So far our probabilistic and quantum systems consist of only one single distribution/state. In the real world, quantum computers often have several different registers (variables).
In theory, we could use a single very big distribution/state to model multiple qubits. For example a 10 qubit system could be modeled with the classical possibilities \(0000000000, 0000000001, \dots, 1111111110, 1111111111\).
Unfortunately the vector for these states gets really big, for 10 qubits, the vector would have the dimension of \(1024\). Since this is very inconvenient to write down, we need to look at a different solution. For this, we compose different probabilistic or quantum systems with each other.
Constructing composite systems
Definition 6.1 (Composite systems / Tensor product) Given two probabilistic or quantum systems \(A\) and \(B\) with the possibilities of \(A\) given by \(x_1, \dots, x_N\) and a distribution/state \(\mu_A\) and with the possibilities of \(B\) given by \(y_1, \dots, y_M\) and a distribution/state \(\mu_B\), the composite system called \(AB\) has the possibilities \[
x_1y_1, x_1y_2, \dots, x_1y_M, x_2y_1, x_2y_2, \dots, x_2y_M,\dots, x_Ny_1, x_Ny_2, \dots, x_Ny_M
\] and the distribution/state \(\mu_{AB}\) of \(AB\) is given by the tensor product \[
\mu_{AB}
\coloneqq \mu_A \otimes \mu_B
= \begin{pmatrix} (\mu_A)_1 \cdot \mu_b \\ \vdots \\ (\mu_A)_N \cdot \mu_b \end{pmatrix}.
\] This vector has the size \(NM\). Here \((\mu_A)_i\) stands for the i-th entry of \(\mu_A\).
The definition of combining a probabilistic and a quantum system are the same.
Notice that the entry corresponding to the possibility \(x_i y_j\) in the composite system is then \((\mu_{AB})_{x_i y_j} = (\mu_A)_{x_i} (\mu_B)_{y_j}\). Here we identify the classical possibility \(x_i\) and \(y_j\) with the indices \(1, \dots, N\) and \(1, \dots, M\), respectively the classical possibility \(x_i y_j\) with the indices \(1, \dots, NM\). (So \((\mu_{AB})_{x_i y_j}\) has just one index, namely \(x_i y_j \in \{1, \dots NM \}\).)
Let the distributions for the system \(A\) with the possibilities \(1,2\) and the system \(B\) with the possibilities \(a,b,c\) be given by \[
\begin{aligned}
\mu_A &= \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix},&
\mu_B &= \begin{pmatrix} \frac{1}{2} \\ 0 \\ \frac{\sqrt{3}}{2} \end{pmatrix}.
\end{aligned}
\] Then the composite system \(AB\) has the possibilities \(1a, 1b, 1c, 2a, 2b, 2c\) and the distribution \[
\mu_{AB}
= \mu_A \otimes \mu_B
= \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix} \otimes \begin{pmatrix} \frac{1}{2} \\ 0 \\ \frac{\sqrt{3}}{2} \end{pmatrix}
= \begin{pmatrix} \frac{1}{\sqrt{2}} \cdot \frac{1}{2} \\[3pt] \frac{1}{\sqrt{2}} \cdot 0 \\[3pt] \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} \\[3pt] -\frac{1}{\sqrt{2}} \cdot \frac{1}{2} \\[3pt] -\frac{1}{\sqrt{2}} \cdot 0 \\[3pt] -\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} \end{pmatrix}.
= \begin{pmatrix} \frac{1}{2\sqrt{2}} \\[3pt] 0 \\[3pt] \frac{\sqrt{3}}{2\sqrt{2}} \\[3pt] -\frac{1}{2\sqrt{2}} \\[3pt] 0 \\[3pt] -\frac{\sqrt{3}}{2\sqrt{2}} \end{pmatrix}.
\]
We now need a way to apply operators on these combined systems. For this we can also construct the tensor product of either two probabilistic processes or two unitary transformations by using the tensor product of two matrices.
Definition 6.2 (Composite matrices / Tensor product) Given two matrices \(S\) and \(T\) with \(S\) of the size \(N \times N\) and \(T\) of the size \(M \times M\). The tensor product \(S \otimes T\) of is given by \[
S \otimes T
= \begin{pmatrix}
S_{11}T & \dots & S_{1N} T\\
\vdots & \ddots & \vdots\\
S_{N1} T & \dots & S_{NN}T
\end{pmatrix}.
\]
Overall we can say: If we apply \(S\) to the system \(A\) and \(T\) to the system \(B\), we apply \(S\otimes T\) to the composite system \(AB\).
If the distribution \(d_{AB}\) of a given probabilistic system \(AB\) can be written as a composite of two distributions \(d_A\) and \(d_B\), we know that \(A\) and \(B\) are independent of each other. If we cannot write \(d_{AB}\) as two separate distributions, the probabilities are depended on each other.
If the quantum state \(\psi_{AB}\) of a given quantum system \(AB\) can be written as a composite of two different quantum states \(\psi_A\) and \(\psi_B\), the quantum states of \(A\) and \(B\) are independent of each other. If we can not write \(\psi_{AB}\) as a tensor product of two quantum systems, the quantum states depend on each other. We call this entangled.
