Constructor theory of probability.

Unitary quantum theory, having no Born Rule, is non-probabilistic. Hence the notorious problem of reconciling it with the unpredictability and appearance of stochasticity in quantum measurements. Generalizing and improving upon the so-called 'decision-theoretic approach', I shall recast that problem in the recently proposed constructor theory of information-where quantum theory is represented as one of a class of superinformation theories, which are local, non-probabilistic theories conforming to certain constructor-theoretic conditions. I prove that the unpredictability of measurement outcomes (to which constructor theory gives an exact meaning) necessarily arises in superinformation theories. Then I explain how the appearance of stochasticity in (finitely many) repeated measurements can arise under superinformation theories. And I establish sufficient conditions for a superinformation theory to inform decisions (made under it) as if it were probabilistic, via a Deutsch-Wallace-type argument-thus defining a class of decision-supporting superinformation theories. This broadens the domain of applicability of that argument to cover constructor-theory compliant theories. In addition, in this version some of the argument's assumptions, previously construed as merely decision-theoretic, follow from physical properties expressed by constructor-theoretic principles.

Introduction

Quantum theory without the Born Rule (hereinafter: unitary quantum theory) is deterministic [1]. Its viability as a universal physical theory has long been debated [1-4]. A contentious issue is how to reconcile its determinism with the unpredictability and appearance of stochasticity in quantum measurements [2,4,5]. Two problems emerge: (i) how unpredictability can occur in unitary quantum theory, as, absent the Born Rule and ‘collapse’ processes, single measurements do not deliver single observed outcomes (§1b), and (ii) how unitary quantum theory, being non-probabilistic, can adequately account for the appearance of stochasticity in repeated measurements (§1c).

These problems also arise in the constructor theory of information [6] (§§2 and 3). In that context unitary quantum theory is one of a class of superinformation theories, most of them yet to be discovered, which are elegantly characterized by a simple, exact, constructor-theoretic condition. Specifically, certain physical systems permitted under such theories—called superinformation media—exhibit all the most distinctive properties of quantum systems. Like all theories conforming to the principles [7] of constructor theory, superinformation theories are expressed solely via statements about possible and impossible tasks, and are necessarily non-probabilistic. So a task being ‘possible’ in this sense means that it could be performed with arbitrarily high accuracy—not that it will happen with non-zero probability. Just as for unitary quantum theory, therefore, an explanation is required for how superinformation theories could account for unpredictable measurement outcomes and apparently stochastic processes.

To provide this, I shall first provide an exact criterion for unpredictability in constructor theory (§4); then I shall show that unpredictability necessarily arises in superinformation theories (including quantum theory) as a result of the impossibility of cloning certain states—thereby addressing problem (i). Then, I shall generalize and improve upon an existing class of proposed solutions to problem (ii) in quantum theory—known as the decision-theoretic approach [1,8-10], by recasting them in constructor theory. This will entail expressing a number of physical conditions on superinformation theories for them to support the decision-theoretic approach—thus defining a class of decision-supporting superinformation theories, which include unitary quantum theory (§§5–7). As I shall outline (§1d), switching to constructor theory widens the domain of applicability of such approaches, to cover certain generalizations of quantum theory; it also clarifies the assumptions on which Deutsch–Wallace-type approaches are based, by revealing that most of the assumptions are not decision-theoretic, as previously thought [1,8], but physical.

Constructor theory

Constructor theory is a proposed fundamental theory of physics [7], consisting of principles that underlie other physical theories (such as laws of motion of elementary particles, etc.), called subsidiary theories in this context. Its mode of explanation requires physical laws to be expressed exclusively via statements about which physical transformations (more precisely, tasks—§2) are possible, which are impossible, and why. This is a radical departure from the prevailing conception of fundamental physics, which instead expresses laws as predictions about what happens, given dynamical equations and boundary conditions in space–time.

As I shall explain (§2), constructor theory is not just a framework (e.g. [11], or category theory [12]) for reformulating existing theories: its principles are proposed physical laws, which supplement the content of existing theories. They express regularities among subsidiary theories, including new ones that the prevailing conception cannot adequately capture. They thereby address some of those theories' unsolved problems and illuminate the theories' underlying meaning, informing the development of successors—just as, for instance, the principle of energy conservation explains invariance properties of the dynamical laws, supplementing their explanatory content without changing their predictive content, providing criteria to guide the search for future theories.

In this work, I appeal to the principles of the constructor theory of information [6]. They express the regularities in physical laws that are implicitly required by theories of information (e.g. Shannon's) as exact statements about possible and impossible tasks, thus giving a physical meaning to the hitherto fuzzily defined notion of information. Notions such as measurement and distinguishability, which are problematic to express in quantum theory, yet are essential to the decision-theoretic approach, can be exactly expressed in constructor theory.

Unpredictability

The distinction between unpredictability and randomness is difficult to pin down in quantum theory, especially unitary quantum theory, but it can be naturally expressed in the more general context of constructor theory. Unpredictability occurs in quantum systems even given perfect knowledge of dynamical laws and initial conditions.[1] When a perfect measurer of a quantum observable —say the x-component of the spin of a spin-1/2 particle—is presented with the particle in a superposition (or mixture) of eigenvectors of , say and , it is impossible to predict reliably which outcome (0 or 1) will be observed. But in unitary quantum theory, a perfect measurement of is merely a unitary transformation on the source (the system to be measured) and the target (the ‘tape’ of the measurer): which implies, by the linearity of quantum theory, for arbitrary complex amplitudes . As no wavefunction collapse occurs, there is no single ‘observed outcome’. All possible outcomes occur simultaneously: in what sense, then, are they unpredictable?

