III AN INTRODUCTION TO QUANTUM INFORMATION THEORY
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Knowledge, information, and (a little about) quantum information
Peierls, knowledge, and information
Among the people who published responses to Bell’s ‘Against “Measurement” ’ paper was Rudolf Peierls , who was certainly a very great physicist. He had been a student of Heisenberg, and spent a considerable amount of time working in Copenhagen with the group of physicists around Bohr. As such, he would always have been among those regarding complementarity as the final answer to the dilemmas of quantum theory. In 1986, in fact, he was to complain  that use of the term ‘Copenhagen interpretation . . . makes it sound as if there were several interpretations of quantum mechanics. There is only one. There is only one way in which you can understand quantum mechanics, so when you refer to the Copenhagen interpretation of the mechanics what you really mean is quantum mechanics.’
Peierls (Figure 15.1) had been born in 1907, and he was studying in Cambridge on a Rockefeller Scholarship when Hitler came to power, so he stayed in England as a refugee. He became Professor of Physics at Birmingham University in 1937, moving to Oxford in 1963, and in both places he would have been regarded as being in charge of the leading department in Britain specializing in what we might call down-to-earth theoretical physics such as solid state and nuclear physics; Cambridge may be pre-eminent in the more esoteric and mathematical fields and in astrophysics.
Peierls himself was perhaps the last theoretical physicist who appreciated and could contribute massively to every area of physics: for example, solid state physics, nuclear physics, superconductivity and liquid helium, quantum field theory and statistical mechanics. In 1940 with Otto Robert Frisch, he crucially showed that, contrary to general belief at that time, an atomic bomb could be made from a comparatively small amount of uranium-235, and thus they kick-started the process of making the bomb, but after the war, concerned about the spread of nuclear weapons, he played a large role in the Pugwash movement, the physicists’ movement to attempt to attain world peace.
Fig. 15.1 Ruudolf Peierls (left) with Francis Simon about 1951. Both Peierls and Simon came to Britain as refugees, both became Professors at Oxford University and both were knighted.
Peierls undoubtedly deserved a Nobel Prize; but perhaps the committee felt that he had a very large number of important achievements, but not the crucial one that might have sent him to Stockholm.
We met him in Chapter 8, where he welcomed Bell to work for a year in Birmingham; he was certainly enormously impressed by him, and helped him greatly in his career. As such a staunch supporter of Copenhagen, clearly he must have been rather irritated by Bell’s dismissal of the work of Bohr and his chief colleagues. Nevertheless he came to appreciate Bell’s theorem itself, and it is pleasant to note that Bell and Peierls were on very good personal terms for the remainder of Bell’s life. Indeed at the conference dinner of the meeting in memory of Bell, which was held at CERN shortly after his death, Mary Bell, of course, was in the central position, and on one side of her was Rudolf Peierls.
It is certainly not surprising that Peierls wished to reply to Bell’s views which were so much opposed to his own. Yet rather strangely the words with which Peierls actually expressed his views were not particularly in accord with those of Bohr and Heisenberg. Rather, perhaps picking up Bell’s prohibition on use of the word ‘knowledge’, he based his approach exactly on that quantity. Mermin suggests, incidentally, that the words ‘knowledge’ and ‘information’ should be regarded as synonymous, and that is exactly what we shall do.
It is quite common for those who think just a little about quantum theory to deduce that the actual formalism is all about information. The reason why the wave-function is said to collapse, they suggest, is that the observer now knows the value, having just measured it, and the wave-function changes to correspond to this value. It is easy to believe that the actual state of the system has not had to change at all, just the information of the observer. Of course that implies that the system does have an ‘actual state’ that may even be broadly classical in nature, and this is totally against everything Bohr stands for. At first sight, though, this general type of approach may make things seem very easy!
And it may seem to be backed up, at least to some extent, by some of the giants of quantum theory. Von Neumann suggested that the collapse is caused by the ‘abstract ego’ of the observer. Wigner, as we have mentioned, argued that it was caused by consciousness. Both of these rather vague ideas were at least rather similar to putting it down to changes in knowledge/information.
Bell’s brief barbs may pull us down to earth a little. ‘Information about what?’ makes the obvious suggestion that, if the information is about something, why do we not concentrate on the something it is about, rather than the information itself? The other question, ‘Whose information?’, leads to the fact that the analysis must be much more complicated then we have suggested so far, and this takes us on to Peierls’ ideas.
Peierls agrees with Bell that a precise formulation of quantum theory is a must, and he also agrees that no textbook explains these things adequately. However, he says that he does not agree with Bell that it is all very difficult! He starts his reply by stating his view that the most fundamental statement of quantum mechanics is that the wave-function, or more generally the density-matrix, represents our knowledge of the system we are trying to describe. If the knowledge is complete, in the sense of being the maximum allowed by the uncertainty principle and the general laws of quantum theory, we use a wave-function; for less knowledge we use a density-matrix.
Uncontrolled disturbances may reduce our knowledge. Measurement may increase it, but if we start off with the wave-function case, so we have maximum allowed knowledge, and if we gain some new knowledge in a measurement, this must be compensated by losing some of the information already there. (In other words, if we know the value of the z-component of spin, and choose to measure the x-component, at the end we will indeed know the value of the x-component but will have lost all knowledge of the z-component.)
He remarks that when our knowledge changes, the density-matrix must change. This, he stresses, is not a physical process, and the change will definitely not follow the Schrodinger equation. Nevertheless our knowledge has changed, and this must be represented by a new density- matrix. For a measurement he remarks that the density-matrix has certainly changed by the time the measurement is complete, but it only reaches its final form, which corresponds to the result of the experiment, when we actually know the result.
He raises the slightly awkward question of how one might apply quantum theory to the early Universe, when there were no observers. His answer is that the observer does not need to be contemporary with the event. When we draw conclusions about the early Universe, we are, in this sense, observers.
On Bell’s second question, ‘Information about what?’, we have already quoted Peierls as saying ‘Knowledge of the system we are trying to describe’. This is a little vague, but it seems to indicate that there is a system, and the knowledge is about it. (This may seem obvious, but we meet a contrary suggestion shortly.)
On the first question, ‘Whose information?’, Peierls is quite forthcoming. Many people, he says, may have some information about the state of a system, but each person’s information may differ, so they may all have their individual density-matrices. While these may be different, they must all be consistent; we cannot allow a situation where one observer has a density-matrix that yields a definite value of the z-component of spin, while the density-matrix of another observer gives a definite value of the x-component, because simultaneous knowledge of both is forbidden by the laws of quantum theory.
Putting this more mathematically, Peierls uses the idea of commutativity, which is actually central in quantum theory. If we write general matrices as A and B, it is an important rule of matrix algebra that, in the general case, AB is not equal to BA; we may say two matrices do not usually commute. However, Peierls says that, if A and B are the density- matrices for the same system as used by two different observers, then AB must equal BA, or in other words the matrices must commute.