The quantum measurement problem,
the role of the observer
and the conditions of knowing:
a non technical exposition
Shantena Augusto Sabbadini www.shantena.com
Since the beginning of quantum theory physicists have felt that the classical appearance of the world had something to do with the role of the observer. That is indeed so, but not because the observer's consciousness collapses the wave function, as it was once hypothesized. The classical appearance of the world is connected with the conditions of our knowing in general. The world appears classical to us because, in order to know, we have to be entangled with what we know.
Superpositions and mixtures
It is generally recognized among physicists that on a microscopic scale (the size of molecules, atoms and smaller systems) the world behaves very differently from the macroscopic world we are accustomed to. The microscopic world is a strange landscape, hard to visualize, because the concepts we have evolved to represent objects are appropriate only for things roughly of our order of magnitude, neither much smaller nor much larger. Those concepts do not apply on a microscopic scale. Quantum physics, the physics of the microscopic world, had to develop entirely new notions, that do not have equivalents in our everyday language or in classical physics, which is the physics of the macroscopic world.
One of these strange notions is the idea of superposition. In the quantum world a proposition and its opposite, "A" and "not A", are not mutually exclusive. Beside "A" and "not A", quantum physics takes into consideration various superpositions of "A" and "not A", in which both possibilities coexist, so to speak, in potentia and one or the other is actualized only when an observation is performed on the system. After the observation the state of the system is either "A" or "not A", the two possibilities are mutually exclusive just as they are in classical physics. But we cannot consistently assume that such was also the case before the observation. The act of observation has a far more active nature in quantum physics compared to classical physics.
And when an observation is performed on the system it is a priori unpredictable whether "A" or "not A"will manifest. Intrinsically unpredictable. This is also a new notion: is not the familiar kind of unpredictability, which we could describe as "we cannot predict this because we don't have enough information about the system ". It is a radical kind of unpredictability, describable as "even with maximum information about the system, we still cannot predict this ". (How we have been led to consider this new notion of unpredictability is a very interesting story, but I am not going into it here because it would be too much of a digression.)
So, in a superposition "A" and "not A" coexist, the system sits astride both possibilities. Let me explain this with an example. Let's imagine that we have a "quantum coin". If we toss an ordinary coin (a "classical coin"), it will land as either head or tail. If we don't look at it (let's say we immediately cover it up with a piece of cloth), we won't know whether it's head or tail – but still we can confidently say that it is either one or the other. Not so with a "quantum coin". The list of the possible states of the the coin under the cloth does not consist just of head and tail, but includes all the superpositions of head and tail, which I shall indicate as
a ´ head + b ´ tail. (1)
In (1) a and b are numbers which tell us whether the superposition is more "weighed" toward head or toward tail, i.e. whether we are more likely to see one or the other when we pull the cloth away. I'll explain this in more detail shortly – but first we have to acquire another concept.
Since in the quantum world the outcome of a single observation is intrinsically unpredictable, a single experiment is not very significant. Quantum physics focuses on statistics, it deals with large numbers of trials. Therefore we have to develop a language to talk about large numbers of trials. This is the language of "sets" or "ensembles". A set is a collection or group of things. Here we shall be concerned with sets of identical systems (e.g. sets of quantum coins or sets of classical coins) in various states, We shall denote sets by putting curly brackets around the state or the states of the systems included in the set. E.g., a set of coins which are all heads will be denoted by
{ head } ,
and a set of quantum coins in the state (1) will be denoted by
{ a ´ head + b ´ tail } . (2)
When a set is a mixture of systems in different states, we shall separate the various states with semicolons, and each state will be preceded by its proportion. E.g., a set of coins a fraction A of which is heads and a fraction B of which is tails will be written
{ A, head ; B, tail } . (3)
A and B are numbers between 0 and 1. E.g., if the set is all heads, A = 1 and B = 0 (so that { 1, head ; 0, tail } is another way of writing { head } ). If the set is half heads and half tails, A = ½ and B = ½, etc. Since we are assuming here that all coins in the set are either heads or tails, A + B = 1.
