Source-linked AI summary
A no-go theorem for observer-independent facts
Caslav Brukner
TL;DR
The paper asks whether Wigner’s and his friend’s directly observed outcomes can be treated as observer-independent facts within one theoretical framework. It reviews the Wigner-friend setup and derives a Bell-type no-go theorem, then connects the result to Frauchiger and Renner’s self-consistency theorem. The supported conclusion is that observational facts may need to be understood relationally, relative to an observer.
Problem
The paper examines whether observational statements by Wigner and his friend can be assigned jointly as observer-independent facts.
Method
It reviews a Bell-type result and derives a no-go theorem using locality, freedom of choice, universal quantum theory, and joint truth-value assignments.
Results
The assumptions are incompatible: Wigner’s and the friend’s observational propositions cannot jointly form a single Boolean algebra of truth values.
Takeaways & Limitations
The result indicates that quantum theory may define facts only relative to an observation and an observer, and links this issue to Frauchiger and Renner’s self-consistency requirement.
Abstract
from arXiv · showhide
In his famous thought experiment, Wigner assigns an entangled state to the composite quantum system made up of Wigner's friend and her observed system. While the two of them have different accounts of the process, each Wigner and his friend can in principle verify his/her respective state assignments by performing an appropriate measurement. As manifested through a click in a detector or a specific position of the pointer, the outcomes of these measurements can be regarded as reflecting directly observable "facts". Reviewing arXiv:1507.05255, I will derive a no-go theorem for observer-independent facts, which would be common both for Wigner and the friend. I will then analyze this result in the context of a newly derived theorem in arXiv:1604.07422, where Frauchiger and Renner prove that "single-world interpretations of quantum theory cannot be self-consistent". It is argued that "self-consistency" has the same implications as the assumption that observational statements of different observers can be compared in a single (and hence an observer-independent) theoretical framework. The latter, however, may not be possible, if the statements are to be understood as relational in the sense that their determinacy is relative to an observer.
A. Introduction
The Wigner’s-friend experiment gives Wigner and the friend different descriptions of a measurement, motivating a no-go theorem for facts jointly objective for both observers. The analysis connects this theorem to Frauchiger and Renner’s self-consistency result and relational interpretations.
- Motivation: Wigner describes the sealed laboratory’s measurement unitarily, while the friend records a definite outcome in an apparatus or memory.The friend projects onto the observed outcome, whereas Wigner assigns a state to the composite system based on information available to him.
- Motivation: Different observer-relative descriptions need not create inconsistency because Wigner and the friend make predictions relative to separate experimental arrangements.When they compare outcomes, the friend’s communication can collapse the state Wigner assigns to the friend and system.
- Contribution: The paper derives a Bell-type no-go theorem showing that Wigner’s and the friend’s facts cannot jointly be local objective properties under locality, freedom of choice, and universal quantum theory.Under these assumptions, joint probabilities for their outcomes cannot be defined.
- Relation to Frauchiger–Renner: The paper relates observer-independent facts to Frauchiger and Renner’s claim that single-world interpretations cannot be self-consistent.It argues that comparing observers’ observational statements within one theoretical framework has equivalent implications.
- Scope: The theorem applies specifically to facts understood as observers’ immediate experiences, including directly observable detector clicks or pointer positions.Other proposed quantum realities are covered only insofar as they generate such directly observable facts.
B. Deutsch’s version of Wigner’s friend experiment
Deutsch’s version models the friend’s measurement as entanglement between a two-level system and macroscopic laboratory states, while allowing Wigner to verify the resulting state. The setup assumes the friend perceives a definite outcome and quantum theory applies universally, including to the laboratory and memory.
- Experimental setup: The experiment uses a quantum two-level system whose two measurement outcomes are recorded by an apparatus and eventually in the friend’s memory.Wigner, outside the isolated laboratory, can measure the combined particle-and-laboratory system.
- Friend’s description: A pointer position lets the friend assign the spin-z observable the value “up” or “down”.The argument requires only that the friend perceives a definite outcome, not that she formally uses quantum theory.
- Wigner’s description: Wigner describes the measurement unitarily, entangling the spin states with orthogonal macroscopic configurations of the apparatus, laboratory, and friend’s memory.The configurations are represented by the orthogonal states |Fz+⟩F and |Fz−⟩F.
- Verification: Wigner can verify his state assignment by performing a Bell-state measurement in a basis combining the spin and friend-laboratory states.The relative phase between the amplitudes is fixed by the measurement interaction under Wigner’s control; without that control, he would use an incoherent mixture.
- Consequence: Under universal quantum theory, Wigner can confirm his state assignment while the message indicates that the friend perceives a definite outcome.This contrasts with collapse models, which predict a breakdown of quantum-mechanical laws at some scale.
