1. Quick check §
Which principle yields \(K_{AB}=K_AK_B\) and the tensor product?
Hint 1
Count the fiducial probabilities of a pair.
Hint 2
Joint states from local statistics.
Worked solution
Local tomography.
Quantum Foundations · Chapter 17
What Was Assumed
Which principles did the work, what would the world be like without each, and what remains open?
In this chapter · 3 sections
Reading guide · preparation and approach
Starting point: The whole course.
Anchor example: The gallery of foil theories, reviewed one last time.
Principle at work: All principles of the course, compared.
Foil: Every foil met in the course.
This lesson includes explanations, worked examples, and problems with solutions. Advanced results whose proofs are outside the chapter are identified explicitly.
We began by refusing to read quantum theory off a table of experimental outcomes. Instead we stated a framework that every testable theory shares, added principles one at a time, and derived finite-dimensional quantum theory. Symmetry of space then fixed position and momentum.
A derivation is only as good as its list of assumptions. What, exactly, did we assume? We first collect every assumption, including the ones we labelled along the way, and ask what each did and which foil shows its content. Experiments can then be read as tests of particular principles against particular foils. Other researchers have derived quantum theory from other starting points, so we ask what their routes take as primitive instead of ours. Finally we ask which questions the derivation leaves open.
17.1 The Principles Reviewed
What did the derivation use, and what did each assumption buy? We first list them all, with their consequences and foils. Then we ask how experiments bear on each one.
17.1.1 Each principle and its foil
Which assumptions did we make, including those we labelled only in passing? The table collects them in the order they entered.
| Assumption | Main consequences | Foil |
|---|---|---|
| Framework: preparations, transformations, measurements, statistics of repeated trials, mixing | Convex state spaces; probabilities linear in states; effects and measurements | None: it is the precondition of a testable theory |
| Causality (framework): later choices do not change earlier statistics; operations on separate systems commute | No-signaling; reduced states of parts | None in this course: it is part of what “earlier” and “separate” mean |
| Finiteness | States as vectors in \(\mathbb R^K\); finite capacity \(N\) | Classical mechanics; infinite-dimensional systems such as a particle on a line |
| Local tomography | \(K_{AB}=K_AK_B\); tensor products; with reversibility, the three-dimensional ball and complex numbers | Real quantum theory |
| Composite capacity, \(N_{AB}=N_AN_B\) (after Hardy) | The power law \(K=N^r\); the count of a pair of bits | None worked out here |
| Counting assumption: \(K\) depends only on \(N\) (after Hardy; later a consequence of equivalence of subspaces) | \(K\) strictly increasing; \(K=N^r\) | Theories whose systems with equal \(N\) differ |
| Continuous reversibility | Intrinsic randomness; round state spaces; unitary dynamics and the Schrödinger equation | Classical probability theory; the square bit and boxworld |
| Closure of the reversible transformations under limits | A compact group, so averaging over it is possible | None worked out here |
| Equivalence of subspaces | \(K=N^2\); density operators on \(\mathbb C^N\) for every \(N\) | Theories whose faces differ from their genuine systems (no worked example) |
| All measurements allowed | Effects are all \(0\le E\le I\); the Born rule \(\tr(\rho E)\) | Theories with restricted measurements |
| Phase choice for time evolution (Bargmann’s result) | \(U(t+s)=U(t)U(s)\), hence a single generator | None: a quoted theorem, not a physical choice |
| Repeatability and minimal disturbance (for ideal measurements) | The Lüders rule | Devices that measure the same observable but disturb more |
| Indistinguishability | Symmetry sectors of identical particles; no detectable coherence between them | Labelled particles |
| Sector assumption: each kind of particle uses one sector | Bosons and fermions; exclusion and bunching | Parastatistics |
| Galilean symmetry | Position, momentum, and \([x,p]=i\hbar\) | Finite-dimensional theories |
What does the list give? The rows up to all measurements allowed give finite-dimensional quantum theory, by theorems proved in the course or cited with their assumptions. The later rows add structure that those principles do not supply. A few physical links were also imported and labelled as such: the identification of the generator of time translations with energy, and the connection between spin and statistics.
