Pusey–Barrett–Rudolph theorem

The Pusey–Barrett–Rudolph (PBR) theorem^[1] is a no-go theorem in quantum foundations due to Matthew Pusey, Jonathan Barrett, and Terry Rudolph (for whom the theorem is named) in 2012. It has particular significance for how one may interpret the nature of the quantum state.

With respect to certain realist hidden variable theories that attempt to explain the predictions of quantum mechanics, the theorem rules that pure quantum states must be "ontic" in the sense that they correspond directly to states of reality, rather than "epistemic" in the sense that they represent probabilistic or incomplete states of knowledge about reality.

The PBR theorem may also be compared with other no-go theorems like Bell's theorem and the Bell–Kochen–Specker theorem, which, respectively, rule out the possibility of explaining the predictions of quantum mechanics with local hidden variable theories and noncontextual hidden variable theories. Similarly, the PBR theorem could be said to rule out preparation independent hidden variable theories, in which quantum states that are prepared independently have independent hidden variable descriptions.

This result was cited by theoretical physicist Antony Valentini as "the most important general theorem relating to the foundations of quantum mechanics since Bell's theorem".^[2]

Theorem

This theorem, which first appeared as an arXiv preprint^[3] and was subsequently published in Nature Physics,^[1] concerns the interpretational status of pure quantum states. Under the classification of hidden variable models of Harrigan and Spekkens,^[4] the interpretation of the quantum wavefunction ${\displaystyle |\psi \rangle }$ can be categorized as either ψ-ontic if "every complete physical state or ontic state in the theory is consistent with only one pure quantum state" and ψ-epistemic "if there exist ontic states that are consistent with more than one pure quantum state." The PBR theorem proves that either the quantum state ${\displaystyle |\psi \rangle }$ is ψ-ontic, or else non-entangled quantum states violate the assumption of preparation independence, which would entail action at a distance.

In conclusion, we have presented a no-go theorem, which—modulo assumptions—shows that models in which the quantum state is interpreted as mere information about an objective physical state of a system cannot reproduce the predictions of quantum theory. The result is in the same spirit as Bell’s theorem, which states that no local theory can reproduce the predictions of quantum theory.

— Matthew F. Pusey, Jonathan Barrett, and Terry Rudolph, "On the reality of the quantum state", Nature Physics 8, 475-478 (2012)

Mathematical Details of the Proof

The PBR theorem demonstrates that under the assumption of preparation independence for the underlying physical or "ontic" states, the quantum state vector ${\displaystyle |\psi \rangle }$ cannot be merely statistical information (epistemic) about that underlying reality, but must be a direct representation of it (ontic). The proof proceeds by showing a contradiction between the predictions of quantum mechanics and the existence of overlapping probability distributions for the ontic states of non-orthogonal quantum states.^[1]

Setup and Assumptions

Let ${\displaystyle \Lambda }$ be the space of possible physical "ontic" states ${\displaystyle \lambda }$ that a system can possess. A preparation procedure corresponding to a quantum state ${\displaystyle |\psi \rangle }$ is described by a probability distribution ${\displaystyle p(\lambda |\psi )}$ over ${\displaystyle \Lambda }$.

The theorem contrasts two views of the quantum state:

1. ${\displaystyle \psi }$-ontic: The quantum state corresponds directly to reality. In this view, if two quantum states ${\displaystyle |\psi _{1}\rangle }$ and ${\displaystyle |\psi _{2}\rangle }$ are distinct, i.e. are not a phase factor multiple of each other, then the probability distributions ${\displaystyle p(\lambda |\psi _{1})}$ and ${\displaystyle p(\lambda |\psi _{2})}$ must be disjoint (i.e., have no overlap in ${\displaystyle \Lambda }$). Knowing the quantum state implies knowing the ontic state (or a restricted set of possible ontic states unique to that quantum state).
2. ${\displaystyle \psi }$-epistemic: The quantum state represents only a state of knowledge or information about the underlying reality. In this view, the distributions ${\displaystyle p(\lambda |\psi _{1})}$ and ${\displaystyle p(\lambda |\psi _{2})}$ for distinct quantum states can overlap. This overlap means there exists a region ${\displaystyle \Delta \subset \Lambda }$ where for any ${\displaystyle \lambda \in \Delta }$, both ${\displaystyle p(\lambda |\psi _{0})>0}$ and ${\displaystyle p(\lambda |\psi _{1})>0}$. An ontic state ${\displaystyle \lambda }$ in this region is consistent with either preparation, so observing ${\displaystyle \lambda }$ would not allow one to perfectly distinguish which state was prepared.^[5]

