Von Neumann Chain

The von Neumann chain extends a measurement interaction to additional apparatuses, memories and observers. At every stage, unitary evolution can propagate correlations without replacing their superposition by one globally selected branch. The chain is a diagnostic for proposed measurement accounts: adding another observer does not by itself identify a new physical law.

Helpful background. Measurement Problem establishes linear premeasurement correlations.

A succession of record correlations

Let a system label $i$ be copied into an apparatus $A$, then a memory $M$. For an ideal calibrated sequence,

$$\begin{aligned} \sum_i c_i|i\rangle|A_r\rangle|M_r\rangle &\longrightarrow\sum_i c_i|i\rangle|A_i\rangle|M_r\rangle\\ &\longrightarrow\sum_i c_i|i\rangle|A_i\rangle|M_i\rangle. \end{aligned}$$

A memory state can be a physical computer register; nothing in the algebra requires consciousness. When record labels are orthogonal, local readouts agree in the correlated branches. This explains consistency of records conditional on a branch, while the global state still contains the full sum.

The von Neumann measurement model derives the apparatus interaction. The present question is what changes when the description includes additional record-bearing systems.

Moving the cut

An operational calculation can place the quantum/classical description boundary after the first apparatus or after its copied memory. If both models represent the same physical instrument and conditioning event, they must give the same accessible probabilities. This equivalence permits flexible modeling; it does not prove that an objective collapse physically occurs at whichever boundary was chosen.

For a two-outcome experiment, suppose the first apparatus records zero and the memory faithfully copies it. Conditioning immediately on the apparatus label or later on the identical memory label gives the same branch weight $|c_0|^2$. If the memory has readout errors, the conditional predictions differ according to its error model. A cut can be shifted only while preserving the actual information and intervention represented.

Worked example: reversing an isolated chain

Take three controllable qubits in the order $S,A,M$, prepare

$$|\psi\rangle_S=\frac{\sqrt3}{2}|0\rangle+\frac{i}{2}|1\rangle, \qquad |A_rM_r\rangle=|00\rangle,$$

and copy first $S$ into $A$, then $A$ into $M$, with controlled-NOT gates. The output and the corresponding nonselective branch mixture are

$$\begin{aligned} |\Psi\rangle&=\frac{\sqrt3}{2}|000\rangle+\frac{i}{2}|111\rangle,\\ \rho_{\rm mix}&=\frac34|000\rangle\langle000|+\frac14|111\rangle\langle111|. \end{aligned}$$

Both predict identical $Z$ records: all three labels agree, with probabilities $3/4$ and $1/4$. To see the limit of replacing the global vector by this mixture, undo the copies in reverse order. The coherent state returns to $|\psi\rangle_S|00\rangle_{AM}$; the mixture becomes

$$\left(\frac34|0\rangle\langle0|+\frac14|1\rangle\langle1|\right)_S \otimes|00\rangle\langle00|_{AM}.$$

A subsequent Pauli $Y$ measurement, with $|+y\rangle=(|0\rangle+i|1\rangle)/\sqrt2$, therefore gives

$$p_{\rm coherent}(+y)=\frac{2+\sqrt3}{4}\simeq0.9330, \qquad p_{\rm mix}(+y)=\frac12.$$

The operations and readout are now sensitive to the phase between record branches. Treating the apparatus as a classical label was sufficient for ordinary record readings, but discarding its coherence is insufficient for this later intervention. More precisely, for orthogonal branches $|b_i\rangle$, a coherent state and its diagonal branch mixture agree on any effect $F$ with $\langle b_j|F|b_i\rangle=0$ for $i\ne j$. Here $F$ includes all later operations pulled back to the time at which the descriptions are compared. The reversal followed by $Y$ readout does not satisfy that condition.

No outcome has been conditioned on in the reversal experiment, and the three qubits are assumed isolated. If a further system retains a record, reversing these two gates alone need not restore the phase. This is the operational distinction between reduced predictions and global coherence discussed by Schlosshauer (2005, Secs. III.B–D); it does not settle which interpretation describes a macroscopic measurement.

Environment and practical irreversibility

A macroscopic chain usually spreads record information into uncontrolled surroundings. Reversing it would require coherent control over all relevant degrees of freedom, including scattered radiation. This practical restriction explains why a later observer normally uses an effectively classical record.

In principle, a deliberately isolated microscopic chain can be undone by inverse operations. The contrast between ordinary record reading and coherent interrogation of the whole laboratory is central to Wigner’s Friend. The thought experiment changes what an outside observer is allowed to measure; it should not be confused with two people merely reading the same detector.

Where an interpretation enters

An objective-collapse account inserts a specified stochastic dynamics at physical scales set by its model, not at an arbitrary observer boundary. A hidden-variable account supplies definite configurations throughout the chain. A relative-state account identifies records within correlated branches. These are proposals about what the state description means or how it evolves; the chain itself selects none of them.

Consciousness-dependent collapse would require a concrete criterion and quantitative law before it became a testable addition. The existence of a chain supplies no evidence for that addition.

Exercises

Erase the last copy

For $(|000\rangle+|111\rangle)/\sqrt2$ on system, apparatus and memory, apply the inverse copying controlled-NOT from apparatus to memory. Has the system measurement been undone?

Solution

The state becomes $(|00\rangle+|11\rangle)|0\rangle/\sqrt2$. The memory has reset, but system and apparatus remain entangled. Undoing the earlier system-to-apparatus coupling is also necessary to restore the initial product state. Erasing one accessible record is insufficient when another record remains.

An uncontrolled record remains

In the worked example, let the normalized environmental states $|E_0\rangle,|E_1\rangle$ carry an additional record, so the total state is $\sqrt3|000\rangle|E_0\rangle/2+i|111\rangle|E_1\rangle/2$. Undo the two copying gates but do not act on the environment. For $\gamma=\langle E_1|E_0\rangle$, find the final system state and $p(+y)$. Evaluate the result for real $\gamma=1/5$, and compare $\gamma=0$ and $\gamma=1$.

Solution

The apparatus and memory return to $|00\rangle$, while the system remains correlated with the environment. Tracing out that environment gives

$$\rho_S=\begin{pmatrix} 3/4&-i\sqrt3\,\gamma/4\\ i\sqrt3\,\gamma^*/4&1/4 \end{pmatrix}, \qquad p(+y)=\frac12+\frac{\sqrt3}{4}\operatorname{Re}\gamma.$$

For $\gamma=1/5$, $p(+y)=1/2+\sqrt3/20\simeq0.5866$. Orthogonal environmental records give $p(+y)=1/2$ and the diagonal system mixture; identical environmental vectors give the fully restored result $0.9330$. These local probabilities do not imply that a component vanished from the global vector. For complex $\gamma$, a $Y$ readout tests only its real part; a suitable phase scan can detect the remaining coherence.

References

• Everett, Hugh, III, “Relative State Formulation of Quantum Mechanics,” Reviews of Modern Physics 29, 454–462 (1957), doi:10.1103/RevModPhys.29.454 — treating correlated observers within the quantum description.
• Schlosshauer, Maximilian, “Decoherence, the measurement problem, and interpretations of quantum mechanics,” Reviews of Modern Physics 76, 1267–1305 (2005), doi:10.1103/RevModPhys.76.1267, Secs. III.B–D; open manuscript — local statistics and environmental records.
• von Neumann, John, Mathematical Foundations of Quantum Mechanics, translated by Robert T. Beyer, Princeton University Press (1955) — measurement-chain analysis.