17.1.2 Experimental tests of the principles
If principles are not justified by experiments, what do experiments do? They expose principles to refutation. Each principle has an observable consequence that its foil does not share.
Local tomography separates complex from real quantum theory. Can a laboratory tell them apart? Marc-Olivier Renou and collaborators showed in 2021 that it can, in a network with two independent sources: real quantum theory cannot reproduce certain correlations that complex quantum theory predicts. Experiments with photons and with superconducting qubits, reported in 2022, observed the complex-theory correlations.
The Born rule can be tested too. Because probabilities are squared magnitudes of sums of amplitudes, the pattern behind three open slits is fixed by the patterns behind each pair of slits and each single slit. A theory with “higher-order interference” would add a genuinely three-slit term. A triple-slit experiment by Urbasi Sinha and collaborators in 2010 bounded any such term.
Bell tests and contextuality tests confirm consequences that exclude local and noncontextual hidden-variable foils. Each such experiment tests a principle against a definite alternative, which is the role experiments play in a principle theory.
Problem 17.1. For each foil, name a principle it violates: classical probability theory, real quantum theory, boxworld, the square bit as a single system.
Worked solution
Classical probability theory violates continuous reversibility. Real quantum theory violates local tomography. Boxworld violates continuous reversibility, since its reversible transformations are only local operations and permutations. The square bit violates continuous reversibility as a single system, since its symmetries are the eight discrete symmetries of a square.
17.2 Other Reconstructions
Ours is one route of several. Which features of it are robust? We compare routes by asking what each takes as primitive instead of our principles. Every successful one needs something like local tomography, something like reversibility or purification, and a way to fix the qubit.
17.2.1 Purification and information
What did the earliest complete route assume? Lucien Hardy’s 2001 derivation used five axioms:
- probabilities as limiting frequencies;
- simplicity, which selects the smallest \(K\);
- the subspace axiom;
- the composite rules \(K_{AB}=K_AK_B\) and \(N_{AB}=N_AN_B\);
- continuity of reversible transformations.
We followed his counting, but replaced simplicity by an argument from reversible interaction.
What if information processing is primitive? Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti treat processes as circuits. In 2011 they derived quantum theory from purification together with five informational axioms:
- causality;
- local discriminability, which is equivalent to local tomography;
- perfect distinguishability;
- ideal compression;
- pure conditioning.
Two shorter routes take information limits as primitive. Borivoje Dakić and Časlav Brukner start from an information capacity of one bit per elementary system, together with local tomography and continuous reversibility. Rob Clifton, Jeffrey Bub, and Hans Halvorson work in an algebraic setting. They characterize quantum theory by three impossibilities: signaling, copying of unknown states, and a certain cryptographic task called unconditionally secure bit commitment.
17.2.2 Quantum logic and other routes
What if we start from yes–no tests instead of probabilities? An older programme does so. It takes as primitive the yes–no tests of a system, called propositions, and the relations of compatibility, implication, and negation among them. Garrett Birkhoff and John von Neumann proposed it in 1936.
How far does it get? Constantin Piron showed that, under certain axioms on how propositions combine, they correspond to the closed subspaces of a Hilbert space. The scalars of that space need not be complex numbers at first. Maria Pia Solèr proved in 1995 that, if the space is infinite-dimensional and contains an infinite orthonormal sequence, the scalars are the real numbers, the complex numbers, or the quaternions. Gleason’s theorem then supplies the probabilities. This route assumes no probabilities at the start. The price is abstract axioms whose physical motivation is debated.
Other routes take still other primitives.
- Algebraic structure: Pascual Jordan, von Neumann, and Eugene Wigner classified in 1934 the algebras of observables that satisfy natural rules for squaring and adding.
- Consistency of a probability calculus: Philip Goyal, Kevin Knuth, and John Skilling derive the complex rules from symmetry requirements on how experimental outcomes combine.
- The rest of quantum theory: David Deutsch and David Wallace argue from decision theory, and Wojciech Zurek from symmetries of entanglement, for the Born rule alone.
