by Jim Baggott
There’s nothing non-local, mysterious, or spooky about any of this. But then, we might ask, how is the correlation between A and B established? That’s easy. Alice and Bob used what they know about the two-particle quantum system from previous experience, and they coded this information in the total wavefunction. Remember, there’s a law of conservation which means that the only possible outcomes are A↑B↓ and A↓B↑. This law specifically excludes the possibilities A↑B↑ and A↓B↓, which is why these were not included in the expression for the total wavefunction. In other words, information about the correlation was ‘preloaded’ into the total wavefunction. That the correlation is indeed observed in the two laboratories—by virtue of a further communication transmitted in a very non-spooky way at speeds no faster than light—simply reflects that the information has been coded correctly. All our researchers have done is take information about past events and used it to predict the outcome of a series of subsequent events.
Rovelli’s interpretation requires that quantum systems enter a relationship before meaningful information about them can be gained. But there are alternatives which focus exclusively on the nature of the information associated with quantum systems. These are generally known as information-theoretic interpretations.
Drawing cues once more from Heisenberg’s positivism, physicist Anton Zeilinger has suggested that quantum mechanics is essentially a theory about information, in which what we call physical properties are actually propositions relating to this information, derived from previous experience. Such propositions can then be determined to be true or false through future observations. ‘In fact, the object therefore is a useful construct connecting observations,’ Zeilinger writes.12
In quantum mechanics, not all propositions can be simultaneously true—‘this system exhibits information characteristic of a linear particle trajectory’ and ‘this system exhibits information characteristic of wave interference’ cannot both be true simultaneously for the same system. This means that the amount of information about the system is necessarily limited or constrained.
Zeilinger identifies an elementary quantum system to be one that carries information sufficient to determine the truth of just one proposition. Now quantum systems may possess a number of physical properties that are classified as polar opposites, such as positive or negative, (+ or −), up or down, (↑ or ↓), left or right. We can think of these as ‘off’ or ‘on’, as binary numbers, 0 or 1, known in computing as ‘bits’. An elementary quantum system therefore carries just one bit of information. What we get now depends on what kind of experimental question we ask of the system.
Consider a quantum system prepared exclusively in an ↑ state. If we now assert the proposition ‘this system exhibits information characteristic of the ↑ state’, we will return the result ‘true’. But what if we assert ‘this system exhibits information characteristic of the + state’? As before, we’re obliged to rewrite the total wavefunction of the system as a superposition of + and − states. But now the information available in the system is insufficient for a simple ‘true’ or ‘false’ pronouncement. Instead the result is completely random. In some instances it will be ‘true’; in others it will be ‘false’, with equal (50:50) probability.
We can obviously scale up to multiple elementary systems, capable of carrying information sufficient to determine the truth of multiple propositions, involving multiple bits. An entangled state is a two-bit system involving two particles, in which the joint correlation between the particles requires more than one bit. Starting from some fairly simple information-theoretic principles, it is possible to reconstruct the entirety of quantum mechanics.
In Zeilinger’s formulation, quantum information is a manifestation of the underlying quantum properties, much like temperature is a manifestation of the underlying motions of atoms and molecules. But the theorist Jeffrey Bub has argued that quantum information is a new physical ‘primitive’, one that cannot be reduced to physical fields or particles.13 Bub’s interpretation of information is not dependent on the existence of observers, but rather represents a fundamental element of reality itself.
As with the relational interpretation, adopting the preconception that the wavefunction represents information about a physical system and not the system itself spares us all of the uncomfortable consequences that quantum mechanics appears to imply. But at the same time it is entirely counterintuitive to separate the physics of a quantum system from our representation of it. We struggle to resist the temptation to read more into our representation than might be warranted. Rovelli sympathizes. The metaphysical preconceptions which flow from Proposition #3 are to a significant degree second nature, developed over a considerable period of acquaintance with classical mechanics. The presumption of the reality of the base concepts of our representations ‘was a philosophical assumption to which science was obviously immensely indebted’.14 But it was always an assumption.
And look at what we get if we’re prepared to reject it. All our problems go away.
But the relational and information-theoretic interpretations demand a significant trade-off, and there is a heavy price to be paid. To gain these advantages we must relax our grip on reality itself, as Rovelli explains: ‘the abandonment of Einstein’s strict realism allows one to exempt himself from…. intellectual acrobatics’.15 We must content ourselves with what we can discover about quantum physical systems and use the mathematical formalism to interpret these in ways that allow us to predict the outcomes of future measurements. We know this works fantastically well. But don’t expect this interpretation to tell us what is actually going on.
What happens physically to an electron on its journey from an electron gun, through a plate with two slits, to a phosphorescent screen on which it is detected as a single bright spot? What happens physically when a quantum system with two measurement possibilities yields just one measurement outcome? What happens physically to Schrödinger’s cat before we lift the lid of the box, and look inside? What happens physically to both particles A and B when Alice detects particle A to be in an ↑ state? In this experiment, is there any kind of physical influence on particle B?
