4. Linear Algebra: Inner Products
Inner Products
The Inner Product is the final, essential piece of single-qubit linear algebra. It is the tool that turns abstract vectors into concrete, measurable probabilities.
1. The Bra Vector and Dirac Notation
To perform the Inner Product, we need the bra vector. The bra $\langle \phi |$ is the Hermitian conjugate (the complex conjugate transpose) of the ket $|\phi\rangle$.
If the ket for $|0\rangle$ is:
$$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$Then the corresponding bra $\langle 0 |$ is a row vector:
$$\langle 0 | = \begin{pmatrix} 1 & 0 \end{pmatrix}$$2. The Inner Product (Bra-Ket)
The Inner Product of two state vectors, $|\psi\rangle$ and $|\phi\rangle$, is the matrix multiplication of the bra $\langle \phi |$ and the ket $|\psi\rangle$, resulting in a single complex number (a scalar):
$$\langle \phi | \psi \rangle = \begin{pmatrix} \phi_0^* & \phi_1^* \end{pmatrix} \begin{pmatrix} \psi_0 \\ \psi_1 \end{pmatrix} = \phi_0^* \psi_0 + \phi_1^* \psi_1$$3. Measurement Probability 🎯
In quantum mechanics, the probability ($P$) of measuring a state $|\psi\rangle$ to be in a specific state $|\phi\rangle$ is given by the squared magnitude of their inner product:
$$P(|\psi\rangle \text{ collapses to } |\phi\rangle) = |\langle \phi | \psi \rangle|^2$$Your Task:
Consider the common superposition state $|+\rangle$:
$$|+\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}$$Calculate the probability $P$ that when we measure the state $|+\rangle$, it collapses to the $|0\rangle$ state.
- First, calculate the inner product $\langle 0 | + \rangle$.
- Then, find the probability $P = |\langle 0 | + \rangle|^2$.