 Solution

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.

Solved by gpt-5.6-sol high.