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 Incoming links: Improper mixture

Past exam of the quantum information course of the University of Cambridge / 2012 / Paper 1 / 2 / Solution 2026-10-20

Include Alice's detector and environment before taking a partial trace. Ideal measurement produces the global state

∣ Ψ_{1} ⟩ = \frac{1}{2} (∣01 ⟩ ∣ d_{0 A} E_{0 A} ⟩ - ∣10 ⟩ ∣ d_{1 A} E_{1 A} ⟩) .

(1)

Orthogonality of the two record states removes their cross terms from the reduced state of the particles:

ρ_{A B}^{(1)} = \frac{1}{2} ∣01 ⟩ ⟨ 01∣ + \frac{1}{2} ∣10 ⟩ ⟨ 10∣ = diag (0, 1/2, 1/2, 0) .

(2)

This nonselective quantum measurement has changed the initially entangled singlet into a separable, classically anticorrelated reduced state. Its entropy is one bit, although the full particles-plus-records state is still pure under the assumed unitary dynamics. The reduced density matrix is an improper mixture; tracing the environment does not by itself assert a literal collapse of that global state.

The Born rule gives Alice outcomes zero and one with equal probability. Conditional on a recorded result, the selective quantum measurement assigns

∣01 ⟩

∣10 ⟩

to the particles. The unitary measurement coupling and orthogonal records explain decoherence in this basis; they do not supply an additional rule selecting a unique outcome. That is part of the measurement problem. In a collapse postulate the individual outcome is stochastic; other interpretations account for outcome experience differently while retaining these operational probabilities.

When Bob makes the same kind of measurement, the global state becomes

∣ Ψ_{2} ⟩ = \frac{1}{2} (∣01 ⟩ ∣ d_{0 A} E_{0 A} ⟩ ∣ d_{1 B} E_{1 B} ⟩ - ∣10 ⟩ ∣ d_{1 A} E_{1 A} ⟩ ∣ d_{0 B} E_{0 B} ⟩) .

(3)

Tracing both laboratories gives the same reduced matrix,

ρ_{A B}^{(2)} = ρ_{A B}^{(1)}

. Bob has added a correlated record, not restored the lost reduced coherence. If Alice's outcome is known, Bob's opposite outcome is certain in this ideal common basis. Without that record, his marginal is

I /2

, so his outcome remains equiprobable. Alice cannot choose her random result or use this correlation for faster-than-light signalling, consistently with quantum no-signalling.

These common-basis measurements alone do not distinguish quantum predictions from a local source sending opposite preassigned bits. To test that explanation, prepare fresh singlets and let the observers independently choose among incompatible measurement axes before recording each result. For singlet spin measurements along unit vectors

a, b

, quantum mechanics predicts

E (a, b) = - a \cdot b

. The CHSH inequality bounds

∣ S ∣ \leq 2

for Bell-local models with measurement-independent settings. Choosing

a = z, a^{'} = x, b = (z + x) / 2, b^{'} = (z - x) / 2

(4)

gives

S = E (a, b) + E (a, b^{'}) + E (a^{'}, b) - E (a^{'}, b^{'}) = - 22

. A suitable Bell test constrains that class of explanations. Independently chosen settings, control of detection bias and separation of the measurement events matter when interpreting an actual experiment.

Figure 1.
Singlet coherence is lost from the reduced density matrix after a local unread measurement; common-axis anticorrelation survives
. **Details**.

If Alice first performs the fixed-axis irreversible measurement described above, the resulting mixed state has

E (a, b) = - a_{z} b_{z}

and cannot supply that singlet violation. The incompatible settings must belong to the Bell measurement itself. Moreover, Bell violations do not choose among all interpretations of quantum mechanics: interpretations sharing the Born predictions remain observationally equivalent for this experiment. Objective-collapse modifications could instead require additional interference tests with specified departures from unitary dynamics.

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