Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 325 / 4

Assume an objective-collapse theory in which collapses occur at localized spacetime events. At an event $x$, define the local quantum state under objective collapse by taking the reduced density operator on spacelike hypersurfaces that approach the past light cone of $x$. A quantum state readout device at $x$ prints a complete classical description of this local density operator, in a fixed basis and here with infinite precision, without disturbing it.

For a localized pure state whose relevant preparation lies in the past light cone, the output is its rank-one projector $|\psi ⟩⟨\psi |$. For an improper mixed state produced by entanglement with an uncollapsed exterior system, it is the reduced density matrix. For a proper mixed state, once the selecting preparation or collapse event is in the past light cone, it is the actually selected component; before that causal information arrives, it is the corresponding local mixture.

A spacelike-separated collapse is excluded from the hypersurface limit and cannot change the output. Its effect appears only on and inside its future light cone. The definition therefore prevents superluminal signalling while allowing light-speed changes in the readout.

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One obtains a logically consistent extension by specifying three rules: ordinary quantum evolution and localized objective collapses determine the global state; the past-light-cone prescription determines a unique local density operator at every readout event; and a readout copies a classical description of that operator into a classical register without changing the quantum state. These rules assign an output to every finite-dimensional input and every spacetime history.

The device is outside the class of ordinary quantum measurements and permits tasks such as learning and then preparing a copy of an unknown pure state, but violating the operational no-cloning theorem for two pure states does not make the enlarged axioms contradictory. Feed-forward operations can be defined from earlier readouts in the same causal region. Because every output depends only on collapse events in its past light cone, no closed causal dependence or spacelike signal is introduced. The extension is therefore mathematically coherent, although it is hypothetical and goes beyond standard quantum theory.

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Let $z_A,z_B\in \{+1,-1\}$ denote the two $Z$ outcomes and $x_C\in \{+1,-1\}$ the $X$ outcome. Measuring $Z_A$ in the GHZ state projects $B$ onto the same $Z$ eigenstate, while the later $X_C$ outcome is unbiased. Hence the possible triples, in the order $(z_A,x_C,z_B)$, areeach with probability $1/4$.

Choose $(z_A,x_C,z_B)=(+,+,+)$. Immediately before $t=0$, each local readout is $I/2$. The collapse at $(t,x)=(0,0)$ is immediately in $A$'s past light cone, reaches $B$ at $t=1$, and reaches $C$ at $t=2$. ThusHere $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. The $X_C$ collapse at $(3,2)$ reaches $B$ at $t=4$ and $A$ at $t=5$, but the state is already a product after the first collapse, so it does not change their local states. Likewise, the $Z_B$ collapse at $(5,1)$ reaches $A$ and $C$ at $t=6$ without changing their states. For another allowed outcome, replace $|0⟩$ by $|1⟩$ according to $z_A=z_B$ and replace $|+⟩$ by $|-⟩$ according to $x_C$.

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