Source
Deutsch & Marletto (2015), LIT-1663, §§1–5; read in NOTE-1219 from the published JATS XML (CC BY 4.0).
What was actually shown
The problem. A non-perturbing measurement (1.1) takes a blank receiver state b to x or y while leaving the message unchanged. But “this does not, in fact, distinguish message x from message y unless the resulting receiver states x and y are themselves distinguishable”. Taken as a definition, (1.1) is “recursive”, and orthogonality would not do as its base, because one would then have to say “what it is about orthogonal states that would make them usable for communication” (§1).
The order of the definitions. A computation variable is a set S of at least two attributes for which every permutation task C_Π(S) = ⋃{x∈S} {x → Π(x)} is possible with side-effects (§3). An information variable is “a clonable computation variable”, with cloning R_S(x0) = ⋃{x∈S} {(x, x0) → (x, x)} on S⊕S (4.1). Only then is X distinguishable, when ⋃_{x∈X} {x → ψ_x} is possible with side-effects for an information variable {ψ_x} (5.1). “That provides the base for the recursive definition (1.1), because we have defined ‘information variable’ above without reference to distinguishability, and only to the possibility or impossibility of tasks (specifically, permutations and cloning)” (§5).
Counterfactual and intrinsic. “A substrate S instantiates classical information if some information variable S of S is sharp, and if giving it any of the other attributes in S was possible. Note that the constructor-theoretic mode of explanation has allowed this to be expressed as an exact, intrinsic property of the substrate” (§4). “Intrinsic” is defined in §2: an attribute “that does not refer to any other specific substrate”.
The construction could have failed. Had information variable needed distinguishability, the circle would have stayed. It does not, and that can be checked on the page.
What this does not say
- That the defined notions are the intuitive ones. “We think that these correspond reasonably closely to the intuitive concepts with those names, but our claims in no way depend on that being so” (§3).
- That information has its expected properties. “The most important properties of information do not follow from the definitions we have given” (§6). Those come from the conjectured principles III–IX, which are not established here.
- That an information medium is individuated with no reference to anything else. Cloning uses a second instance of the substrate, and (5.1) uses an information variable of some other medium. These are generic others, so the property is intrinsic only in the §2 sense. It is a modal property, not a relational constitution, and not a non-modal intrinsic nature.
- That possibility requires an actual constructor. “No perfect constructors exist in nature” (§2); a possible task is one the laws allow to be approximated without limit.