# Wigner's friend

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Wigner's friend is a thought experiment in theoretical quantum physics, proposed by the physicist Eugene Wigner in 1961.[1] The scenario involves an indirect observation of a quantum measurement: An observer W observes another observer F who performs a quantum measurement on a physical system. The two observers then formulate a statement about the physical system's state after the measurement according to the laws of quantum theory. However, in most of the interpretations of quantum theory, the resulting statements of the two observers contradict each other. This reflects a seeming incompatibility of two laws in quantum theory: the deterministic and continuous time evolution of the state of a closed system and the probabilistic, discontinuous collapse of the state of a system upon measurement. Wigner's friend is therefore directly linked to the measurement problem in quantum mechanics with its famous Schrödinger's cat paradox.

## The thought experiment

The thought experiment posits a friend of Wigner in a laboratory and lets him perform a quantum measurement on a physical system (this could be a spin system or also Schrödinger's cat). This system is assumed to be in a superposition of two distinct states, say, state 0 and state 1 (or "dead" and "alive", in the case of Schrödinger's cat). When Wigner's friend measures the system in the 0/1 - basis, according to quantum mechanics, he will get one of the two possible outcomes (0 or 1) and the system collapses into the corresponding state.

Now Wigner himself models the scenario from outside the laboratory, knowing that inside, his friend will at some point perform the {0, 1} - measurement on the physical system. According to the linearity of the quantum mechanical equations, Wigner will assign a superposition state to the whole laboratory (i.e. the joint system of the physical system together with the friend): The superposition state of the lab is then a linear combination of "system is in state 0/ friend has measured 0" and "system is in state 1/ friend has measures 1".

Let Wigner now ask his friend what he had obtained as a measurement result: Whichever answer the friend gives (0 or 1), in each case, Wigner would then assign the state "system is in state 0/ friend has measured 0" or "system is in state 1/ friend has measured 1" to the laboratory. Therefore, it is only at the time when he learns about his friend's result that the superposition state of the laboraty collapses.

However, unless Wigner is considered in a "priviliged position as ultimate observer"[1], the friend's point of view must be regarded as equally valid, and this is where an apparent paradox comes into play: From the point of view of the friend, the measurement result was determined long before Wigner had asked about it, and the state of the physical system has already collapsed. When now exactly did the collapse occur? Was it when the friend had finished his measurement, or when the information of its result entered Wigner's consciousness?

## Mathematical description

Assume for simplicity that the physical system is a two-state spin system S with states ${\displaystyle |0\rangle _{S}}$and ${\displaystyle |1\rangle _{S}}$, corresponding to measurement results 0 and 1.

Initially, S is in a superposition state

${\displaystyle \alpha |0\rangle _{S}+\beta |1\rangle _{S}}$

and gets measured by F in the {${\displaystyle |0\rangle _{S}}$, ${\displaystyle |1\rangle _{S}}$} - basis. Then, with probability ${\displaystyle |\alpha |^{2}}$, F will measure 0 and with probability ${\displaystyle |\beta |^{2}}$, he will measure 1.

From the friend's point of view, the spin has collapsed into one of its basis states upon his measurement, and hence, he will assign to the spin the state corresponding to his measurement result: If he got 0, he will assign the state ${\displaystyle |0\rangle _{S}}$to the spin, if he got 1, he will assign the state ${\displaystyle |1\rangle _{S}}$to the spin.

Wigner (W) now models the combined system of the spin together with his friend (the joint system is given by the tensor product ${\displaystyle S\otimes F}$). He thereby takes a viewpoint outside of F's laboratory, which is considered isolated from the environment. Hence, by the laws of quantum mechanics for isolated systems, the state of the whole laboratory evolves unitarily in time. Therefore, the correct description of the state of the joint system as seen from outside is the superposition state

${\displaystyle \alpha (|0\rangle _{S}\otimes |0\rangle _{F})+\beta (|1\rangle _{S}\otimes |1\rangle _{F})}$,

where ${\displaystyle |0\rangle _{F}}$denotes the state of the friend when he has measured 0, and ${\displaystyle |1\rangle _{F}}$denotes the state of the friend when he has measured 1.

