# Correct way of expressing a measurement in a different computational basis

Sometimes we find that the result we want from a quantum algorithm is expressed in terms of a basis that is different from the usual computational basis, which I will call

$$B_C = \left\{ \lvert 0 \rangle, \lvert 1 \rangle \right\} = \left\{ \left(\begin{array}{c}1\\0\end{array}\right),\left(\begin{array}{c}0\\1\end{array}\right) \right\}.$$

For example, at the end of the Deutsch's algorithm, the first qubits is in the state

$$\lvert\psi\rangle = (-1)^{f(0)} \dfrac{\lvert 0 \rangle + (-1)^{f(0) \oplus f(1)} \lvert 1 \rangle}{\sqrt{2}},$$

which can be expressed in terms of the Hadamard's basis

$$B_H = \{ \lvert + \rangle, \lvert - \rangle \} = \left\{ \dfrac{1}{\sqrt{2}} \left( \begin{array}{c}1\\1\end{array} \right), \dfrac{1}{\sqrt{2}} \left( \begin{array}{c}1\\-1\end{array} \right) \right\}$$

$$\begin{cases} f(0) \oplus f(1) = 0 \quad\Rightarrow\quad \lvert\psi\rangle = (-1)^{f(0)} \lvert + \rangle \\ f(0) \oplus f(1) = 1 \quad\Rightarrow\quad \lvert\psi\rangle = (-1)^{f(0)} \lvert - \rangle \end{cases}$$

Therefore, we can get the value of $$f(0) \oplus f(1)$$ just by measuring $$\lvert\psi\rangle$$ on $$B_H$$. Since we can only perform physical measurements on $$B_C$$, this can be achieved performing a change of basis.

Given two basis $$A$$ and $$B$$, if the matrix that transforms the elements of $$A$$ to the elements of $$B$$ is the matrix $$M$$, this is also the matrix that maps the coordinates of vectors with respect to $$B$$ to their coordinates with respect to $$A$$. Hence, which of these propositions is the correct to express the change of basis we must do to perform the measurement on $$B_H$$ knowing that physical measurements are actually performed in $$B_C$$?

1. We must apply the Hadamard's gate, because the matrix $$H$$ maps $$\left\{ \lvert 0 \rangle,\lvert 1 \rangle \right\}$$ to $$\left\{ \lvert + \rangle, \lvert - \rangle \right\}$$, and therefore it transforms the components $$\lvert\psi\rangle_{B_H}$$ to $$\lvert\psi\rangle_{B_C}$$.

2. We must apply the Hadamard's gate, because the matrix $$H$$ maps $$\left\{ \lvert + \rangle,\lvert - \rangle \right\}$$ to $$\left\{ \lvert 0 \rangle, \lvert 1 \rangle \right\}$$, and therefore it transforms the components $$\lvert\psi\rangle_{B_C}$$ to $$\lvert\psi\rangle_{B_H}$$.

• This is boiling down to a confusion about when to use $U$ vs $U^{-1}$. You should give an example where $U \neq U^{-1}$ in order to get a case where you can actually tell the difference. Apr 17 '19 at 21:01
• That example would actually be the answer to this question, so if I had it, I wouldn't have come here Apr 18 '19 at 8:32

Say you have two bases $$B_1$$ and $$B_2$$. Let $$U$$ be the unitary that transforms the orthonormal basis vectors in $$B_1$$ to the basis vectors in $$B_2$$. So, it is obvious that $$U^{\dagger}$$ will be the unitary that transforms the basis vectors in $$B_2$$ to that in $$B_1$$. For instance let $$B_1 = \{|0\rangle, |1\rangle\}$$ (the computational basis) and let $$B_2 = \{|\uparrow\rangle, |\downarrow\rangle\}$$ where $$|\uparrow\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ i\end{pmatrix}$$ and $$|\downarrow\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -i\end{pmatrix}$$. Notice that the matrix $$U = \frac{1}{\sqrt{2}}\begin{bmatrix}1& 1\\ i & -i\end{bmatrix}$$ is the matrix that transforms the basis vectors in $$B_1$$ to that in $$B_2$$, i.e., $$U|0\rangle = |\uparrow\rangle$$ and $$U|1\rangle = |\downarrow\rangle$$. So we also have $$U^{\dagger}|\uparrow\rangle = |0\rangle$$ and $$U^{\dagger}|\downarrow\rangle = |1\rangle$$. Also notice that $$U\neq U^{\dagger}$$.
For notational convenience, let $$|0\rangle = \begin{pmatrix}1\\ 0\end{pmatrix}_{B_1} ~~\text{ and }~~~~ |1\rangle = \begin{pmatrix}0\\ 1\end{pmatrix}_{B_1}.$$ Also let, $$|\uparrow\rangle = \begin{pmatrix}1\\ 0\end{pmatrix}_{B_2} ~~\text{ and }~~~~ |\downarrow\rangle = \begin{pmatrix}0\\ 1\end{pmatrix}_{B_2}.$$ Now, if you have a state $$|\psi\rangle$$ in basis $$B_1$$, say $$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$$ we can represent this as $$|\psi\rangle = \begin{pmatrix}\alpha \\ \beta\end{pmatrix}_{B_1}$$. Now notice that the state $$|\psi\rangle$$ can also be given as $$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle = \alpha U^{\dagger}|\uparrow\rangle + \beta U^{\dagger}|\downarrow\rangle = U^{\dagger}(\alpha|\uparrow\rangle + \beta|\downarrow\rangle).$$ This in vector notation can be given as $$|\psi\rangle = U^{\dagger} \begin{pmatrix}\alpha\\ \beta\end{pmatrix}_{B_1} = \begin{pmatrix}\alpha'\\ \beta'\end{pmatrix}_{B_2}.$$
So for a state that is given in basis $$B_1$$, to represent the state in $$B_2$$ basis, we apply $$U^{\dagger}$$ on the state vector corresponding to the basis $$B_1$$. Similarly for a state that is given in basis $$B_2$$, to represent the state in $$B_1$$ basis, we apply $$U$$ on the state vector corresponding to the basis $$B_2$$ where $$U$$ is the matrix that transforms the basis vectors of basis $$B_1$$ to the basis vectors of the basis $$B_2$$.
If you want to measure $$|\phi\rangle$$ in some basis $$U|b_1\rangle,...,U|b_n\rangle$$ instead of $$|b_1\rangle,...,|b_n\rangle$$, then you need to rotate the state "backward", i.e. measure $$U^{-1}|\phi\rangle$$ in $$|b_1\rangle,...,|b_n\rangle$$.
The simple rule to find the direction of rotation is to consider the state and the required measurement basis together $$\{|\phi\rangle, U|b_1\rangle,...,U|b_n\rangle\}$$. Then If you what to perform the same measurement in a different basis you need to rotate the whole system altogether.