Is a System’s Wave Function in One-to-One Correspondence with Its Elements of Reality?

Although quantum mechanics is one of our most successful physical theories, there has been a long-standing debate about the interpretation of the wave function—the central object of the theory. Two prominent views are that (i) it corresponds to an element of reality, i.e., an objective attribute that exists before measurement, and (ii) it is a subjective state of knowledge about some underlying reality. A recent result [M. F. Pusey, J. Barrett, and T. Rudolph, arXiv:1111.3328] has placed the subjective interpretation into doubt, showing that it would contradict certain physically plausible assumptions, in particular, that multiple systems can be prepared such that their elements of reality are uncorrelated. Here we show, based only on the assumption that measurement settings can be chosen freely, that a system’s wave function is in one-to-one correspondence with its elements of reality. This also eliminates the possibility that it can be interpreted subjectively.

Figure 1

Simple example illustrating the ideas. Two meteorologists attempt to predict tomorrow’s weather (whether it will be sunny or cloudy in a particular location). Both have access to historical data giving the joint distribution of the weather on successive days. However, only the meteorologist on the left has access to today’s weather, and consequently the two make different probabilistic forecasts, $F$ and $F^{\prime }$. Assuming that the processes relevant to the weather are accurately described by classical mechanics and thus deterministic, the list of elements of reality, $\Lambda$, may include tomorrow’s weather, $X$. Such a list $\Lambda$ would then necessarily satisfy $\Gamma \leftrightarrow \Lambda \leftrightarrow X$ (for any arbitrary $\Gamma$) and therefore be complete [cf. Eq. (1)]. However, the analogue of Eq. (2), $\Lambda \leftrightarrow F\leftrightarrow X$, would imply $X\leftrightarrow F\leftrightarrow X$. This Markov chain cannot hold for the nondeterministic forecasts $F$ and $F^{\prime }$, which are hence not complete. This is unlike the quantum-mechanical wave function, which gives a complete description for the prediction of measurement outcomes. Note that this difference explains why, in contrast to the quantum-mechanical wave function, $F$ and $F^{\prime }$ need not be included in $\Lambda$ and can therefore be considered subjective.

Figure 2

Illustration of the setup. A system is prepared in a particular quantum state (specified by a wave function $\Psi$). The elements of reality, $\Lambda$, may depend on this preparation. A measurement setting $A$ is then randomly chosen, and the system is measured, producing an outcome $X$. We assume that $\Lambda$ is complete for the description of the system, in the sense that there does not exist any other parameter that provides additional information (beyond that contained in $\Lambda$) about the outcome of any chosen measurement. In particular, $\Psi$ cannot provide more information than $\Lambda$. Conversely, the nonextendibility of quantum theory [16] implies that $\Lambda$ cannot provide more information (about the outcome) than $\Psi$. Taken together, these statements imply that $\Psi$ and $\Lambda$ are informationally equivalent. From this and the fact that different quantum states generally lead to different measurement statistics, we conclude that $\Psi$ must be included in the list $\Lambda$ and is therefore an element of reality of the system.