A trace-preserving operation on subsystem cannot change the reduced density matrix of a disjoint subsystem when its unobserved outcomes are averaged. Local measurements on therefore cannot reveal which operation was chosen on .
For local Kraus operators with , let . For every remote operator , cyclicity of the trace gives . Hence the remote reduced density matrix is unchanged. In particular every remote POVM has the same outcome probabilities, proving the no-communication theorem from the measurement rule rather than assuming it as a physical postulate.
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The No-communication theorem is a concept in quantum mechanics that pertains to the behavior of entangled particles. It states that quantum entanglement cannot be used to transmit information or communicate faster than the speed of light, even though the measurement of one entangled particle can instantaneously affect the state of another, distant entangled particle.