The last group derives the Born rule from the rest of quantum theory, rather than the theory from principles.
Problem 17.2. Hardy’s simplicity axiom and this course’s reversible-interaction argument both lead to \(r=2\). What is the advantage of the second?
Worked solution
Simplicity is a methodological preference with no direct empirical content: a world with \(r=3\) would not violate it observably in any single experiment. Reversible interaction between two bits is a physical property that a foil violates, so the derivation rests on a claim about the world rather than about economy of description.
17.3 Open Questions
Which questions has the derivation closed, and which has it only sharpened? We first return to definite outcomes, which the framework took as primitive. Then we look beyond the finite-dimensional theory that the principles fix.
17.3.1 The measurement problem
Can a theory whose primitives are outcomes also describe the devices that produce them? The framework takes definite outcomes as primitive, and the principles then determine the theory of their statistics. Applied to the measuring apparatus itself, the theory describes an entangled superposition of outcomes. Decoherence makes such superpositions locally indistinguishable from mixtures, but it does not make one outcome occur. So the reconstruction sharpens the measurement problem into that precise question. Interpretations answer it differently, and some, such as objective-collapse models, make predictions that experiments are beginning to test.
Do the principles constrain the answers? They do. Any interpretation must reproduce intrinsic randomness for the theory’s states, contextuality for any deeper variables, and nonlocality for deeper variables of pairs of systems. These are not features of particular interpretations. They are consequences of the principles.
17.3.2 Beyond finite dimensions and beyond quantum theory
How far does the derivation reach? No reconstruction of infinite-dimensional quantum theory, or of quantum field theory, from operational principles is as complete as the finite-dimensional ones. Symmetry fixes position and momentum for finitely many particles. A principle-based derivation of field theory, including its inequivalent representations, remains open. So does the relation between quantum theory and gravity. There the assumption of a fixed background of space and time, used implicitly in every laboratory description, may fail.
What use is the larger framework, now that it has selected quantum theory? It contains every foil we met, and many more. That makes it a tool for testing quantum theory. Any proposed deviation, such as higher-order interference, real amplitudes, or stronger correlations, can be written as a theory in the framework. Its violated principle can be identified, and an experiment designed to look for it. Quantum theory is the unique theory satisfying a short list of principles. Whether nature satisfies them exactly is, in the end, an empirical question.
Problem 17.3. The 2022 network experiments observed correlations that real quantum theory cannot reproduce. Does this prove local tomography?
Worked solution
No. It excludes one specific foil, real quantum theory with the standard rule for composing independent sources, which violates local tomography. Other theories that violate local tomography in different ways are not tested by this experiment. As in every principle theory, experiments refute definite alternatives; the principle remains a general claim that survives each test it could have failed.
Summary
- Five operational principles—causality, local tomography, subspace equivalence, continuous reversibility, and purification—uniquely single out finite-dimensional complex quantum theory.
- Relaxing local tomography yields real-amplitude quantum mechanics (where \(K_{AB} > K_A K_B\)), while relaxing continuous reversibility or subspace equivalence leads to classical or post-quantum foil theories.
- Open foundational frontiers include reconstructing quantum field theory and understanding how causal structure behaves when spacetime geometry itself is in quantum superposition.
Questions and worked problems
These exercises draw on the whole chapter. Attempt each question before opening a hint, explain your reasoning, then compare it with the worked solution.
2. Quick check §
Which theory is the foil for the purification principle?
Hint 1
In which theory do mixed states have no pure extension?
Hint 2
Recall pure joint states with mixed parts.
Worked solution
Classical probability theory: a pure classical joint state has pure marginals, so no mixed classical state can be purified.
3. Quick check §
Name two physical links that the course imported rather than derived.
Hint 1
One concerns energy.
Hint 2
One concerns which particles are fermions.
Worked solution
The identification of the generator of time translations with energy and the spin–statistics connection. Galilean symmetry is a further input beyond the finite-dimensional principles.