Preparation Independence: If two or more systems are prepared independently, the probability distribution over the joint ontic state space is the product of the individual distributions. For a composite system prepared in the state ${\displaystyle |\psi \rangle \otimes |\phi \rangle }$, the ontic distribution is ${\displaystyle p(\lambda _{1},\lambda _{2}|\psi \otimes \phi )=p(\lambda _{1}|\psi )p(\lambda _{2}|\phi )}$. This is a crucial assumption about the nature of reality, implying no "spooky" correlations between the underlying states of independently prepared systems.^[1]

The Contradiction via Antidistinguishable States

The proof constructs a measurement that is impossible for a set of quantum states, and then shows that a ψ-epistemic model would predict this impossible outcome could occur.

Consider two non-orthogonal states, for instance ${\displaystyle |\psi _{0}\rangle =\cos \left({\frac {\theta }{2}}\right)|0\rangle +\sin \left({\frac {\theta }{2}}\right)|1\rangle }$ and ${\displaystyle |\psi _{1}\rangle =\cos \left({\frac {\theta }{2}}\right)|0\rangle -\sin \left({\frac {\theta }{2}}\right)|1\rangle }$. Now, consider preparing ${\displaystyle n}$ systems, where each is independently prepared in either the ${\displaystyle |\psi _{0}\rangle }$ or ${\displaystyle |\psi _{1}\rangle }$ state. This results in one of ${\displaystyle 2^{n}}$ possible tensor product states, such as ${\displaystyle |\psi _{0}\rangle \otimes |\psi _{1}\rangle \otimes \dots \otimes |\psi _{0}\rangle }$.

The key is to find a measurement that can distinguish these preparations in a specific way. This is achieved using the concept of antidistinguishability.

Antidistinguishable POVMs: A set of quantum states ${\displaystyle \{|\phi _{i}\rangle \}}$ is said to be antidistinguishable if there exists a POVM ${\displaystyle \{E_{k}\}}$ such that the probability of obtaining outcome ${\displaystyle k=i}$ is zero for every state in the set:

${\displaystyle \langle \phi _{i}|E_{i}|\phi _{i}\rangle =0\quad \forall i}$

^[6]

For large enough n's (satisfying ${\displaystyle 2\arctan \left(2^{1/n}-1\right)\leq \theta }$^[1]), there exists an antidistinguishable POVM for the tensor products of n ${\displaystyle |\psi _{0}\rangle }$ or ${\displaystyle |\psi _{1}\rangle }$.

Now, let's see the contradiction

1. Assume a ψ-epistemic model is true. There is an overlap region ${\displaystyle \Delta }$ for the ontic distributions of ${\displaystyle |\psi _{0}\rangle }$ and ${\displaystyle |\psi _{1}\rangle }$. Let the probability of the ontic state ${\displaystyle \lambda }$ being in this region be ${\displaystyle q_{0}=\int _{\Delta }p(\lambda |\psi _{0})d\lambda }$ (and similarly for ${\displaystyle |\psi _{1}\rangle }$).

2. Because of preparation independence, if we prepare ${\displaystyle n}$ systems, the probability that all ${\displaystyle n}$ systems have an ontic state ${\displaystyle \lambda _{i}\in \Delta }$ is at least ${\displaystyle q^{n}}$. This is a non-zero probability.

