Wigner’s Friend

Wigner’s friend asks how quantum theory applies when a measuring observer is included inside another observer’s quantum system. The tension concerns the compatibility of a definite internal record with a coherent external description. It is not resolved merely by saying that observers have different information: the proposed external measurement can distinguish a coherent laboratory from a collapsed mixture.

Helpful background. The measurement problem identifies the conflict between universal unitary dynamics, complete state descriptions, and unique outcomes.

A laboratory with two record states

Let a qubit start in $\lvert+\rangle$ and let the friend’s measurement correlate its computational basis with orthogonal laboratory states. Write

$$\lvert\Psi\rangle= \frac{\lvert0,F_0\rangle+\lvert1,F_1\rangle}{\sqrt2}, \qquad \lvert\Phi_\pm\rangle= \frac{\lvert0,F_0\rangle\pm\lvert1,F_1\rangle}{\sqrt2}.$$

The symbols $F_0,F_1$ include the friend’s memory and all degrees of freedom needed for a reversible description. Wigner could, in principle, measure the projectors onto $\Phi_+$ and $\Phi_-$. Universal unitary evolution predicts $p(+)=1$. Replacing the laboratory state by the outcome mixture

$$\rho_{\rm mix}=\tfrac12\lvert0,F_0\rangle\langle0,F_0\rvert +\tfrac12\lvert1,F_1\rangle\langle1,F_1\rvert$$

instead gives $p(+)=p(-)=1/2$. The difference lies in off-diagonal coherence, not in the friend’s ability to report either record when questioned in the record basis.

Reading a record and measuring interference

If Wigner reads the memory in the $\{F_0,F_1\}$ basis, ordinary conditional probabilities are consistent with the friend’s report. The interference experiment is a different intervention. Its projectors do not commute with the record projectors within the two-dimensional laboratory subspace. One cannot append “the record is left intact and publicly copied” to the coherent protocol without modeling those copies.

For example, suppose an uncontrolled environment carries states $E_0,E_1$. Tracing it out multiplies the laboratory coherence by $\gamma=\langle E_1|E_0\rangle$. Then

$$p(+)=\frac{1+\operatorname{Re}\gamma}{2}.$$

A durable external record with orthogonal environment states removes the proposed interference unless Wigner also controls that environment. This explains the severity of the laboratory-control assumption; decoherence alone does not choose a unique global outcome.

What the scenario constrains

A collapse theory can predict loss of coherence through an additional physical law. Everettian accounts retain the entangled state and interpret the friend’s result within a branch. Relational and agent-centered accounts restrict how outcome or state assignments are transferred among observers. Hidden-variable accounts add variables and must evaluate the actual later measurement dynamics. These responses differ in ontology and assumptions, even when an ordinary record-reading experiment has identical statistics.

Extended friend theorems add further assumptions, such as rules for combining certainty claims or locality conditions. Frauchiger–Renner is one such argument. Bong and collaborators’ local-friendliness theorem is another; their proof-of-principle photonic realization does not establish coherent control of a conscious laboratory. Neither scenario proves that consciousness causes collapse or that experience can be altered at will.

Exercise

In the dephased laboratory above, let $\gamma=0.6i$. Wigner may instead test $(\lvert0,F_0\rangle+e^{i\theta}\lvert1,F_1\rangle)/\sqrt2$. Can a suitable phase reveal coherence?

Solution

The probability is $(1+\operatorname{Re}[\gamma e^{i\theta}])/2$. Choosing $\theta=-\pi/2$ gives $0.8$, although the unshifted test gives $0.5$. Measuring only one interference quadrature can miss coherence with a different phase.

References

• Bong, K.-W., et al., “A strong no-go theorem on the Wigner’s friend paradox,” Nature Physics 16, 1199–1205 (2020), doi:10.1038/s41567-020-0990-x — explicit assumptions for an extended scenario.
• Wigner, E. P., “Remarks on the mind–body question,” in I. J. Good, ed., The Scientist Speculates, Heinemann, pp. 284–302 (1961) — historical motivation, whose consciousness proposal is not an assumption of the operational model here.