Quantum limits to the flow of information and entropy
Journal of Physics A: Mathematical and General
The quantum limits to information flow are explored developing the central concept of a medium comprising several channels through which the information flows. In each channel there is an inequality between information flow i and energy flow E i 2 < E ' r r / ( 3 h~n 2 2 ) a relationship which, though speculated on before or derived for only a restricted class of systems, is proven here for a wide range of systems. Applications are made to the energy cost of computing and to the maximum rate of
... the maximum rate of cooling, Q, which in any one channel is Q S l r k i T 2 / ( 3 h ) . The medium is the message When information is signalled from one place to another it must be conveyed by some medium. Despite information being an abstract concept, when we come to process it there is always a specific context: a flow of electric current, radio waves, light waves or sound waves, Identifying information with entropy (Shannon 1948 , Brillouin 1956, there is an analogy to be made between a medium carrying information and a fluid which conveys entropy away from the interior of a refrigerator. This is the analogy we shall pursue. The medium will remain unchanged after the information transfer, but we shall require that it obeys the known laws of physics. Already the laws of relativity tell us that information cannot travel faster than the speed of light. Recently there has been much interest in exploring whether quantum mechanics places further constraints on the transfer of information. Broadly speaking, work falls into two categories: the search for a very general law which holds under every conceivable circumstance and more limited studies dealing with specific systems. The work of Bremermann (1967) and Bekenstein (1981) falls into the former category. They find a relationship between information flow rate, 1, and the energy flow rate E. However, the extreme difficulty of making a general argument leads to ambiguities and some scepticism has been expressed about the results (Deutsch 1982 , Landauer 1981 . On the other hand Lebedev and Levitin (1966) considered transmission of an electromagnetic field in one dimension, drawing the analogy with a single-channel communication system. They found an energy cost per bit of information depending on the flow rate in the same qualitative way that Bremermann's result does. Marko (1965) pointed out a relationship of this general form but without deriving the exact form of the coefficients. Further references may be found in the book by Yu (1976) .