Wigner's friend

Wigner's friend is a thought experiment proposed by the physicist Eugene Wigner; it is an extension of the Schrödinger's cat experiment.

The thought experiment

The thought experiment posits a friend of Wigner who performs the Schrödinger's cat experiment after Wigner leaves the laboratory. Only when he returns does Wigner learn the result of the experiment from his friend, that is, whether the cat is alive or dead. The question is raised: was the state of the system a superposition of "dead cat/sad friend" and "live cat/happy friend," only determined when Wigner learned the result of the experiment, or was it determined at some previous point?

Mathematical description

The standard algebraic description of the experiment is by means of a tensor product of three states: the state of the cat, the state of Wigner's friend, and the state of Wigner himself. Let $|1\rangle _{C}$ represent the state of the living cat, and $|0\rangle _{C}$ represent the dead cat. The initial cat state is then the quantum superposition

|0\rangle _{C}+|1\rangle _{C}

This is sometimes written as

|\psi _{\text{dead}}\rangle _{C}+|\psi _{\text{alive}}\rangle _{C}

with or without the C subscript. These two are alternative notations for the same state; otherwise they mean the same thing. Wigner's friend, upon observing the cat, enters into one of two states: happy or sad, which are denoted by $|1\rangle _{F}$ and $|0\rangle _{F}$ . The combined state of the cat plus Wigner's friend is denoted by

|0\rangle _{C}\otimes |0\rangle _{F}+|1\rangle _{C}\otimes |1\rangle _{F}

This state is a maximally-entangled Bell state. Wigner, upon entering the room, observes either a happy friend, or a sad one; Wigner's state is denoted by $|1\rangle _{W}$ and $|0\rangle _{W}$ . The combined state is then

|0\rangle _{C}\otimes |0\rangle _{F}\otimes |0\rangle _{W}+|1\rangle _{C}\otimes |1\rangle _{F}\otimes |1\rangle _{W}

This combined state, of Wigner, his friend and the cat, is known as a Greenberger–Horne–Zeilinger state.

It is very important to observe that the above description is, in a certain way, a very simplified version of the original problem. It posits that the state of each of the three beings can be described by a single qubit. This is not an unreasonable thing to do, as it allows certain types of quantum-mechanical analysis to be performed; yet it assumes a coherency of dynamics that comes from the box being a perfectly closed quantum system. The basis of $|0\rangle _{C}$ and $|1\rangle _{C}$ is not arbitrary, but rather eigenstates of the unitary evolution imposed by the Hamiltonian.

In 2016, Frauchiger and Renner^[1] demonstrated that single-world interpretations of quantum mechanics were inconsistent with a fully unitary dynamics.

Consciousness and measurement

Wigner designed the experiment to illustrate his belief that consciousness is necessary to the quantum mechanical measurement process. If a material device is substituted for the conscious friend, the linearity of the wave function implies that the state of the system is in a linear sum of possible states. It is simply a larger indeterminate system.

However, a conscious observer (according to his reasoning) must be in either one state or the other, hence conscious observations are different, hence consciousness is not material. Wigner discusses this scenario in "Remarks on the mind-body question", one in his collection of essays, Symmetries and Reflections, 1967.^[2] The idea has become known as the consciousness causes collapse interpretation.

A counterargument is that the superimposition of two conscious states is not paradoxical — just as there is no interaction between the multiple quantum states of a particle, so the superimposed consciousnesses need not be aware of each other.^[3]

The state of the observer's perception is considered to be entangled with the state of the cat. The perception state 'I perceive a live cat' accompanies the 'live-cat' state and the perception state 'I perceive a dead cat' accompanies the 'dead-cat' state. [..] It is then assumed that a perceiving being always finds his/her perception state to be in one of these two; accordingly, the cat is, in the perceived world, either alive or dead.[..] I wish to make clear that, as it stands, this is far from a resolution of the cat paradox. For there is nothing in the formalism of quantum mechanics that demands that a state of consciousness cannot involve the simultaneous perception of a live and a dead cat.
— Roger Penrose

Alternative interpretations

Wigner's friend in Many Worlds

The many worlds interpretation avoids the need to postulate that consciousness causes collapse — indeed, that collapse occurs at all. Systems split (decohere) when there is an irreversible difference between their state in the world where the cat survived or will survive and their counterpart in the other case. In the example below this happens with the cyanide device and the telephone.

According to “Many worlds”, when Wigner's friend (the investigator) finds out the result of the Schrödinger's cat experiment, the world has split in two. In one world, the friend observes a live cat. In the other, the friend observes a dead cat.

According to many-worlds the device was split into two states—cyanide released or not (…) As the cyanide/no-cyanide interacts with the cat the cat is split into two states (dead or alive). From the surviving cat's point of view it occupies a different world from its deceased copy. The onlooker is split into two copies only when the box is opened and they are altered by the states of the cat. The cat splits when the device is triggered, irreversibly. The investigator splits when they open the box. The alive cat has no idea that the investigator has split, any more than it is aware that there is a dead cat in the neighbouring split-off world. The investigator can deduce, after the event, by examining the cyanide mechanism, or the cat's memory, that the cat split prior to opening the box.^[4]

Objective collapse theories

According to objective collapse theories, wave function collapse occurs when a superposed systems reaches a certain objective threshold of size, complexity etc. Objective collapse proponents would expect a system as macroscopic as a cat to have collapsed before the box was opened, so the question of observation-of-observers does not arise for them.

Sources

Wigner's original remarks about his friend appeared in his article "Remarks on the Mind-Body Question", published in the book "The Scientist Speculates", edited by I. J. Good. The article is reprinted in Wigner's own book Symmetries and Reflections.

In fiction

Stephen Baxter's novel, Timelike Infinity, discusses a variation of Wigner's friend thought experiment through a refugee group of humans self-named "The Friends of Wigner". They believe that an ultimate observer at the end of time may collapse all possible entangled wave-functions generated since the beginning of the universe, hence choosing a reality without oppression.

References

↑ Frauchiger, D. and Renner, R., "Single-world interpretations of quantum theory cannot be self-consistent"
↑ E.P. Wigner (1961), "Remarks on the mind-body question", in: I.J. Good, "The Scientist Speculates", London, Heinemann
↑ R. Penrose, Road to Reality, section 29.8
↑ http://www.hedweb.com/manworld.htm#splitsh

THE EVERETT FAQ, by Michael Clive Price

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