# How do we perform a measurement of an arbitrary 1-qubit quantum state in any arbitrary orientation?

Let's imagine we have an arbitrary 1-qubit quantum system $$\alpha \vert 0 \rangle + \beta \vert 1 \rangle$$ Making a measurement in the +/- basis is equivalent to performing a Hadamard gate and then making a measurement in the standard computational basis.

Can we extend this notion generally for measurement of an arbitrary 1-qubit state in any arbitrary orientation? So, if we want to make a measurement of an arbitrary 1-qubit quantum state in some arbitrary orientation in the 2-dimensional Bloch sphere, can we perform some unitary transformation $$U$$ (supposing such $$U$$ can be composed theoretically) that transforms the basis state into orthonormal states of that orientation and then perform a measurement in the computational basis?

Let's say we have unitary transformation $$U$$ that maps $$\vert 0 \rangle$$ to $$\vert a \rangle$$ and $$\vert 1 \rangle$$ to $$\vert b \rangle$$ with $$\langle a \vert b \rangle = 0$$ (where $$\vert a \rangle$$ and $$\vert b \rangle$$ have certain orientation with respect to the standard basis states) the same way $$H$$ transforms $$\vert 0 \rangle$$ to $$\vert + \rangle$$ and $$\vert 1 \rangle$$ to $$\vert - \rangle$$ state. One difference could be that $$H$$ is Hermitian too while $$U$$ may not be. Another sub question is whether $$U$$ needs to be Hermitian also to make this generalization.

• Afaik we have no \ket, but we have | and \rangle: $|\alpha \rangle$. Jul 31 '20 at 18:33
• Thanks for the edit. Aug 1 '20 at 3:58
• @RabinsWosti No problem, and welcome to QCSE. Aug 1 '20 at 5:40
• you can use \ket, you just need to include $\newcommand{\ket}[1]{\lvert #1\rangle}$ to the beginning of the post
– glS
Aug 3 '20 at 12:07

Yes, this observation can be generalized. To start with, let's notice why is Hadamard the transformation required to measure a state $$| \psi \rangle$$ in the $$\sigma_{x}$$ basis. This is because it is the unitary intertwiner'' connecting the $$\sigma_{x}$$ basis to the $$\sigma_{z}$$ basis (a.k.a. computational basis). Recall that the $$\sigma_{x}$$ eigenvectors are $$\{ | + \rangle, | - \rangle \}$$ and the $$\sigma_{z}$$ eigenvectors are $$\{ | 0 \rangle, | 1 \rangle \}$$. The unitary operator connecting these basis is: $$\mathcal{U}_{\sigma_{x} \rightarrow \sigma_{z}} = | 0 \rangle \langle + | + | 1 \rangle \langle - | = H.$$
Let's take a moment to interpret the action of this intertwiner: when it acts on the $$| + \rangle$$ state, it sends it to the $$| 0 \rangle$$ state and when it acts on the $$| - \rangle$$ state, it sends it to the $$| 1 \rangle$$ state, thereby connecting the basis elements (and by linearity, any other vector expressed in these bases). Therefore, measuring $$| \psi \rangle$$ in the $$\sigma_{z}$$ basis is the same as applying $$\mathcal{U}_{\sigma_{x} \rightarrow \sigma_{z}}$$ and then measuring in the $$\sigma_{z}$$ basis.
For connecting to an arbitrary basis, we simply replace $$\{ | \pm \rangle \}$$ with the new basis vectors, say $$\mathbb{B} = \{ | \phi_{+} \rangle, | \phi_{-} \rangle \}$$, giving us, $$\mathcal{U}_{\mathbb{B} \rightarrow \sigma_{z}} = | 0 \rangle \langle \phi_{+} | + | 1 \rangle \langle \phi_{-} |$$
In the most general case, where you want to connect a basis $$\mathbb{B}_{0} = \{ \phi_{j} \}$$ with $$\mathbb{B}_{1} = \{ \chi_{j} \}$$, the intertwiner is defined as, $$\mathcal{U}_{\mathbb{B}_{0} \rightarrow \mathbb{B}_{1}} = \sum\limits_{j=1}^{d} | \chi_{j} \rangle \langle \phi_{j} | .$$