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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.

Solved by gpt-5.6-sol high.