von Neumann Measurement Model

The von Neumann model describes a measurement interaction as unitary evolution of a system and a pointer. It explains how an apparatus records information and how ignoring that apparatus changes the reduced state. To obtain selective probabilities and states, it still uses the Born rule and conditioning for the pointer; it does not derive a unique observed result from unitarity alone.

Helpful background. Tensor products describe joint states, and the partial trace gives the state of the system when its pointer is ignored.

Discrete Correlation Model

Let $\{|a\rangle\}$ be orthonormal eigenstates of the system observable being measured, and let $|A_0\rangle$ be the ready state of the apparatus pointer. An ideal premeasurement unitary satisfies

$$U_{\mathrm{meas}} \left( |a\rangle|A_0\rangle \right) = |a\rangle|A_a\rangle,$$

where normalized, mutually orthogonal pointer states $|A_a\rangle$ represent perfectly distinguishable outcomes. The initial system and pointer are assumed uncorrelated. Orthogonality is a sharp-readout idealization; partial overlaps are considered below.

By linearity, an input superposition evolves as

$$\left(\sum_a c_a|a\rangle\right)|A_0\rangle \longrightarrow \sum_a c_a|a\rangle|A_a\rangle.$$

The result is not a product state. It is an entangled system-apparatus state, unless only one coefficient $c_a$ is nonzero.

The von Neumann premeasurement step correlates system alternatives $|a\rangle$ with pointer states $|A_a\rangle$. Reading the pointer gives a selective state; ignoring or tracing over the pointer gives the nonselective measurement channel.

Recovering Outcome Probabilities

If the pointer states are orthonormal or effectively distinguishable,

$$\langle A_b|A_a\rangle=\delta_{ab},$$

then measuring the pointer in the $\{|A_a\rangle\}$ basis gives outcome $a$ with probability

$$p(a)=|c_a|^2.$$

For a general density operator $\rho$, the same ideal measurement is represented by projectors

$$P_a=|a\rangle\langle a|,$$

and the probability is

$$p(a)=\operatorname{Tr}(P_a\rho).$$

Applying the Born rule to the pointer reproduces the system PVM probabilities when its states faithfully distinguish the alternatives. This is a consistency check of the apparatus model, not a derivation of the Born rule from unitary dynamics.

Selective Update from Reading the Pointer

Suppose the pointer is read and outcome $a$ is obtained. Conditioning the joint state on the pointer result gives the system state

$$\rho_a = \frac{P_a\rho P_a} {\operatorname{Tr}(P_a\rho)}.$$

This is the selective projective update. In the premeasurement picture, it is not applied before the apparatus interaction. It is the state assignment after the pointer record is known.

Nonselective Update from Ignoring the Pointer

If the pointer outcome is not read or not retained, the system state is obtained by tracing over the pointer. Starting from a pure input

$$|\Psi\rangle = \sum_a c_a|a\rangle|A_a\rangle,$$

the reduced system state is

$$\rho_S = \operatorname{Tr}_A|\Psi\rangle\langle\Psi| = \sum_{a,b} c_a c_b^* \langle A_b|A_a\rangle |a\rangle\langle b|.$$

If the pointer states are orthogonal, this becomes

$$\rho_S = \sum_a |c_a|^2 |a\rangle\langle a|.$$

For a general density operator, the unread ideal measurement channel is

$$\rho \longmapsto \sum_a P_a\rho P_a.$$

Thus the nonselective update is reduced dynamics: the total system-apparatus state may evolve unitarily, while the system alone loses coherence in the measured basis.

Imperfect Pointers and Unsharp Measurements

The ideal projective limit requires pointer states that are reliably distinguishable. If the pointer states have nonzero overlaps,

$$\langle A_b|A_a\rangle\ne0 \qquad (a\ne b),$$

then the reduced state retains some coherence between $|a\rangle$ and $|b\rangle$:

$$\rho_{ab} \longmapsto \rho_{ab}\langle A_b|A_a\rangle.$$

The measurement is then not fully projective. It may be better described as an unsharp measurement or a POVM, depending on how the pointer is read out. This is the operational meaning of finite resolution: if the apparatus cannot reliably separate alternatives, it cannot implement a sharp PVM.

Continuous Pointer Model

The textbook von Neumann model often uses a continuous pointer coordinate $Q$ with conjugate momentum $P_Q$ satisfying

$$[Q,P_Q]=i\hbar.$$

Let the measured system observable be

$$A=\sum_a aP_a.$$

During a short measurement interval, the interaction Hamiltonian is idealized as

$$H_{\mathrm{int}}(t) = g(t)\,A\otimes P_Q.$$

Neglect the free system and pointer evolution during the pulse (the impulsive approximation), or work in a setting where they commute with this coupling and factor out. With

$$\kappa=\int dt\,g(t),$$

Hamiltonians at different pulse times commute, so the interaction unitary is

$$U_{\mathrm{int}} = \exp\!\left( -\frac{i}{\hbar} \kappa A\otimes P_Q \right).$$

