5.3.1 Two-qubit states

A straightforward way to define the two-qubit states is to build the two-qubit basis states from product states:

There are four possible product states of the usual single-qubit basis states:

A general two-qubit state, therefore can be expressed in terms the product states as

where is normalised in the usual way:

To specify the state of the two-qubit system, six real numbers must be given. Counting them in the same way as for a single qubit, these are the eight numbers specifying the real and imaginary parts of each complex number (where the indices are mn = 00, 01, 10, 11). Eight is reduced by one because of the normalisation condition, given in Equation 15 and by one more, to six because the phase of one of the basis states can be set to zero without changing anything physical.

In general, an n-qubit system requires real numbers to specify the state, which is an exponential scaling in the number of qubits.

If the two-qubit gate is an operation, an equation can be written which represents the transformation from a two-qubit state into a new two-qubit state :

Multiple qubits are represented using the tensor product, which combines their states into a larger system. The resulting matrices represent the full system and can be used to calculate outputs by multiplying them with quantum state vectors. For instance the example above can be written as

This representation shows that a two-qubit gate can be expressed as a 4 × 4 matrix with elements , which act on a 4 × 1 column vector representing the quantum state. This allows you to compute the output state using matrix multiplication. However, we won’t be using this formalism in our discussions, as we will focus on other intuitive approaches to understanding multi-qubit systems and gates.