What a Physical Theory Must Say

1.1 Principles and Construction

A natural first answer is that experiments justify a theory. If that were true, we could stop: quantum theory fits every experiment done so far. We first test that answer and find it too weak. Then we ask what other kind of justification a theory can have, and find a model in the way thermodynamics and relativity were built.

1.1.1 Why outcomes underdetermine theories

Can a table of outcomes, however large, force us to one theory? Suppose we try to read the theory off the data. We meet three obstacles.

The first is finiteness. Every experiment produces finitely many records, and infinitely many rules agree with any finite list. A lookup table that stores every observed outcome “explains” the data perfectly, yet it predicts nothing. So choosing between rules always uses something beyond the data, such as simplicity, symmetry, or consistency with other domains.

The second is that descriptions of experiments are already theoretical. To say that a device is “turned by \(\theta\),” we assume we know how turning acts on whatever passes through it. The data arrive shaped by the concepts we brought to them. So the data cannot justify those concepts independently.

The third is universality. A theory claims to apply to every system of a given kind, in every arrangement, including arrangements nobody has built. The two-device experiment concerns one kind of system in one family of arrangements. Bell’s model shows what this leaves open. It reproduces every single-particle measurement, yet it predicts different correlations for pairs, and experiments on pairs favor quantum theory. So the experiments that usually introduce quantum mechanics make it plausible, not necessary.

1.1.2 Principle theories and constructive theories

If data cannot select a theory, what can? Physics already has an answer. In a 1919 article Albert Einstein distinguished two kinds of theory.

A constructive theory builds complex phenomena from simpler parts that it assumes. The kinetic theory of gases is an example: it builds pressure and temperature from molecules in motion. A principle theory starts instead from general regularities, raised to principles that every process must satisfy. Thermodynamics is the standard example. From the impossibility of perpetual-motion machines it derives limits on every engine, whatever the engine is made of. Special relativity is another. It derives how lengths and times transform from two principles: all inertial observers find the same laws, and all measure the same speed of light.

Why should a principle theory carry more authority than a good fit to data? We can give three reasons. Its principles are few and general. Each can be stated without the formalism it will produce, so we can judge whether it is reasonable before seeing its consequences. And each has empirical content, because a violation would be observable. The theory then follows by mathematics, so accepting the principles means accepting the theory.

Can quantum theory be presented this way? Since 2001 several research programmes have shown that its finite-dimensional part can. Lucien Hardy derived it from five “reasonable axioms.” Lluís Masanes and Markus Müller gave a complete derivation from operational principles. Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti independently did the same from a different set. We follow mainly the route of Masanes and Müller. We borrow Hardy’s method of counting parameters, and we meet the purification principle of Chiribella, D’Ariano, and Perinotti as an alternative route when we study systems that lose information to their surroundings.

1.2 The Operational Language

To derive a theory from principles, we must state the principles in a language that does not already assume the theory. That language cannot presuppose quantum theory, classical mechanics, or any picture of what systems are made of. So we restrict it to what is done in a laboratory and what is recorded. We first name the kinds of laboratory procedure, then ask what a theory must predict about their records.

1.2.1 Preparations, transformations, and measurements

What does every experiment have in common, whatever the theory behind it? Something produces a system, something may act on it, and something records a result. We name these three kinds of recipe.

A preparation is a recipe that produces a system. Heating an oven of silver and selecting atoms in a narrow beam is one. A transformation acts on a system and returns a system, possibly of a different type. Letting the system fly through a field for a fixed time is one. A measurement acts on a system and produces one of a set of records, its outcomes. A spot in the upper half of a screen, or a click in one of two detectors, is an outcome.

An experiment is a preparation, followed by transformations, ending in a measurement. The records are classical in a plain sense: they can be copied, read by several people, and written in a notebook. Everything else in the description is a recipe. The language does not say what a system is, whether it has properties before measurement, or whether its behavior is determined. Those are questions a theory answers, and different theories answer them differently.

Suppose two recipes differ in their details, yet no experiment ever gives them different statistics. Then no theory can tell them apart, and we identify them. A state will be a class of preparations identified in this way, and the same holds for transformations and outcomes. This is not a hypothesis about nature. It is the decision to describe a system by what it can do.

The recipes in an experiment also come in a time order, so we must ask: can a choice made later change the statistics of a record written earlier? Suppose it could. Then the frequencies already in the notebook would depend on a choice not yet made, for example by a coin tossed after the record was written. We exclude this as part of what “earlier” means.

Causality (framework assumption). The probabilities of the outcomes of a measurement do not depend on which transformations or measurements are chosen after it. Operations on separate systems may be performed in either order without changing any statistics.

