Quantum Error Correction, an informal introduction. Start. <-- [prev] . [next] --> |
3. Mathematical notation for quantum bits
This is meant to be a friendly introduction and I want to avoid abstract mathematics as much as possible. However, I hope most readers will enjoy at least some of the mathematical ideas, therefore we will need to introduce some notation. I think it best to stick to the standard notation in this field, so readers familiar with that can skip this section.
When referring to classical bit strings, i.e. strings of ones and zeros,
we just write the string, for example 1001010. When
referring to quantum states, we use a bracket symbol
| x >
(introduced by Dirac).
The vertical line should be read as the left side of the bracket, the angled line
should be read as the right side of the bracket, and the
x symbol in the middle is a label.
There is a good reason to use non-matching symbols for the bracket (rather than, say,
|x| or < x >):
it is because the mathematics of quantum systems
is such that
| x >
is a column vector, and a related
symbol
< x |
refers to the corresponding row vector.
The simplest possible quantum system can be described in terms of two quantum states, and it is convenient to label these using labels 0 and 1. Therefore the two basic states are |0> and |1>. The general state of the system is
a |0> + b |1>
for complex numbers a, b. This is just like writing a general vector in terms of two components (the components are |0> and |1>, the numbers a and b say how much of each component is used to make the state). A quantum system described fully by just two basic states (and combinations thereof) is called a qubit.
When both a and b are non-zero, we say we have a "superposition". It is sometimes claimed that superposition is a remarkable and strange quantum phenomenon, but really it is just like the fact that the direction north-east is a combination of north and east, and a step up a steep hill is a combination of along and up.
Two important superposition states are
( |0> + |1> ) /√2
and
( |0> - |1> ) /√2.
These will come up repeatedly. They are often referred to as the
"|+>" and "|->"
states, for obvious reasons. If |0> and
|1> are vectors
pointing along the x and y axes, then |+>
and |-> are the diagonal
vectors (see figure). Note that |+> and
|-> are orthogonal to one another and act
in all respects like an alternative pair of basis states. For this reason, and
especially in order to bring out the behaviour of states involving
multiple qubits, it will be useful to use the notation
|0"> =
( |0> + |1> ) /√2,
|1"> =
( |0> - |1> ) /√2.
where the " after the zero or one indicates that the state label is understood to refer to this pair of states. The reader should verify that
|0> =
( |0"> + |1"> ) /√2,
|1> =
( |0"> - |1"> ) /√2.
When we have several such systems (qubits), the joint states are written using combinations of |0> and |1>, such as for example |0>×|0>×|1> for a case where the first two qubits are in state |0>, and the third in state |1>. It is convenient to compress this notation to |001>. Now we have a quantum state of three qubits, labelled by a classical bit string. It should now be clear how the notation works: the surrounding |...> bracket signals that we are dealing with a quantum state, and the bit string inside tells us the situation for each of the qubits present.
To get some practice, you might like to check the following results, and note the overall pattern:
|00"> =
( |00> + |01> +
|10> + |11> ) /2,
|01"> =
( |00> - |01> +
|10> - |11> ) /2,
|10"> =
( |00> + |01> -
|10> - |11> ) /2,
|11"> =
( |00> - |01> -
|10> + |11> ) /2.