Now, the quantum superposition (2) presents some similarities to the classical mixture (3), but also some significant differences. The most significant difference is that the set (3) can in principle be divided in two subsets, one of which is all heads and the other all tails, while no such separation is possible in (2). In (2) each single coin, as we have noticed before, sits astride both possibilities: the set is homogeneous, i.e. it consists of only one state, but this state is neither head nor tail (or, if you like, it is both).
The similarity consists in the fact that, when we look at each coin, (2) turns into (3), and the numbers a and b contain all the information we need to predict A and B, i.e. to predict which fraction of the total will turn out to be heads and which franction will turn out to be tails. But, and here is another significant difference, a and b contain more information than that. The additional information, which has no classical equivalent, is an angle or a phase. I.e., a is the same thing as A plus an angle, b is the same thing as B plus an angle.
Mathematicians have developed a very convenient language to talk about objects like that: it's the language of complex numbers. While ordinary numbers ("real" numbers) correspond to points on a line, complex numbers correspond to points in a plane. Each complex number corresponds to two real numbers (coordinates of that point along two perpendicular axes) or to a positive real number (distance from the origin) plus an angle (direction). The distance from the origin is called the modulus of the complex number z and is indicated by ½ z ½ . The coefficients a and b in (2) are complex numbers. The math of quantum physics is most conveniently formulated in the language of complex numbers, instead of that of real numbers, like classical physics.
As I said, a and b contain all the information we need to predict A and B. The relationship is simply the following: A = ½ a ½ 2 (modulus square of a ) and B = ½ b ½ 2 (modulus square of b ). I.e., when we look at each coin in the quantum superposition (2), it turns into the classical mixture
{ ½ a ½ 2 , head ; ½ b ½ 2 , tail } . (4)
We can also say that ½ a ½ 2 is the probability of observing a head when we look at a single coin and ½ b ½ 2 is the probability of observing a tail when we look at a single coin.
We are now in a position to define more precisely in which sense the act of observation in quantum physics has a far more active role than in classical physics. Looking at a coin in the classical mixture set (4) in order to find out whether it's a head or a tail doesn't change anything in the coin itself. It simply amounts to finding out whether the coin belongs to the heads subset or to the tails subset. That's what is called a mere increase in information. Not so with the quantum superposition (2). In (2) there are no subsets, and finding a head or a tail is no mere increase in information. It is an actual change in the state of the coin.
There is a whole lot of experimental evidence that the microscopic world cannot be described in terms of classical mixtures. It requires the introduction of the specifically quantum notion of superposition. Atoms and molecules and smaller things are not either-or creatures, they are and-and creatures. They are like quantum coins, not like classical coins.
As a paradox, Erwin Schrödinger once imagined a famous quantum thought-experiment. Instead of a quantum coin he had a cat and instead of "head" and "tail" he had "alive" and "dead". The experiment would cause the cat to end up in a superposed state of life and death. This was a kind of provocation, something like: if we consistently follow through with the notions of quantum physics, look where we get.
Now we are confronted with a question. In the macroscopic world we inhabit we never see coins being simultaneously head and tail or cats being simultaneously dead and alive. Yet we have good reasons to believe that cats and coins are made of atoms and molecules and that atoms and molecules behave in the strange way that has just been described. The question then is: how does the well-behaved world of cats and coins arise out of the weird world of atoms and molecules?
The quantum measurement problem
This question becomes particularly critical when we analyze in detail what happens in a "quantum measurement", i.e. in the process of observation of a quantum system.
We have no way to observe microscopic systems directly through our senses. The observation always involves a process of amplification. A chain of correlated interactions takes place, starting at the atom and molecule level (or smaller), with the observed system interacting with some microscopic systems in our measuring apparatus and changing their state. These microscopic systems in turn interact with a larger number of microscopic systems, also changing their state, and these with many more – and so on, until the process reaches macroscopic proportions and becomes visible to our eyes.
This chain of interactions involved in a measurement is called a "von Neumann chain" (from John von Neumann, who first mathematically described it). The essential requirement, of course, is the fact that the changes transmitted along the chain should all be correlated, so that they carry information from the microscopic eventually all the way to the macroscopic level.
