Past exam of the computer science course of the University of Oxford / 2019 / Quantum information / 2 / e

For every nontrivial dimension $d\ge 2$, no such pure state exists. The literal claim needs this qualification: for $d=1$, the only measurement outcome is necessarily the same for both parties. The contradiction below requires allowing all complex orthonormal bases, as quantum theory does.

If equal outcomes were certain in the computational basis, all amplitudes with different labels would vanish, so the normalized joint state would have the formChoose distinct labels $j,k$ and replace their computational vectors in both local bases by $|+{⟩}_{jk}=(|j⟩+|k⟩)/\sqrt{2}$ and $|-{⟩}_{jk}=(|j⟩-|k⟩)/\sqrt{2}$, leaving all other vectors fixed. The joint amplitude for the different outcomes $+,-$ is $(c_j-c_k)/2$. Certainty of equal outcomes would force $c_j=c_k$.

Now use the complex pair $|+i{⟩}_{jk}=(|j⟩+i|k⟩)/\sqrt{2}$ and $|-i{⟩}_{jk}=(|j⟩-i|k⟩)/\sqrt{2}$ in the same way on both systems. The amplitude for different outcomes $+i,-i$ is $(c_j+c_k)/2$, because the conjugated bra factors on the $|kk⟩$ term multiply to $(-i)(i)=1$. Thus equal outcomes would also force $c_j=-c_k$. Combining the two requirements gives $c_j=c_k=0$. Every label belongs to such a pair when $d\ge 2$, forcing the zero vector and contradicting normalization. This proves impossibility of universal identical-basis perfect correlations.

The maximally entangled state $d^{-1/2}{\sum }_j|jj⟩$ is not a counterexample: it correlates matching labels when one party uses a basis and the other its coordinatewise complex conjugate basis. Requiring the two complex bases to be identical is different. This distinction also follows from conjugate-state steering from a maximally entangled pair; restricting attention to real bases would miss the contradiction.