For an eigenstate $|a\rangle$ and pointer wavefunction $\phi(q)$, this unitary shifts the pointer packet:

$$|a\rangle\phi(q) \longrightarrow |a\rangle\phi(q-\kappa a).$$

The displacement $\kappa a$ has the same units as $q$. A superposition produces differently shifted pointer packets. For a real normalized Gaussian

$$\phi(q)=(2\pi\sigma^2)^{-1/4}e^{-q^2/(4\sigma^2)},$$

$\sigma$ is the standard deviation of the position probability density, not of the amplitude. Completing the square in the overlap integral gives

$$\langle A_b|A_a\rangle =\int dq\,\phi(q-\kappa b)\phi(q-\kappa a) =e^{-\kappa^2(a-b)^2/(8\sigma^2)}.$$

Thus a separation of one $\sigma$ retains a coherence factor $e^{-1/8}\simeq0.882$, while six $\sigma$ leaves $e^{-9/2}\simeq0.011$. The continuous position readout has measurement-operator density $M(q)=\phi(q-\kappa A)$ and $\int dq\,M(q)^\dagger M(q)=I$. Its probability density is $p(q)=\operatorname{Tr}[M(q)\rho M(q)^\dagger]$; it is not the probability of an exact point outcome.

Both pointer packets have standard deviation $\sigma$. Increasing their center separation $d$ makes the readout clearer and reduces the unread system coherence by $e^{-d^2/(8\sigma^2)}$. The curves are probability densities; the coherence factor is an overlap of amplitudes.

Decoherence of Pointer States

Macroscopic pointers are not isolated. They interact with many environmental degrees of freedom. A more realistic correlation chain is

$$\sum_a c_a|a\rangle|A_a\rangle|E_0\rangle \longrightarrow \sum_a c_a|a\rangle|A_a\rangle|E_a\rangle.$$

After tracing over the environment, interference between different pointer alternatives is multiplied by environmental overlaps:

$$\langle E_b|E_a\rangle.$$

For macroscopic records, these overlaps are often extremely small on practical time scales. In a model whose interaction also preserves the relevant pointer alternatives, this suppresses observable interference between records. Small overlaps alone do not prove that an arbitrary pointer basis is dynamically stable.

This does not, by itself, settle every interpretation question about definite outcomes. It explains the suppression of interference in reduced descriptions and the robustness of pointer records.

Example: Stern–Gerlach as a Pointer Model

In a Stern–Gerlach measurement of spin along $z$, the spin degree of freedom becomes correlated with spatial wavepackets. Schematically,

$$\left( \alpha|+\rangle_z + \beta|-\rangle_z \right) \phi_0(\mathbf r) \longrightarrow \alpha|+\rangle_z\phi_+(\mathbf r) + \beta|-\rangle_z\phi_-(\mathbf r).$$

The spatial packet is the pointer. If $\phi_+$ and $\phi_-$ are well separated at the detection screen, reading the spot position approximately measures $\sigma_z$. If the packets overlap substantially, the measurement is not an ideal sharp spin measurement.

Exercises

1. Suppose

$$|\psi\rangle = \alpha|0\rangle+\beta|1\rangle$$

and the apparatus unitary maps $|0\rangle|A_0\rangle$ to $|0\rangle|A_0'\rangle$ and $|1\rangle|A_0\rangle$ to $|1\rangle|A_1'\rangle$. Write the final system-apparatus state.

Solution

By linearity,

$$|\psi\rangle|A_0\rangle \longrightarrow \alpha|0\rangle|A_0'\rangle + \beta|1\rangle|A_1'\rangle.$$

For normalized input amplitudes, the state is entangled if both coefficients are nonzero and the pointer states are not proportional. Pointer states differing only by an overall phase represent the same physical ray and do not generate entanglement.

2. For the state in Exercise 1, compute the reduced system density matrix when $\langle A_1'|A_0'\rangle=\gamma$.

Solution

The reduced state is

$$\rho_S = |\alpha|^2|0\rangle\langle0| + |\beta|^2|1\rangle\langle1| + \alpha\beta^*\gamma |0\rangle\langle1| + \alpha^*\beta\gamma^* |1\rangle\langle0|.$$

If $\gamma=0$, the system is fully dephased in the $\{|0\rangle,|1\rangle\}$ basis. If $|\gamma|=1$, its magnitude is unchanged, although a complex phase of $\gamma$ rotates the coherence. For $\gamma=1$ the full original system state is recovered.

3. In the continuous pointer model, why does large separation between pointer packets make the measurement closer to projective?

Solution

Large separation makes the shifted pointer wavefunctions nearly orthogonal. Orthogonal pointer states can be reliably distinguished and suppress off-diagonal reduced-system terms after the pointer is ignored. Overlapping pointer states give ambiguous readout and leave residual coherence, so the measurement is weaker or unsharp.

4. Does the unitary premeasurement step by itself produce a selected outcome?

Solution

No. The premeasurement unitary produces a correlated superposition $\sum_a c_a|a\rangle|A_a\rangle$. A selective state is assigned only after conditioning on a pointer record. If the pointer is ignored, the system is described by a nonselective reduced state. Interpreting how a single definite outcome is selected is a foundations question beyond the unitary correlation step alone.

References

• P. Busch and P. J. Lahti, “The standard model of quantum measurement theory: History and applications,” Foundations of Physics 26, 875–893 (1996). Author preprint.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010). Publisher.
• W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Reviews of Modern Physics 75, 715–775 (2003). DOI.