Pusey–Barrett–Rudolph theorem

Key takeaways

Pusey–Barrett–Rudolph theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pusey–Barrett–Rudolph theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pusey–Barrett–Rudolph theorem from memory before moving on to harder problems.

Reference excerpt

The Pusey–Barrett–Rudolph (PBR) theorem 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.

Theorem This theorem concerns the interpretational status of pure quantum states. Under the classification of hidden variable models of Nicholas Harrigan and Robert Spekkens, the interpretation of the quantum wavefunction | ψ ⟩ {\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" or 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 | ψ ⟩ {\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. More specifically, the theorem applies to models that treat quantum states as probability distributions over hidden variables, or ontic states. In such a model, writing the space of ontic states as Λ {\displaystyle \Lambda } , a quantum state ψ {\displaystyle \psi } is a probability distribution p ψ ( λ ) {\displaystyle p_{\psi }(\lambda )} defined on the set Λ {\displaystyle \Lambda } . An observable A {\displaystyle A} is represented as a set of response functions, or conditional probability densities: A ( S , λ ) {\displaystyle A(S,\lambda )} is the probability that the measurement A {\displaystyle A} has the outcome S {\displaystyle S} if the ontic state of the system being measured is λ {\displaystyle \lambda } . In order to reproduce the predictions of quantum mechanics, the probability of obtaining an outcome S {\displaystyle S} given a state ψ {\displaystyle \psi } as calculated by the Born rule must satisfy

P ( S , ψ ) = ∫ Λ A ( S , λ ) p ψ ( λ ) d λ . {\displaystyle P(S,\psi )=\int _{\Lambda }A(S,\lambda )p_{\psi }(\lambda )\,d\lambda .}

The theorem concludes that if two quantum states are distinct, they must correspond to probability distributions that do not overlap. The PBR theorem employs the concept of an "antidistinguishable" set of quantum states. A finite set of quantum states { ρ i : i = 1 , … , N } {\displaystyle \{\rho _{i}:i=1,\ldots ,N\}} , written as density matrices to include the possibility of mixed states, is antidistinguishable if there exists a generalized measurement (a POVM) such that, for each value of i {\displaystyle i} , some outcome of the POVM is assigned probability zero by the state ρ i {\displaystyle \rho _{i}} . In other words, for an antidistinguishable set of density matrices, there exists a POVM { E i } {\displaystyle \{E_{i}\}} such that

t r ( E i ρ i ) = 0 {\displaystyle \mathrm {tr} (E_{i}\rho _{i})=0}

for all i = 1 , … , N {\displaystyle i=1,\ldots ,N} . This concept was introduced by Carlton M. Caves, Christopher A. Fuchs and Rüdiger Schack under the name "post-Peierls incompatibility", as it generalizes a condition proposed by Rudolf Peierls. An antidistinguishable, or post-Peierls incompatible, set is also sometimes termed a set that allows "conclusive exclusion".

See also Quantum foundations Bell's theorem Kochen–Specker theorem

References

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Worked examples

Example 1 — a first encounter with Pusey–Barrett–Rudolph theorem

Start with the simplest possible case. Write down what Pusey–Barrett–Rudolph theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pusey–Barrett–Rudolph theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pusey–Barrett–Rudolph theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pusey–Barrett–Rudolph theorem

In research: Pusey–Barrett–Rudolph theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pusey–Barrett–Rudolph theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pusey–Barrett–Rudolph theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Hidden variable theory, No-go theorems, Quantum information science, so understanding it makes those chapters shorter.
In everyday life: Look for Pusey–Barrett–Rudolph theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Pusey–Barrett–Rudolph theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pusey–Barrett–Rudolph theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pusey–Barrett–Rudolph theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pusey–Barrett–Rudolph theorem in simple terms?

The Pusey–Barrett–Rudolph (PBR) theorem 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.

Why does Pusey–Barrett–Rudolph theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pusey–Barrett–Rudolph theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pusey–Barrett–Rudolph theorem.