Additional explanations are needed—e.g. Everett's [13] is that the observer differentiates into multiple instances, each observing a different outcome, whence the impossibility of predicting which one [1,5]. Such accounts, however, can only ever be approximate in quantum theory, as they rely on emergent notions such as observed outcomes and ‘universes’. Also, unpredictability is a counterfactual property: it is not about what will happen, but what cannot be made to happen. So, while the prevailing conception struggles to accommodate it, constructor theory does so naturally. Just as the impossibility of cloning a set of non-orthogonal quantum states is an exact property [14], I shall express unpredictability exactly as a consequence of the impossibility of cloning certain sets of states under superinformation theories (§4). This distinguishes it from (apparent) randomness, which, as I shall explain, requires a quantitative explanation.

The appearance of stochasticity

Another key finding of this paper is a sufficient set of conditions for superinformation theories to support a generalization of the decision-theory approach to probability, thereby explaining the appearance of stochasticity, namely that repeated identical measurements not only have different unpredictable outcomes but are also, to all appearances, random. Specifically, consider the frequencies of each observed outcome[2] x in multiple measurements of a quantum observable on N systems each prepared in a superposition or mixture of -eigenstates . The appearance of stochasticity is that, for sufficiently large N, the frequencies do not differ significantly (according to some a priori fixed statistical test) from the numbers (equality occurring in the limiting case of an ensemble (§6)).

To account for this, the Born Rule states that the probability that x is the outcome of any individual -measurement is , thus linking, by fiat, with the frequencies in finite sequences of experiments. In unitary quantum theory no such link can, prima facie, exist, as all possible outcomes occur in reality. How can that theory inform an expectation about finite sequences of experiments, as its Born-Rule-endowed counterpart can?

The decision-theoretic approach claims to explain how [1,8-10,15-17]. It models measurements as deterministic games of chance: is measured on a superposition or mixture of -eigenstates; the reward is equal (in some currency) to the observed outcome. Thus, the problem is recast as that of how unitary quantum theory can inform decisions of a hypothetical rational player of that game, satisfying only non-probabilistic axioms of rationality. The decision-theory argument shows that the player, knowing unitary quantum theory (with no Born Rule) and the state , reaches the same decision, in situations where the Born Rule would apply, as if they were informed by a stochastic theory with Born-Rule probabilities . This explains how can inform expectations in single measurements under unitary quantum theory. One must additionally prove, from this, that unitary quantum theory is as testable as its Born-Rule-endowed counterpart [1,15,18].

Thus, the decision-theoretic approach claims to explain the appearance of stochasticity in unitary quantum theory without invoking stochastic laws (or axioms), rather as Darwin's theory of evolution explains the appearance of design in biological adaptations without invoking a designer. It has been challenged, especially in regard to testability [4,19], and defended in, for example, [1,15,18]. Note that this work is not a defence of that approach; rather, it aims at clarifying and illuminating its assumptions (showing that most of them are physical) and at broadening its domain of applicability to more general theories than quantum theory.

In my generalized version of the decision-theoretic approach, I shall define a game of chance under superinformation theories (§7) and then identify a sufficient set of conditions for them to support decisions (under that approach) in the presence of unpredictability (§6). These conditions define the class of decision-supporting superinformation theories (including unitary quantum theory). Specifically, they include conditions for superinformation theories to support the generalization of the numbers (§5) corresponding to Born-Rule probabilities. That is to say, my version of the decision-theory argument explains how the numbers can inform decisions of a player satisfying non-probabilistic rationality axioms under certain superinformation theories (§7). Those theories would account for the appearance of stochasticity at least as adequately as unitary quantum theory.

Summary of the main results

Switching to constructor theory yields three interrelated results:

(1) The unpredictability of measurements in superinformation theories is exactly distinguished from the appearance of stochasticity, and proved to follow from the constructor-theory generalization of the quantum no-cloning theorem (§4).

(2) A sufficient set of conditions for superinformation theories to support the decision-theoretic argument (§§5–7) is provided, defining a class of decision-supporting superinformation theories, including unitary quantum theory. Constructor theory emancipates the argument from formalisms and concepts specific to (Everettian) quantum theory—such as ‘observed outcomes’ or ‘relative states’.

(3) Most premises of the decision-theory argument are no longer controversial decision-theoretic axioms, as in existing formulations, but follow from physical properties implied by exact principles of constructor theory.

In §§2 and 3, I summarize as much of constructor theory as is needed; in §4, I present the criterion for unpredictability; in §§5 and 6, I give the condition for superinformation theories to permit the constructor-theoretic generalization of the numbers ; in §7, I present the decision-theory argument in constructor theory.

Constructor theory

In constructor-theoretic physics the primitive notion of a ‘physical system’ is replaced by the slightly different notion of a substrate—a physical system some of whose properties can be changed by a physical transformation. Constructor theory's primitive elements are tasks (as defined below), which intuitively can be thought of as the specifications of physical transformations affecting substrates. Its laws take the form of conditions on possible/impossible tasks on substrates allowed by subsidiary theories. As tasks involving individual states are rarely fundamental, more general descriptors for substrates are convenient.

Attributes and variables

The subsidiary theory must provide a collection of states, attributes and variables, for any given substrate. These are physical properties of the substrate and can be represented in several interrelated ways. For example, a traffic light is a substrate, each of whose eight states (of three lamps, each of which can be on or off) is labelled by a binary string , where, say, indicates that the red lamp is off, and that it is on. Similarly, for (amber) and (green). Thus, for instance, the state where the red lamp is on and the others off is .

An attribute is any property of a substrate that can be formally defined as a set of all the states in which the substrate has that property. So, for example, the attribute red of the traffic light, denoted by , is the set of all states in which the red lamp is on: .

