6. Multi-Qubit: Tensor Products
The Tensor Product
This is the mathematical machine that allows us to scale from one qubit to a million. It is the reason quantum simulation is so hard for classical computers.
1. The Physics: Merging Universes
When you have two separate classical coins, you describe them separately: "Coin A is Heads, Coin B is Tails."
In Quantum Mechanics, when you bring two systems together, they cease to be separate mathematical entities. They merge into a single, larger state vector residing in a larger Hilbert Space.
The tool we use to glue these spaces together is the Tensor Product (denoted by the symbol $\otimes$).
If system A is in state $|\psi\rangle$ and system B is in state $|\phi\rangle$, the combined system is:
$$|\Psi_{AB}\rangle = |\psi\rangle \otimes |\phi\rangle$$2. The Math: The Kronecker Product
How do we actually calculate this? We use the Kronecker Product rule. It takes a vector of size $M$ and a vector of size $N$ and creates a new vector of size $M \times N$.
$$\begin{pmatrix} a \\ b \end{pmatrix} \otimes \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a \cdot \begin{pmatrix} c \\ d \end{pmatrix} \\ b \cdot \begin{pmatrix} c \\ d \end{pmatrix} \end{pmatrix} = \begin{pmatrix} ac \\ ad \\ bc \\ bd \end{pmatrix}$$3. The New Basis (The Computational Basis)
For two qubits, we tensor their individual basis states ($|0\rangle, |1\rangle$) to form the new 4-dimensional basis for the combined system.
- $|00\rangle = |0\rangle \otimes |0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}$
- $|01\rangle = |0\rangle \otimes |1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$
- $|10\rangle = |1\rangle \otimes |0\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}$
- $|11\rangle = |1\rangle \otimes |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$
Notice the pattern? The vector has a $1$ in the position corresponding to the binary value (00 is index 0, 11 is index 3).
Your Task: The Product State
You have two qubits.
- Qubit 1 is in state $|1\rangle$.
- Qubit 2 is in the superposition state $|+\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}$.
Calculate the state vector of the combined system $|\Psi\rangle = |1\rangle \otimes |+\rangle$.
- Set up the tensor product of the column vectors.
- Perform the multiplication to find the resulting 4-element column vector.
- Rewrite this vector in Dirac notation (e.g., $\alpha|00\rangle + \beta|01\rangle...$).
(Hint: You will see that only the bottom half of the vector is populated.)