Quantum information

In quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a two-state quantum system. However, unlike classical digital states (which are discrete), a two-state quantum system can actually be in a superposition of the two states at any given time. Quantum information differs from classical information in several respects, among which we note the following:

• It cannot be read without the state becoming the measured value,
• An arbitrary state cannot be cloned,
• The state may be in a superposition of basis values.

However, despite this, the amount of information that can be retrieved in a single qubit is equal to one bit. It is in the processing of information (quantum computation) that a difference occurs. The ability to manipulate quantum information enables us to perform tasks that would be unachievable in a classical context, such as unconditionally secure transmission of information. Quantum information processing is the most general field that is concerned with quantum information. There are certain tasks which classical computers cannot perform "efficiently" (that is, in polynomial time). However, a quantum computer can compute the answer to some of these problems in polynomial time; one well-known example of this is Shor's factoring algorithm. Other algorithms can speed up a task less dramatically - for example, Grover's search algorithm which gives a quadratic speed-up over the best possible classical algorithm. Quantum information, and changes in quantum information, can be quantitatively measured by using an analogue of Shannon entropy. Given a statistical ensemble of quantum mechanical systems with the density matrix S, it is given by

<math> -\operatorname{Tr}(S \ln S). </math>

Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as the conditional quantum entropy.

Quantum Information Theory

The theory of quantum information is a result of the effort to generalise classical information theory to the quantum world. Quantum information theory aims to answer the following question:

What happens if information is stored in a state of a quantum system?

It is a strength of classical information theory that it does not need to ask the question about the physical representation of information: There is no need for a 'ink-on-paper' information theory or a 'DVD information' theory. This is due to that fact that it is always possible to efficiently transform information from one representation to another representation. For this reason, one might be tempted to believe that it is not important whether information is stored in classical systems or in quantum systems. However this is not the case: it is not possible, for example, to write down the previously unknown information contained in the polarisation of a photon of ink on a paper. In general quantum mechanics does not allow us to read out the state of a quantum system with arbitrary precision. The existence of Bell correlations between quantum systems cannot be converted into classical information. It is possible to transform quantum information between quantum systems of sufficient quantum information capacity. The quantum information content of a quantum message <math>\mathcal{M}</math> can for this reason be measured in terms of the minimum number <math>n</math> of two-level systems which are needed to store the message: <math>\mathcal{M}</math> consists of <math>n</math> qubits. In its original quantum information theoretical sense, the term qubit is thus a measure for the amount of information. A two-level quantum system can carry at most one qubit, in the same sense a classical binary digit can carry at most one classical bit. The term qubit is used as a synonym for a two-level quantum system. A pure one qubit state is specified by two real parameters, in this sense quantum information is similar to analog (in contrast to digital) classical information. Analog information processing seems to be much more efficient than digital information processing on a first sight, since an analog information carrier could contain an infinite amount of information. However, analog information processing is being, or is already been, replaced by digital information processing. From this one can see, that in practise analog information processing performs more than digital information processing. In the presence of noise, which is responsible for this gap between the theoretical promise and the practical application of analog information. In the case of noise, the information content of an analog information carrier is no longer infinite, but finite. This is a consequence of Shannon's noisy coding theorem. It is very difficult to protect the remaining finite information content of analog information carriers against noise. The example of classical analog information shows that quantum information processing schemes must necessary be tolerant against noise, otherwise there would be a chance for them to be useful. It was a big break through for the theory of quantum information, when quantum error correction codes and fault-tolerant quantum computation schemes were discovered.

See also

• Quantum computing
• Quantum statistical mechanics
• POVM (positive operator value measure)

External links and references

• Center for Quantum Computation - The CQC, part of Cambridge University, is a group of researchers studying quantum information, and is a useful portal for those interested in this field.
• Quantum Information Group The quantum information research group at the University of Nottingham.
• Qwiki - A quantum physics wiki devoted to providing technical resources for practicing quantum information scientists.
• Quantiki - A wiki portal for quantum information with introductory tutorials.
• Charles H. Bennett and Peter W. Shor, "Quantum Information Theory," IEEE Transactions on Information Theory, Vol 44, pp 2724-2742, Oct 1998
• Institute for Quantum Computing - The Institute for Quantum Computing, based in Waterloo, ON Canada, is a research institute working in conjunction with the University of Waterloo and the Perimeter Institute on the subject of Quantum Information.
• Quantum information can be negative
• Gregg Jaeger's new book on Quantum Information(published by Springer, New York, 2007, ISBN 0-387-35725-4)
• Pluch, P. Theory for Quantum Probability, PhD Thesis, Klagenfurt University (2006)
• The International Conference on Quantum Information (ICQI)