A3/Bit and Qubit
In the last section we introduced the concept of quantifiable information. Next we will see, how to model this for classical and quantum information. For this we will introduce the basic unity of information – the "bit" and the "qubit".
One of the fundamentals of information theory is the task of quantifying information. The intuition here would be, that information should be additive. This means, that two letters will carry twice the amount of information then one, or that in two minutes time twice as much information can be communicated than in one minute. The simplest example for an information recording device is a switch: It has two possible positions that can be labelled with "On" and "Off" or with the numbers "0" and "1". One defines that the amount of information stored in a switch is 1bit of information. This also implies that two switches carry 2bit of information.
Now, one can consider how many different position two switches can be set to. If one switch can be set either to 0 or to 1, then two switches can be set to four different positions: 00, 01, 10 and 11. Likewise, three switches allow eight positions: 000, 001, 010, 011, 100, 101, 110 and 111. If a system consists of N switches, i.e., it is carrying N bits of information, there are 2^N different settings. This means that the number of different settings grows exponentially with the number of bits. From this it follows, that if one considers a memory that can be in set to one of k different settings, the system carries \log_2 k bits of information.
Coding of an alphabet
Consider another example: In the English language there are 26 basic letters. If one wants to code a letter in positions of switches, this would require at least 5 switches, as \Log_2 26 \approx 4,7. If one wants to distinguish upper and lower case letters one needs to add another bit. In computing it is common to encode a letter using 8 bit, which gives the possibility to encode 2^8=256 different letters. This means that not only the letters of the alphabet but also punctuation and control characters. A symbol consisting of 8bit is also called one byte. This represents the basic unit of memory space of hard drives or USB-memory devices.
The notion of one bit as such is independent from the physical realisation, e.g., whether it is realises as magnetization information on a hard drive, as electrical current in a wire or as optical information in an optical fibre.
The basic unit of quantum information is called the qubit. Again there are a number of possible realisations, for example as polarisation of photons, as spin of an electron or as energy state of an atom. We call a quantum system a qubit, if it most two states of the system can be distinguished with certainty by a measurement. Mathematically this means that the corresponding complex Hilbert space is two dimensional. We call these two states |0> and |1>, where the notation |.> (also called Dirac notation) is introduced to distinguish quantum states from classical information. Mathematically, each state represents a vector in the system Hilbert space. One of the fundamental properties of quantum physics is, that thy qubit can be in an arbitrary superposition of these basis states. We will discuss the properties for a prominent example: polarisation.
Consider a qubit that is realised by the polarisation degree of freedom of light. In this case, the state |0> could be represented by horizontally (0°) polarised light, while |1> could be realised by vertically (90°) polarised light. Quantum mechanics allows forming of a coherent superposition of these states. Depending of the orientation and phase of the two light beams this will lead to any lead to e.g. +45° polarised light, circular polarized or any other fully polarised light state. It is further more possible to mix two light beams incoherently, which might result in un-polarised light. A suitable representation of polarisation (and hence any qubit) is the Bloch sphere.
On the Bloch sphere any two states that can be distinguished with certainty lie antipodal, e.g., left circular and right circular polarised light may lie on opposing poles of the sphere. The surface of the sphere represents the maximally polarised states, while the centre is the un-polarised state. We will discuss more examples later in section ???.