Symmetry and global independence in classical and quantum theories

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Renner\'s global quantum de Finetti theorem establishes that if the state of a quantum system is invariant under permutations of its systems, then almost all of its subsystems are almost in the same state and independent of each other. Motivated by this result, we show that the most straightforward classical analogue of Renner\'s theorem is false.
Joint work with Matthias Christandl (Cambridge).