Quantum no-signalling

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

Let $U={\sum }_b{U}_b^A\otimes |b⟩⟨b{|}_B\otimes V_b^C$, where the conditional target operators are unitary. For product input ${\rho }_A\otimes {\rho }_B\otimes {\rho }_C$,Tracing over $B$ removes cross terms with unequal control labels. Tracing over $A$ leaves $\mathrm{Tr}(U_b{\rho }_A{U}_b^{†})=1$, proving the formula and its independence of ${\rho }_A$. Exchanging $A$ and $C$ proves the corresponding reverse independence. Thus the two targets cannot signal to each other through this fixed common-control interaction, despite phase kickback allowing a target to affect its control.

Two controlled-NOT gates with a common control and distinct targets are an example. They commute; their chronological order can be exchanged, giving an equivalent causal proof. Intervening control rotations or additional interactions can change the conclusion.

A projective measurement on one half of a Bell state prepares the complex-conjugate basis on the other half, conditional on the outcome. Measuring Alice in the columns of $U^{\ast }$ prepares Bob in the columns of $U$. Ignoring the equally likely outcomes leaves Bob's reduced density matrix equal to $I/2$ in either basis.

Suppose two orthonormal qubit bases have disjoint physical rays. Alice chooses which conjugate basis to measure on her half of a Bell state. A device that perfectly labels Bob's individual member of either basis would reveal Alice's choice without her outcome being communicated. Spacelike operation would therefore violate relativistic causality, even though ordinary projective measurements see the same reduced density matrix $I/2$.

With a fixed reference $|0⟩$, exact overlap readout on Bob's remotely prepared state distinguishes the computational basis from the diagonal basis in one use, by the displayed disjoint squared-overlap sets. This would reveal Alice's spacelike basis choice. The argument rules out even deterministic exact overlap magnitude, avoiding the additional unphysical global-phase dependence of a raw complex inner product.

The nonselective channel of a hypothetical spacelike nonlocal measurement in quantum mechanics 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.