Abstract
This paper improves on the result in my (Bacciagaluppi in Eur. J. Philos. Sci. 3: 87–100, 2013), showing that within the framework of the unitary Schrödinger equation it is impossible to reproduce the phenomenological description of quantum mechanical measurements (in particular the collapse of the state of the measured system) by assuming a suitable mixed initial state of the apparatus. The result follows directly from the nosignalling theorem applied to the entangled state of measured system and ancilla. As opposed to many other ‘insolubility theorems’ for the measurement problem of quantum mechanics, it focuses on the impossibility of reproducing the phenomenological collapse of the state of the measured system.
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Notes
Note this might or might not coincide with the upper or lower half of the screen, as it depends on both the sign of the gradient and the polarity of the magnetic field.
For a comprehensive discussion in the language of POV measures, see e.g. [2].
Page references are to the 1955 translation (which I have mostly tacitly amended).
The 1955 translation has ‘I was in a (unique) state’, but the German ‘einheitlich’ is used by von Neumann explicitly as a synonym for ‘rein’, i.e. ‘pure’, in which case it is generally translated as ‘homogeneous’ (see e.g. the definition of a ‘homogeneous or pure’ expectation functional on p. 307). Clearly, von Neumann here means a pure state, represented by a projection ψ〉〈ψ.
Since the evolution of the total system is unitary, the total system needs to be closed, and we need to include in the ‘observer’ anything with which the system will interact directly or indirectly. That is, we need to include any ancillas, pieces of apparatus, and relevant parts of the environment (including human observers, if applicable). For simplicity, we shall usually talk of the ‘apparatus’ or the ‘screen’, but we shall tacitly include in it everything else of relevance.
Presumably, the microscopic states corresponding to different readings need to be orthogonal, but will not form a basis of the (very highdimensional) Hilbert space.
Note also that, arguably, the claim von Neumann is explicitly ruling out has the form

(a)
There exist a measurement scheme \((\mathcal{H}, U, \rho)\) and a finest ignoranceinterpretable decomposition of ρ that explain the phenomenological collapse for any ψ〉.
However, von Neumann’s assumption that the ignoranceinterpretable decomposition of the final state has the form (17) essentially constrains this decomposition uniquely, so that allowing the decomposition to depend on ψ〉 does not make the claim more general. That is, his proof rules out also the a priori weaker

(b)
There exists a measurement scheme \((\mathcal{H}, U, \rho )\) such that for any ψ〉 there exists a finest ignoranceinterpretable decomposition of ρ that explains the phenomenological collapse.
Both Brown [5] and myself [1] have argued that, with the exception of [4, 5, 7], the later generalisations of von Neumann’s result rule out claims that are essentially equivalent to (b), and thus establish something perhaps unnecessarily stronger than insolubility in the original sense.

(a)
More generally, we can assume that the total system evolves to states that are indistinguishable from the states on the righthand side of (21), even though they might contain some classical or quantum correlations. These states need not be the same on each run of the experiment, as long as they are recognisable as indicating the given result. We can also allow for slight variation (even covariation) from one run of the experiment to another both in the initial states Ψ〉 and in the measurement scheme \((\mathcal{H}, U,\rho)\), as long as for the correct proportion of runs a condition of the form (21) holds. We can even include a further component Λ _{0} in (20), giving rise to ‘dud runs’ of the experiment.
Please be aware of a confusing amalgam of text and quotation on pp. 95–96 (introduced after the proofs were corrected).
Einstein to Schrödinger, 19 June 1935, AHQP, mf. 92, Sect. 2107 (in German). Also in [19], vol. 2, pp. 537–539. All translations of passages from this letter are by E. Crull and myself.
In Howard’s [17] analysis, it corresponds to the conjunction of ‘separability’, i.e. the existence of separate states for the two subsystems, and ‘locality’, i.e. nosignalling. Howard points out that Einstein starts making this distinction explicit by 1946 or 1947.
In the classical analogue that precedes the quantum mechanical discussion in the letter, Einstein talks about ‘insufficiently known factors foreign to the described system’.
There is a tradeoff, however: the argument works already with the manipulations Alice can perform in practice (measuring this or that observable).
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Acknowledgements
I wish to thank audiences at Aberdeen, Berlin, Cagliari and Oxford, as well as Arthur Fine and Max Schlosshauer, who heard or read and commented on previous versions of this and connected material. I am particularly indebted to Alex Blum, Martin Jähnert and especially Christoph Lehner for discussions of Einstein’s argument and of how it might relate (or not) to von Neumann’s. These discussions also helped me redirect my use of the nosignalling theorem to the general case of measurements with ancillas, whether or not there is spatial separation. Finally, I wish to thank Elise Crull, my collaborator on the Leverhulme Trust Project Grant ‘The Einstein Paradox’: The Debate on Nonlocality and Incompleteness in 1935 (project grant nr. F/00 152/AN), during the tenure of which this paper was written.
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Bacciagaluppi, G. Insolubility from NoSignalling. Int J Theor Phys 53, 3465–3474 (2014). https://doi.org/10.1007/s107730131821y
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DOI: https://doi.org/10.1007/s107730131821y