21.1 An isolated laboratory inside a superobserver’s apparatus
21.1.1 Internal record formation versus external unitary evolution
Eugene Wigner sharpened Schrödinger’s cat scenario in 1961 by replacing the inanimate pointer with an observer—Wigner’s friend \(F\)—inside a completely isolated laboratory \(L = S \otimes F\). The friend measures a spin/qubit \(S\) prepared in \(\alpha|0\rangle_S + \beta|1\rangle_S\) in the \(\{|0\rangle_S, |1\rangle_S\}\) basis and records a definite outcome in memory register \(F\). From the friend’s internal perspective upon seeing an outcome, the state appears updated to \(|0\rangle_S|0\rangle_F\) (with probability \(|\alpha|^2\)) or \(|1\rangle_S|1\rangle_F\) (with probability \(|\beta|^2\)). From the external perspective of Wigner (a “superobserver” whose laboratory is completely isolated from \(L\)), the interaction inside \(L\) is a unitary premeasurement \(U_{\mathrm{meas}} = \mathrm{CNOT}_{S\to F}\) that prepares the entangled state
21.1.2 Reading the friend’s record versus performing a joint interference test
So long as Wigner simply opens the door and asks the friend which outcome occurred—measuring in the record basis \(\{|00\rangle_{SF}, |11\rangle_{SF}\}\)—both descriptions predict the exact same probabilities \(P(00) = |\alpha|^2\) and \(P(11) = |\beta|^2\). A physical discrepancy between the pure entangled state \(|\Psi_{SF}\rangle\) and the incoherent mixture \(\rho_{\mathrm{mix}} = |\alpha|^2|00\rangle\langle 00| + |\beta|^2|11\rangle\langle 11|\) arises only if Wigner instead performs a coherent joint measurement across the entire laboratory \(S \otimes F\) in a complementary superobserver basis.
21.1.3 Modeling the friend as a reversible quantum memory register
In circuit language, treating the friend’s apparatus and memory register as a qubit \(F\) initialized to \(|0\rangle_F\) models the internal measurement as \(\mathrm{CNOT}_{S\to F}((\alpha|0\rangle_S + \beta|1\rangle_S)|0\rangle_F) = \alpha|00\rangle_{SF} + \beta|11\rangle_{SF}\). Testing whether coherence between \(|00\rangle_{SF}\) and \(|11\rangle_{SF}\) survives requires either measuring the two-qubit parity operator \(X_S \otimes X_F\) or applying the inverse unitary \(\mathrm{CNOT}_{S\to F}^\dagger = \mathrm{CNOT}_{S\to F}\) to uncompute the friend’s memory register.
Visual animation · Section 21.1
An isolated laboratory inside a superobserver’s apparatus
Interactive visual model for 21.1 An isolated laboratory inside a superobserver’s apparatus (Internal record formation versus external unitary evolution · Reading the friend’s record versus performing a joint interference test · Modeling the friend as a reversible quantum memory register). Use Animate or Step to follow the physical state evolution and quantitative predictions.
21.2 Uncomputing a CNOT premeasurement and joint interference
21.2.1 Superobserver basis states and the coherent visibility witness
Suppose the system starts in \(|+\rangle_S = (|0\rangle_S + |1\rangle_S)/\sqrt{2}\) (\(\alpha = \beta = 1/\sqrt{2}\)) so that unitary premeasurement yields \(|\Phi^+\rangle_{SF} = (|00\rangle_{SF} + |11\rangle_{SF})/\sqrt{2}\). Define Wigner’s joint interference basis
An irreversible collapse inside \(L\) (\(W = 0\)) yields a flat line \(P_+(\phi) = 1/2\) for all \(\phi\), whereas reversible unitary evolution (\(W = 1\)) yields full-contrast fringes \(P_+(\phi) = \cos^2(\phi/2)\), reaching \(P_+(0) = 1\).
21.2.2 Reversing the system–memory CNOT to test interference
David Deutsch noted in 1985 that Wigner can implement the \(\phi = 0\) interference test by applying a second \(\mathrm{CNOT}_{S\to F}\) to uncompute the friend’s record:
Measuring \(S\) in the \(\{|+\rangle_S, |-\rangle_S\}\) basis then yields \(|+\rangle_S\) with probability \(P(+_S) = (1 + W)/2 = 1\) when \(W = 1\), versus \(P(+_S) = 1/2\) if the friend’s measurement collapsed the state to \(\rho_{\mathrm{mix}}\). Crucially, however, uncomputing \(F\) back to \(|0\rangle_F\) erases the friend’s outcome record: Wigner can never simultaneously retain the friend’s which-outcome record and observe \(P(+_S) = 1\).
