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The concept of a quantum state is one of the most fundamental in quantum physics. However, despite its importance, qualitative interpretive explanations for this mathematical tool are still the subject of intense debate. First, I review the basic concept of the wavefunction from the point of view of an undergraduate physics student, before considering the arguments for ontic and epistemic interpretations of quantum states.

The question of what the wavefunction (and by extension the quantum state) represents, has existed since the origins of quantum theory. Schro ̈dinger and de Broglie, two true quantum pioneers, originally sought to view the wavefunction as a real physical wave  with some modifications to allow for quantum behaviour. Since those early days, the Copenhagen interpretation has emerged

This depth of insight tends to suffice most undergraduate students of physics, though there is much metaphysics to be explored in the question of the true nature of the quantum state.

Put most simply, an ontic quantum state is one which is a state of reality. It is a state that describes something that exists in reality, independently of any possible observers. Leifer gives a simple analogy of this in , in terms of a single particle with position x and momentum p. In this simple one-dimensional system, the ontic state of such a particle would be described by the exact point (x, p) it occupies in phase space.

In contrast to ontic states, an epistemic quantum state specifies knowledge or information about some aspect of a physical system. In terms of Leifer’s one-dimensional particle analogy , the corresponding epistemic state would be the probability density f(x,p) over a phase space, where the particle’s exact position and momentum are unknown. In this way, epistemic states are fundamen- tally probabilistic by nature and not intrinsic properties of a physical system. Interestingly, the probabilistic na- ture of epistemic states implies that a single ontic state may exist in more than one epistemic state.

In , an epistemic state like that described by Leifer is utilised, and it is shown that where two distributions f1,f2 overlap such a system contradicts quantum predictions. Pusey, Barrett and Rudolph propose a quantum system with states |ψ0⟩ and |ψ1⟩ such that | ⟨ψ0|ψ1⟩ | =

1 . These states have a basis in Hilbert space so that 2

|ψ⟩=|0⟩,|ψ⟩=|+⟩= |0⟩+|1⟩ and|−⟩= |0⟩−|1⟩. 0 1 √2 √2

Because the distributions f1 (λ) and f2 (λ) overlap, there will be a probability q of the physical state λ being in the overlap region of the two distributions.

To generate the contradiction, they propose a scenario where there exists two devices capable of independently generating the quantum states described above. Given that each device can generate either state independently of the other device, it follows that there is a probability q2 that the physical states of both are from the overlap region. The physical state of the devices is compatible with four possible quantum states:

If the two devices are brought together, the resulting entangled measurement would project onto four orthogonal states:

The first outcome |ζ1⟩ is orthogonal to the basis Equation (1a), so will have zero probability when the state is |0⟩ ⊗ |0⟩. This condition is replicated for |ζ2⟩ with the state |0⟩ ⊗ |+⟩, for |ζ3⟩ with the state |+⟩ ⊗ |0⟩, and for |ζ4⟩ with the state |+⟩ ⊗ |+⟩. Therefore, at least q2 of the time, the measuring device will be uncertain which of the four preparation methods was used. This means that, at least q2 of the time, the measured outcome will have zero probability. Pusey, Barrett and Rudolph therefore show that the distributions for |0⟩ and |+⟩ cannot overlap without violating quantum predictions. They cite this as proof that an “epistemic model cannot reproduce the pre- dictions of quantum theory” and that the quantum state must therefore be ontic.

Leifer notes in [2] that the most controversial of the assumptions made by Pusey, Barrett and Rudolph is the preparation independence postulate (PIP). The PIP is an assumption about how composite systems should be treated when the subsystems have been prepared independently of one another. It essentially argues that the two composite systems prepared in a product state should be independent of each other. To illustrate the consequences of the PIP, Leifer gives the example of two quantum states ρA,ρB whose joint state is the product ρA ⊗ ρB . If ρA exist in an ontic state space defined by (ΛA,ΣA), and similarly (ΛB,ΣB) for ρB, then the PIP ensures that the joint system exists in the ontic space (ΛA ×ΛB,ΣA ⊗ΣB).

