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For two systems of equal dimension $d\ge 2$, no joint pure state guarantees equal outcome labels whenever both systems undergo the same arbitrary basic measurement. Equal outcomes in the computational basis would force $|\Psi ⟩={\sum }_j{c}_j|jj⟩$. On any two coordinates j,k, changing both bases to the real sum/difference pair makes the different-outcome amplitude $(c_j-c_k)/2$, so all coefficients must agree. Changing both to the imaginary-phase pair makes a different-outcome amplitude $(c_j+c_k)/2$, so those coefficients must also be negatives. They must all vanish, contradicting normalization. The one-dimensional case is an exception.

The requirement of all complex bases matters. A maximally entangled vector $d^{-1/2}{\sum }_j|jj⟩$ does produce matching labels if one party uses a basis and the other its coordinatewise complex conjugate basis. Conjugate-state steering from a maximally entangled pair gives this conjugation explicitly. Identical real bases are a special case, not evidence for identical arbitrary complex bases.

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.

Measure the first qutrit in the computational basis and the second in the orthonormal basis $\{|{\varphi }_0⟩,|{\varphi }_1⟩,|{\varphi }_2⟩\}$ from Question 1(a), with those labels as outcomes. For first-party outcome 0, the unnormalized conditional vector of the second party is $|0⟩/\sqrt{3}$, so this outcome occurs with probability $1/3$ and leaves the normalized state $|0⟩$.

The second-party Born rule probabilities are then $|⟨{\varphi }_0|0⟩{|}^2=1/2$, $|⟨{\varphi }_1|0⟩{|}^2=0$, and $|⟨{\varphi }_2|0⟩{|}^2=1/2$. HenceThe relevant joint probabilities are $1/6,0,1/6$, each divided by $\mathrm{Pr}(A=0)=1/3$. This distinguishes the conditional statement from an unconditional outcome probability. More generally, conjugate-state steering from a maximally entangled pair explains how the first local measurement determines the second party's conditional vector; the basis choice here makes that vector especially simple.