21.2.3 Conditioned internal states versus unconditioned global states
The mathematical difference between the friend’s state assignment (\(|0\rangle_S\) or \(|1\rangle_S\)) and Wigner’s state assignment (\(|\Phi^+\rangle_{SF}\)) is the difference between a state conditioned on an internal branch record and an unconditioned joint state of system plus memory. Both assignments agree on every measurement that preserves and commutes with the friend’s memory register \(Z_F\); they differ only on superobserver operations that do not commute with \(Z_F\) and therefore overwrite the friend’s memory.
Visual animation · Section 21.2
Uncomputing a CNOT premeasurement and joint interference
Interactive visual model for 21.2 Uncomputing a CNOT premeasurement and joint interference (Superobserver basis states and the coherent visibility witness · Reversing the system–memory CNOT to test interference · Conditioned internal states versus unconditioned global states). Use Animate or Step to follow the physical state evolution and quantitative predictions.
21.3 Environmental record leakage and observer assumptions
21.3.1 How interpretations resolve the friend–superobserver assignment gap
Different interpretations of quantum mechanics locate the resolution of Wigner’s friend in different places:
Textbook collapse / objective collapse: A sufficiently macroscopic record collapses \(\rho_{SF}\) to \(W = 0\), predicting \(P_+(\phi) = 1/2\).
Everett / relative-state / relational / QBism: Unitary evolution holds universally (\(W = 1\) for an isolated lab), and state assignments are indexed relative to the physical record system or agent; reversing the CNOT erases the relative record.
Bohmian mechanics: The universal wavefunction evolves unitarily (\(W = 1\)) while actual configuration coordinates occupy one branch until the branches are re-overlapped by \(\mathrm{CNOT}_{S\to F}\).
21.3.2 Scope of few-qubit photonic and superconducting friend demonstrations
Recent experiments on entangled photons and superconducting circuits implement the Wigner’s friend circuit by using a single ancillary qubit or photon path as the “friend” register \(F\). These experiments confirm \(W \approx 1\) and verify the two-qubit circuit identities cleanly, but because a one-qubit register is not a macroscopic observer with thermodynamically irreversible records, they test two-qubit unitary mechanics rather than macroscopic observer collapse.
If even a single environmental mode \(E\) outside Wigner’s control correlates with the friend’s record (\(|00\rangle|E_0\rangle\) vs.\ \(|11\rangle|E_1\rangle\) with \(\langle E_0|E_1\rangle = 0\)), applying \(\mathrm{CNOT}_{S\to F}\) to \(S\) and \(F\) alone produces \((|0\rangle_S|E_0\rangle + |1\rangle_S|E_1\rangle)|0\rangle_F/\sqrt{2}\), whose reduced state on \(S\) is \(\frac{1}{2}I_S\) with \(P(+_S) = 1/2\). Reversing a measurement requires reversing every physical copy of the record.
Visual animation · Section 21.3
Environmental record leakage and observer assumptions
Interactive visual model for 21.3 Environmental record leakage and observer assumptions (How interpretations resolve the friend–superobserver assignment gap · Scope of few-qubit photonic and superconducting friend demonstrations · Why unreversed environmental copies destroy superobserver interference). Use Animate or Step to follow the physical state evolution and quantitative predictions.
Explore the Apparatus
Prepare the system state \(\alpha|0\rangle_S + \beta|1\rangle_S\), run the internal \(\mathrm{CNOT}_{S\to F}\) premeasurement, and adjust the environmental record leakage \(1 - W\) outside the isolated laboratory. Compare Wigner’s outcomes when reading the friend’s record (\(Z_F\)) versus uncomputing the record and measuring interference in the superobserver basis \(|\Phi_{\pm}(\phi)\rangle_{SF}\).
Interactive model · Idealized predictions
Inside the laboratory; outside the laboratory
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
Read the values as a table
Can an observer and an outside experimenter assign different states consistently? The text and worked examples remain available without JavaScript. Animation starts only when requested; Step gives a motion-free way to follow the model.
Worked Check
Problem: System \(S\) is initialized in \(|+\rangle_S = (|0\rangle_S + |1\rangle_S)/\sqrt{2}\) and the friend’s memory register \(F\) starts in \(|0\rangle_F\). After the internal measurement \(\mathrm{CNOT}_{S\to F}\), residual coupling to an unreversed environment mode leaves coherence \(W = |\langle E_0|E_1\rangle| = 0.76\). Wigner then applies \(\mathrm{CNOT}_{S\to F}\) to uncompute the friend’s register and measures \(S\) in the \(\{|+\rangle_S, |-\rangle_S\}\) basis. Find the probabilities \(P(+_S)\) and \(P(-_S)\), and compare them with the case of complete collapse (\(W = 0\)).
