Source
Deutsch & Marletto (2015), LIT-1663, §§2, 6–8; read in NOTE-1219. Equations (8.1)–(8.3) checked against the arXiv preprint 1405.5563.
What was actually shown
The prohibition. A superinformation medium M is an information medium with at least two information observables, with mutually disjoint attributes, whose union is not an information observable (§7). “From here on, we propose no further principles.” For a qubit, any two orthogonal states form an information observable, and no union of two or more such sets does.
The features, each argued from the principles (§8, with X, Y such a pair):
- Indistinguishable attributes (8a). If every x ∈ X were distinguishable from every y ∈ Y, then IV makes X∪Y measurable, VI makes its attributes preparable, and measure–compute–reprepare gives every permutation. Pairwise measurement with III gives cloning. Then X∪Y would be an information variable, which is a contradiction.
- No-cloning (8c). Repeated cloning of {x, y} would prepare z(∞), and IX would make x(∞) and y(∞) distinguishable, so x and y would be.
- Undetectable sharpness (8b) and no joint sharpness or measurement (8d).
- Deterministic unpredictability (8e). For maximal X and input y, no prediction “the outcome will be x” is true, yet {χ̄̄_y, χ̄_y} is sharp, so it is predictable that the outcome is one of the x not distinguishable from y. This is the constructor-theoretic counterpart of the projector onto fewer than three photons.
- Irreducible disturbance (8f), the impossibility (8.1).
- Consistent repeated measurement (8g), through (8.2)–(8.3).
- Discrete and continuous (8h): each quantum information observable is discrete, and there is a continuum of them with no union an information observable.
- Locally inaccessible information (8i): under coherence, the controlled-not (8.4) and its extension (8.5) yield an information variable {(0,1), ψ(1)} of M1⊕M2 not measurable through products of local variables, while locality II holds.
Each step could have failed: a feature not reachable from the prohibition and the principles would have needed a further prohibition. The paper needs none.
What this does not say
- That principles II–IX hold. They are conjectured (§6). The authors say IV and V need hold only “for a special class of substrates”, and that VI and VII are simplifying assumptions.
- That superinformation is quantum theory. Quantum information is shown to be an instance. Whether other theories satisfy the prohibition is not asked.
- That quantum theory is derived. “We are not trying to derive quantum theory, or any other existing law, from anything” (§1).
- That the proofs are formal. They are prose arguments, and some cite principles loosely: 8c names “assumption VII” for preparability, which is VI.
- What explains the unpredictability. “It is up to the subsidiary theories to explain this deterministic unpredictability” (§8e); the Everettian account is offered as one such explanation, not derived.