The second sentence matters only when there are several systems. It will let us show that distant laboratories cannot send each other messages through shared systems. Like the rest of the framework, causality is a condition for testing, not a finding about nature. We list it with the other assumptions when we review what the derivation used.

1.2.2 Repetition and relative frequencies

A single run produces a single outcome. What, then, can a theory predict that a record could contradict? It must say how often each outcome occurs when the same experiment is repeated.

Suppose an experiment with outcomes \(k=1,\dots,m\) is repeated \(n\) times with the same recipes, and outcome \(k\) occurs \(n_k\) times. Its relative frequency is \(n_k/n\). We assume that for each experiment there are numbers \(p_k\), the probabilities, such that the frequencies of long, independent runs cluster around them.

How closely do they cluster? Suppose each run produces outcome \(k\) with probability \(p\), independently of the others. Then the count \(n_k\) has mean \(np\) and standard deviation \(\sqrt{np(1-p)}\). Dividing by \(n\), the relative frequency has standard deviation \(\sqrt{p(1-p)/n}\). For \(p=1/2\) and \(n=1000\) this is \(\sqrt{0.25/1000}\approx0.016\), so observed fractions between about \(0.47\) and \(0.53\) are typical. Probabilities are the theory’s predictions, and frequencies are what we record. The comparison between them is statistical.

Does assuming probabilities assume that nature is random? No. A deterministic theory satisfies the assumption by predicting only probabilities \(0\) and \(1\) for completely specified preparations. Probabilities strictly between \(0\) and \(1\) then arise only from incomplete specification, such as an oven that emits atoms with random properties. Classical mechanics with uncertain initial conditions is a theory of this kind. Whether a completely specified preparation can still give probabilities strictly between \(0\) and \(1\), which would be randomness without ignorance, is something we will derive, not assume.

1.3 The Plan of the Course

We now have a language in which every testable theory can be stated. How do we get from it to one theory? A derivation can easily smuggle in what it claims to derive, and the usual route is to mix what every theory must say with what one theory happens to say. So we sort our ingredients into three kinds, and then ask how a principle can be judged when data alone cannot judge it.

1.3.1 Framework, principles, and theorems

What does every testable theory provide? The framework: preparations, transformations, measurements, probabilities for the outcomes of repeated experiments, the causality assumption, and the freedom to choose a preparation at random. We will see how much structure follows from the framework alone. States form a convex set, outcome probabilities are linear functions of states, and two systems can be considered together. Classical probability theory, quantum theory, and many other theories all fit.

What separates quantum theory from the others? The principles: general claims that a theory may or may not satisfy. We use five, each introduced where it does work. Stated in plain words, they are these.

1. Finiteness. A system has only finitely many states that one measurement can tell apart with certainty, and finitely many probabilities fix its state.
2. Local tomography. The state of two systems together is fixed by measurements made on each part separately, together with the correlations between their outcomes.
3. Continuous reversibility. Any state that is not a random mixture of others can be turned into any other such state by a continuous, reversible process.
4. Equivalence of subspaces. Systems that can hold the same number of perfectly distinguishable states behave alike, including those that sit inside larger systems.
5. All measurements allowed. Every rule for assigning probabilities that is consistent with the states is realized by some measurement.

Two further assumptions are needed along the way: how many distinguishable states a pair of systems has, and, for identical particles, which joint states occur. Each is labelled as an assumption where it enters.

What do the principles give us? The theorems. The states of the simplest system that is not classical form a three-dimensional ball. Every finite system is then described by matrices of complex numbers, and a fixed rule turns those matrices into probabilities. Reversible change is a continuous rotation of the states. Familiar consequences follow, from uncertainty relations to correlations between distant systems that no classical theory can produce. Some theorems are proved in full. Others rest on mathematical results that we state precisely and cite.

1.3.2 Foils: theories that break one principle

How can a principle be justified if not by data? Part of the answer is its generality, and part is that it has testable content. The rest comes from the alternatives. For each principle we present a foil: a consistent theory that satisfies the framework but violates that principle.

Each foil pairs with one principle. Classical probability theory violates continuous reversibility: it has no continuous path between two certain outcomes. Real quantum theory, a version of quantum theory with real numbers in place of complex ones, violates local tomography. Boxworld, a theory whose pairs of systems can be more strongly correlated than any quantum pair, also violates continuous reversibility: its pairs cannot be correlated by any reversible process. Theories with restricted measurements violate the last principle.

What do foils buy us? First, they show that each principle is independent, so none can be dropped without changing the theory. Second, they show what the world would be like without it. A classical world has no continuous change between distinguishable states. A real-number world lets two systems hold information that no local measurements reveal. Judging a principle then becomes judging which world is ours, and that is a question experiments can address.