Solution:
1. Before Wigner’s reversal, the reduced state of \(S\otimes F\) is
2. Applying \(\mathrm{CNOT}_{S\to F}\) maps \(|00\rangle \mapsto |00\rangle\) and \(|11\rangle \mapsto |10\rangle\), factoring out \(|0\rangle_F\langle 0|_F\) and leaving system \(S\) in
\[\rho_S^{\mathrm{rev}}(W) = \frac{1}{2}\begin{pmatrix} 1 & W \\ W & 1 \end{pmatrix} = \frac{1+W}{2}|+\rangle\langle +| + \frac{1-W}{2}|-\rangle\langle -|.\]
In Wigner’s friend scenario, an internal observer \(F\) describes her measurement by a projective state collapse, whereas an external superobserver \(W\) describes the isolated laboratory unitarily as an entangled state \(|\Phi\rangle_{SF}\).
As long as \(W\) does not perform coherent interference measurements that overwrite \(F\)’s memory register, both descriptions predict identical marginal probabilities for \(F\)’s recorded outcomes.
Testing \(|\Phi\rangle_{SF}\) in a coherent superobserver basis requires reversing the entire laboratory unitary—an operation that erases \(F\)’s classical record.
Practice Problems
Problem 21.1: Expectation value of the joint parity witness \(X_S \otimes X_F\)
For the partially decohered laboratory state \(\rho_{SF}(W) = \frac{1}{2}(|00\rangle\langle 00| + |11\rangle\langle 11| + W|00\rangle\langle 11| + W|11\rangle\langle 00|)\), calculate \(\langle Z_S\otimes Z_F\rangle\) and \(\langle X_S\otimes X_F\rangle\).
Hint
Use \((X_S\otimes X_F)|00\rangle = |11\rangle\) and \((X_S\otimes X_F)|11\rangle = |00\rangle\).
Solution
Because both \(|00\rangle\) and \(|11\rangle\) have \(Z_S Z_F = +1\), \(\langle Z_S\otimes Z_F\rangle = +1\) for every \(W \in [0,1]\). By contrast, \(\mathrm{Tr}[\rho_{SF}(W)(X_S\otimes X_F)] = \frac{W}{2}\langle 11|11\rangle + \frac{W}{2}\langle 00|00\rangle = W\). Thus \(Z_S Z_F\) verifies record correlation while \(X_S X_F\) directly measures the surviving coherence \(W\).
Problem 21.2: Why retaining a copy of the friend’s record blocks interference
Suppose that before Wigner applies \(\mathrm{CNOT}_{S\to F}\) to uncompute \(F\), the friend copies the record into a notebook qubit \(N\) via \(\mathrm{CNOT}_{F\to N}\), preparing \((|000\rangle_{SFN} + |111\rangle_{SFN})/\sqrt{2}\). If Wigner reverses only \(S\) and \(F\) (leaving \(N\) untouched) and measures \(S\) in the \(\{|+\rangle_S, |-\rangle_S\}\) basis, what is \(P(+_S)\)?
Hint
Apply \(\mathrm{CNOT}_{S\to F}\) to \((|000\rangle_{SFN} + |111\rangle_{SFN})/\sqrt{2}\) and trace over \(N\).
Solution
Applying \(\mathrm{CNOT}_{S\to F}\) produces \((|000\rangle_{SFN} + |101\rangle_{SFN})/\sqrt{2} = \frac{1}{\sqrt{2}}(|0\rangle_S|0\rangle_N + |1\rangle_S|1\rangle_N)\otimes|0\rangle_F\). Because \(S\) remains maximally entangled with the unreversed notebook \(N\), \(\rho_S = \frac{1}{2}I_S\) and \(P(+_S) = 1/2\).
Problem 21.3: Noncommutativity of record readout and interference readout
Compute the commutator \([Z_F,\, X_S\otimes X_F]\) between reading the friend’s record \(Z_F = I_S\otimes Z_F\) and measuring Wigner’s joint coherence witness \(X_S\otimes X_F\).
Hint
Use \([Z, X] = 2iY\) on the second tensor factor.
Solution
\([I_S\otimes Z_F,\, X_S\otimes X_F] = X_S\otimes [Z_F, X_F] = 2i(X_S\otimes Y_F) \neq 0\). Because the two observables do not commute, no measurement can simultaneously read the friend’s definite outcome \(Z_F\) and certify nonvanishing superobserver coherence \(X_S\otimes X_F\).
Key Takeaways
Inside an isolated laboratory \(S\otimes F\), a reversible measurement \(\mathrm{CNOT}_{S\to F}\) maps \((\alpha|0\rangle_S + \beta|1\rangle_S)|0\rangle_F\) to \(\alpha|00\rangle_{SF} + \beta|11\rangle_{SF}\).
In the record basis \(\{|00\rangle, |11\rangle\}\), the entangled state and the collapsed mixture \(\rho_{\mathrm{mix}}\) give identical statistics; they differ only on superobserver interference observables such as \(X_S\otimes X_F\) or after CNOT uncomputation.
Observing superobserver interference after uncomputation requires erasing the friend’s memory register and requires that zero unreversed copies of the outcome have leaked into the environment.
Sources and Further Reading
E. P. Wigner, “Remarks on the mind-body question,” in I. J. Good (ed.), The Scientist Speculates, Heinemann, pp. 284–302 (1961).