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# 5.1 Defining a qubit

A qubit is defined by the equation

Equation label:(11)

where is a two-state quantum system and and are logical states. Note that has a logical value of 0 and has a logical value of 1. and are the probability amplitudes which may be complex numbers and satisfy the normalisation condition .

Examples of qubits are the general spin states of spin-½ particles as described in Section 3.2. You can see that the spin-up state has been replaced by logical state and the spin-down state by logical state and you can see that Equation 9 and Equation 11 have a similiar form.

To fully specify the two complex amplitudes or four real numbers are required: the real and imaginary parts of each. However, the number of values can be reduced by two, one because of the normalisation condition, and one because the phase of one basis state can be set to zero without changing anything.

This leads to the equation,

where and are real numbers with and .

A qubit state can therefore also be represented as a column vector

This representation is like the spinor representation introduced in Section 3.2. The column vector representation is useful because the operators corresponding to single-qubit gates and observables may be written as 2 × 2 matrices. The qubit basis states are defined as

Equation label:(12)

The column vectors of Equations 12 are eigenstates of a 2 × 2 matrix operator known as the Pauli-Z operator or , which as mentioned earlier is defined as

## Exercise 12

Show that the qubit basis states and are eigenvectors of the Pauli-Z operator and find the corresponding eigenvalues. Determine the relationship between the eigenvalues and the logical values of the basis states. Use the symbol, m to represent the logical value.

### Answer

Noting and has logical value, m = 0; and and has logical value, m = 1 and that

First, for basis state, the eigenvalue equation (Equation 2) becomes

Doing the matrix multiplication gives

showing that

Similarly, for basis state

showing that

Comparing the eigenvalues with the logical values: when , m = 0, and when , m = 1. The relationship between the eigenvalue and the logical value is therefore

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