To see what this means, let's build a little model of a quantum measurement in terms of quantum coins. We shall consider a very simplified von Neumann chain, consisting at one end of a quantum coin (microscopic observed system, or "object system") and at the other end of a measuring apparatus which has only three possible states: NEUTRAL, HEAD and TAIL. NEUTRAL is the state of the apparatus before the measurement. HEAD is the state of the apparatus when it has registered the "head" result, and TAIL is the state of the apparatus when it has registered the "tail" result.
Then the above correlation requirement simply translates into the following. If the initial state of the observed system is head, i.e. the initial state of the whole system object + apparatus is "headNEUTRAL", the interaction must produce the endstate "headHEAD":
headNEUTRAL ® headHEAD.
If the initial state of the observed system is tail, i.e. the initial state of the system object + apparatus is "tailNEUTRAL", the interaction must produce the endstate "tailTAIL":
tailNEUTRAL ® tailTAIL.
What happens then if the initial state of the quantum coin is the superposition (1), i.e. the initial state of the system object + apparatus is
a ´ headNEUTRAL + b ´ tailNEUTRAL? (5)
A fundamental property of the equations of quantum physics is that they are linear, i.e. if we know what happens to each term of a superposition, we know what happens to the whole superposition. In particular the superposition (5) turns into
a ´ headHEAD + b ´ tailTAIL. (6)
To look at this in statistical terms, i.e. in the language of sets, we only need to wrap curly brackets around (5) and (6):
{ a ´ headNEUTRAL + b ´ tailNEUTRAL } ®
{ a ´ headHEAD + b ´ tailTAIL } (7)
The trouble is that this does not resemble at all the result we expect (the result corresponding to laboratory experience), which, as previously mentioned, is that the set separates into two subsets, a fraction ½ a ½ 2 heads and a fraction ½ b ½ 2 tails, a situation which would be described by the classical mixture
{ ½ a ½ 2 , headHEAD ; ½ b ½ 2 , tailTAIL } . (8)
The root cause of the trouble is the linearity of the equations of quantum physics: linear equations are incapable of generating a superposition from a mixture. They can only turn a superposition into another superposition.
It is interesting to notice here also another characteristics of the superposition (7). It involves the observed system and the measuring apparatus in such a way that their states are inextricably connected: it is what is called an entangled state. When two systems are thus entangled, they are like psycho twins, even if they move apart, they are never really separate: whatever is done to one of them instantly affects the other, however far it may be. This is one of the strangest and most powerful notions of quantum physics (it is the notion which at the base of quantum teleportation and of the hopes for developing quantum computers).
Let's take stock of where we have got so far. When we try to describe a measurement process by applying the equations of quantum physics we get a strange result: instead of the classical mixture (8) we get the entangled superposition (7): Schrödinger's cat simultaneously dead and alive… Paradoxically quantum physics is able to describe everything, except the measurements on which the theory itself is built!
How do we deal with that? Quantum physics is eighty years old, and the paradox has been with us for that long. The 'orthodox' or 'Copenhagen' interpretation of quantum theory takes a very practical approach, which basically consists in ignoring the problem. It introduces an ad hoc postulate declaring the outcome of a measuring process to be (8) and not (7). This is known as "collapse of the state vector", or "collapse of the wave function". It assumes that something happens in the measurement process which collapses the entangled situation (7) into the classical situation (8).
But of course the theory is fundamentally incomplete unless we are able to say what is that something that collapses the wave function. In other words, what is it that makes the macroscopic world, sitting on top of all this weird quantum stuff, appear classical, either-or, well-behaved, solid (kind of), instead of being an entangled mess of head and tail coins and dead and alive cats?
Proposed solutions
Many proposals have been brought forth. They fall into two broad categories:
(i) either they propose a mechanism to describe the collapse;
(ii) or they propose that the superposition (7) and the mixture (8) are in some sense indistinguishable.
