= Solution
Include Alice's detector and environment before taking a <partial trace>. Ideal measurement produces the global state
$$
|\Psi_1\rangle=\frac1{\sqrt2}\bigl(|01\rangle|d_{0A}E_{0A}\rangle-|10\rangle|d_{1A}E_{1A}\rangle\bigr).
$$
<Orthogonality> of the two record states removes their cross terms from the reduced state of the particles:
$$
\boxed{\rho_{AB}^{(1)}=\frac12|01\rangle\langle01|+\frac12|10\rangle\langle10|
=\operatorname{diag}(0,1/2,1/2,0).}
$$
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 $|01\rangle$ or $|10\rangle$ 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
$$
|\Psi_2\rangle=\frac1{\sqrt2}\bigl(
|01\rangle|d_{0A}E_{0A}\rangle|d_{1B}E_{1B}\rangle
-|10\rangle|d_{1A}E_{1A}\rangle|d_{0B}E_{0B}\rangle\bigr).
$$
Tracing both laboratories gives \b[the same reduced matrix], $\rho_{AB}^{(2)}=\rho_{AB}^{(1)}$. 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 \b[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 $\mathbf a,\mathbf b$, quantum mechanics predicts $E(\mathbf a,\mathbf b)=-\mathbf a\cdot\mathbf b$. The <CHSH inequality> bounds $|S|\leq2$ for Bell-local models with measurement-independent settings. Choosing
$$
\mathbf a=\widehat z,\quad\mathbf a'=\widehat x,\quad
\mathbf b=(\widehat z+\widehat x)/\sqrt2,\quad
\mathbf b'=(\widehat z-\widehat x)/\sqrt2
$$
gives $S=E(\mathbf a,\mathbf b)+E(\mathbf a,\mathbf b')+E(\mathbf a',\mathbf b)-E(\mathbf a',\mathbf b')=-2\sqrt2$. 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.
\Image[/past-exam-of-the-quantum-information-course-of-the-university-of-cambridge/2012/paper-1-singlet-dephasing-correlation.png]
{title=Singlet coherence is lost from the reduced density matrix after a local unread measurement; common-axis anticorrelation survives}
If Alice first performs the fixed-axis irreversible measurement described above, the resulting <mixed state> has $E(\mathbf a,\mathbf b)=-a_zb_z$ 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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