Wigner's friend

Wigner's friend is a thought experiment in theoretical quantum physics, first published by the Hungarian-American physicist Eugene Wigner in 1961, and further developed by David Deutsch in 1985. However, the use of photons as observers has been criticized by quantum physicist Lev Vaidman of Tel Aviv University as "ridiculous; the friend has to be macroscopic". Philosopher of physics Tim Maudlin of New York University says that "Nobody thinks a photon is an observer".

Original paradox

Wigner introduced the thought experiment in a 1961 article "Remarks on the Mind-Body Question".

Responses in different interpretations of quantum mechanics

Many-worlds interpretations

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" 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. (RQM) was developed in 1996 by Carlo Rovelli and is one of the more recent interpretations of quantum mechanics. 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 <math>z</math>, where <math>z</math> takes the possible outcome values 0 or 1, the above Wigner's friend situation is modelled in the RQM context as follows: <math>F</math> models the situation as the before-after-transition

<math display="block">\alpha|0\rangle_S + \beta|1\rangle_S \to |1\rangle_S</math>

of the state of <math>S</math> relative to him (here it was assumed that <math>F</math> received the outcome <math>z = 1</math> in his measurement of <math>S</math>).

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

A different way to model the same situation is again an outside (Wigner's) perspective. From that viewpoint, a measurement by one system (<math>F</math>) of another (<math>S</math>) 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 (<math>W</math>) has the information that the physical variable <math>z</math> of <math>S</math> is being measured by <math>F</math>, but not knowing what <math>F</math> received as result, <math>W</math> must model the situation as

where <math>|\bot\rangle_F</math> is considered the state of <math>F</math> before the measurement, and <math>|1\rangle_F</math> and <math>|0\rangle_F</math> are the states corresponding to <math>F</math>'s state when he has measured 1 or 0 respectively. This model is depicting the situation as relative to <math>W</math>, so the assigned states are relative states with respect to the Wigner system. In contrast, there is no value for the <math>z</math> outcome that actualizes with respect to <math>W</math>, 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 <math>z</math> on the system <math>S</math> by <math>F</math>) 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 and Bayesian interpretations

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. Jaynes expresses this as follows: "There is a paradox only if we suppose that a density matrix (i.e. a probability distribution) is something 'physically real' and 'absolute'. But now the dilemma disappears when we recognize the 'relativity principle' for probabilities. A density matrix (or, in classical physics, a probability distribution over coordinates and momenta) represents, not a physical situation, but only a certain state of knowledge about a range of possible physical situations". And 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.

De Broglie–Bohm theory

The De Broglie-Bohm theory, also known as Bohmian mechanics or pilot wave theory, postulates, in addition to the wave function, an actual configuration of particles that exists even when unobserved. This particle configuration evolves in time according to a deterministic law, with the wave function guiding the motion of the particles. The particle configuration determines the actual measurement outcome—e.g., whether Schrödinger's cat is dead or alive or whether Wigner's friend has measured 0 or 1—even if the wave function is a superposition. Indeed, according to the De Broglie-Bohm theory, the wave function never collapses on the fundamental level. There is, however, a concept of effective collapse, based on the fact that, in many situations, "empty branches" of the wave function, which do not guide the actual particle configuration, can be ignored for all practical purposes.

The De Broglie-Bohm theory does not assign any special status to conscious observers. In the Wigner's-friend situation, the first measurement would lead to an effective collapse. But even if Wigner describes the state of his friend as a superposition, there is no contradiction with this friend having observed a definite measurement outcome as described by the particle configuration. Thus, according to the De Broglie–Bohm theory, there is no paradox because the wave function alone is not a complete description of the physical state.

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. 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 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: <math>F_1</math> with <math>W_1</math> and <math>F_2</math> with <math>W_2</math>. 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

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 (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".

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.

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 question 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".

QBism, relational quantum mechanics and the De Broglie–Bohm theory 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.