An intrinsic attribute is one that can be specified without referring to any other specific system. For example, ‘having the same colour lamp on’ is an intrinsic attribute of a pair of traffic lights, but ‘having the same colour lamp on as the other one in the pair’ is not an intrinsic attribute of either of them. In quantum theory, ‘being entangled with each other’ is, likewise, an intrinsic attribute of a qubit pair; ‘having a particular density operator’ is an intrinsic attribute of a qubit that has, for instance, undergone an entangling interaction. The rest of the quantum state, in the Heisenberg picture, describes entanglement with the other systems that the qubit has interacted with, and so is not an intrinsic attribute of the qubit.

A physical variable is defined in a slightly unfamiliar way as any set of disjoint attributes of the same substrate. In quantum theory, this includes not only all observables (which are representable as Hermitian operators), but many other constructs, such as any set where and are the attributes of being, respectively, in distinct non-orthogonal states and of a quantum system—i.e. the eigenvalues of two non-commuting observables. Whenever a substrate is in a state in an attribute , where is a variable, we say that is sharp (on that system), with the value x—where the x are members of the set X of labels[3] of the attributes in . As a shorthand, ‘ is sharp in ’ shall mean that the attribute is a subset of some attribute in . In the case of the traffic light, ‘whether some lamp is on’ is the variable , where I have introduced the attributes and , which contains all the states where at least one lamp is on. So, when the traffic light is, say, in the state where only the red lamp is on, we say that ‘ is sharp with value on’. Also, we say that is sharp in the attribute (red, defined above), with value on—which means that . In quantum theory, a substrate can be a quantum spin-1/2 particle—e.g. an electron. The z-component of the spin is a variable, represented as the set of two intrinsic attributes: that of the z-component of the spin being 1/2 and −1/2. That variable is sharp when the qubit is in a pure eigenstate of the observable corresponding to the z-component of the spin and is non-sharp otherwise.

Tasks

A task is the abstract specification of a physical transformation on a substrate, which is transformed from having some physical attribute to having another. It is expressed as a set of ordered pairs of input/output attributes of the substrates. I shall represent it as:[4]

The are the legitimate input attributes; the are the output attributes. A constructor for the task is defined as a physical system that would cause to occur on the substrates and would remain unchanged in its ability to cause that again. Schematically, where constructor and substrates jointly are isolated. This scheme draws upon two primitive notions that must be given physical meanings by the subsidiary theories, namely: the substrates with the input attribute are presented to the constructor, which delivers the substrates with the output attribute. A constructor is capable of performing if, whenever presented with the substrates with a legitimate input attribute of (i.e. in any state in that attribute), it delivers them in some state in one of the corresponding output attributes, regardless of how it acts on the substrate with any other attribute. A task on the traffic light substrate is ; and a constructor for it is a device that must switch off all its lamps whenever presented when any of the states in . In the case of the task , it is enough that, when the traffic light as a whole is switched off (in the state ), it delivers some state in the attribute —say by switching on the red lamp only, delivering the state —not necessarily all of them. In quantum information, for instance, a unitary quantum gate can be thought of as implementing a one-to-one possible task on the qubits that it acts on—its substrates. The physical system implementing the gate and the substrates constitute an isolated system: the gate is the same after the task as before. (Impossible tasks do not have constructors and thus cannot be thought of as corresponding to gates obeying quantum theory.)

The fundamental principle

A task is impossible if there is a law of physics that forbids it being carried out with arbitrary accuracy and reliability by a constructor. Otherwise, is possible, which I shall denote by . This means that a constructor capable of performing can be physically realized with arbitrary accuracy and reliability (short of perfection). Catalysts and computers are examples of approximations to constructors. So, ‘ is possible’ means that can be brought about with arbitrary accuracy, but it does not imply that it will happen, as it does not imply that a constructor for it will ever be built and presented with the right substrate. Conversely, a prediction that will happen with some probability would not imply 's possibility: that ‘rolling a seven’ sometimes happens when shooting dice does not imply that the task ‘roll a seven under the rules of that game’ can be performed with arbitrarily high accuracy.

Non-probabilistic, counterfactual properties—i.e. about what does not happen, but could—are central to constructor theory's mode of explanation, as expressed by its fundamental principle: The radically different mode of explanation employed by this principle permits the formulation of new laws of physics (e.g. constructor information theory's ones). Thus, constructor theory differs in motivation and content from existing operational frameworks, such as resource theory [11], which aims at proving theorems following from subsidiary theories, allowing their formal properties to be expressed in the resource-theoretic formalism. Constructor theory, by contrast, proposes new principles, not derivable from subsidiary theories, to supplement them, elucidate their physical meaning and impose severe restrictions ruling out some of them. In addition, while resource theory focuses on allowed/forbidden processes under certain dynamical laws, constructor theory's main objects are impossible and possible tasks; the latter are not just allowed processes: they require a constructor to be possible.

I. All (other) laws of physics are expressible solely in terms of statements about which tasks are possible, which are impossible, and why.

However, resource theory might be applicable to express certain constructor-theoretic concepts. For instance, as remarked, the notion of a chemical catalyst, as recently formalized in resource theory [21], is related to that of a constructor. A constructor is distinguished among general catalyst-type objects in that it is required to be capable of performing the task reliably, repeatedly and with perfect accuracy. Hence an ideal constructor is not itself a physical object, but the limiting case of an infinite sequence of possible physical objects. Each element of the sequence is an approximation to a constructor, performing the task to some finite accuracy. For example, the task of copying a string of letters is a possible task; a perfect copier is never realized in reality; but one can in principle build increasingly accurate approximations to it: a template copier is inaccurate; for higher accuracies, one would need to include some error-correction mechanism. Thus, that a task is possible is a manner of speaking about the realizability of each element of such a sequence, except for the limiting, perfect constructor that is never physically realizable, because of the inevitability of errors and deterioration under our physical laws.