(Note that for an initial state ${\displaystyle |0\rangle _{S}}$of S, the state for ${\displaystyle S\otimes F}$would be ${\displaystyle |0\rangle _{S}\otimes |0\rangle _{F}}$after the interaction, and for an initial state ${\displaystyle |1\rangle _{S}}$, the state of ${\displaystyle S\otimes F}$ would be ${\displaystyle |1\rangle _{S}\otimes |1\rangle _{F}}$. Now, by the linearity of the quantum mechanical equations of motion, an initial state ${\displaystyle \alpha |0\rangle _{S}+\beta |1\rangle _{S}}$for S results in the superposition${\displaystyle \alpha (|0\rangle _{S}\otimes |0\rangle _{F})+\beta (|1\rangle _{S}\otimes |1\rangle _{F})}$for ${\displaystyle S\otimes F}$.

## Discussion

### Consciousness and Wigner's Friend

E. Wigner designed the thought experiment to illustrate his belief that consciousness is necessary to the quantum mechanical measurement process (and therefore, that consciousness in general must be an "ultimate reality"[1] according to Descartes's "Cogito ergo sum" philosophy): "All that quantum mechanics purports to provide are probability connections between subsequent impressions (also called 'apperceptions') of the consciousness"[1].

Here, "impressions of the consciousness" are understood as specific knowledge about a (measured) system, i.e., the result of an observation. This way, the content of one's consciousness is precisely all knowledge of one’s external world and measurements are defined as the interactions which create the impressions in our consciousness. Since the knowledge about any quantum mechanical wave function is based on such impressions, the wave function of a physical system is modified once the information about the system enters our consciousness. This idea has become known as the consciousness causes collapse interpretation.

In the Wigner's Friend thought experiment, this (E. Wigner's) view comes in as follows:

The friend's consciousness gets "impressed" by his measurement of the spin, and therefore he may assign a wave function to it according to the nature of his impression. Wigner, having no access to that information, can only assign the wave function ${\displaystyle \alpha (|0\rangle _{S}\otimes |0\rangle _{F})+\beta (|1\rangle _{S}\otimes |1\rangle _{F})}$to the joint system of spin and friend after the interaction. When he then asks his friend about the measurement outcome, Wigner's consciousness gets "impressed" by the friend's answer: As a result, Wigner will be able to assign a wave function to the spin system, i.e., he will assign to it the wave function corresponding to the friend's answer.

So far, there is no inconsistency in the theory of measurement. However, Wigner then learns (by asking his friend again) that the feelings/ thoughts of his friend about the measurement outcome had been in the friend's mind long before Wigner had asked about them in the first place. Therefore, the correct wave function for the joint system of spin and friend just after the interaction must have been either ${\displaystyle |0\rangle _{S}\otimes |0\rangle _{F}}$or ${\displaystyle |1\rangle _{S}\otimes |1\rangle _{F}}$, and not their linear combination. Hence, there is a contradiction.

E. Wigner then follows that "the being with a consciousness must have a different role in quantum mechanics than the inanimate measuring device":[1] If the friend were replaced by some measuring device without a consciousness, the superposition state would describe the joint system of spin and device correctly. In addition, E. Wigner considers a superposition state for a human being to be absurd, as the friend could not have been in a state of "suspended animation"[1] before he answered the question. This view would need the quantum mechanical equations to be non-linear. It is E. Wigner's belief that the laws of physics must be modified when allowing conscious beings to be included.

The above and other of E. Wigner's original remarks about his friend appeared in his article "Remarks on the Mind-Body Question", published in the book The Scientist Speculates (1961), edited by I. J. Good. The article is reprinted in E. Wigner's own book Symmetries and Reflections (1967).

#### A counterargument

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.[2]

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.

### 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.[3] 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]

In 2016, Frauchiger and Renner used an elaboration of the Wigner's-friend scenario to argue that "single-world" interpretations of quantum mechanics cannot be consistent with fully unitary time evolution of quantum states.[5] As of 2018, the implications of their argument are still under debate.[6]

### 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.

## In fiction

Stephen Baxter's novel Timelike Infinity (1992) 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

1. E.P. Wigner (1961), "Remarks on the mind-body question", in: I.J. Good, "The Scientist Speculates", London, Heinemann
2. ^ R. Penrose, The Road to Reality, section 29.8.
3. ^ Everett, Hugh III. "'Relative State' Formulation of Quantum Mechanics". Reviews of Modern Physics. 29: 454–462. doi:10.1103/RevModPhys.29.454.
4. ^ "The Everett Interpretation". www.hedweb.com. Retrieved 2017-12-31.
5. ^ Frauchiger, Daniela; Renner, Renato (2016-04-25). "Single-world interpretations of quantum theory cannot be self-consistent". arXiv: [quant-ph].
6. ^ Responses taking various positions include the following: