Quantum no-signalling

An uncommunicated local trace-preserving operation cannot change the other subsystem's reduced state.

The nonselective channel of a hypothetical spacelike nonlocal quantum measurement must not allow a distant input change to alter local output statistics. For a rank-one measurement in the rotated even and odd two-qubit bases with angle $\theta$, Alice's output satisfiesBob can change the last correlation using only his local Pauli Z gate while Alice keeps the input $|+⟩$. The resulting signal excludes $0<\theta <\pi /4$. Degenerate observables that do not distinguish this basis are not covered by the rank-one argument. Locally recording a jointly encoded result, which is decoded later by local operations and classical communication, is compatible with quantum no-signalling.

A rank-one projective measurement whose Bob basis is computational when Alice is zero and rotated by $\theta$ when Alice is one can signal. Bob starts in zero. Alice zero leaves Bob zero; Alice one dephases Bob in the rotated basis, giving the displayed nonzero flip probability for $0<\theta \le \pi /4$. This violates quantum no-signalling. The conclusion applies to a measurement distinguishing the four product-basis states, rather than a degenerate observable that does not resolve them.

The binary Lüders rule measurement of singlet versus triplet preserves the entire triplet subspace, but its nonselective channel permits signalling. Starting with $|00⟩$, Bob retains $|0⟩$. If Alice instead applies a local Pauli X gate, the input is $|10⟩$ and the channel outputs the equal mixture of $|{\Psi }^{+}⟩$ and $|{\Psi }^{-}⟩$; Bob has reduced density matrix $I/2$. Thus no spacelike local implementation can realize this channel. A Bell-state nondemolition measurement has a finer triplet readout and a different nonselective channel, so this obstruction does not exclude it.