In, Emerson, Serbin, Sutherland and Veitch crit- icise that the “assumption of preparation independence relies on an assumption of ‘local causality’ which is the likely source of the conflict they obtain with quantum the- ory” and assert “ assumption of preparation indepen- dence encodes an assumption of local causality, but we already know from Bell’s theorem that, if we are going to consider the hidden variable models framework in the first place, then we must reject that assumption.” Fundamentally, they argue that hidden variables are the root cause of the contradiction discussed by Pusey, Barrett and Rudolph, not the epistemic nature of the quantum state. Therefore, in the view of Emerson, Serbin, Suther- land and Veitch, an epistemic quantum state is not impossible.

Also criticising PIP, Lewis et al argue in  that, with- out the assumption that independent preparations pro- duce uncorrelated ontic states, it is possible to produce epistemic quantum states. They provide modifications to the work of Bell in  as proof of the validity of this view.

In , Rovelli proposes that the notion of the quan- tum state being epistemic arises from situations in which different observers measure different results from a se- quence of events. Rovelli’s argument considers a system S with observer O carrying out measurements on it. If the quantity measured, q, can be equal to 1 or 2, then the two eigenstates obtained by measurement of q will be |1⟩ and |2⟩. Hence, S is described by the state vector |ψ⟩ = α |1⟩ + β |2⟩ in a two-dimensional Hilbert Space HS,where|α|2+|β|2=1.

Rovelli focuses on the specific measurement E which obtains q = 1. From an initial time ti to a time tf after measurement has taken place, Rovelli defines E as the sequence of events:

Rovelli then defines a second observer P that describes a larger S-O system formed by S and O. Such a system is described by the Hilbert space HSO = HS ⊗ HO. The process during which O measures q implies that there exists a physical interaction between O and S. Such an interaction would evolve the initial state of O to |O1⟩ when the initial state of S is |1⟩. The same convention is described for |O2⟩. The interaction between O and S implies that the total state vector is entangled. So, if O has state vector |Oi⟩ at the time ti, P describes E as

Rovelli notes that “this is the conventional description of a measurement as a physical process”, citing the work of von Neumann in [8]. The point of measuring E by these two methods is that O and P yield two distinctly different yet equally correct descriptions. Rovelli cites the work of Zureck in [9] as drawing similar conclusions. From this observation, Rovelli concludes that “Thus, a quantum mechanical description of a certain system (state and/or values of physical quantities) cannot be taken as an ‘ab- solute’ (observer-independent) description of reality, but rather as a formalization, or codification, of properties of a system relative to a given observer. Quantum mechan- ics can therefore be viewed as a theory about the states of systems and values of physical quantities relative to other systems.”

A strong criticism of Rovelli’s work is given by Marchildon in [10]. Marchildon observes that the phrase “conventional description” is vague, and suggests it could either refer to unitary Schro ̈dinger evolution or to Schr ̈odinger evolution and collapse. It is Marchildon’s view that “the issue is that...with any precise definition there is a specific problem.” In the case of strict Schr ̈odinger evolution, the specific problem is quantum mechanics contradicting the experiment. In the case of Schro ̈dinger evolution with collapse, the specific problem is the need to provide a mechanism for collapse. As such, Marchildon judges that Rovelli has not established a convincing argument for the epistemic quantum state. It should however be noted that, in private response to Marchildon’s criticisms, Rovelli “stressed that the rela- tional character of quantum theory is the solution of the apparent contradiction between discarding Schr ̈odinger evolution without collapse and not providing an explicit collapse mechanism.”

Fundamentally, the question of whether the wavefunc- tion specifies the state or just our knowledge remains without a satisfactory answer. In spite of the criticisms discussed, general feeling within the scientific community currently appears to support the work of Pusey, Barrett and Rudolph. This may partly be explained by the high profile nature of their work, but the also by the fact that for every letter criticising their work there is another of support.

Although rigorous scientific analysis is the bedrock of any proof, it is easy for one to lose sight of the wider significance of reaching a conclusive answer. The importance of seeking such an answer is plain to see; both the ontic and epistemic viewpoints imply deep and far- reaching truths about the wider interpretations of quantum theories as a whole. Definitive proof of either being the true nature of the quantum state would undoubtedly provide clues to the solutions of many existing problems within quantum physics.

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