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A No-Go Result for QBism

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In QBism the wave function does not represent an element of physical reality external to the agent, but represent an agent’s personal probability assignments, reflecting his subjective degrees of belief about the future content of his experience. In this paper, I argue that this view of the wave function is not consistent with protective measurements. The argument does not rely on the realist assumption of the ψ-ontology theorems, namely the existence of the underlying ontic state of a quantum system.

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  1. There are two known schemes of protection. The first one is to introduce a protective potential, and the second one is via the quantum Zeno effect. It should be pointed out that the protection requires that some information about the measured system should be known before a PM, and PMs cannot measure an arbitrary unknown wave function. In some cases, the information may be very little. For example, we only need to know that a quantum system such as an electron is in the ground state of an external potential before we make PMs on the system to find its wave function, no matter what form the external potential has.

  2. Note that PMs are different from quantum non-demolition measurements. In a quantum non-demolition measurement, the measured observable is required to commute with the total Hamiltonian so that it is a constant of the motion. This implies that the measurement is repeatable, but it does not imply that the wave function of the measured system is unchanged during the measurement. By comparison, a PM does not require that the measured observable must commute with the total Hamiltonian, and the wave function of the measured system does not change during the measurement.

  3. An example of how to measure B is given by [14, p. 4622]. In the gedanken experiment, a charged particle Q is in a thin circular tube enclosing a magnetic but with the magnetic field vanishing inside the tube. A protective measurement of each eigenstate can be made by shooting electrons near the tube and observing their trajectories; from the accelerations of the electrons the charge density \(Q\rho\) and the current density Qj can be determined.

  4. In most cases the measured wave function can be reconstructed only in principle. For a spatial wave function like \(\psi (x)\), since one needs to measure the observables A and B in infinitely many points in space, this is an impossible task in practice.

  5. But this does not mean that the wave function must be a direct representation of the ontic state of the measured system even if the ontic state exists [19]. More work still needs to be done to prove this stronger result [20].


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I am grateful to the editors and reviewers of this journal for their useful comments and suggestions. This work is supported by the National Social Science Foundation of China (Grant No. 16BZX021).

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Correspondence to Shan Gao.

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Gao, S. A No-Go Result for QBism. Found Phys 51, 103 (2021).

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