# Probability of measuring the first qubit in the state $\frac{1}{\sqrt 2}(|0⟩+|1⟩)$ in a two-qubit state

If I consider $$a|00\rangle + b|01\rangle +c|10\rangle +d|11\rangle$$ as a valid two-qubit system, what is the probability of measuring the first qubit in state $$\frac{1}{\sqrt 2}(|0⟩+|1⟩)$$?

I know that the probability of measuring the first qubit in state $$|0\rangle$$ is $$|a|^2 + |b|^2$$, but I couldn't figure out how it works here. Thanks in advance.

• Hint: You need to perform a basis change from $\{|0\rangle, |1\rangle\}$ to $\{|+\rangle, |-\rangle\}$ where $|+\rangle$ is $\frac{1}{\sqrt 2}(|0\rangle + |1\rangle)$ and $|-\rangle$ is $\frac{1}{\sqrt 2}(|0\rangle - |1\rangle)$. If this terminology is unclear to you, it's advisable that you learn some linear algebra first. Nov 21 '19 at 16:40

Given an arbitrary state $$|\psi\rangle$$, if it is expressed in the computational basis as $$|\psi\rangle=\sum_k c_k |k\rangle$$, then it will give the $$k$$-th result (when measuring in the computational basis) with probability $$|c_k|^2$$.

Note that here by "computational basis" I simply mean the measurement basis under consideration.

If you consider another type of measurement, corresponding to a different basis, say $$\{|u_k\rangle\}_k$$, then to figure out the outcome probabilities in this new basis you need to express $$|\psi\rangle$$ in terms of the $$|u_k\rangle$$. Say that doing this you get something of the form $$|\psi\rangle = \sum_k d_k |u_k\rangle.$$ That means that the outcome corresponding to $$|u_k\rangle$$ is obtained with probability $$|d_k|^2$$.

If you start with a description of the state in the $$|k\rangle$$ basis, and want to switch to a description in terms of the $$|u_k\rangle$$ one, you simply need to compute the inner products $$\langle u_k|\psi\rangle=d_k$$.

More generally, given a state $$|\psi\rangle$$, the probability of finding it in a state $$|\phi\rangle$$ is given by $$|\langle\phi|\psi\rangle|^2$$.

There is, however, a slightly more general way to do measurements, that doesn't (necessarily) involve a full collapse of the state. You can ask questions of the form "is the state in a given subspace?". For example, you can ask whether a state is in one of first $$3$$ (or any other subset of) computational basis states. You model this situation by using a projector $$P$$ that projects onto the required basis, and the associated measurement probability is then given by $$\|P|\psi\rangle\|^2$$.

Taking as an example your specific case, you have a bipartite state expressed in terms of the computational basis $$\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$$ with coefficients $$a,b,c,d$$. Let's call this state $$|\psi\rangle$$. You are asking about the probability of the state being found in the subspace $$\{|+,0\rangle, |+,1\rangle\}$$. To compute this probability you therefore write down the projector $$P$$ defined as $$P = |+,0\rangle\!\langle +,0| + |+,1\rangle\!\langle +,1| \equiv |+\rangle\!\langle +| \otimes I_2,$$ and you compute $$\|P|\psi\rangle\|^2$$. Equivalently, you just rewrite $$|\psi\rangle$$ expressing the first qubit in the $$|\pm\rangle$$ basis, and then sum the probabilities of finding it in either $$|+,u\rangle$$ or $$|+,v\rangle$$, with $$\{|u\rangle,|v\rangle\}$$ the basis chosen for the second qubit (it doesn't matter which one is used).

• A hyperlink seems to be missing in the first sentence... Nov 22 '19 at 12:59
• @SanchayanDutta Fixed, thanks! Nov 22 '19 at 14:21