# Deutsch’s Constructors

Deutsch & Marletto’s Constructors

David Deutsch, Chiara Marletto *Constructor theory of information *Roy. Soc. Proc. A February 2015 Volume: 471 Issue: 2174 Published 17 December 2014. DOI: 10.1098/rspa.2014.0540 Roy. Soc.

## Abstract

We propose a theory of information expressed solely in terms of which transformations of physical systems are possible and which are impossible—i.e. in constructor-theoretic terms. It includes conjectured, exact laws of physics expressing the regularities that allow information to be physically instantiated. Although these laws are directly about information, independently of the details of particular physical instantiations, information is not regarded as an a priori mathematical or logical concept, but as something whose nature and properties are determined by the laws of physics alone. This theory solves a problem at the foundations of existing information theory, namely that information and distinguishability are each defined in terms of the other. It also explains the relationship between classical and quantum information, and reveals the single, constructor-theoretic property underlying the most distinctive phenomena associated with the latter, including the lack of in-principle distinguishability of some states, the impossibility of cloning, the existence of pairs of variables that cannot simultaneously have sharp values, the fact that measurement processes can be both deterministic and unpredictable, the irreducible perturbation caused by measurement, and locally inaccessible information (as in entangled systems.

## 1 Information

[Our] theory of information consists of proposed principles of physics that explain the regularities in physical systems that are associated with information, including quantum information.

## 2 Constructor Theory

The basic principle of constructor theory is that

I. All other laws of physics are expressible entirely in terms of statements about which physical transformations are possible and which are impossible, and why.

…

That individual physical systems (and not just the entire physical world) have states and attributes in the sense we have described is guaranteed by Einstein’s principle of locality [8], which has a precise expression in constructor-theoretic form:

II. There exists a mode of description such that the state of the combined system S1⊕S2 of any two substrates S1 and S2 is the pair (x, y) of the states x of S1 and y of S2, and any construction undergone by S1 and not S2 can change only x and not y.

…

[Principle I] also rules out theories that are stochastic at a fundamental level. For instance, it implies that ‘generalized probabilistic theories’ [12], can at most be descriptions at an emergent level, because, translated into constructor-theory-like terms, they would include statements about something being possible or impossible with a given probability. …

…

A constructor-theoretic statement is one that refers only to substrates and which tasks on them are possible or impossible—not to constructors. Constructor theory is the theory that the (other) laws of physics can be expressed without referring explicitly to constructors.

## 5 Measurement

… A variable X of a substrate S is distinguishable if

(⋃ x∈X {x→ψ x }) ✓ − − 5.1,

where the {ψx} constitute an information variable—which … implies the possibility of a subsequent non-perturbing measurement, as required for communication. [And the tick indicates that this is not ruled out by the particular ‘laws of nature’ being considered.]

## 6 Conjectured principles of physics bearing on information

Crucially, the most important properties of information do not follow from the definitions we have given. Here, we seek the constructor-theoretic principles of physics that determine those properties. Of these, perhaps the most fundamental one cannot even be stated in the prevailing conception of fundamental physics, but it has an elegant expression in constructor theory. It is the interoperability principle

III. The combination of two substrates with information variables S1 and S2 is a substrate with information variable S1×S2, where multiplication symbol denotes the Cartesian product of sets.

…

We shall conjecture that this is a special case of a deeper principle: we expect that whenever there is a regularity among observable phenomena in a substrate (such as a set of its attributes being pairwise distinguishable), that is always because the phenomena are related by a unifying explanation—i.e. that they can all be distinguished by measuring some variable … . This principle, too, has an elegant, purely constructor-theoretic expression:

IV. If every pair of attributes in a variable X is distinguishable, then so is X.

And similarly,

V. If every state with attribute y is distinguishable from an attribute x, then so is y.

…

So if that cannot tell the difference between x (∞) and y (∞) [denoting unlimited supplies of instances of x, y] , nothing (that uses only generic resources) can. If something can, we call x and y ensemble distinguishable. Thus, we propose the principle:

IX. Any two disjoint, intrinsic attributes are ensemble distinguishable.

## My Comments

- Constructor theory does not constrain ‘real systems’ as such, but rather constrains theories of systems. It does not consider actual regularities in real systems, but rather regularities expressed by theories of systems. I consider this an important distinction.
- Constructor theory could be thought of as an application of category theory.
- The physical transformations are morphisms of the ‘physical states’.
- The substrates could be represented by categories whose morphisms map the physical states to names of substrates, defining inclusion.
- Attributes could be represented by morphisms from substrates to structured spaces.
- Constructors would then be functors on attributes.
- Constructor theory would then add constraints like those of the standard ‘natural transformations’.
- Classical physics would then correspond to cases where the some attributes constrain the other possible attributes, up to a form of equivalence.
- Otherwise, there would be no such constraints.
- From a category theory perspective, it would be more natural to consider a physical ‘space’ that did not necessarily comprise states.

- The concept of distinguishability is vital to the theory. Distinguishability is defined relative to a given theory and concerns theoretical possibilities, not practicalities. Thus even if a particular constructor theory denies the possibility that two things are distinguishable, experience might come to show that they are different, leading to an advance in the theory.
- The theory excludes explicit probabilities. Implicit probabilities arise through long-run behaviours, concerning what would happen in theory (not in practice – e.g. if the sun explodes). Valid probabilistic statements should be seen as ways of expressing constructor-theoretic statements. (These might be logical – as in ‘the probability of two Heads for a fair coin’ – or frequentist.)
- Einstein’s principle of locality may not be ‘really’ true for complex systems such as aggregate economies. But a variant in which we suppose only that an action can change only x and not y’ in the short term’ or ‘unless something exceptional happens’ could still lead to a useful constructor theory of complex system ‘in the short run’ or conditionally. One would then want to distinguish between more states to develop theory that was more local, albeit never absolutely local.
- Thus a physics-like theory of a system that is actually complex (and hence liable to surprise us) is at best a rationalisation of our concrete understanding of the system, and liable to be falsified by experience in the long-run.
- Hence in addition to being able to apply ‘the best available theory’ to a situation, it is may also be important to understand when the theory is more likely to be violated, and to look out for new (emerging?) things that need to be distinguished.(Which long-run may be rather soon, if we are ‘pushing the bounds’.)