Hence principle I requires subsidiary theories to have two crucial properties (holding in unitary quantum theory): (i) they must support a topology on the set of physical processes they apply to, which gives a meaning to a sequence of approximate constructions, converging to an exact performance of ; (ii) they must be non-probabilistic—as they must be expressed exclusively as statements about possible/impossible tasks. For instance, the Born-Rule-endowed versions of quantum theory, being probabilistic, do not obey the principle.

Principle of locality

is the substrate consisting of substrates and . Constructor theory requires subsidiary theories to provide the following support for such a combination. First, if subsidiary theories designate any task as possible which has as the input substrate, they must provide a meaning for presenting and to the relevant constructor as the substrate . Second, they must conform to Einstein's [22] principle of locality in the form:

II. There exists a mode of description such that the state of is the pair of the states[5] of and of , and any construction undergone by and not can change only and not .

Unitary quantum theory satisfies principle II, as is explicit in the Heisenberg picture [23,24]. In that picture, the state of a quantum system is, at any one time, a minimal set of generators for the algebra of observables of that system, plus the Heisenberg state [25,26]. As the latter never changes, it can be abstracted away when specifying tasks: any residual ‘non-locality’ in that state [27] does not prevent quantum theory from satisfying principle II.

The parallel composition of two tasks and is the task whose net effect on a substrate is that of performing on M and on N. When is possible for some task on some generic, naturally occurring substrate (as defined in [6]), is possible with side effects, which is written . ( represents the side effect.) Note that in quantum theory the constructor performing a possible task on may generate entanglement between the two substrates. For example, consider two tasks and including only one pair of input/output attributes on some qubit, and let be the output attribute of , b that of . Let be the quantum projector for a qubit to have the attribute and for attribute . The constructor performing is required to deliver the two qubits in a quantum state in the +1-eigenspace of the projector (where the small denotes here the tensor product between quantum operators). Such eigenspaces in general include states describing entangled qubits.

Constructor theory of information

I shall now summarize the principles of the constructor theory of information [6]. These express exactly the properties required of physical laws by theories of (classical) information, computation and communication—such as the possibility of copying—as well as the exact relation between what has been called informally ‘quantum information’ and ‘classical information’.[6]

First, one defines computation media.[7] A computation medium with computation variable (at least two of whose attributes have labels in a set V) is a substrate on which the task of performing a permutation defined via the labels V is possible (with or without side effects), for all . is a reversible computation.[8]

Information media are computation media on which additional tasks are possible. Specifically, a variable is clonable if for some attribute of S the computation on namely cloning is possible (with or without side effects).[9] An information medium is a substrate with at least one clonable computation variable, called an information variable (whose attributes are called information attributes). For instance, a qubit is a computation medium with any set of two pure states, even if they are not orthogonal [6]; with orthogonal states it is an information medium. Information media must also obey the principles of constructor information theory, which I shall now recall.

Interoperability

Let and be variables of substrates and , respectively, and be the variable of whose attributes are labelled by the ordered pair , where and are the sets of labels of and , respectively, and × denotes the Cartesian product of sets. The interoperability principle is elegantly expressed as a constraint on the composite system of information media (and on their information variables):

III. The combination of two information media with information variables and is an information medium with information variable .

Distinguishing and measuring

These are expressed exactly in constructor theory as tasks involving information variables—without reference to any a priori notion of information. A variable of a substrate S is distinguishable if where is an information variable (whereby if ). I write if is a distinguishable variable. Information variables are distinguishable, by the interoperability principle III.[10]

A variable is measurable if a special case of the distinguishing task (3.2) is possible (with or without side effects)—namely, when the original source substrate continues to exist[11] in some attribute and the result is stored in a target substrate: where is a generic, ‘receptive’ attribute and is an information variable of the target substrate, called the output variable (which may, but need not, contain ). When is sharp on the source with any value x, the target is changed to having the information attribute , meaning 《S had attribute 》.

A measurer of is any constructor capable of performing the task (3.3) for some choice of its output variable, labelling and receptive state.[12] Thus, it also is a measurer of other variables: for example, it measures any subset of , or any coarsening of (a variable whose members are unions of attributes in ). Two notable coarsenings of are: , where the attributes are re-labelled with numbers (and combined accordingly), and , where the attributes are re-labelled with numbers (and likewise combined). I shall consider only non-perturbing measurements, i.e. in (3.3). Whenever the output variable is guaranteed to be sharp with a value ‘’, I shall say, with a slight abuse of terminology, that the measurer of “delivers a sharp output ‘”.

The ‘bar’ operation

Given an information attribute , define the attribute (-bar) as the union of all attributes that are distinguishable from . With this useful tool one can construct a Boolean information variable, defined as (which, as explained below, is a generalization of quantum projectors). Also, for any variable , define the attribute . The attribute is the constructor-theoretic generalization of the subspace spanned by a set of quantum states. For example, consider an information variable where and are the attributes of being in particular eigenstates of a non-degenerate quantum observable (which also has other eigenstates). Then, is the attribute of being in any of the possible superpositions and mixtures (prepared by any possible preparation[13]) of those two eigenstates of .

Consistency of measurement

In quantum theory repeated measurements of physical properties are consistent in the following sense. Consider the variable defined above. Let be the attribute of being in a particular eigenstate of orthogonal to both and . All measurers of the variable are then also measurers of the variable , so that all measurers of the former, when given any attribute , will give the same sharp output ‘’. The principle of consistency of measurement requires all subsidiary theories to have this property:

IV. Whenever a measurer of a variable would deliver a sharp output when presented with an attribute , all other measurers of would too.

It follows [6] that they would all deliver the same sharp output.

