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Hi Guys
Quantum Darwinism came up in another thread so I thought I would go through Zurecks paper on it.
http://arxiv.org/pdf/0903.5082v1.pdf
Went through it years ago but know quite a bit more about the foundations of QM now.
What struck me this time was his postulates:
'Quantum physics is based on several “textbook” postulates The first two; (i) States are represented by vectors in Hilbert space, and; (ii) Evolutions are unitary – give complete account of mathematics of quantum theory, but make no connection with physics. For that one needs to relate calculations made possible by the superposition principle of (i) and unitarity of (ii) to experiments. Postulate (iii) Immediate repetition of a measurement yields the same outcome starts this task. This is the only uncontroversial measurement postulate (even if it is difficult to approximate in the laboratory): Such repeatability or predictability is behind the very idea of “a state”. In contrast to (i)-(iii), collapse postulate (iv) Outcomes correspond to eigenstates of the measured observable, and only one of them is detected in any given run of the experiment, is inconsistent with (i) and (ii). Conflict arises for two reasons: Restriction to a preferred set of outcome states seems at odds with with the egalitarian principle of superposition, embodied in (i). This restriction prevents one from finding out unknown quantum states, so it is responsible for their fragility.'
My concern has to do with Gleason's theorem. Starting from Von Neumann measurements (the probability of outcome i is described by a resolution of the identity Ei, such that outcome i depends only on Ei) one applies Gleason and you get; first, states must exist (and they obey by fiat the principle of superposition since pure states form a vector space), and secondly, the Born rule he asserts is in conflict with the first. That however is logically not possible. I am not worried about the unitary thing since that falls out of Wigners Theorem.
Really scratching my head here.
What do others think?
Thanks
Bill
Quantum Darwinism came up in another thread so I thought I would go through Zurecks paper on it.
http://arxiv.org/pdf/0903.5082v1.pdf
Went through it years ago but know quite a bit more about the foundations of QM now.
What struck me this time was his postulates:
'Quantum physics is based on several “textbook” postulates The first two; (i) States are represented by vectors in Hilbert space, and; (ii) Evolutions are unitary – give complete account of mathematics of quantum theory, but make no connection with physics. For that one needs to relate calculations made possible by the superposition principle of (i) and unitarity of (ii) to experiments. Postulate (iii) Immediate repetition of a measurement yields the same outcome starts this task. This is the only uncontroversial measurement postulate (even if it is difficult to approximate in the laboratory): Such repeatability or predictability is behind the very idea of “a state”. In contrast to (i)-(iii), collapse postulate (iv) Outcomes correspond to eigenstates of the measured observable, and only one of them is detected in any given run of the experiment, is inconsistent with (i) and (ii). Conflict arises for two reasons: Restriction to a preferred set of outcome states seems at odds with with the egalitarian principle of superposition, embodied in (i). This restriction prevents one from finding out unknown quantum states, so it is responsible for their fragility.'
My concern has to do with Gleason's theorem. Starting from Von Neumann measurements (the probability of outcome i is described by a resolution of the identity Ei, such that outcome i depends only on Ei) one applies Gleason and you get; first, states must exist (and they obey by fiat the principle of superposition since pure states form a vector space), and secondly, the Born rule he asserts is in conflict with the first. That however is logically not possible. I am not worried about the unitary thing since that falls out of Wigners Theorem.
Really scratching my head here.
What do others think?
Thanks
Bill