Qubits are usually measured in the computational basis, but we can change the basis by a unitary $U$ to measure in the basis formed by the columns of $U$.

For example, if $| \psi \rangle = | 0 \rangle$, we can convert $| \psi \rangle$ to the basis $U$:

$U | \psi \rangle = \frac{1}{\sqrt{2}} | a \rangle + \frac{1}{\sqrt{2}} | a_\perp \rangle$.

In this case, our basis is $\{ | + \rangle, | - \rangle \}$. But what if we entangle two qubits:

$ | \psi \rangle = | + \rangle$, $ | \phi \rangle = | 0 \rangle$

$$ \texttt{CNOT} | \psi \rangle | \phi \rangle = \frac{1}{\sqrt{2}}\texttt{CNOT} | 00 \rangle + \frac{1}{\sqrt{2}}\texttt{CNOT} | 10 \rangle = \frac{1}{\sqrt{2}} | 00 \rangle + \frac{1}{\sqrt{2}} | 11 \rangle $$

And of course, this state is entangled. Now, we send the first qubit (originally storing the state $| \psi \rangle$) to another party. This party wants to measure $\psi$ in a different basis, $X$. How can we represent this basis change mathematically? We can't write an ket for the first qubit anymore, as the state is non-separable.

My assumption is that we can represent the other party measuring through the basis $X$ as the unitary transformation: $X \otimes I$, where $I$ is the identity matrix. But I would like some confirmation if this is correct; and if it is, is there some physical interpretation?



1 Answer 1


Yes, and no!

If you have some basis change unitary $U$ for a single qubit, then the person holding the first qubit can perform that unitary on their qubit, using the mathematical description $U\otimes I$. If the person were holding the second qubit, they would describe it as $I\otimes U$. So, that's the "yes" part!

The "no" part is what the unitary looks like. If you want to measure in the $X$ basis, you need a unitary that transforms $X$ into $Z$ (the standard measurement basis), i.e. $U^\dagger XU=Z$. These things are easy enough to write down: you have $U^\dagger |+\rangle=|0\rangle$ and $U^\dagger|-\rangle=|1\rangle$, but you might happen to already know that Hadamard does this job.


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