6  Composite Systems

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.

6.1 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 \(1,\dots,N\)¹ and a distribution/state \(a\) and with the possibilities of \(B\) given by \(1,\dots,M\) and a distribution/state \(b\), the composite system called \(AB\) has the possibilities \[ 11,12,\dots,1M,21,22,\dots,2M,\dots,N1,N2,\dots,NM \]

and the distribution/state \(ab\) of \(AB\) is given by the tensor product \[ ab= a \otimes b = \begin{pmatrix} a_1 \cdot b\\ \vdots \\ a_N \cdot b \end{pmatrix} \] This vector has the size \(NM\).

Note that the definition of combining a probabilistic and a quantum system are the same.

Example: Tensor product

Given two vectors \(a\) and \(b\), with \[ \begin{aligned} a &= \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} & b &= \begin{pmatrix} 10 \\ 100 \end{pmatrix} \end{aligned} \] the tensor product of those vectors is given by \[ a\otimes b = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} \otimes \begin{pmatrix} 10 \\ 100 \end{pmatrix} = \begin{pmatrix} 1\cdot 10 \\ 1 \cdot 100 \\ 2\cdot 10 \\ 2 \cdot 100 \\ 4\cdot 10 \\ 4 \cdot 100 \end{pmatrix} = \begin{pmatrix} 10 \\ 100 \\ 20 \\ 200 \\ 40 \\ 400 \end{pmatrix} \]

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 can not write \(d_{AB}\) as two septate systems, 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.

6.2 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 \(A_1,\dots,A_N \subseteq \{1,\dots,N\}\) and \(B_1,\dots,B_M \subseteq \{1,\dots,M\}\). 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 \(C_1, \dots ,C_N := A \otimes \{1,\dots, M \}\).

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1. Note that the possibilities are labels \(1,\dots,N\) and do not have be the numbers \(1,\dots,N\). We could also label these e.g. red, green and blue (or any other label).