Without going into a detailed discussion of the various solutions, I will just outline a few of them to give you a glimpse of the variety of philosophical positions implied.
An example of a type (i) proposal is the idealistic solution suggested by Wigner in the 60's. He proposed that the collapse of the state vector is not due to the interaction of the microscopic system with the measuring apparatus, but to the observer's consciousness taking in the measurement's results. The measuring apparatus, after interacting with the microscopic system, does indeed end up in the entangled state (7). And we can assume that the observer's sense organs and brain, after interacting with the measuring apparatus, also end up in an entangled state with all the rest. We can extend the von Neumann chain to include the observer's retina, optic nerve, brain, etc. Let's call all this just "brain", for short, and indicate its state by capital Monotype Corsiva font. Then the resultant entangled state is
{ a ´ headHEAD HEAD + b ´ tailTAIL TAIL } , (9)
(where HEAD denotes the state of the observer's brain when he/she has read HEAD on the apparatus, and TAIL denotes the state of the observer's brain when he/she has read TAIL). At this point a non-physical element, i.e. the observer's consciousness comes in. The observer becomes aware that his/her brain has registered HEAD or that it has registered TAIL. Consciosness, says Wigner, cannot be in a superposed state. Therefore it causes the whole chain of correlated systems (observed system-apparatus-retina-optic nerve-brain) to collapse into one or the other possibility. The outcome is therefore the mixture:
{ ½ a ½ 2 , headHEAD HEAD ; ½ b ½ 2 , tailTAIL TAIL } . (10)
This view of the role of consciousness in quantum measurement is still popular in New Age literature. But it leads to serious difficulties, which have been very well emphasized by Schrödinger's thought-experiment with the cat, and not many physicists would vouch for it at present (Wigner himself later abandoned this proposal).
A type (ii) proposal is Everett's "many-worlds" idea. Everett assumes that the superposition (7) correctly describes the outcome of a measurement process. But he claims that each term of the superposition exists in a different world. With every quantum measurement the universe branches out and the various measurement results exist simultaneously in different worlds, together with different copies of ourselves, the experimenters. We are unaware of these other replicas of ourselves and of their worlds, and therefore we see only one result. A science-fiction idea, which nevertheless still has a non-negligible following.
A different type (ii) proposal which is at present popular among physicists is the so-called "de-coherence" approach, to which many people have contributed. The work of the Milano school, to which I gave a small contribution with my tesi di laurea, laid the foundation for it in the late 60's. The idea is that, although the superposition (7) and the mixture (8) represent in principle profoundly different situations, when they are applied to macroscopic systems their difference becomes unobservable. Various factors contribute to this. For one thing all kinds of random processes take place in a macroscopic body. Furthermore a macroscopic body can never be thought of as being isolated: it constantly interacts with its environment. Both these factors have the effect of smearing out the difference between (7) and (8), making it practically unobservable. The actual outcome of a measuring process is still the superposition (7): but it can be replaced FAPP, "for all practical purposes", as John Bell used to say, by the mixture (8).
Is the de-coherence approach a true solution of the measurement problem? How one answers this question depends on one's evaluation of FAPP and opinions in this respect vary. The predictions derived from the superposition (7) are never shown to fully coincide with those derived from the mixture (8). The de-coherence results are always approximate – but the approximation is better than any conceivable experimental error. De-coherence is a FAPP-solution of the measurement problem.
A new proposal
The approach I am proposing is also of type (ii), i.e. it takes the consequences of quantum physics seriously and assumes that the world is indeed an entangled web of potentialities. Within the framework of our present understanding, i.e. within the model provided by quantum physics, that's the nature of reality. Therefore my question is not how the entangled web of potentialities collapses into a solid, classical world, but how comes that this entangled web appears to us as a solid, classical world.
What I am doing is in a way similar to de-coherence. The difference is that it is much easier and it is exact. De-coherence is hard work: it finds a difficult answer, and one which moreover is only approximately true. I believe that there is a simple and exact answer.
As a first approximation the simple answers can be stated as follows:
the quantum world appears classical to us not because of the macroscopic nature of our measuring devices, but beacause we, as observers, are part of the world we observe.