Observables

As (from the definition of ‘bar’) , attributes with have a useful property, whence the following constructor-theoretic generalization of quantum information observables: an information observable really has the attribute .[14] A necessary and sufficient condition for a variable to be an observable is that for all its attributes . For example, the above-defined variable is not an observable (a Z-measurer delivers a sharp output even when presented with a state , where ‘’ denotes set exclusion), but is.

Superinformation media

A superinformation medium S is an information medium with at least two information observables, and , that contain only mutually disjoint attributes and whose union is not an information observable. and are called complementary observables. For example, any pair of orthogonal states of a qubit constitutes an information observable, but no union of two or more such pairs does: its members are not all distinguishable. Superinformation theories are subsidiary theories obeying constructor theory and permitting superinformation media.

From that simple property it follows that superinformation media exhibit all the most distinctive properties of quantum systems [6]. In particular, the attributes in are the constructor-theoretic generalizations of what in quantum theory is called ‘being in a superposition or mixture’ of states in the complementary observable .

Generalized mixtures

Consider an attribute and define the observable . (In quantum theory could be the photon number observable in some cavity, , and the attribute of being in some superposition or mixture of some of its eigenstates, e.g. . In that case would contain two attributes, namely those of being in the states and , respectively.) One proves [6] that:

(1)  is non-sharp in as (where ‘’ denotes the empty set), and contains at least two attributes.

(2) Some coarsenings of are sharp in , just as in quantum theory—where the state is in the +1-eigenspace of the projector . The observable , , is the constructor-theoretic generalization of such a projector, and it is sharp in , with value . As in quantum theory, any measurer of presented with , followed by a computation whose output is «whether the outcome was one of the ‘’ with , will provide a sharp output , corresponding to «». (I adopt the convention that a ‘quoted’ attribute is the one that would be delivered by a measurement of the un-quoted one, with suitable labelling. Likewise for variables.) In summary: For any observable , I call an information variable a generalized mixture of (the attributes in) if either is in (then it is a trivial mixture) or and is sharp in with value . (In quantum theory, the could be attributes of being eigenstates of some non-degenerate quantum observable, and a generalized mixture of those attributes would be a quantum superposition or mixture of those eigenstates.)

(3) Let be the attribute[15] delivered by an X-measurer (with substrates ), when presented with (figure 1). (In quantum theory, could be an entangled state resulting from a measurement of , where was ). One can show that each of the local descriptors and has the same properties (1) and (2) as does, as follows:

Figure 1.

A measurer of X when X is not sharp in input.

Let . (i) is not sharp in . (If it were, with value ‘’, that would imply, via the property of observables, that , contrary to the defining property that .) (ii) Also, (for if , then could be distinguished from , contrary to assumption). (iii) . For a measurer of applied to the target substrate of an X-measurer is also a measurer of ; hence, when presented with , it must deliver a sharp output . By the property of observables, must be sharp in , with value . By the same argument, is not sharp in , i.e. ; also ; and is sharp in with value .

Intrinsic parts of attributes

The attributes and are not intrinsic, for each depends on the history of interactions with other systems. (In quantum theory, and are entangled.) However, because of the principle of locality, given an information observable , one can define the of the attribute as follows. Consider the attribute prepared by measuring on system using some particular substrate as the target substrate. In each such preparation, will have the same intrinsic attribute , which I shall call the X-intrinsic part of , which is therefore the union of all the attributes preparable in that way. The same construction defines the ‘X’-intrinsic part of .

It follows, from the corresponding property of : that is sharp in with value ; that ; that . Similarly for the ‘quoted’ variables and attributes. In quantum theory, and are attributes of having the reduced density matrices on and . Unlike in [31], they are not given any probabilistic interpretation. They are merely local descriptors of locally accessible information (defined deterministically in constructor theory [6]).

Successive measurements

In unitary quantum theory, the consistency of measurement (see above) is the feature that when successive measurers of are applied to the same source initially in the state , with two systems and as targets: the projector for «whether the two target substrates hold the same value» is sharp with value 1. In constructor theory, the generalization of that property is required to hold. Define a useful device, the -comparer[16] . It is a constructor for the task of comparing two instances of a substrate in regard to an observable defined on each: where if ′ (i.e. if the first two substrates (sources) hold attributes with the same label) and otherwise. is an information observable of a third substrate (target). In quantum theory if has eigenstates , is realized by a unitary that delivers (respectively ) whenever the state of the sources is in or a mixture thereof (respectively ). The equivalent holds under constructor theory, by the principle of consistency of measurement IV: delivers the output if and only if the sources hold an attribute in . Thus, in particular, if one of the sources has the attribute then is a measurer of , i.e. of whether the other source also has the attribute (a property used in §4).

The fact that a quantum would deliver a sharp if presented with the target substrates of successive measurements of an observable on the same source is what makes ‘relative states’ and ‘universes’ meaningful in Everettian quantum theory, because it makes the notion of ‘observed outcome in a universe’ meaningful even when the input variable of the measurer is not sharp. The same holds in superinformation theories (figure 2).

Figure 2.

Consistency of repeated measurements.

Unpredictability in superinformation media

I can now define unpredictability exactly in constructor theory, and show how it arises in superinformation media.

X-predictor

An X-predictor for the output of an X-measurer whose input attribute is drawn from some variable (in short: ‘X-predictor for ’) is a constructor for the task: where is an information observable whose attributes —each representing the prediction «the outcome of the X-measurer will be ‘’ given the attribute as input»—are required to satisfy the network of constructions in figure 3. first prepares with the information attribute specified by some information attribute ; then the X-measurer is applied to ; and then its target and the output of the predictor, , are presented to an ‘X’-comparer . If that delivers a sharp , the prediction is confirmed. If it would be confirmed for all , then is an X-predictor for . The exact definition of unpredictability is then:

Figure3.

