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From: Stephen Wolfram, A New Kind of Science
Notes for Chapter 9: Fundamental Physics
Section: Quantum Phenomena
Page 1062

Quantum measurement. The basic mathematical formalism used in standard quantum theory to describe pure quantum processes deals just with vectors of probability amplitudes. Yet our everyday experience of the physical world is that we observe definite things to happen. And the way this is normally captured is by saying that when an observation is made the vector of amplitudes is somehow replaced by its projection s into a subspace corresponding to the outcome seen - with the probability of getting the outcome being taken to be determined by s.Conjugate[s].

At the level of pure quantum processes, the standard rules of quantum theory say that amplitudes should be added as complex numbers - with the result that they can for example potentially cancel each other, and generally lead to wave-like interference phenomena. But after an observation is made, it is in effect assumed that a system can be described by ordinary real-number probabilities - so that for example no interference is possible. (At a formal level, results of pure quantum processes are termed pure quantum states, and are characterized by vectors of probability amplitudes; results of all possible observations are termed mixed states, and are in effect represented as mixtures of pure states.)

Ever since the 1930s there have been questions about just what should count as an observation. To explain everyday experience, conscious perception presumably always must. But it was not clear whether the operation of inanimate measuring devices of various kinds also should. And a major apparent problem was that if everything - including the measuring device - is supposed to be treated as part of the same quantum system, then all of it must follow the rules for pure quantum processes, which do not explicitly include any reduction of the kind supposed to occur in observations.

One approach to getting around this suggested in the late 1950s is the many-worlds interpretation (see page 1040): that there is in a sense a universal pure quantum process that involves all possible outcomes for every conceivable observation, and that represents the tree of all possible threads of history - but that in a particular thread, involving a particular sequence of tree branches, and representing a particular thread of experience for us, there is in effect a reduction in the pure quantum process at each branch point. Similar schemes have been popular in quantum cosmology since the early 1990s in connection with studying wave functions for the complete universe.

A quite different - and I think much more fruitful - approach is to consider analyzing actual potential measurement processes in the context of ordinary quantum mechanics. For even if one takes these processes to be pure quantum ones, what I believe is that in almost all cases appropriate idealized limits of them will reproduce what are in effect the usual rules for observations in quantum theory. A key point is that for one to consider something a reasonable measurement it must in a sense yield a definitive result. And in the context of standard quantum theory this means that somehow all the probability amplitudes associated with the measuring device must in effect be concentrated in specific outcomes - with no significant interference between different outcomes.

If one has just a few quantum particles - governed say by an appropriate Schrödinger equation - then presumably there can be no such concentration. But with a sufficiently large number of particles - and appropriate interactions - one expects that there can be. At first this might seem impossible. For the basic rules for pure quantum processes are entirely reversible (unitary). So one might think that if the evolution of a system leads to concentration of amplitudes, then it should equally well lead to the reverse. But the crucial point is that while this may in principle be possible, it may essentially never happen in practice - just like classical reversible systems essentially never show behavior that goes against the Second Law of thermodynamics. As suggested by the main text, the details in the quantum measurement case are slightly more complicated - since to represent multiple outcomes measuring devices typically have to have the analogs of multiple equilibrium states. But the basic phenomena are ultimately very similar - and both are in effect based on the presence of microscopic randomness. (In a quantum system the randomness serves to give collections of complex numbers whose average is essentially always zero.)

This so-called decoherence approach was discussed in the 1930s, and finally began to become popular in the 1980s. But to make it work there needs to be some source of appropriate randomness. And almost without exception what has been assumed is that this must come through the first mechanism discussed in Chapter 7: that there is somehow randomness present in the environment that always gets into the system one is looking at. Various different specific mechanisms for this have been suggested, including ones based on ambient low-frequency photons, background quantum vacuum fluctuations and background spacetime metric fluctuations. (A somewhat related proposal involves quantum gravity effects in which irreversibility is assumed to be generated through analogs of the black hole processes mentioned in the previous note.) And indeed in recent practical experiments where simple pure quantum states have carefully been set up, they seem to be destroyed by randomness from the environment on timescales of at most perhaps microseconds. But this does not mean that in more complicated systems more characteristic of real measuring devices there may not be other sources of randomness that end up dominating.

One might imagine that a possibility would be the second mechanism for randomness from Chapter 7, based on ideas of chaos theory. For certainly in the standard formalism, quantum probability amplitudes are taken to be continuous quantities in which an arbitrary number of digits can be specified. But at least for a single particle, the Schrödinger equation is in all ways linear, and so it cannot support any kind of real sensitivity to initial conditions, or even to parameters. But when many particles are involved the situation can presumably be different, as it definitely can be in quantum field theory (see page 1066).

I suspect, however, that in fact the most important source of randomness in most cases will instead be the phenomenon of intrinsic randomness generation that I first discovered in systems like the rule 30 cellular automaton. Just like in so many other areas, the emphasis on traditional mathematical methods has meant that for the most part fundamental studies have been made only on quantum systems that in the end turn out to have fairly simple behavior. Yet even within the standard formalism of quantum theory there are actually no doubt many closed systems that intrinsically manage to produce complex and seemingly random behavior even with very simple parameters and initial conditions. And in fact some clear signs of this were already present in studies of so-called quantum chaos in the 1980s - although most of the specific cases actually considered involved time-independent constraint satisfaction, not explicit time evolution. Curiously, what the Principle of Computational Equivalence suggests is that when quantum systems intrinsically produce apparent randomness they will in the end typically be capable of doing computations just as sophisticated as any other system - and in particular just as sophisticated as would be involved in conscious perception.

As a practical matter, mechanisms like intrinsic randomness generation presumably allow systems involving macroscopic numbers of particles to yield behavior in which interference becomes astronomically unlikely. But to reproduce the kind of exact reduction of probability amplitudes that is implied by the standard formalism of quantum theory inevitably requires taking the limit of an infinite system. Yet the Principle of Computational Equivalence suggests that the results of such a limit will typically be non-computable. (Using quantum field theory to represent infinite numbers of particles presumably cannot help; after appropriate analysis of the fairly sophisticated continuous mathematics involved, exactly the same computational issues should arise.)

It is often assumed that quantum systems should somehow easily be able to generate perfect randomness. But any sequence of bits one extracts must be deduced from a corresponding sequence of measurements. And certainly in practice - as mentioned on pages 303 and 972 - correlations in the internal states of measuring devices between successive measurements will tend to lead to deviations from randomness. Whatever generates randomness and brings measuring devices back to equilibrium will eventually damp out such correlations. But insofar as measuring devices must formally involve infinite numbers of particles this process will formally require infinitely many steps. So this means that in effect an infinite computation is actually being done to generate each new bit. But with this amount of computation there are many ways to generate random bits. And in fact an infinite computation could even in principle produce algorithmic randomness (see page 1071) of the kind that is implicitly suggested by the traditional continuous mathematical formalism of quantum theory. So what this suggests is that there may in the end be no clear way to tell whether randomness is coming from an underlying quantum process that is being measured, or from the actual process of measurement. And indeed when it comes to more realistic finite measuring devices I would not be surprised if most of the supposed quantum randomness they measure is actually more properly attributed to intrinsic randomness generation associated with their internal mechanisms.

Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), page 1062.
© 2002, Stephen Wolfram, LLC