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5.1 Defining a qubit

A qubit is defined by the equation

absolute value of psi mathematical right angle bracket equals a sub zero times zero mathematical right angle bracket prefix plus of a sub one vertical line one mathematical right angle bracket
Equation label:(11)

where vertical line psi mathematical right angle bracket is a two-state quantum system and vertical line zero mathematical right angle bracket and vertical line one mathematical right angle bracket are logical states. Note that vertical line zero mathematical right angle bracket has a logical value of 0 and vertical line one mathematical right angle bracket has a logical value of 1. a sub zero and a sub one are the probability amplitudes which may be complex numbers and satisfy the normalisation condition absolute value of a sub zero squared plus absolute value of a sub one squared equals one.

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 You can see that the spin-up state has been replaced by logical state vertical line zero mathematical right angle bracket and the spin-down state by logical state vertical line one mathematical right angle bracket 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,

absolute value of psi mathematical right angle bracket equals cosine of theta solidus two times zero mathematical right angle bracket prefix plus of sine of theta solidus two times normal e super i phi vertical line one mathematical right angle bracket full stop

where theta and phi are real numbers with zero less than or equals theta less than or equals pi and zero less than or equals phi less than or equals two times pi.

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

vertical line psi mathematical right angle bracket equals vector element 1 cosine of theta solidus two element 2 sine of theta solidus two times normal e super i phi full stop

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

absolute value of zero mathematical right angle bracket equals vector element 1 one element 2 zero and times one mathematical right angle bracket equals vector element 1 zero element 2 one
Equation label:(12)

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

sigma hat sub z equals matrix row 1column 1 10 row 2column 1 zero minus minus one

Exercise 12

Show that the qubit basis states vertical line zero mathematical right angle bracket and vertical line one mathematical right angle bracket 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 vertical line zero mathematical right angle bracket equals vector element 1 one element 2 zero and has logical value, m = 0; and vertical line one mathematical right angle bracket equals vector element 1 zero element 2 one and has logical value, m = 1 and that sigma hat sub z equals matrix row 1column 1 10 row 2column 1 zero minus minus one

First, for basis state, vertical line zero mathematical right angle bracket the eigenvalue equation (Equation 2) becomes

matrix row 1column 1 10 row 2column 1 zero minus minus one times vector element 1 one element 2 zero equals lamda sub zero times vector element 1 one element 2 zero

Doing the matrix multiplication gives

equation sequence part 1 matrix row 1column 1 10 row 2column 1 zero minus minus one times vector element 1 one element 2 zero equals part 2 vector element 1 one element 2 zero equals part 3 lamda sub zero times vector element 1 one element 2 zero

showing that lamda sub zero equals one

Similarly, for basis state vertical line one mathematical right angle bracket

equation sequence part 1 matrix row 1column 1 10 row 2column 1 zero minus minus one times vector element 1 zero element 2 one equals part 2 vector element 1 zero element 2 negative one equals part 3 negative vector element 1 zero element 2 one equals part 4 lamda sub one times vector element 1 zero element 2 one

showing that lamda sub one equals negative one

Comparing the eigenvalues with the logical values: when lamda sub zero equals one, m = 0, and when lamda sub one equals negative one, m = 1. The relationship between the eigenvalue and the logical value is therefore

m equals one minus lamda sub m divided by two
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OpenLearn - Introduction to quantum computing
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