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On the reality of the quantum state
Matthew F. Pusey, Jonathan Barrett, Terry Rudolph
TL;DR
The paper asks whether a quantum state can be merely informational about an underlying physical state while independently prepared systems have independent physical states. It formalizes this model and constructs multi-system measurements, showing that the model’s predictions contradict quantum theory and that the construction appears to use the smallest possible number of systems.
Problem
The paper addresses whether quantum states can represent mere information about underlying physical states under independent preparation, rather than correspond directly to reality.
Source: Appendix A · pp. 5–6Method
It constructs joint measurements on independently prepared copies, analyzes their circuit implementation, and numerically optimizes the required number of systems.
Source: Appendix A · pp. 5–6, Appendix C · p. 8Results
The constructed measurement gives zero probability to an outcome for each alternative preparation when n is sufficiently large, and numerical tests indicate it uses the smallest possible number of systems.
Source: Appendix A · pp. 5–6, Appendix C · p. 8Takeaways & Limitations
The results support a no-go theorem against models in which quantum states are merely informational and independently prepared systems have independent physical states.
Source: Appendix A · p. 5, Appendix C · p. 8
Abstract
from arXiv · showhide
Quantum states are the key mathematical objects in quantum theory. It is therefore surprising that physicists have been unable to agree on what a quantum state truly represents. One possibility is that a pure quantum state corresponds directly to reality. However, there is a long history of suggestions that a quantum state (even a pure state) represents only knowledge or information about some aspect of reality. Here we show that any model in which a quantum state represents mere information about an underlying physical state of the system, and in which systems that are prepared independently have independent physical states, must make predictions which contradict those of quantum theory.
Appendix A: The measurement circuit
Appendix A constructs a joint measurement on n independently prepared systems such that every possible preparation has an outcome assigned zero probability by quantum theory. The circuit parameters can be chosen whenever n is sufficiently large relative to the states’ overlap.
Appendix A: The measurement circuit
- For any distinct states, sufficiently many independent copies admit a joint measurement whose every outcome has zero probability for at least one of the 2^n preparations.The required measurement is a quantum circuit followed by computational-basis measurements.Source: Appendix A · pp. 5–6
- The systems can be restricted to qubits, with the joint circuit applying Z_β rotations, an entangling gate R_α, and Hadamard gates before measurement.R_α applies a phase to |0⋯0⟩ and acts as the identity on other computational-basis states.Source: Appendix A · pp. 5–6
- The outcome probability is evaluated for the basis state matching the preparation string, and the chosen parameters force this probability to zero.The n-system preparation is indexed by x_i ∈ {0,1}, specifying whether each system is |ψ0⟩ or |ψ1⟩.Source: Appendix A · p. 6
- By continuity of f, the proof establishes an intersection with the unit circle by exhibiting one point outside it and noting that 0 ≤ f(π) ≤ 1, so f(π) is inside or on the circle.Source: Appendix A · p. 6
Appendix B: Formal, noise-tolerant version of the argument
Appendix B formalizes the no-go argument for models with underlying physical states and measurement responses determined by those states. It extends the result to approximate quantum predictions and relates the resulting bounds to overlap between preparation distributions.
Appendix B: Formal, noise-tolerant version of the argument
- The formal model assigns each preparation a distribution μ_i(λ) over real physical states and each measurement outcome a response ξ_M,k(λ), reproducing quantum probabilities through the corresponding operator expression.The model assumes measurement behavior depends only on the system’s physical state and the measurement procedure.Source: Appendix B · p. 7
- The argument remains valid approximately: if every model prediction is within ϵ of quantum theory, the resulting bound constrains the total variation distance between distributions.The appendix identifies the no-noise case as a mathematical version of the main-text argument.Source: Appendix B · p. 7
- For any pair of pure states, Eq. (B2) holds when n satisfies Eq. (A3), linking the overlap of their physical-state distributions to the multi-system measurement argument.The overlap is defined as a quantity associated with μ0(λ) and μ1(λ).Source: Appendix B · p. 7
- The n-system measurement has zero quantum probability for the outcome matching each preparation string, while the model assigns at most ϵ to that outcome, yielding the overlap bound.Independence makes the joint physical-state distribution factor across the n preparations.Source: Appendix B · pp. 7–8
- Figure 5 shows how increasing n rules out progressively larger regions of overlap versus state distinguishability, eventually forcing zero overlap for every pair of states.The figure states that the boundaries are generally not ruled out, except for the orthogonal-state case described there.Source: Figure 5 · p. 8
Appendix C: Numerical results
Appendix C tests whether fewer systems can achieve the no-go measurement and studies bounds when the analytic condition is not met. The numerical results indicate that the proposed measurement uses the smallest possible number of systems and remains useful for bounding overlap outside that regime.
Appendix C: Numerical results
- The numerical study asks whether measurements using fewer systems exist by solving a semidefinite program for different values of θ and n.The tested objective is the quantity σ associated with the candidate POVM.Source: Appendix C · p. 8
- Numerical optimization found σ = 0 exactly when condition (C1) holds, indicating that the Section A measurement uses the smallest possible number of systems.A measurement proves the no-go theorem precisely when the nonnegative objective σ vanishes.Source: Appendix C · p. 8
- When (C1) fails, the optimal measurement retains the Section A form with α = π and β = 0, and it yields the bound (ω(μ0, μ1))^n ≤ σ.This measurement therefore constrains overlap even when exact zero-overlap exclusion is unavailable.Source: Appendix C · p. 8
- The optimization problem has an operational interpretation as maximum-error discrimination of the 2^n product states with equal priors.For two states, the analogous problem becomes minimum-error discrimination after swapping outcome labels.Source: Appendix C · p. 8