Lemma 6.1 For the (unitary) matrices \(A, B, C\) and \(D\), the vectors (quantum states) \(\psi, \phi\) and \(\chi\) and the constant \(c\), the following applies
- \((A \otimes B)(\psi \otimes \phi) = A\psi \otimes B\phi\),
- \((A \otimes B)(C \otimes D) = AC \otimes BD\),
- \(\psi \otimes (\phi + \chi) = (\psi \otimes \phi) + (\phi \otimes \chi)\),
- \((\psi + \phi) \otimes \chi = (\psi \otimes \chi) + (\phi \otimes \chi)\),
- \(A \otimes (B + C) = (A \otimes B) + (A \otimes C)\),
- \((A + B) \otimes C = (A \otimes C) + (B \otimes C)\),
- \(c\phi \otimes \psi = c(\phi \otimes \psi)\),
- \(\phi \otimes c\psi = c(\phi \otimes \psi)\),
- \(A \otimes cB = c(A \otimes B)\) and
- \(cA \otimes B = c(A \otimes B)\).
These rules only apply if the dimensions of the matrices and vectors match.
Measuring composite systems
To perform a (partial) observation or (partial) measurement on a composite system \(AB\), we can compose two separate measurements on the systems \(A\) and \(B\) similar as we constructed the tensor product.
Definition 6.3 (Composite measurements) Given two systems \(A\) and \(B\) with possibilities \(1,\dots, N\) and \(1,\dots, M\) and two partial measurements \(M_A\) and \(M_B\) on systems \(A\) and \(B\) with alternatives \(A_1, \dots, A_N \subseteq \{x_1, \dots, x_n\}\) and \(B_1, \dots, B_M \subseteq \{y_1, \dots, y_M\}\). The measurement \(M_A \otimes M_B\) on \(AB\) is a measurement with the alternatives \(C_{11},C_{12},\dots,C_{NM}\) where \(C_{ij} = A_i \times B_j\).
If we only have a set of alternatives for system \(A\), we can do a measurement \(M_A \otimes I\) with alternatives \(C_1, \dots ,C_N \coloneqq A \otimes \{y_1, \dots, y_M\}\).
Let \(A\) be a system with the states \(1,2,3\) and \(\mu_A\) a measurement with the two alternatives \(A_{\text{low}} = \{1,2\}\), \(A_{\text{high}} = \{3\}\). The quantum state is \[
\psi_A = \begin{pmatrix} \frac{2}{3} \\[2pt] \frac{1}{3} \\[2pt] -\frac{2}{3} \end{pmatrix}.
\]
Another system \(B\) has the states \(a, b, c\). The measurement \(\mu_B\) the two alternatives \(B_{\text{vocal}} = \{a\}\), \(B_{\text{consonant}} = \{b, c\}\). The quantum state is \[
\psi_B = \begin{pmatrix} \frac{1}{2} \\[2pt] \frac{1}{2}i \\[2pt] \frac{1}{\sqrt{2}} \end{pmatrix}.
\]
So the composite system \(C = AB\) has the classical possibilities \(1a, 1b, 1c, 2a, 2b, 2c, 3a, 3b\) and \(3c\). The measurement \(\mu_C \coloneqq \mu_A \otimes \mu_B\) has the alternatives \[
\begin{aligned}
C_{\text{low, vocal}} &= \{1a, 2a\}, && C_{\text{low, consonant}} &= \{1b, 1c, 2b, 2c\}, \\
C_{\text{high, vocal}} &= \{3a\}, && C_{\text{high, consonant}} &= \{3b, 3c\}.
\end{aligned}
\]
The quantum state is \[
\psi_C
= \psi_A \otimes \psi_B
= \begin{pmatrix} \frac{2}{3} \\[2pt] \frac{1}{3} \\[2pt] -\frac{2}{3} \end{pmatrix} \otimes \begin{pmatrix} \frac{1}{2} \\[2pt] \frac{1}{2}i \\[2pt] \frac{1}{\sqrt{2}} \end{pmatrix}
= \begin{pmatrix} \frac{2}{6} \\[4pt] \frac{2}{6}i \\[4pt] \frac{2}{3\sqrt{2}} \\[4pt] \frac{1}{6} \\[4pt] \frac{1}{6}i \\[4pt] \frac{1}{3\sqrt{2}} \\[4pt] -\frac{2}{6} \\[4pt] -\frac{2}{6}i \\[4pt] -\frac{2}{3\sqrt{2}} \end{pmatrix}.
\]
We get for the alternative \(C_{\text{low, vocal}}\) the probability \[
\left|\frac{2}{6}\right|^2 + \left|\frac{1}{6}\right|^2 = \frac{4}{36} + \frac{1}{36} = \frac{5}{36}
\] and thus the post-measurement-state is \[
\begin{pmatrix} \frac{2}{6} \\ 0 \\ 0 \\ \frac{1}{6} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} / \sqrt{\frac{5}{36}}
= \begin{pmatrix} \frac{2}{6} \\ 0 \\ 0 \\ \frac{1}{6} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} / \frac{\sqrt{5}}{6}
= \begin{pmatrix} \frac{2}{\sqrt{5}} \\ 0 \\ 0 \\ \frac{1}{\sqrt{5}} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}.
\]