9,095 research outputs found
Conway-Kochen and the Finite Precision Loophole
Recently Cator & Landsman made a comparison between Bell's Theorem and Conway
& Kochen's Strong Free Will Theorem. Their overall conclusion was that the
latter is stronger in that it uses fewer assumptions, but also that it has two
shortcomings. Firstly, no experimental test of the Conway-Kochen Theorem has
been performed thus far, and, secondly, because the Conway-Kochen Theorem is
strongly connected to the Kochen-Specker Theorem it may be susceptible to the
finite precision loophole of Meyer, Kent and Clifton. In this paper I show that
the finite precision loophole does not apply to the Conway-Kochen Theorem
A Verified Certificate Checker for Finite-Precision Error Bounds in Coq and HOL4
Being able to soundly estimate roundoff errors of finite-precision
computations is important for many applications in embedded systems and
scientific computing. Due to the discrepancy between continuous reals and
discrete finite-precision values, automated static analysis tools are highly
valuable to estimate roundoff errors. The results, however, are only as correct
as the implementations of the static analysis tools. This paper presents a
formally verified and modular tool which fully automatically checks the
correctness of finite-precision roundoff error bounds encoded in a certificate.
We present implementations of certificate generation and checking for both Coq
and HOL4 and evaluate it on a number of examples from the literature. The
experiments use both in-logic evaluation of Coq and HOL4, and execution of
extracted code outside of the logics: we benchmark Coq extracted unverified
OCaml code and a CakeML-generated verified binary
Non-Contextual Hidden Variables and Physical Measurements
For a hidden variable theory to be indistinguishable from quantum theory for
finite precision measurements, it is enough that its predictions agree for some
measurement within the range of precision. Meyer has recently pointed out that
the Kochen-Specker theorem, which demonstrates the impossibility of a
deterministic hidden variable description of ideal spin measurements on a spin
1 particle, can thus be effectively nullified if only finite precision
measurements are considered. We generalise this result: it is possible to
ascribe consistent outcomes to a dense subset of the set of projection valued
measurements, or to a dense subset of the set of positive operator valued
measurements, on any finite dimensional system. Hence no Kochen-Specker like
contradiction can rule out hidden variable theories indistinguishable from
quantum theory by finite precision measurements in either class.Comment: Typo corrected. Final version: to appear in Phys. Rev. Let
Finite precision measurement nullifies the Kochen-Specker theorem
Only finite precision measurements are experimentally reasonable, and they
cannot distinguish a dense subset from its closure. We show that the rational
vectors, which are dense in S^2, can be colored so that the contradiction with
hidden variable theories provided by Kochen-Specker constructions does not
obtain. Thus, in contrast to violation of the Bell inequalities, no
quantum-over-classical advantage for information processing can be derived from
the Kochen-Specker theorem alone.Comment: 7 pages, plain TeX; minor corrections, interpretation clarified,
references update
Robust explicit MPC design under finite precision arithmetic
We propose a design methodology for explicit Model Predictive Control (MPC) that guarantees hard constraint satisfaction in the presence of finite precision arithmetic errors. The implementation of complex digital control techniques, like MPC, is becoming increasingly adopted in embedded systems, where reduced precision computation techniques are embraced to achieve fast execution and low power consumption. However, in a low precision implementation, constraint satisfaction is not guaranteed if infinite precision is assumed during the algorithm design. To enforce constraint satisfaction under numerical errors, we use forward error analysis to compute an error bound on the output of the embedded controller. We treat this error as a state disturbance and use this to inform the design of a constraint-tightening robust controller. Benchmarks with a classical control problem, namely an inverted pendulum, show how it is possible to guarantee, by design, constraint satisfaction for embedded systems featuring low precision, fixed-point computations
Noncontextuality, Finite Precision Measurement and the Kochen-Specker Theorem
Meyer recently queried whether non-contextual hidden variable models can,
despite the Kochen-Specker theorem, simulate the predictions of quantum
mechanics to within any fixed finite experimental precision. Clifton and Kent
have presented constructions of non-contextual hidden variable theories which,
they argued, indeed simulate quantum mechanics in this way. These arguments
have evoked some controversy. One aim of this paper is to respond to and rebut
criticisms of the MCK papers. We thus elaborate in a little more detail how the
CK models can reproduce the predictions of quantum mechanics to arbitrary
precision. We analyse in more detail the relationship between classicality,
finite precision measurement and contextuality, and defend the claims that the
CK models are both essentially classical and non-contextual. We also examine in
more detail the senses in which a theory can be said to be contextual or
non-contextual, and in which an experiment can be said to provide evidence on
the point. In particular, we criticise the suggestion that a decisive
experimental verification of contextuality is possible, arguing that the idea
rests on a conceptual confusion.Comment: 27 pages; published version; minor changes from previous versio
Kochen-Specker Theorem for Finite Precision Spin One Measurements
Unsharp spin 1 observables arise from the fact that a residual uncertainty
about the actual orientation of the measurement device remains. If the
uncertainty is below a certain level, and if the distribution of measurement
errors is covariant under rotations, a Kochen-Specker theorem for the unsharp
spin observables follows: There are finite sets of directions such that not all
the unsharp spin observables in these directions can consistently be assigned
approximate truth-values in a non-contextual way.Comment: 4 page
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