87 research outputs found
Bounding the plausibility of physical theories in a device-independent setting via hypothesis testing
The device-independent approach to physics is one where conclusions about
physical systems (and hence of Nature) are drawn directly and solely from the
observed correlations between measurement outcomes. This operational approach
to physics arose as a byproduct of Bell's seminal work to distinguish, via a
Bell test, quantum correlations from the set of correlations allowed by
local-hidden-variable theories. In practice, since one can only perform a
finite number of experimental trials, deciding whether an empirical observation
is compatible with some class of physical theories will have to be carried out
via the task of hypothesis testing. In this paper, we show that the
prediction-based-ratio method---initially developed for performing a hypothesis
test of local-hidden-variable theories---can equally well be applied to test
many other classes of physical theories, such as those constrained only by the
nonsignaling principle, and those that are constrained to produce any of the
outer approximation to the quantum set of correlations due to
Navascu\'es-Pironio-Ac\'{\i}n. We numerically simulate Bell tests using
hypothetical nonlocal sources of correlations to illustrate the applicability
of the method in both the independent and identically distributed (i.i.d.)
scenario and the non-i.i.d. scenario. As a further application, we demonstrate
how this method allows us to unveil an apparent violation of the nonsignaling
conditions in certain experimental data collected in a Bell test. This, in
turn, highlights the importance of the randomization of measurement settings,
as well as a consistency check of the nonsignaling conditions in a Bell test.Comment: 10 pages, 1 figure, 3 tables (essentially the published version with
simplified discussion and clearer presentation of results
Quantum Probability Estimation for Randomness with Quantum Side Information
We develop a quantum version of the probability estimation framework
[arXiv:1709.06159] for randomness generation with quantum side information. We
show that most of the properties of probability estimation hold for quantum
probability estimation (QPE). This includes asymptotic optimality at constant
error and randomness expansion with logarithmic input entropy. QPE is
implemented by constructing model-dependent quantum estimation factors (QEFs),
which yield statistical confidence upper bounds on data-conditional normalized
R\'enyi powers. This leads to conditional min-entropy estimates for randomness
generation. The bounds are valid for relevant models of sequences of
experimental trials without requiring independent and identical or stationary
behavior. QEFs may be adapted to changing conditions during the sequence and
trials can be stopped any time, such as when the results so far are
satisfactory. QEFs can be constructed from entropy estimators to improve the
bounds for conditional min-entropy of classical-quantum states from the entropy
accumulation framework [Dupuis, Fawzi and Renner, arXiv:1607.01796]. QEFs are
applicable to a larger class of models, including models permitting measurement
devices with super-quantum but non-signaling behaviors and semi-device
dependent models. The improved bounds are relevant for finite data or error
bounds of the form , where is the number of random bits
produced. We give a general construction of entropy estimators based on maximum
probability estimators, which exist for many configurations. For the class of
Bell-test configurations we provide schemas for directly optimizing
QEFs to overcome the limitations of entropy-estimator-based constructions. We
obtain and apply QEFs for examples involving the Bell-test
configuration to demonstrate substantial improvements in finite-data
efficiency.Comment: v2: Clarified soundness discussion and other edits, see the
explanation after the references. v3: Clarified discussion of examples and
comparisons. Parts of this paper have been published as Physical Review
Research, 2, 013016, 2020,
https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.01301
Device-independent point estimation from finite data and its application to device-independent property estimation
The device-independent approach to physics is one where conclusions are drawn
directly from the observed correlations between measurement outcomes. In
quantum information, this approach allows one to make strong statements about
the properties of the underlying systems or devices solely via the observation
of Bell-inequality-violating correlations. However, since one can only perform
a {\em finite number} of experimental trials, statistical fluctuations
necessarily accompany any estimation of these correlations. Consequently, an
important gap remains between the many theoretical tools developed for the
asymptotic scenario and the experimentally obtained raw data. In particular, a
physical and concurrently practical way to estimate the underlying quantum
distribution has so far remained elusive. Here, we show that the natural
analogs of the maximum-likelihood estimation technique and the
least-square-error estimation technique in the device-independent context
result in point estimates of the true distribution that are physical, unique,
computationally tractable and consistent. They thus serve as sound algorithmic
tools allowing one to bridge the aforementioned gap. As an application, we
demonstrate how such estimates of the underlying quantum distribution can be
used to provide, in certain cases, trustworthy estimates of the amount of
entanglement present in the measured system. In stark contrast to existing
approaches to device-independent parameter estimations, our estimation does not
require the prior knowledge of {\em any} Bell inequality tailored for the
specific property and the specific distribution of interest.Comment: Essentially published version, but with the typo in Eq. (E5)
correcte
The statistical strength of experiments to reject local realism with photon pairs and inefficient detectors
Because of the fundamental importance of Bell's theorem, a loophole-free
demonstration of a violation of local realism (LR) is highly desirable. Here,
we study violations of LR involving photon pairs. We quantify the experimental
evidence against LR by using measures of statistical strength related to the
Kullback-Leibler (KL) divergence, as suggested by van Dam et al. [W. van Dam,
R. Gill and P. Grunwald, IEEE Trans. Inf. Theory. 51, 2812 (2005)].
Specifically, we analyze a test of LR with entangled states created from two
independent polarized photons passing through a polarizing beam splitter. We
numerically study the detection efficiency required to achieve a specified
statistical strength for the rejection of LR depending on whether photon
counters or detectors are used. Based on our results, we find that a test of LR
free of the detection loophole requires photon counters with efficiencies of at
least 89.71%, or photon detectors with efficiencies of at least 91.11%. For
comparison, we also perform this analysis with ideal unbalanced Bell states,
which are known to allow rejection of LR with detector efficiencies above 2/3.Comment: 18 pages, 3 figures, minor changes (add more references, replace the
old plots, etc.)
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