According to the relational and information-theoretic interpretations, there are simply no answers to these questions. This is not because we lack the wit to discover them, but because the questions themselves are meaningless. The quantum state of the electron has no significance until it establishes a relation with the phosphorescent screen, and until this happens we can say nothing meaningful about it. We can say nothing meaningful about a quantum system with two measurement possibilities until it establishes a relation with a measuring device, at which point we see one outcome. We can say nothing meaningful about the state of Schrödinger’s cat until we lift the lid, and establish a relation with it. Irrespective of the result Alice gets for particle A, we can say nothing meaningful about the state of particle B until Bob establishes a relation with it. We can say nothing meaningful about the state of the two-particle system AB until Alice establishes a relation with A, and Bob establishes a relation with B, and they then go on to establish a relationship between themselves.
There’s nothing to see here.
John Wheeler, every ready with an apt turn of phrase or a clever epithet, called it the ‘great smoky dragon’ (see Figure 10). We appear to have a handle on a quantum system at the start of some physical transformation—we can see the tail of the dragon—and at the finish we know the outcome—we can see the dragon’s head. But between start and finish it seems that we can say nothing meaningful about the physics. The body of the dragon is inaccessible, as though clouded in some obscure quantum fog.
Figure 10 Wheeler’s ‘great smoky dragon’.
Do you remember what Bohr is quoted as saying? ‘It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.’16
This compares rather neatly with a quote from Alfred J. Ayer, the British spokesperson for
the Vienna Circle’s particular brand of positivism:17
The originality of the logical positivists lay in their making the impossibility of metaphysics depend not upon the nature of what could be known but upon the nature of what could be said.
If we’re also mindful of Ludwig Wittgenstein’s famous caution: ‘Whereof one cannot speak, thereof one must be silent’,18 we’re led inexorably to a rather infamous conclusion. If we can say nothing meaningful about the physics but we have a perfectly satisfactory representation which works wonderfully well, then perhaps we should just shut up and calculate.
Although this last phrase is frequently attributed to Richard Feynman, it appears to have been coined by N. David Mermin. As a research student studying quantum mechanics in the 1950s, Mermin’s questions about meaning and interpretation were rebuffed by his professors:19
‘You’ll never get a PhD if you allow yourself to be distracted by such frivolities,’ they kept advising me, ‘so get back to serious business and produce some results.’ ‘Shut up,’ in other words, ‘and calculate.’ And so I did, and probably turned out much the better for it. At Harvard, they knew how to administer tough love in those olden days.
In my view, the relational and information-theoretic interpretations are firmly anti-realist in the sense of Proposition #3. But, mindful of Hacking’s wariness of judgements based just on representation, let’s ask a different question. Are the relational and information-theoretic interpretations simply passive, empirically adequate (and therefore anti-realist) interpretations, or are they something more active, in the spirit of Proposition #4?
My view is that if the wavefunction is coded information, gathered from previous experience of quantum phenomena, then these are surely rather passive representations. As we’ve seen, they can provide no basis for saying anything meaningful about the physics that gives rise to the information, and so they provide no deeper insight or understanding. Arguably, these approaches provide no real incentive to do anything differently, because there really is nothing to see here.
In these interpretations, we load the Ship of Science with all the empirical data we have gathered, we codify this in our passive representations, and we head straight for the rock shoal of Scylla, content with a rather empty instrumentalism.
* You can find out more about Rovelli, Smolin, and loop quantum gravity in Jim Baggott, Quantum Space: Loop Quantum Gravity and the Search for the Structure of Space, Time, and the Universe, published by Oxford University Press in 2018.
* Unless the two particles somehow communicate with each other at speeds faster than light, violating one of the fundamental postulates of Einstein’s special theory of relativity (which nobody in their right mind wants to do).
* We’ll see in Chapter 7 how real measurements in real laboratories have been performed specifically to test quantum mechanics in this way.
6
Quantum Mechanics is Complete
But We Need to Reinterpret What it Says
Revisiting Quantum Probability: Reasonable Axioms, Consistent Histories, and QBism
Where do we go from here?
Despite the implications of the Copenhagen interpretation, relational quantum mechanics, and information-theoretic interpretations, we might still have no wish to suggest that the quantum formalism is in any way incomplete. Is there nevertheless some solace to be gained by seeking to reinterpret what the theory says? This offers the advantage that we avoid messing about too much with the equations, as we know that these work perfectly well. Instead, we look hard at what some of the symbols in these equations might actually mean. This doesn’t necessarily lead us to adopt a more realist position, but it might help us to say something more meaningful about the underlying physics that the symbols are supposed to represent.
As a starting point, let’s just acknowledge that quantum mechanics appears to be an inherently probabilistic theory, and that it is founded on a set of axioms. Instead of wrestling endlessly with the interpretation of the theory, is it possible to reconstruct it completely using a different set of axioms, in a way that allows us to attach greater meaning to its concepts?
