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arXiv:2202.07111

Wigner's Friend Depends on Self-Contradictory Quantum Amplification

Andrew Knight

Published Feb 15, 2022• 0 citations
arXivHTMLPDF

Abstract

In a recent paper, Żukowski and Markiewicz showed that Wigner's Friend (and, by extension, Schrödinger's Cat) can be eliminated as physical possibilities on purely logical grounds. I validate this result and demonstrate the source of the contradiction in a simple experiment in which a scientist S attempts to measure the position of object $|O\rangle = |A\rangle_S + |B\rangle_S$ by using measuring device M chosen so that $|A\rangle_M \approx |A\rangle_S$ and $|B\rangle_M \approx |B\rangle_S$. I assume that the measurement occurs by quantum amplification without collapse, in which M can entangle with O in a way that remains reversible by S for some nonzero time period. This assumption implies that during this "reversible" time period, $|A\rangle_M \neq |A\rangle_S$ and $|B\rangle_M \neq |B\rangle_S$ -- i.e., the macroscopic pointer state to which M evolves is uncorrelated to the position of O relative to S. When the scientist finally observes the measuring device, its macroscopic pointer state is uncorrelated to the object in position $|A\rangle_S$ or $|B\rangle_S$, rendering the notion of "reversible measurement" a logical contradiction.

DOI
10.48550/arxiv.2202.07111

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In a recent paper, Żukowski and Markiewicz showed that Wigner's Friend (and, by extension, Schrödinger's Cat) can be eliminated as physical possibilities on purely logical grounds. I validate this result and demonstrate the source of the contradiction in a simple experiment in which a scientist S attempts to measure the position of object ∣O⟩=∣A⟩S+∣B⟩S|O\rangle = |A\rangle_S + |B\rangle_S∣O⟩=∣A⟩S​+∣B⟩S​ by using measuring device M chosen so that ∣A⟩M≈∣A⟩S|A\rangle_M \approx |A\rangle_S∣A⟩M​≈∣A⟩S​ and ∣B⟩M≈∣B⟩S|B\rangle_M \approx |B\rangle_S∣B⟩M​≈∣B⟩S​. I assume that the measurement occurs by quantum amplification without collapse, in which M can entangle with O in a way that remains reversible by S for some nonzero time period. This assumption implies that during this "reversible" time period, ∣A⟩M≠∣A⟩S|A\rangle_M \neq |A\rangle_S∣A⟩M​=∣A⟩S​ and ∣B⟩M≠∣B⟩S|B\rangle_M \neq |B\rangle_S∣B⟩M​=∣B⟩S​ -- i.e., the macroscopic pointer state to which M evolves is uncorrelated to the position of O relative to S. When the scientist finally observes the measuring device, its macroscopic pointer state is uncorrelated to the object in position ∣A⟩S|A\rangle_S∣A⟩S​ or ∣B⟩S|B\rangle_S∣B⟩S​, rendering the notion of "reversible measurement" a logical contradiction.
"The unambiguous account of proper quantum phenomena must, in principle, include a description of all relevant features of experimental arrangement" (Bohr). The measurement process is composed of premeasurement (quantum correlation of the system with the pointer variable) and an irreversible decoherence via interaction with an environment. The system ends up in a probabilistic mixture of the eigenstates of the measured observable. For the premeasurement stage, any attempt to introduce an "outcome" leads, as we show, to a logical contradiction, 1=i. This nullifies claims that a modified concept of Wigner's friend, who just premeasures, can lead to valid results concerning quantum theory.
I show in this paper why the universality of quantum mechanics at all scales, which implies the possibility of Schrodinger's Cat and Wigner's Friend thought experiments, cannot be experimentally confirmed, and why macroscopic superpositions in general cannot be observed or measured, even in principle. Through the relativity of quantum superposition and the transitivity of correlation, it is shown that from the perspective of an object that is in quantum superposition relative to a macroscopic measuring device and observer, the observer is already sufficiently well correlated to the measuring device that once the object correlates to the measuring device, there is no time period in which the observer can perform an appropriate interference experiment to show that the measuring device is in a superposition.
By assuming (i) a universal, observer-independent quantum state, by considering (ii) measurement processes with wave function collapse as part of quantum mechanical time evolution (also within isolated systems), and by (iii) clearly distinguishing the state of a quantum system from the knowledge of conscious observers about this state, we suggest a modified Copenhagen interpretation of quantum mechanics that resolves Wigner's Friend and extended Wigner's Friend paradoxes. The suggested interpretation leads to differences in some outcome probabilities compared to previous analyses which, in principle, make it possible to falsify the suggested modifications or the previous analyses (or both). The fundamental problem of the incompleteness of quantum mechanics in the Copenhagen interpretation regarding the lack of a precise definition and a mechanistic description of the measurement process (including the collapse of the wave function) is not addressed in this study. However, the argumentation regarding the resolution of Wigner's Friend paradoxes also applies to attempts for such a mechanistic description using collapse models with stochastic extensions to the Schrödinger equation or trying to model wave function collapse with unitary time evolution and irreversibility of the measurement process resulting from quantum statistical mechanics.
Quantum theory in its conventional formulation is notoriously subject to various measurement-related paradoxes, as exemplified by the “Schrödinger’s Cat” and “Wigner’s Friend” thought experiments. It has been shown, for example, by Frauchiger and Renner, that nested measurements such as those occurring in the Wigner’s Friend experiment can lead to inconsistencies concerning the putative outcomes of measurements. Such inconsistencies are commonly presumed to remain private and incommensurable, but this is not the case. A counterexample, in which the inconsistencies can be revealed among the observers, is reviewed. The implications for a recent attempt to shield Relational Quantum Mechanics from such inconsistencies are considered, and it is concluded that the attempt is not successful. Further implications for the state of the debate concerning the viability of quantum theory in its various formulations are discussed.
This paper offers a comprehensive analysis of Wigner’s Friend paradox (1961), a foundational thought experiment in quantum mechanics that highlights the tension between the internal perspective of a measuring observer (who experiences a definite outcome) and the external perspective of a super-observer who treats the entire laboratory as a quantum system in superposition. It traces the historical lineage from Schrödinger’s cat and von Neumann’s measurement theory, provides a formal mathematical description of the state evolution and logical structure of the paradox, examines implications for major interpretations of quantum mechanics (Copenhagen, Everettian many-worlds, QBism, relational quantum mechanics, and objective collapse theories), and discusses modern extensions such as the Frauchiger–Renner paradox. The work explores profound philosophical and epistemological consequences for the nature of observation, objectivity, intersubjectivity, and the role of consciousness in quantum theory (2026).