# Wigner's friend

Wigner's friend is a thought experiment in theoretical quantum physics, first conceived by the physicist Eugene Wigner in 1961, and developed into a thought experiment by David Deutsch in 1985. 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 nondeterministic, 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.

Generalizations and extensions of Wigner's friend have been proposed. Two such scenarios involving multiple friends have been implemented in a laboratory, using photons to stand in for the friends.

## The thought experiment

The thought experiment posits a friend of Wigner in a laboratory, and Wigner lets the friend perform a quantum measurement on a physical system (this could be a spin system or something analogous to 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, they 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 measured 1".

Let Wigner now ask his friend for the result of the measurement: whichever answer the friend gives (0 or 1), 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 laboratory collapses.

However, unless Wigner is considered in a "privileged position as ultimate observer", 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 exactly did the collapse occur? Was it when the friend had finished their 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 $|0\rangle _{S}$ and $|1\rangle _{S}$ , corresponding to measurement results 0 and 1.

Initially, $S$ is in a superposition state

$\alpha |0\rangle _{S}+\beta |1\rangle _{S}$ and gets measured by Wigner's friend ($F$ ) in the $\{|0\rangle _{S},|1\rangle _{S}\}$ -basis. Then, with probability $|\alpha |^{2}$ , $F$ will measure 0 and with probability $|\beta |^{2}$ , 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, they will assign to the spin the state corresponding to their measurement result: If they got 0, they will assign the state $|0\rangle _{S}$ to the spin, if they got 1, they will assign the state $|1\rangle _{S}$ 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 $S\otimes F$ ). 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

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

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

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

## Discussion

### Consciousness and Wigner's friend

Eugene 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" 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".

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 (Wigner's) view comes in as follows:

The friend's consciousness gets "impressed" by their measurement of the spin, and therefore they may assign a wave function to it according to the nature of their impression. Wigner, having no access to that information, can only assign the wave function $\alpha (|0\rangle _{S}\otimes |0\rangle _{F})+\beta (|1\rangle _{S}\otimes |1\rangle _{F})$ 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 $|0\rangle _{S}\otimes |0\rangle _{F}$ or $|1\rangle _{S}\otimes |1\rangle _{F}$ , and not their linear combination. Hence, there is a contradiction, specifically in the "consciousness causes collapse" interpretation.

Wigner then follows that "the being with a consciousness must have a different role in quantum mechanics than the inanimate measuring device": 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, 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" before they answered the question. This view would need the quantum mechanical equations to be non-linear. It is Wigner's belief that the laws of physics must be modified when allowing conscious beings to be included.

The above and other of 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 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.

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 the many-worlds interpretation

The various versions of the many worlds interpretation avoid the need to postulate that consciousness causes collapse – indeed, that collapse occurs at all.

Hugh Everett III's doctoral thesis "'Relative state' formulation of quantum mechanics" serves as the foundation for today's many versions of many-worlds interpretations. In the introductory part of his work, Everett discusses the "amusing, but extremely hypothetical drama" of the Wigner's friend paradox. Note that there is evidence of a drawing of the scenario in an early draft of Everett's thesis. It was therefore Everett who provided the first written discussion of the problem four or five years before it was discussed in "Remarks on the mind-body question" by Wigner, of whom it received the name and fame thereafter. However, Everett being a student of Wigner's, it is clear that they must have discussed it together at some point.

In contrast to his teacher Wigner, who held the consciousness of an observer to be responsible for a collapse, Everett understands the Wigner's friend scenario in a different way: Insisting that quantum states assignments should be objective and nonperspectival, Everett derives a straightforward logical contradiction when letting $F$ and $W$ reason about the laboratory's state of $S$ together with $F$ . Then, the Wigner's Friend scenario shows to Everett an incompatibility of the collapse postulate for describing measurements with the deterministic evolution of closed systems. In the context of his new theory, Everett claims to solve the Wigner's Friend paradox by only allowing a continuous unitary time evolution of the wave function of the universe. However, there is no evidence of any written argument of Everett's on the topic.