Let me now try to make this statement a bit more precise. We are physical beings: our gathering information about the world happens through the interaction of our body with other systems, and we can analyze this interaction in terms of the von Neumann chain of systems described in the previous section. The important point is that, in order for us to know something, a trace must remain somewhere along the line of such a von Neumann chain. We know only insofar as such a physical trace exists. About processes that don't leave any trace we cannot say anything, because we never experience them! Our knowing is anchored in traces happening along von Neumann chains of interacting systems. And the way the world appears to us is conditioned by that general condition of knowing.
To explain why that is so is much easier in mathematical terms than in words. The difficulty with ordinary language is that it not meant to deal with things like superpositions. Our ancestors didn't need them for their survival. An explanation in words is bound to be only suggestive, rather than a precise statement of the argument. But here it goes, anyway.
To focus our thoughts, let's stay with the quantum coin measurement described above. In this context saying that the world appears as classical to us amounts to saying that every possible observation we can make is correctly predicted by the mixture (10) (i.e. outcomes of the quantum coin measurement can be subdivided into heads and tails, one or the other, no funny in-between business).
But the requirement that every possible observation should be correctly predicted by the mixture (10) is unnecessarily restrictive. Such requirement includes also observations that destroy all traces of the measurement performed on the quantum coin, and about these we don't need to say anything, because nobody ever experiences a "measurement" that leaves no traces. If no trace of the measurement process is left, it is not really a measurement. It is an ordinary quantum process, not involving an observer or a recording of any kind. For such processes we have strong experimental evidence that the ordinary quantum rules should apply, and we should expect a superposition, not a mixture, to correctly describe the outcome.
As I said, all our knowing is based on interactions which do leave a trace along a von Neumann chain. So, in the context of the quantum coin example, when we say that the world appears as classical to us we mean that every possible observation that preserves a trace of the quantum coin measurement is correctly predicted by the mixture (10). For that restricted class of observations we require the superposition (9) to be equivalent to the mixture (10).
Now comes the tricky part, the part that is more easily explained in mathematical language. It has to do with the notion of entanglement. In the superposition (9) all the systems involved (observed system, measuring apparatus, observer's brain) are entangled together. As I said, that means that they are no longer independent: whatever you do to one of them, you do to them all. Now the interesting thing for our purpose is that, in a certain sense, entanglement works also the other way around: if don't do something to one system, that also has consequences for the whole chain. If you leave a trace of the outcome of the quantum measurement in one system, i.e. if you allow the result of the quantum measurement to be in principle recoverable by an observation performed on that one system, then all the other systems in the chain will behave accordingly. I.e. the outcome of any observation performed on any of them will split up into a subset corresponding to the head result and one corresponding to the tail result. In other words, if you allow a trace of the measurement to persist in one of the entangled systems, it is as if it persisted in the whole chain, in spite of the fact that all other traces of the measurement may have been destroyed. The memory of one system is shared by the whole entangled chain. Under this condition the predictions derived from the superposition (9) are exactly the same as those derived from the mixture (10).
This result is exact and is independent of the microscopic or macroscopic nature of the systems involved. The preserved trace of the quantum measurement can be at any level, microscopic or macroscopic. In simple words, the general statement is: the outcome of any observation process that leaves a trace can be described in classical terms.
But our knowing happens only through processes that leave a trace (if nowhere else, at least in us!). That's why the world appears classical to us. Not because it is classical, in an ultimate sense (ultimate always within the context of our models). But because the conditions of our knowing make it appear that way.
Since the beginning of quantum theory physicists have felt that the classical appearance of the world had something to do with the role of the observer. That is indeed so, but not because the observer's consciousness collapses the wave function, as it was once hypothesized. The classical appearance of the world is connected with the conditions of our knowing in general. The world appears classical to us because, in order to know, we have to be entangled with what we know.
For an exact mathematical formulation of the above argument, see:
Shantena Augusto Sabbadini, "Persistence of Information in the Quantum Measurement Problem", Physics Essays,