Abstract
The difference between classical and quantum information arises because of the different distinguishability properties of classical and quantum pure states: only orthogonal quantum states are reliably distinguishable with zero probability of error. Classical information is that sort of information represented in a set of distinguishable states and so can be regarded as a subcategory of quantum information. The transition from classical to relativistic physics rests on the recognition that space-time structurally different than we thought. In the transition from classical to quantum physics, what we have discovered is that information in the physical sense is structurally different than we thought. The claim about information and quantum mechanics is that the puzzling and seemingly paradoxical features of the theory, including the measurement problem, are to be understood as arising from this structural difference.
This chapter is dedicated to Bill Demopoulos , my oldest philosophical friend. We began talking about quantum mechanics in the 1960’s, when it seemed to us that quantum logic was the key. Over the years we have had countless conversations about quantum mechanics. Our positions have evolved, sometimes differently, but at the heart of it there is a continuity.
This is also the title of a paper by Gilles Brassard with similar ideas on quantum information. See (Brassard 2000).
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Notes
- 1.
See (Short and Wehner 2010), where the authors show how to define a general measure of information for a broad class of theories, including nonlocal box theories (see below) that reduces to von Neumann entropy for quantum theories and to Shannon entropy for classical theories.
- 2.
It is convenient to change units here to relate the probability to the usual expression for the Clauser-Horne-Shimony-Holt correlation , where the expectation values are expressed in terms of ±1 values for x and y (corresponding to the relevant observables). Note that ‘outputs same’ or ‘outputs different’ mean the same thing whatever the units, so the probabilities \(p\,({\textrm{outputs}}\,{\textrm{same}}\,|\,xy)\) and \(p\,({\textrm{outputs}}\,{\textrm{different}}\,|\,xy)\) take the same values whatever the units, but the expectation value \(\langle xy \rangle\) depends on the units for x and y.
- 3.
A polytope is the analogue of a polygon in many dimensions. A convex set is, roughly, a set such that from any point in the interior it is possible to ‘see’ any point on the boundary.
- 4.
‘Outputs same’ = parity 0; ‘outputs different’ = parity 1.
- 5.
There is also the ‘many worlds’ option of the Everett interpretation (Saunders et al. 2010). This essentially involves a multiplicity of simplices, one for each ‘world.’
- 6.
Note that Bell ’s objection to decoherence as a solution to the measurement problem (Bell 1990) concerns (rightly) the inadequacy of decoherence as a ‘for all practical purposes’ solution to the first problem. It is not an objection to the second problem.
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Acknowledgments
This paper was written during the tenure of a University of Maryland RASA semester research award.
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Bub, J. (2012). Is Information the Key?. In: Frappier, M., Brown, D., DiSalle, R. (eds) Analysis and Interpretation in the Exact Sciences. The Western Ontario Series in Philosophy of Science, vol 78. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2582-9_12
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