The scheme defining an X-predictor .

A substrate exhibits unpredictability if, for some observable , there is a variable such that an X-predictor for is impossible.

Hence unpredictability is the impossibility of an X-predictor for a variable . Note the similarity to ‘no-cloning’, i.e. the impossibility of a constructor for the cloning task (3.2) on the variable .

No-cloning implies unpredictability

Indeed, I shall now show that superinformation theories (and thus unitary quantum theory) exhibit unpredictability as a consequence of the impossibility of cloning certain sets of attributes.

Consider two complementary observables and of a superinformation medium and define the variable . I show that there cannot be an X-predictor for . For suppose there were. The predictor's output information observable would have to include the observable . For, if =  for some , has to be sharp on the target of the measurer with value ‘’; so the ‘’-comparer yielding a sharp would require . (See §3: is a measurer of when is sharp on one of its sources with value ‘’.)

When = , the X-predictor's output attribute must still cause the X-comparer to output the sharp outcome ; also, is required to be an information variable: hence either for some ; or . In the former case, again by considering as a measurer of , would have to be sharp on the target of the X-measurer, with the value ‘’; whence , contrary to the definition of superinformation. In the latter, as , would have the attribute (§3) so that the ‘X’-comparer would have to output a sharp , again contradicting the assumptions. So, there cannot exist an X-predictor for , just as there cannot be a cloner for , because is not an information variable.

Thus, unpredictability is predicted by the superinformation theory's deterministic[17] laws. Its physical explanation is given by the subsidiary theory. In Everettian quantum theory, it is that there are different ‘observed outcomes’ across the multiverse. But constructor theory has emancipated unpredictability from ‘observers’, ‘relative states’ and ‘universes’, stating it as a qualitative information-theoretic property, just as no-cloning is.

X-indistinguishability equivalence classes

Quantum systems exhibit the appearance of stochasticity, which is more than mere unpredictability. Consider a quantum observable of a d-dimensional system S, with eigenstates and eigenvalues x. Successive measurements of on N instances of S, each identically prepared in a superposition or mixture of -eigenstates, display the following convergence property: (i) for large N, the fraction of replicas delivering the observed outcome[18] ‘x’ when is measured can be expected not to differ significantly (according to some a priori fixed statistical test) from the number ; (ii) in an ensemble (infinite collection) of such replicas, each prepared in state (a ‘-ensemble’, for brevity), the fraction of instances that would give rise to an observed outcome ‘x’ equals [32].

But what justifies the expectation in (i)? A frequentist approach to probability would simply postulate that the number from (ii) is the ‘probability’ of the outcome x when is measured on —which would imply (via ad hoc methodological rules; e.g. [18,33]) that could inform decisions about finitely many measurements. By contrast, in unitary quantum theory, it is the decision-theoretic approach that establishes the same conclusion—without probabilistic assumptions. Absent that argument, the numbers are just labels of equivalence classes within the set of superpositions and mixtures of the states . Each class, labelled by the d-tuple , , , is the set of states with . For instance, belongs to the class labelled by .

I shall now give sufficient conditions on superinformation theories for a generalization of those equivalence classes, which I shall call X-indistinguishability classes, to exist on the set of all generalized mixtures of attributes of a given observable . One of the conditions for a superinformation theory to support the decision-theoretic argument will be that they allow such classes (§6).

In quantum theory, the equivalence classes are labelled by the d-tuple , where . As the ‘trace’ operator need not be available in superinformation theories, to construct such equivalence classes I shall deploy fictitious ensembles. This is a novel mathematical construction; properties of such ensembles will be used to define properties of single systems without, of course, any probabilistic or frequentist interpretation.

At this stage, the are only labels of equivalence classes. Additional conditions will therefore be needed for the to inform decisions in the way that probabilities are assumed to do in stochastic theories (including traditional quantum theory via the Born Rule). I shall give these in §7 via the decision-theory argument. No conclusion about decisions could possibly follow merely from what the results of measurements on an infinite ensemble would be, which is what my formal definition of the is about.

X-indistinguishability classes

I denote by a substrate , consisting of N replicas of a substrate S. Let us fix an observable of S, whose attributes I suppose with no relevant loss of generality to be labelled by integers: , where . Let be the set of strings of length N whose digits can take values in X, each denoted by . is an observable of . In quantum theory, supposing that is an observable of a d-dimensional system S, might be , whose non-degenerate eigenstates are the strings of length N: .

Fix an N. For any attribute in , I define a constructor for the task of counting the number of replicas that hold a sharp value of : where the numbers label the attributes of the output information variable , with denoting the set of fractions with denominator N: . Thus, whenever presented with the substrate on which is sharp with value (i.e. is in a state ), outputs the number of instances of S on which is sharp with value . It could be realized, for instance, by measuring the observable ‘’ on each of the N substrates in , and then adding one unit to the output substrate, initially at 0, for each ‘’ detected (figure 4). In quantum theory, it effects a unitary operation:

Figure 4.

The constructor .

I shall now use to define attributes of the substrate , whose limit for will be used to define the X-indistinguishability classes.

Consider the observable containing the strings where a fraction of the replicas of S hold a sharp attribute . They have the property that when presented to they make the observable sharp with value . For example, for N = 3 and , and contains the quantum states . Now define the attribute as the union of all attributes of that when presented to make sharp in output, with value . Next, consider the observable . It follows from the consistency of measurement (§3) that:

Therefore, crucially, for a given , can be sharp even if the observable is not. In the example above, for N = 3 and , is the set of all superpositions and mixtures of the eigenstates of contained in , : is not sharp in most such mixtures and superpositions.