In many worlds interpretations, measurements are modelled as interactions between subsystems of the universe and manifest themselves as a branching of the universal state. The different branches account for the different possible measurement outcomes and are seen to exist as subjective experiences of the corresponding observers. In this view, the friend's measurement of the spin results in a branching of the world into two parallel worlds, one, in which the friend has measured the spin to be 1 and another, in which the friend has received the measurement outcome 0. If then Wigner measures at a later time the combined system of friend and spin system, the world again splits into two parallel parts.

### Objective collapse theories

According to objective collapse theories, wave function collapse occurs when a superposed system reaches a certain objective threshold of size or complexity. 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. If the measured system were much simpler (such as a single spin state) then once the observation was made the system would be expected to collapse since the larger system of the scientist, equipment, and room would be considered far too complex to become entangled in the superposition.

### Wigner's Friend in Relational Quantum Mechanics

Relational Quantum Mechanics (RQM) was developed in 1996 by Carlo Rovelli and is one of the more recent Interpretations of Quantum Theory. In RQM, any physical system can play the role of an observing system, to which any other system may display "facts" about physical variables. This inherent relativity of facts in RQM provides a straightforward ”solution” to the seemingly paradoxical situation in Wigner's Friend scenario: The state that the Friend assigns to the spin is a state relative to himself as Friend, whereas the state that Wigner assigns to the combined system of friend and spin is a state relative to himself as Wigner. By construction of the theory, these two descriptions do not have to match, because both are correct assignments of states relative to their respective system.

If the physical variable that is measured of the spin system is denoted by z, where z takes the possible outcome values 0 or 1, the above Wigner's Friend situation is modelled in the RQM context as follows: $F$ models the situation as the before-after-transition

$\alpha |0\rangle _{S}+\beta |1\rangle _{S}\rightarrow |1\rangle _{S}$ of the state of $S$ relative to him (here it was assumed that $F$ received the outcome z=1 in his measurement of $S$ ).

In RQM language, the fact z = 1 for the spin of $S$ actualized itself relative to $F$ during the interaction of the two systems.

A different way to model the same situation is again an outside (Wigner-) perspective. From that viewpoint, a measurement by one system ($F$ ) of another ($S$ ) results in a correlation of the two systems. The state displaying such a correlation is equally valid for modelling the measurement process. However, the system with respect to which this correlated state is valid changes. Assuming that Wigner ($W$ ) has the information that the physical variable z of $S$ is being measured by $F$ , but not knowing what $F$ received as result, $W$ must model the situation as

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

where $|\bot \rangle _{F}$ is considered the state of $F$ before the measurement, and $|1\rangle _{F}$ and $|0\rangle _{F}$ are the states corresponding to $F$ ’s state when he has measured 1 or 0, respectively. This model is depicting the situation as relative to $W$ , so the assigned states are relative states with respect to the Wigner system. In contrast, there is no value for the z-outcome that actualizes with respect to $W$ , as he is not involved in the measurement.

In this sense, two accounts of the same situation (process of the measurement of the physical variable z on the system $S$ by $F$ ) are accepted within RQM to exist side by side. Only when deciding for a reference system, a statement for the ”correct” account of the situation can be made.

### QBism

In the interpretation known as QBism, advocated by N. David Mermin among others, the Wigner's-friend situation does not lead to a paradox, because there is never a uniquely correct wavefunction for any system. Instead, a wavefunction is a statement of personalist Bayesian probabilities, and moreover, the probabilities that wavefunctions encode are probabilities for experiences that are also personal to the agent who experiences them. As von Baeyer puts it, "Wavefunctions are not tethered to electrons and carried along like haloes hovering over the heads of saints—they are assigned by an agent and depend on the total information available to the agent." Consequently, there is nothing wrong in principle with Wigner and his friend assigning different wavefunctions to the same system. A similar position is taken by Brukner, who uses an elaboration of the Wigner's-friend scenario to argue for it.

