State Vectors and Dirac Notation

We wish to know exactly how the behavior of our qubit differs from a that of a classical bit. Recall that a classical bit can store either a 1 or a 0, and when measured the value observed will always be the value stored. Quantum physics states that when we measure the qubit we will determine that it is in the 1 or the 0 state. In this manner our qubit is not different from a classical bit. The differences between the qubit and the bit come from what sort of information a qubit can store when it is not being measured.

According to quantum physics we may describe that state of our qubit by a state vector in a Hilbert Space. In general the mathematical term space refers to a something which depends on many independent coordinates which can be defined by a set of perpendicular axes, one for each independent variable. For example you are probably familiar with the $x$, $y$, $z$ coordinate system where the $x$, $y$, and $z$ axes are mutually perpendicular real number lines, which coincide at the point $x = 0$, $y = 0$, $z = 0$.

A Hilbert Space is a special kind of space, it has the properties that it is a complex vector space, and it is a linear vector space. A complex vector space is one where the lengths of the vectors within the space are described by complex numbers. Complex numbers are numbers which take the form $a + i*b$, where $a$ and $b$ are real numbers, and $i$ is the square root of negative one. A linear vector space is one where you may add and multiply vectors that lie within the space and the resulting vector will still lie within the space. (Williams, Clearwater)

In the Hilbert Space for a quantum system's state vector, we choose to define these perpendicular axes to correspond to each possible state that the system can be measured in. Our Hilbert Space for a single qubit will have two perpendicular axes, one corresponding to the qubit being in the 1 state, and the other corresponding the qubit being in the 0 state. These states which the vector can be measured to be are referred to as ``eigenstates.'' The vector which exists somewhere in this space which represents the state of our qubit is called the ``state vector.'' The projection of the state vector onto one of the axes shows the contribution of that axis' eigenstate to the whole state.

In general, the state of a qubit can be any combination of the base states. In this manner a qubit it totally unlike a bit, for a bit can exist in only the 0 or 1 state, but the qubit can exist, in principle, in any combination of the 0 and 1 state, and is only constrained to be in the 0 or 1 state upon measurement.

Now I will introduce some standard notation for state vectors in Quantum physics. The state vector is written with the following way and called a ``ket vector'' $\left\vert\psi \right\rangle$. Where $\psi$ is a list of numbers which contain information about the projection of the state vector onto its base states. The term ket and this notation come from the physicist Paul Dirac who wanted a concise shorthand way of writing formulas that occur in Quantum physics. These formulas frequently took the form of the product of a row vector with a column vector. Thus he referred to row vectors as ``bra vectors'' represented as $\left\langle y\right\vert$. The product of a ``bra'' and a ``ket'' vector would be written $\left\langle y\vert x\right\rangle$, and would be referred to as a ``bracket.'' (Williams, Clearwater)

There is nothing special about this vector notation, you may think of any state vector being written as a letter with a line over it, or as a bold letter, as vectors are normally denoted in mathematical literature. If you do further reading in this area you will assuredly come across the bra and ket notation, which is why it is presented.