C. The no-go theorem
The paper formulates observer-independent facts as jointly assignable truth values and proves their incompatibility with universal quantum validity, locality, and freedom of choice. A Bell-type construction yields the contradiction, while a GHZ extension makes the discrepancy deterministic.
- Observer-independent facts: Observer-independent facts require jointly assigning truth values to different observers’ outcome propositions within a Boolean algebra with probabilities.The weaker conjunction requirement already permits joint probabilities for noncommuting friend- and Wigner-type observables.
- The incompatibility: The no-go theorem identifies universal validity of quantum theory, locality, freedom of choice, and observer-independent facts as mutually incompatible assumptions.Universality applies quantum predictions even when measured systems include observers, laboratories, and memories.
- Bell-type proof: Alice and Bob test a Bell inequality using settings corresponding respectively to Charlie’s or Debbie’s measurements and to super-observer measurements on their laboratories.The construction uses entangled spin-1/2 systems and a rotation chosen so measured observables are of friend or Wigner type.
- Bell-type proof: The inequality violation shows that the conjunction of assumptions used in its derivation is untenable.Because noncommuting observables receive joint truth assignments, the framework effectively introduces hidden variables for the Bell argument.
- Extensions: A three-Wigner, three-friend GHZ version strengthens the discrepancy between quantum theory and theories respecting assumptions 2–4 from probabilistic to deterministic.The appendix presents this extension as a GHZ-type theorem.
- Conclusion: Under the retained assumptions, Wigner cannot jointly assign a specific friend outcome with his directly observed outcome, suggesting facts are relative to observers.The stated scope concerns directly observable facts such as detector clicks or pointer positions.
D. Relation to the paper by Frauchiger and Renner, arXiv: 1604.07422
The paper relates Frauchiger and Renner’s self-consistency theorem to the impossibility of jointly comparing observational statements from different observers. It argues that self-consistency effectively reproduces an observer-independent assignment of truth values, conflicting with locality, freedom of choice, and universal quantum theory.
- Frauchiger–Renner theorem: Frauchiger and Renner argue that no physical theory can simultaneously satisfy single-world, self-consistency, and compliance with quantum theory.The paper presents these as informal versions of the three properties underlying their incompatibility proof.
- Frauchiger–Renner theorem: The argument combines nested statements about outcomes and conclusions made by observers F1, F2, A, and W.The chain begins with statements S1–S4 and iteratively produces W’s conclusion about what other observers conclude.
- Frauchiger–Renner theorem: W’s iterated conclusion that another observer concludes that F1 concludes w ≠ ok contradicts W’s directly observed outcome w = ok.The contradiction concerns W’s conclusion about other observers’ reasoning, not W’s direct observation itself.
- Relation to observer-independent facts: Self-consistency promotes other observers’ observation-based knowledge into W’s knowledge, allowing their statements to be logically compared with W’s direct observation.The paper identifies this promotion as having the same predictive power as jointly assigning truth values to different observers’ statements.
- Relation to observer-independent facts: A single Boolean algebra for different observers’ observational statements is incompatible with locality, freedom of choice, and unconstrained application of quantum theory.The paper interprets this as suggesting that the strong Frauchiger–Renner conclusion may depend on a restrictive self-consistency requirement.
- Interpretive implications: The paper distinguishes direct observations from knowledge about others’ knowledge, emphasizing this distinction for future Bayesian inference in Wigner-friend situations.Observer-dependent statements may be meaningful only relative to the measurement procedures that define them.
I. APPENDIX
The appendix extends the Bell-theorem argument to a GHZ scenario involving three Wigners and their friends. Its deterministic perfect-correlation constraints imply an inconsistency among the predefined measurement values.
- GHZ setup: The construction uses three spatially separated Wigners, each measuring a subsystem containing a friend who previously measured spin along x.Alice, Bob, and Cleve measure Debbie, Eric, and Fiona together with their respective spin particles.
- GHZ setup: The GHZ test assigns each Wigner two observables, with x and y settings chosen for the corresponding measurement operators.For Alice, the settings are ˆA1 = ˆAx and ˆA2 = ˆAy, with analogous choices for Bob and Cleve.
- Deterministic contradiction: The predefined values must satisfy AxByCy = AyBxCy = AyByCx = 1 to reproduce the GHZ state’s perfect correlations.The argument assumes that Alice, Bob, and Cleve perform these measurements on a shared GHZ state.
- Deterministic contradiction: Multiplying those constraints implies AxBxCx = 1, producing the deterministic incompatibility at the heart of the GHZ theorem.Unlike a probabilistic contradiction, this version bypasses the notion of probability, similarly to Frauchiger and Renner’s formulation.