## An extension of the Wigner's friend experiment

In 2016, Frauchiger and Renner used an elaboration of the Wigner's-friend scenario to argue that quantum theory cannot be used to model physical systems that are themselves agents who use quantum theory. They provide an information-theoretic analysis of two specifically connected pairs of "Wigner's friend" experiments, where the human observers are modelled within quantum theory. By then letting the four different agents reason about each other’s measurement results (using the laws of quantum mechanics), contradictory statements are derived.

The resulting theorem highlights an incompatibility of a number of assumptions that are usually taken for granted when modelling measurements in quantum mechanics.

In the title of their published version of September 2018, the authors' interpretation of their result is apparent: Quantum theory as given by the textbook and used in the numerous laboratory experiments to date "cannot consistently describe the use of itself" in any given (hypothetical) scenario. The implications of the result are currently subject to many debates among physicists of both theoretical and experimental quantum mechanics. In particular, the various proponents of the different interpretations of quantum mechanics have challenged the validity of the Frauchiger–Renner argument.

The experiment was designed using a combination of arguments by Wigner (Wigner's friend), Deutsch and Hardy (see Hardy's paradox). The setup involves a number of macroscopic agents (observers) performing predefined quantum measurements in a given time order. Those agents are assumed to all be aware of the whole experiment and to be able to use quantum theory to make statements about other people’s measurement results. The design of the thought experiment is such that the different agents' observations along with their logical conclusions drawn from a quantum theoretical analysis yields inconsistent statements.

The scenario corresponds roughly to two parallel pairs of "Wigners" and friends: $F_{1}$ with $W_{1}$ and $F_{2}$ with $W_{2}$ . The friends each measure a specific spin system, and each Wigner measures "his" friend's laboratory (which includes the friend). The individual agents make logical conclusions that are based on their measurement result, aiming at predictions about other agent's measurements within the protocol. Frauchiger and Renner argue that an inconsistency occurs if three assumptions are taken to be simultaneously valid. Roughly speaking, those assumptions are

(Q): Quantum theory is correct.

(C): Agent's predictions are information-theoretically consistent.

(S): A measurement yields only one single outcome.

More precisely, assumption (Q) involves the probability predictions within quantum theory given by the Born rule. This means that an agent is allowed to trust this rule being correct in assigning probabilities to other outcomes conditioned on his own measurement result. It is however sufficient for the Extended Wigner's friend experiment to assume the validity of the Born rule for probability-1 cases, i.e., if the prediction can be made with certainty.

Assumption (S) specifies that once an agent has arrived at a probability-1 assignment of a certain outcome for a given measurement, they could never agree to a different outcome for the same measurement.

Assumption (C) invokes a consistency among different agents' statements in the following manner: The statement "I know (by the theory) that they know (by the same theory) that x" is equivalent to "I know that x".

Assumptions (Q) and (S) are used by the agents when reasoning about measurement outcomes of other agents, and assumption (C) comes in when an agent combines other agent's statements with their own. The result is contradictory, and therefore, assumptions (Q), (C) and (S) cannot all be valid, hence the no-go theorem.

### Reflection

The meaning and implications of the Frauchiger–Renner thought experiment are highly debated. A number of assumptions taken in the argument are very foundational in content, and therefore cannot be given up easily. However, the questions remains whether there are "hidden" assumptions that do not explicitly appear in the argument. The authors themselves conclude that "quantum theory cannot be extrapolated to complex systems, at least not in a straightforward manner". On the other hand, one presentation of the experiment as a quantum circuit models the agents as single qubits and their reasoning as simple conditional operations.

QBism and relational quantum mechanics have been argued to avoid the contradiction suggested by the extended Wigner's-friend scenario of Frauchiger and Renner.

## 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.