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A guaranteed exact overlap device would also distinguish two remotely prepared ensembles and permit a forbidden signal. Feed it a known reference $|{\psi }_1⟩=|0⟩$ and Bob's unknown state $|{\psi }_2⟩$, obtained from Alice's projective measurement on a shared Bell state. A device promised to work for arbitrary unknown inputs must work on this fixed-reference subset.

Alice chooses either the computational basis or the diagonal basis $|+⟩,|-⟩$. By the same remote preparation in conjugate bases, Bob receives a uniformly random member of the chosen basis. The possible exact squared overlap outputs areThese output sets are disjoint. Bob would infer Alice's basis choice from one run, without her communicating the random measurement result. Spacelike operation would contradict special relativity. Thus exact pure-state overlap readout would allow superluminal signalling, and no guaranteed exact overlap device is compatible with the stated assumptions.

This argument rules out even the phase-invariant magnitude squared. The raw complex inner product has an additional issue: replacing either pure state representative by a global phase changes that complex number without changing the physical input. The no-signalling contradiction therefore remains valid even if some phase convention is supplied. Statistical estimation from many identically prepared copies is not the promised exact single-pair readout.

Share the Bell state $|{\Phi }_{+}⟩$ between Alice and Bob, and allow their actions to be spacelike separated. The projective measurement postulate gives the remote preparation in conjugate bases identityIf Alice measures in the computational basis, Bob is prepared in $|0⟩$ or $|1⟩$, each with probability one half. If she instead measures in the conjugate basis $U^{\ast }|0⟩,U^{\ast }|1⟩$, Bob is prepared in $U|0⟩$ or $U|1⟩$, again equally likely. The conjugation is essential for a general complex unitary operator.

Interpreting distinct states as distinct physical rays, the two candidate pairs are disjoint. A hypothetical perfect identifier would tell Bob which member he received and therefore which basis Alice chose. Alice could transmit a bit by that basis choice, without sending her outcome or any other message. Bob could read it while spacelike separated from Alice, violating the no-superluminal-signalling requirement of special relativity. This establishes that perfect two-basis discrimination would allow superluminal signalling and rules out the device using the requested physical assumptions.

Ordinary measurement in quantum measurements do not signal: the two equally weighted ensembles have identical reduced density matrixThe contradiction arises from the hypothesized ability to label the ensemble's individual pure members. Global-phase variants of a vector do not count as distinct physical pure states; treating them as distinguishable labels would already introduce an unphysical extra readout.