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Suppose I have the m-dimensional unitary B to prepare the ancillary state:

\begin{align} B|0\rangle=\frac{1}{\sqrt s}\sum_{j=0}^{m-1}\sqrt{\beta_j}|j\rangle, \text{ where } s\equiv\sum_{j=0}^{m-1}\beta_j \end{align}

Suppose there's some $|\Phi\rangle$ whose ancillary state is supported in the subspace orthogonal to $|0\rangle$, I'm wondering is there a way I can infer/calculate the representation of $B|\Phi\rangle$? Do $B|0\rangle$ orthogonal to $B|\Phi\rangle$? Thanks!!

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is there a way I can infer/calculate the representation of 𝐵|Φ⟩?

Not in general without more information.

Do 𝐵|0⟩ orthogonal to 𝐵|Φ⟩?

Yes, since unitary transformations preserve the inner product between two states.

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  • $\begingroup$ Thanks for the answer! Which kind of information is needed to figure out $B|\Phi\rangle$? $\endgroup$
    – ZR-
    Commented Jan 16, 2021 at 20:21
  • $\begingroup$ What you are basically asking is, "can I infer all the components of a unitary matrix given one row of that matrix?" A unitary matrix from the group $U(N)$ (the group of all $N\times N$ unitary matrices) has $N^2$ real parameters. A row vector has $2N$ parameters (since the elements can be complex). So you need $N^2-2N$ additional independent constraints to fix the matrix. $\endgroup$
    – Andrew
    Commented Jan 16, 2021 at 20:27
  • $\begingroup$ One column. (And to get B|phi>, only one additional column is needed.) $\endgroup$
    – Norbert Schuch
    Commented Jan 16, 2021 at 21:43
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In general, there is not enough information to construct $B$, you really only know the first column of its matrix representation in the computational basis.

If $\langle\Phi|0\rangle = 0$, then $\langle\Phi|\underbrace{B^\dagger B}_{=1} |0\rangle = \langle\Phi|0\rangle = 0$, so the orthogonality is preserved after applying $B$.

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