplement their analysis with
some new techniques and results. We first show that existing graphical
techniques due to Evans can be used to confirm by inspection that many graphs
are interesting without having to explicitly search for inequality violations.
For three exceptional cases -- the graphs numbered 15,16,20 in HLP -- we show
that there exist non-Shannon type entropic inequalities that imply these graphs
are interesting. In doing so, we find that existing methods of entropic
inequalities can be greatly enhanced by conditioning on the specific values of
certain variables.Comment: 13 pages, 9 figures, 1 bicycle. Added an appendix showing that
e-separation is strictly more general than the skeleton method. Added journal
referenc
Security against eavesdropping in quantum cryptography
In this article we deal with the security of the BB84 quantum cryptography
protocol over noisy channels using generalized privacy amplification. For this
we estimate the fraction of bits needed to be discarded during the privacy
amplification step. This estimate is given for two scenarios, both of which
assume the eavesdropper to access each of the signals independently and take
error correction into account. One scenario does not allow a delay of the
eavesdropper's measurement of a measurement probe until he receives additional
classical information. In this scenario we achieve a sharp bound. The other
scenario allows a measurement delay, so that the general attack of an
eavesdropper on individual signals is covered. This bound is not sharp but
allows a practical implementation of the protocol.Comment: 11 pages including 3 figures, contains new results not contained in
my Phys. Rev. A pape
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