Dirac Notation for Qubit States

The Language of Quantum States: Kets, Bras, and Hilbert Space

The notation we've been using, $|\psi ⟩$, is called a ket. It represents the quantum state of a system as a vector in an abstract mathematical space called a Hilbert space. For our purposes, and given your background in linear algebra, you can think of a ket as a column vector.

A key benefit of this abstract notation is that it separates the concept of the state itself from its specific numerical representation, much like a geometric vector exists independently of the coordinate system (e.g., Cartesian, polar) we use to describe its components.

To explore this idea further, the following video provides an excellent conceptual introduction to Dirac notation, explaining why this abstract approach is so powerful.

Every ket has a corresponding bra. If a ket $|\psi ⟩$ is a column vector, its corresponding bra, written as $⟨\psi |$, is its Hermitian adjoint (or conjugate transpose). This means you take the transpose of the column vector to get a row vector, and then take the complex conjugate of each element.

The names "bra" and "ket" come from how they combine. When you place a bra and a ket together, you form a "bra-ket" or bracket, like $⟨\varphi |\psi ⟩$. This operation is the inner product.

Let's solidify these definitions with a clear, practical guide.

Representing a Qubit with Dirac Notation

Now, let's apply this formalism to the single qubit. A single qubit's state space is a 2-dimensional complex Hilbert space. The two computational basis states, which correspond to the classical bits 0 and 1, are represented by orthonormal basis vectors.

• The ket $|0⟩$ is the column vector: $$|0⟩=\left(\begin{array}{c}1\\ 0\end{array}\right)$$
• The ket $|1⟩$ is the column vector: $$|1⟩=\left(\begin{array}{c}0\\ 1\end{array}\right)$$

The corresponding bras are the conjugate transposes of these kets. Since the elements are real, this is simply the transpose:

• The bra $⟨0|$ is the row vector: $$⟨0|=\left(\begin{array}{cc}1& 0\end{array}\right)$$
• The bra $⟨1|$ is the row vector: $$⟨1|=\left(\begin{array}{cc}0& 1\end{array}\right)$$

A general qubit state $|\psi ⟩$ in superposition is a linear combination of these basis kets, as we saw in the previous lesson:

where $\alpha$ and $\beta$ are complex numbers satisfying the normalization condition $|\alpha {|}^2+|\beta {|}^2=1$.

Using the vector representations, we can see this is just vector addition:

This explicitly connects the Dirac notation to the column vector representation from our last lesson.

The following video demonstrates this conversion from the matrix (column vector) form to the more conventional Dirac notation.

The Inner Product: Extracting Amplitudes and Probabilities

The inner product is where Dirac notation truly shines. As mentioned, it takes a bra and a ket to produce a single complex number. In linear algebra terms, this is matrix multiplication of a row vector with a column vector (i.e., a dot product).

Let's consider two kets, $|\varphi ⟩=\left(\begin{array}{c}a\\ b\end{array}\right)$ and $|\psi ⟩=\left(\begin{array}{c}c\\ d\end{array}\right)$. The bra $⟨\varphi |$ is $\left(\begin{array}{cc}{a}^{\ast }& b^{\ast }\end{array}\right)$, where ${}^{\ast }$ denotes the complex conjugate.

Their inner product $⟨\varphi |\psi ⟩$ is:

A crucial property of the computational basis is that it is orthonormal.

• Orthogonal: The inner product of different basis vectors is zero. $$⟨0|1⟩=\left(\begin{array}{cc}1& 0\end{array}\right)\left(\begin{array}{c}0\\ 1\end{array}\right)=(1)(0)+(0)(1)=0$$ Physically, this means the states $|0⟩$ and $|1⟩$ are perfectly distinguishable.
• Normalized: The inner product of a basis vector with itself is one. $$⟨0|0⟩=\left(\begin{array}{cc}1& 0\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)=(1)(1)+(0)(0)=1$$ $$⟨1|1⟩=\left(\begin{array}{cc}0& 1\end{array}\right)\left(\begin{array}{c}0\\ 1\end{array}\right)=(0)(0)+(1)(1)=1$$

This orthonormality provides a powerful way to find the amplitudes $\alpha$ and $\beta$ for a given state $|\psi ⟩=\alpha |0⟩+\beta |1⟩$. If we want to find $\alpha$, we take the inner product of $|\psi ⟩$ with $⟨0|$:

Similarly, to find $\beta$, we take the inner product with $⟨1|$:

This confirms that the amplitude $\alpha$ is the projection of the state vector $|\psi ⟩$ onto the $|0⟩$ basis vector, and $\beta$ is the projection onto the $|1⟩$ basis vector. This neatly formalizes the Born rule from our last lesson: the probability of measuring the state $|0⟩$ is $|⟨0|\psi ⟩{|}^2=|\alpha {|}^2$.

The following reading provides concrete examples of these calculations.

Conclusion

In this lesson, we have formalized the mathematical language used to describe quantum states. This notation, while abstract, provides a compact and powerful framework that you will use throughout your study of quantum computing.

Here are the key takeaways:

• Dirac notation represents quantum states as kets (e.g., $|\psi ⟩$), which are column vectors in a Hilbert space. The corresponding bras (e.g., $⟨\psi |$) are their conjugate transposes (row vectors).
• A single qubit state is written as a linear combination of basis kets: $|\psi ⟩=\alpha |0⟩+\beta |1⟩$, where $|0⟩=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and $|1⟩=\left(\begin{array}{c}0\\ 1\end{array}\right)$.
• The inner product $⟨\varphi |\psi ⟩$ projects the state $|\psi ⟩$ onto the state $|\varphi ⟩$, yielding a complex amplitude.
• The amplitudes of a state $|\psi ⟩$ in the computational basis are found using inner products: $\alpha =⟨0|\psi ⟩$ and $\beta =⟨1|\psi ⟩$. The measurement probabilities are $|\alpha {|}^2$ and $|\beta {|}^2$.

So far, we have primarily focused on the algebraic representation of a qubit. In our next lesson, we will introduce a beautiful geometric tool for visualizing the state of a single qubit: the Bloch sphere. This will give you a powerful visual intuition for how qubit states behave and transform.