The observable is key to generalizing quantum theory's convergence property, for the latter is due to the fact that there exists the limit of the sequence of attributes for . Let me now recall the formal expression of the convergence property in quantum theory [32].[19]

Consider a state with the property that is a superposition of states each having a different . The convergence property is that for any positive, arbitrarily small : where . In other words, there exists an N such that the quantum implementation of , when asked whether the fraction of observed outcomes ‘x’ obtained when measuring on is within of the value , is in a state as close (in the natural Hilbert space norm provided by quantum theory) as desired to one in which it answers . Thus, the proportion of instances delivering the observed outcome ‘x’ when is measured on the ensemble state is equal to (where ). Therefore, all states with the property will be grouped by in the same set, as , which can thus be labelled by . This set is an attribute of a single system, containing all quantum states with the property that . The set of all superpositions and mixtures of eigenstates of is thus partitioned equivalence classes, labelled by the d-tuple , , .

A sufficient condition on a superinformation theory for a generalization of these X-indistinguishability classes to exist under it, on the set of all generalized mixtures of attributes of a given observable , is that it satisfies the following requirements: where ; is the limiting set of —which must exist, and its elements, which are real numbers, must have the property .

(E1) For each , there exists the attribute of the ensemble of replicas of S defined as

The existence of the limit implies that those attributes do not intersect, i.e. the set is a (formal) variable of the ensemble—a limiting case of , generalizing its quantum analogue.

Given an attribute of S, define (in quantum theory, this is the attribute of being in the quantum state ); introduce the auxiliary variable and let —which, unlike its ensemble counterpart , is an attribute of a single substrate .

(E2) For any generalized mixture of the attributes in , there exists a d-tuple with , , such that . I shall call the X-partition of unity for the attribute .[20]

If has such a partition of unity, it must be unique because of (E1). An X-indistinguishability equivalence class is defined as the set of all attributes with the same X-partition of unity: any two attributes within that class cannot be distinguished by measuring only the observable on each individual substrate, even in the limit of an infinite ensemble. In quantum theory, contains all states with . A superinformation theory admits X-partitions of unity (on the set of generalized mixtures of attributes in X) if conditions (E1) and (E2) are satisfied.

A key innovation of this paper is showing how the mathematical construction of an abstract infinite ensemble (culminating in the property (E1)) can define structure on individual systems (via property (E2))—the attributes and the X-partition of unity—without recourse to the frequency interpretation of probability or any other probabilistic assumption.

X-partition of unity of the X-intrinsic part

Consider now the attribute of being in the quantum state whose X-partition of unity is . In quantum theory, the reduced density matrices of the source and target substrate as delivered by an X-measurer acting on still have the same partition of unity. The same holds in constructor theory. Consider the X-intrinsic part (§4) of the attribute generated by measuring on the attribute . By definition of , prepending a measurer of to each of the input substrates of will still give a , with the same labellings. Thus, the construction that would classify as being in a certain X-partition of unity can be reinterpreted as providing a classification of , under the same labellings: the two classifications must coincide. If , the X-intrinsic part of must have the same one. Likewise for the intrinsic part of (obtained as the output attribute on the target of the X-measurement applied to ): the ‘X’-partition of unity of is numerically the same as the X-partition of unity of .

For a given observable and generalized mixture of attributes in the labels of the X-partition of unity defined in §5 are not probabilities. Even though they are numbers between 0 and 1, summing to unity, they need not satisfy other axioms of the probability calculus: for instance, in quantum interference experiments they do not obey the axiom of additivity of probabilities of mutually exclusive events [35]. It is the decision-theory argument (§7) that explains under what circumstances the numbers can inform decisions in experiments on finitely many instances as if they were probabilities. I shall now establish sufficient conditions on superinformation theories to support the decision-theory argument, thus characterizing decision-supporting superinformation theories.

I shall introduce one of the conditions via the special case of quantum theory. Consider the x- and y-components of a qubit spin, and . There exist eigenstates of , i.e. , and of , i.e. that are ‘equally weighted’, respectively, in the x- and y-basis—in other words, are invariant under the action of a unitary that swaps with ; and are invariant under a unitary that swaps with . Moreover, there exist quantum states on the composite system of two qubits, which are likewise ‘equally weighted’ and have the special property that

I shall now require that the analogous property holds in superinformation theories. While in quantum theory it is straightforward to express this via the powerful tools of linear superpositions, in constructor theory expressing the same conditions will require careful definition in terms of ‘generalized mixtures’.

The conditions for decision-supporting information theories is that there exist two complementary information observables and such that:

(T1) The theory admits X-partitions of unity (on the information attributes of S that are generalized mixtures of attributes of ) and -partitions of unity (on the information attributes of the substrate that are generalized mixtures of the attributes in the observable ).

(T2) There exist observables , satisfying the following symmetry requirements:

R1. are generalized mixtures of and are generalized mixtures of attributes of .

Consequently, as is sharp in both and with value , it follows that for all x other than and . Similarly, . Note also that, by definition of complementary observables, —i.e. the attributes are non-trivial generalized mixtures.

Defining the computation which swaps the attributes of S:

R2. ; .

In quantum theory, are two distinguishable equally-weighted quantum superpositions or mixtures of the attributes in , such as . Similarly for . The principle of consistency of measurement (§3) implies that if an attribute has an X-partition of unity with element , for any permutation on , has X-partition of unity such that . This is because presenting to (defined in §5) is equivalent to presenting to the constructor (measurer) obtained prepending the computation to —i.e. the constructor . Therefore, implies . The same, mutatis mutandis, holds for .

Let the attributes be the X-intrinsic parts of the attributes , where is the attribute of prepared by measuring on holding the attribute , with as a target.

R3. , with

This requirement is satisfied in quantum theory: a measurer of the observable acting on a substrate in the state generates the states: whose reduced density operators on are the same, , and are still ‘equally weighted’, thus invariant under swapping and .