4. Worked problem §
Drop continuous reversibility and keep the other principles. Which theories become possible, and what would a world described by them be like?
Hint 1
Classical probability theory satisfies the others.
Hint 2
What could not happen in such a world?
Worked solution
Classical probability theory, among others, becomes possible. In such a world every pure state would assign definite outcomes to sharp tests, randomness would always be ignorance, and a finite system could not change continuously and reversibly between distinguishable states. There would be no interference and no entanglement.
5. Worked problem §
Rafael Sorkin defined the third-order interference term \(I_3=P_{ABC}-P_{AB}-P_{AC}-P_{BC}+P_A+P_B+P_C\), where \(P_S\) is the probability at a detector with the slits in \(S\) open. Show that \(I_3=0\) when \(P_S=|\sum_{s\in S}\psi_s|^2\).
Hint 1
Expand \(|a+b+c|^2\).
Hint 2
Each cross term appears once in \(P_{ABC}\) and once in one pair term.
Worked solution
\(P_{ABC}=|a|^2+|b|^2+|c|^2+2\mathrm{Re}(ab^*+ac^*+bc^*)\). The pair terms sum to \(2(|a|^2+|b|^2+|c|^2)+2\mathrm{Re}(ab^*+ac^*+bc^*)\). So \(I_3=(\Sigma-2\Sigma+\Sigma)+(\text{cross}-\text{cross})=0\). The Born rule allows only pairwise interference, which triple-slit experiments test.
6. Worked problem §
Match principles of this course with axioms of other reconstructions: (a) Chiribella–D’Ariano–Perinotti’s local discriminability; (b) Dakić–Brukner’s information capacity of one bit per elementary system; (c) Hardy’s subspace axiom.
Hint 1
Local discriminability concerns composite states.
Hint 2
One bit means \(N=2\).
Worked solution
(a) Local tomography. (b) The elementary system is a bit, \(N=2\), which with finiteness starts the derivation of the ball and the qubit. (c) Equivalence of subspaces. Different programs share the same structural ingredients under different names.
7. Worked problem §
Solèr’s theorem leaves three possible fields: real, complex, and quaternionic. Which principles of this course exclude the real and quaternionic options?
Hint 1
Recall how the dimension of the ball was fixed.
Hint 2
Consider composite systems.
Worked solution
The real case is excluded by local tomography, since real quantum composites have more parameters than local statistics fix. The quaternionic case is excluded by counting (\(K=6\) for a bit) and by the failure of a well-behaved tensor product for composites. Both exclusions come from how systems combine.
8. Misconception §
A student says: “Since quantum theory has been derived from principles, it is now proven to be true.” Correct the statement.
Hint 1
What is the logical form of the derivation?
Hint 2
Recall why nature satisfying the principles is an empirical question.
Worked solution
The derivation shows that if the principles hold, the theory is quantum theory. Whether the principles hold is an empirical question, and each has foils that experiments could favor. The derivation makes quantum theory understandable and exposes it to sharper tests; it does not make it necessarily true.
9. Extension §
Describe two experimental outcomes that would refute a principle of the course, and say which principle each refutes.
Hint 1
Consider interference patterns and correlations.
Hint 2
Each foil predicts something quantum theory forbids.
Worked solution
A nonzero third-order interference term \(I_3\) would refute the Born rule, and with it the derivation that produces it, including the shape of the state space. Correlations between separated systems with CHSH value above \(2\sqrt2\) would refute the Hilbert-space structure of composites derived from local tomography and continuous reversibility. Observing composite states not determined by local statistics would refute local tomography directly.
10. Extension §
List the features that any interpretation of quantum theory must reproduce, according to the course, and explain why they are not features of a particular interpretation.
Hint 1
Think of randomness, contextuality, and Bell inequalities.
Hint 2
They follow from the principles.
Worked solution
Intrinsic randomness for the theory’s pure states, contextuality of any deeper variables, and nonlocality of any deeper variables for composites. They follow from the principles and from theorems about all hidden-variable completions, so every interpretation that reproduces quantum predictions must accommodate them, whatever it says about the measurement problem.