The theorist Lucien Hardy certainly thought so. In 2001, he posted a paper on the arXiv preprint archive in which he set out what he argued were ‘five reasonable axioms’ from which all of quantum mechanics can be deduced.1 These look nothing like the axioms I presented towards the end of Chapter 4. Gone is the completeness or ‘nothing to see here’ axiom. Gone are the ‘right set of keys’ and the ‘open the box’ axioms. There is no assumption of the Born rule, as such.
Hardy argued that the singular feature which distinguishes quantum mechanics from any theory of physics that has gone before is indeed its probabilistic nature. So, why not forget all about wave–particle duality, wavefunctions, operators, and observables and reconstruct it as a generalized form of probability theory? In fact, the first four of Hardy’s reasonable axioms serve to define the structure of classical probability, of the kind we would use quite happily to describe the outcomes we would anticipate from tossing a coin. It is the fifth axiom, which assumes that transformations between quantum states are continuous and reversible, which extends the foundations to include the possibility of quantum probability.* The rest of quantum mechanics then flows from these, including the Born rule.
At first sight, Hardy’s fifth axiom appears rather counterintuitive. In a theory that is characterized by discontinuities, it seems odd to assume that transformations between quantum states happen in a smooth, continuously incremental fashion. But this is necessary to set up the kind of scenario which just can’t happen in classical physics. ‘Heads’ can’t continuously and reversibly transform into ‘tails’. But the quantum states ↑ and ↓ can. Hardy’s fifth axiom allows for the possibility of quantum superposition, entanglement, and all the fun that follows. Quantum discontinuity is then interpreted straightforwardly as the transformation of our knowledge, from two possibilities (either ↑ or ↓) to one actuality, in just the same way we see that the coin has landed with ‘heads’ facing up.
Hardy’s paper follows something of a tradition in attempts to reconstruct quantum mechanics, and its publication sparked renewed interest in this general approach. Note, however, that any reconstruction of quantum mechanics as a general theory of probability might allow us to say some meaningful things about what goes into it and what comes out, but it tells us nothing whatsoever about what happens in between. ‘What the physical system is is not specified and plays no role in the results,’ explains Giulio Chiribella. Such probability theories ‘are the syntax of physical theories, once we strip them of the semantics’.2
There’s still nothing to see here.
We might be tempted to conclude that rushing to embrace an entirely probabilistic structure risks throwing the baby out with the bathwater, losing sight of whatever physics is contained within the conventional theory. Instead of discarding all the conventional axioms, is it possible just to be a little more selective?
Look back at the axioms detailed at the end of Chapter 4, and listed in the Appendix. If we’re accepting of the assumption that the wavefunction provides a complete description (Axiom #1), then we need to discover how we feel about the others. There doesn’t seem much to be gained by questioning the ‘right set of keys’, the ‘open the box’, or the ‘how it gets from here to there’ axioms, as these are most certainly necessary if we are to retain some predictability and extract the right kind of information from the wavefunction. Inevitably, our attention turns to Axiom #4, the Born rule or ‘What might we get?’ axiom, as this is where we sense some vulnerability.
In quantum mechanics, we tend to interpret the Born rule in terms of quantum probabilities that are established at the moment of measurement. The reason for this is quite simple and straightforward. Whether we interpret the wavefunction realistically or not, when we apply Schrödinger’s wave equation we get a description of the motion that is smooth and continuous, according to Axiom #5. The form o
f the wavefunction at some specific time can be used to predict the form of the wavefunction at some later time. In this sense, the Schrödinger equation works in much the same way as the classical equations of motion. It is only when we introduce an interaction or a transition of some kind that changes the state of a quantum system that we’re confronted with discontinuity—an electron ‘jumps’ to some higher-energy orbit, or the wavefunction collapses to one measurement outcome or the other, as God once more rolls the dice. This discontinuity does not—it simply cannot—appear anywhere in the Schrödinger equation.
This is the reason the Born rule is introduced as an axiom. There is nothing in the formalism itself that tells us unambiguously that this is how nature works. We apply the Born rule because this is the way we try to make sense of the inherent unpredictability of quantum physics. A quantum system has two possible measurement outcomes, but we can’t predict with certainty which outcome we will get in each individual measurement. We use the Born rule in an attempt to disguise our ignorance and to pretend that we really do know what’s going on. The only way we can do this is to assume it’s true.
In conventional quantum mechanics, we assume that quantum probability arises as a direct consequence of the measurement process. But what if we don’t do this? What if we reject the conventional interpretation of the Born rule, or find another, deeper, explanation for the seemingly unavoidable and inherent randomness of the quantum world?
Let’s be clear. Calculating the probability of getting a particular measurement outcome from the square of the total wavefunction is deeply ingrained in the way physicists use quantum mechanics, and nobody is suggesting that this should stop. What we’re suggesting instead is that the Born rule is seen not simply as a calculating device that has to be assumed because of the way quantum systems interact with our classical apparatus, but rather as an inevitable consequence of the underlying quantum physics, or of the way that we as human beings perceive this physics. Either way, this means changing the way we think about quantum probability.