3. If the joint ontic state ${\displaystyle (\lambda _{1},\dots ,\lambda _{n})}$ is in this product overlap region ${\displaystyle \Delta ^{n}}$, then this single ontic state is consistent with any of the ${\displaystyle 2^{n}}$ possible quantum preparations. For this ontic state, the physical theory must assign a probability distribution over measurement outcomes which sum up to 1. So, there has to be at least one measurement outcome with a nonzero probability. But by antidistinguishability, it's incompatible with one possible preparation state.

4. This leads to a direct contradiction. The existence of such an ontic state is therefore incompatible with quantum theory.

The only way to resolve the contradiction is to conclude that the initial premise—that the ontic distributions overlap—must be false. Therefore, ${\displaystyle \Delta }$ must be empty. The distributions for any two non-orthogonal states cannot overlap, meaning the quantum state ${\displaystyle |\psi \rangle }$ must correspond to a unique set of ontic states ${\displaystyle \lambda }$. This establishes the reality of the quantum state.^[1]

Questioning the Preparation Independence Postulate

While the PBR theorem's conclusion is mathematically sound, its physical implications hinge entirely on the validity of its assumptions. The preparation independence postulate (PIP), though appearing commonsensical from a classical viewpoint, is the most frequently challenged of these assumptions and represents the primary "escape hatch" for models that seek to maintain a ${\displaystyle \psi }$-epistemic view of the quantum state.^[5] Questioning this postulate involves embracing a form of holism or a radical revision of causality, suggesting that the universe may be non-separable at a much deeper level than even quantum entanglement initially suggested.

The core of the postulate is an assumption of separability and locality at the level of the ontic states: that the hidden physical reality ${\displaystyle \lambda _{A}}$ of a system A is independent of the choice of preparation for a distant system B.^[7] Critics argue that since quantum mechanics, via Bell's theorem, has already forced us to abandon local realism, we should not be surprised if this locality assumption also fails for the underlying ontic states themselves. If the postulate is dropped, the PBR proof no longer holds, as the joint ontic distribution is no longer a simple product, allowing the system to evade the contradiction. This leads to two main avenues for violating the postulate:

1. Holism and Non-Separability

This view posits that the ontic state of a composite system is not reducible to the individual ontic states of its subsystems. The reality of the combined system ${\displaystyle (\lambda _{1},\cdots ,\lambda _{n})}$ is not just a pair of independent realities. Instead, there is one irreducible, "holistic" ontic state ${\displaystyle \lambda _{1\cdots n}}$ for the joint system. In this framework, the preparation of ${\displaystyle |\psi _{x_{i}}\rangle }$ for system i can constrain the possible ontic states for the entire system, and therefore the marginal state of system k is no longer independent of the choice of ${\displaystyle |\psi _{i}\rangle }$.^[5] The universe, at its fundamental level, might not be composed of independent parts.

2. Retrocausality

A more radical way to violate preparation independence is to allow for influences from the future to affect the past. In a retrocausal model, the choice of a future measurement setting on one particle could influence the ontic state of that particle (or its entangled partner) at the moment of its creation.^[8] In the context of the PBR experiment, the choice of the final entangled measurement ${\displaystyle \{E_{k}\}}$ could be seen as retrocausally influencing the ontic states ${\displaystyle (\lambda _{1},\cdots ,\lambda _{n})}$ that were possible during the preparation stage. This creates a correlation between the systems that violates the simple product rule of PIP. Proponents argue that such models can explain quantum correlations without invoking non-locality (in the sense of faster-than-light action), instead positing a less intuitive causal structure where the past is not fully fixed.

In summary, denying the preparation independence postulate provides a pathway to rescue the ${\displaystyle \psi }$-epistemic view of the quantum state, but it comes at a significant conceptual cost. It requires abandoning the deeply ingrained classical intuition of separability and accepting that the fundamental constituents of reality may be linked in holistic, non-local, or even retrocausal ways that are far stranger than previously imagined.^[5]

See also

• Quantum foundations
• Bell's theorem
• Kochen–Specker theorem