No return to classical reality

Abstract

At a fundamental level, the classical picture of the world is dead, and has been dead now for almost a century. Pinning down exactly which quantum phenomena are responsible for this has proved to be a tricky and controversial question, but a lot of progress has been made in the past few decades. We now have a range of precise statements showing that whatever the ultimate laws of nature are, they cannot be classical. In this article, we review results on the fundamental phenomena of quantum theory that cannot be understood in classical terms. We proceed by first granting quite a broad notion of classicality, describe a range of quantum phenomena (such as randomness, discreteness, the indistinguishability of states, measurement-uncertainty, measurement-disturbance, complementarity, non-commutativity, interference, the no-cloning theorem and the collapse of the wave-packet) that do fall under its liberal scope, and then finally describe some aspects of quantum physics that can never admit a classical understanding – the intrinsically quantum mechanical aspects of nature. The most famous of these is Bell’s theorem, but we also review two more recent results in this area. Firstly, Hardy’s theorem shows that even a finite-dimensional quantum system must contain an infinite amount of information, and secondly, the Pusey–Barrett–Rudolph theorem shows that the wave function must be an objective property of an individual quantum system. Besides being of foundational interest, results of this sort now find surprising practical applications in areas such as quantum information science and the simulation of quantum systems.

Keywords:

• quantum foundations
• nonclassicality
• statistical theories
• wavefunction
• quantum state

Acknowledgements

DJ thanks Sania Jevtic for useful comments on an earlier draft.

Notes

No potential conflict of interest was reported by the authors.

1 Ultimately the formal definition of ‘classical’ will be that the theory is a ‘local, non-contextual theory in which non-orthogonal pure states are represented by overlapping statistical distributions defined on some state space $\mathrm{\Lambda }$’.

2 In Appendices 2–4, we review those aspects of quantum mechanics that are relevant to this article.

3 We give a short account of Gaussian states and operations in Appendix 4. See [14] for a more in depth review.

4 The rough idea is that if Alice repeatedly clones one half of an entangled state that has been collapsed by a remote measurement made by Bob on the other half of the state, then she can magnify the information as to what type of state she possesses (e.g. whether it is a momentum eigenstate or a position eigenstate). Bob can use this to signal faster than light by choosing which type of state to collapse Alice’s system to (e.g. momentum eigenstates = ‘yes’, position eigenstates = ‘no’).

5 Note that, although the delta functions make this state unphysical, the same argument can be run with properly normalised Gaussian states that approximate them [12]. We use the idealised version for simplicity.

6 Even more precisely, probability distributions should be associated to the procedures for preparing quantum states rather than the states themselves to account for a subtlety called preparation contextuality [50]. However, this subtlety does not affect any of the results presented here. See [30] for a more rigorous treatment that does deal with this topic.

7 Measure-theoretic qualifications are needed to deal with the general case. See [30] for details.

Additional information

Funding

DJ is supported by the Royal Society. ML is supported by the Foundational Questions Institute (FQXi). Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.