Abstract
I present a very general and simple argument—based on the no-signalling theorem—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 thrust of the argument is thus similar to that of the ‘insolubility theorems’ for the measurement problem of quantum mechanics (which, however, focus on the impossibility of reproducing the macroscopic measurement results). Although I believe this form of the argument is new, I argue it is essentially a variant of Einstein’s reasoning in the context of the EPR paradox—which is thereby illuminated from a new angle.
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Notes
Page references are to the 1955 translation (which I have mostly tacitly amended).
It is less apparent to whom von Neumann attributes such an explanation. Who are the ones who ‘often proposed’ it? (See also the discussion in Section 4.)
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 \(\vert \psi\rangle\langle \psi\vert\).
Since the evolution of the total system is to be unitary, the total system needs to be closed, and we need to include in the apparatus anything with which the system will interact directly or indirectly. That is, we need to include in the apparatus also all the relevant parts of the environment. For the sake of brevity, however, we shall usually just talk about the ‘apparatus’.
Presumably, the microscopic states of the apparatus corresponding to different readings need to be orthogonal, but will not form a basis of the (very high-dimensional) apparatus Hilbert space.
It is interesting to consider whether such apparently conspiratorial dependencies might be explainable in retrocausal terms (see e. g. Price 1996), but this is clearly not a strategy von Neumann was considering.
A probably not quite exhaustive list includes Wigner (1963), d’Espagnat (1966), Earman and Shimony (1968), Fine (1970), d’Espagnat (1971, Sections 14–1, 14–2, 15–3 and 15–5), Fehrs and Shimony (1974), Shimony (1974), Brown (1986), Busch and Shimony (1996), Stein (1997), and Bassi and Ghirardi (2000). Most of these theorems concern measurements of discrete projection-valued (PV) observables (possibly including errors and disturbance). The only exceptions are the two theorems in Busch and Shimony (1996), which establish insolubility also for measurements of continuous PV observables and for measurements of arbitrary positive-operator-valued (POV) observables, respectively. (For a general reference to POV observables, see Busch et al. (1995).)
Von Neumann assumes the orthogonality of the states \(\vert \psi_n\rangle\otimes\vert \varphi_n\rangle\) in the relevant decomposition of the final state. Thus, he could have easily phrased his result in terms of the orthogonal pointer states \(\vert \varphi_n\rangle\), rather than the collapsed states \(\vert \psi_n\rangle\) of the system. Indeed, if one does so, von Neumann’s proof generalises to the case of disturbing measurements (i. e. where the system is collapsed into some disturbed states \(\vert \tilde{\psi}_n\rangle\)). As a matter of fact, von Neumann’s proof generalises even to the case of approximate measurements (where the final distribution is only approximately given by the squared moduli of the coefficients in the initial state \(\vert \psi\rangle\))—since even in an approximate measurement one requires some dependence of the final distribution on the initial state of the system.
Depending on the theorem in question, perhaps the triple \(({\cal H}, U, \rho)\) satisfies some further minimal constraint to justify labelling it indeed a ‘measurement scheme’; perhaps the result holds only for almost all \(\vert \psi\rangle\); perhaps the apparatus has a definite reading only with probability close to 1; etc.
If the initial state of the system is not pure, then in general we can no longer trace an ignorance in the final state to ignorance of the initial apparatus state alone.
One could be charitable of course, and point out that it might be of independent interest to rule out also putative models with such ‘conspiratorial’ dependencies on \(\vert \psi\rangle\) of the relevant decomposition of ρ.
See also footnote 18 below.
Extracts of this correspondence are well-known from the discussions by Fine (1981) and by Howard (1985, 1990). The correspondence has now been published in full (in the original German) as part of the extensive selection of Schrödinger’s correspondence edited by von Meyenn (2011). The following translations of passages from Einstein’s letter of 19 June are by Elise Crull and myself. We are currently preparing a volume including commentary and translations of original materials relating to the 1935 debate on the EPR paper, focusing especially on some of its lesser known aspects (Bacciagaluppi and Crull, in preparation).
Einstein to Schrödinger, 19 June 1935, Archive for the History of Quantum Physics, microfilm 92, Section 2–107 (in German); von Meyenn (2011), Vol. 2, pp. 537–539.
That is, it describes an ensemble of systems, half of which are in one state and half in the other.
One can perhaps say that a complete theory needs to describe not only the real states of a system, but also its dynamics (both when the system is isolated and when it is subject to measurement).
I am not claiming that this is the correct historical reading of Einstein’s argument. Indeed, it is not an aspect that returns explicitly in later presentations of his incompleteness argument (although I see nothing in the later presentations that would exclude this reading). I am only suggesting that there is a possible analogy between certain aspects of Einstein’s incompleteness argument and certain aspects of von Neumann’s insolubility argument (and by extension certain aspects of our argument in Section 3).
According to Howard (1985), Einstein’s separation principle is to be analysed in terms of separability (the existence of separate states) and locality (no superluminal influence), a distinction which Einstein starts making explicitly by 1946 or 1947. In terms of this distinction, another way of expressing the above disanalogy is to say that if one focuses on what Alice can do in principle, one can derive a contradiction with the weaker locality principle, while if one focuses on what Alice can do in practice, the contradiction will be only with the full separation principle. This may be a less anachronistic way of making the comparison, since as far as I am aware the no-signalling theorem was formulated explicitly only in 1980 by Ghirardi et al.
The result is presented perhaps even more strikingly in Gisin (1990). It should be emphasised that (as Gisin realises perfectly well) the assumption from which signalling is derived, namely that a given theory with a nonlinear Schrödinger equation reproduce the standard predictions of quantum mechanics, is extremely strong and perhaps impossible to satisfy. See Doebner and Goldin (1996) and Bacciagaluppi (2012), respectively, for two explicit examples of nonlinear theories that do not exhibit signalling, or at least not for all entangled states.
At the conference on ‘Emergent Quantum Mechanics’, University of Vienna, 11–13 November 2011.
Under an alternative ‘minimal’ or ‘statistical’ interpretation of the theory—in the sense adopted by the above authors—the measurement problem could be said not to arise. Such an interpretation, however, throws open further weighty questions—of why and how in the context of macroscopic measurements we appear to have observational access to an individual description, and of whether and how this individual description extends also to a ‘sub-quantum’ level.
In particular one that incorporates the intuition that actual measuring apparatuses are typically macroscopic systems with typically thermal environments, and as such should be normally described by highly mixed states.
Note that also in the classic model by Daneri et al. (1962)—which relies on the same intuitive mechanism—the initial state of the apparatus is explicitly taken to be pure.
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Acknowledgements
I wish to thank in particular Arthur Fine for very perceptive comments on a previous draft of this paper. Many thanks also to Theo Nieuwenhuizen for inspiration, to Max Schlosshauer for correspondence, to two anonymous referees for shrewd observations, and to audiences at Aberdeen, Cagliari and Oxford (in particular to Harvey Brown, Elise Crull, Simon Saunders, Chris Timpson and David Wallace) for stimulating questions. This paper was written during my tenure of a Leverhulme Grant on ‘The Einstein Paradox’: The Debate on Nonlocality and Incompleteness in 1935 (Project Grant nr. F/00 152/AN), and it was revised for publication during my tenure of a Visiting Professorship in the Doctoral School of Philosophy and Epistemology, University of Cagliari (Contract nr. 268/21647).
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Bacciagaluppi, G. Insolubility Theorems and EPR Argument. Euro Jnl Phil Sci 3, 87–100 (2013). https://doi.org/10.1007/s13194-012-0057-7
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DOI: https://doi.org/10.1007/s13194-012-0057-7