Consider now the observables of : and R4. There exists an attribute that is both a generalized mixture of attributes in and a generalized mixture of attributes in : with the property that: where the swap S on a pair of substrates acts in parallel in each separately. In quantum theory, is the attribute , because of the property (6.1).

These conditions define the decision-supporting superinformation theories. (See table 1 for a summary, and footnote 26 in appendix A.) They are satisfied by quantum theory, as I said, via the existence of states such as those for which (6.1) holds.

Table 1.

Conditions for decision-supporting superinformation theories.

A superinformation theory is decision-supporting if there exist complementary observables X and Y such that:

(1) The theory admits X-partitions of unity (conditions (E1), (E2) in §5) on the substrate S and (X_a+X_b)-partitions of unity on the substrate S_a⊕S_b.

(2) There exist attributes x_1,x_2∈X that are generalized mixtures of the attributes {y_+,y_−}; and attributes y_+,y_−∈Y that are generalized mixtures of attributes {x_1,x_2} with the property that:

— S_x_1,x_2(y_±)⊆y_±; S_y_+,y_−(x_i)⊆x_i,i=1,2.

— [a_y+]_X=[a_y−]_X and S_x_1,x_2([a_y±]_X)⊆[a_y±]_X.

(3) There exists an attribute q that is a generalized mixture of attributes in S_x={(x_1,x_1),(x_2,x_2)} and a generalized mixture of attributes in S_y={(y_+,y_+),(y_−,y_−)}, such that

S_x_1x_1,x_2x_2(q)⊆q, S_y_+y_+,y_−y_−(q)⊆q.

Games and decisions with superinformation media

The key step in the decision-theoretic approach in quantum theory is to model the physical processes displaying the appearance of stochasticity as games of chance, played with equipment obeying unitary (hence non-probabilistic) quantum theory. In the language of the partition of unity, applied to quantum theory, the game is informally characterized as follows. An -measurer is applied with an input attribute whose X-partition of unity is . The game-device is such that, if the observable is sharp in with value x (so, under quantum theory, is an eigenvalue-x eigenstate of ), the reward is ‘x’ (in some currency); otherwise unpredictability arises (§4). The player may pay for the privilege of playing the game, knowing , the rules of the game and unitary quantum theory (i.e. a full description of the physical situation). One proves that a player satisfying the same non-probabilistic axioms of rationality as in classical decision theory [38], but without assuming the Born Rule or any other rule referring to probability, would place bets as if he had assumed both the identification between the and the Born-Rule probabilities, and the ad hoc methodological rule connecting probabilities with decisions [18,33].

I shall now recast the decision-theoretic approach in constructor theory. The key differences from previous versions are as follows. (i) They involved notions of observed outcomes and relative states, which are powerful tools in Everettian quantum theory. In constructor theory, instead, none of those will be relied upon. (ii) Some axioms that were previously considered decision-theoretic follow from properties of information media and measurements, as expressed in the constructor theory of information.

Games of chance

From now on, I shall assume that we are dealing with a decision-supporting superinformation theory—i.e. a superinformation theory satisfying the conditions given in §6, with complementary observables and . For simplicity, I shall assume and to be real-valued, whereby X and Y are the set of real numbers. I shall require a slightly more detailed model of a ‘game of chance’, whose centrepiece is an X-adder. It is defined as the constructor performing the following computation on an information medium :

An X-adder is a measurer of the observable of labelled so that, when is sharp in input with value x and is sharp in input with value 0, is sharp on the output pay-off substrate with value x. For any fixed , this is also a measurer (as defined in (3.3)) of the observable on .[21] In quantum theory, it is realized by a unitary .

An X-game[22] of chance is a construction defined as follows:

(G1) The game substrate (e.g. a die), with game observable , is prepared with some legitimate game attribute , defined as any information attribute that admits an X-partition of unity (where may be non-sharp in it) .

(G2)  is presented to an adder together with two other pay-off substrates , representing the player's records of the winnings, with pay-off observable . The first instance of (the input pay-off substrate) contains a record of the initial (pre-game) assets, in some units; the other instance (the output pay-off substrate), initially at 0, contains the record of the winnings at the end of the game and it is set during the game to its pay-off under the action of the adder. Keeping track of both of those records is an artefact of my model, superfluous in real life, but making it easier to analyse composite games.

(G3) The composition of two games of chance , with game substrates and , is defined as the construction where the output pay-off substrate of is the input pay-off substrate of (figure 5, top). In other words, the composite game is an ()-adder realized by measuring the observable and the observable separately. Thus, the composition of two games is the game with game substrate (figure 5, bottom).

Figure 5.

The composition of two games (top) is the game (bottom).

The player

I model the player for as a programmable constructor (or automaton) whose legitimate inputs are: the specification of the game attribute , the (deterministic) rules of the game (with the game observable ) and the subsidiary theory. Its program must also satisfy the following axioms:

(A1) Ordering. Given and ’, the automaton orders any two games and —the ordering is transitive and total.

In this constructor-theoretic version of the decision-theory argument, this is the only classical decision-theoretic axiom required of the automaton. It corresponds to the transitivity of preferences in [1,8]. Its effect is to require to be a constructor for the task of providing a real number —the value of the game. Specifically, I define the value of the game , , as the unique with the property that is indifferent between playing the game and the game . As the reader may guess, the key will turn out to be that attributes with the same partition of unity have the same value.

(A2) Game of chance. The only observables allowed to condition[23] the automaton's output are: (i) the observable for whether the rules of the game are followed; (ii) the observable , defined as the difference between the observable of the first reward substrate before the game and the observable of the second reward substrate after the game, as predicted by the automaton (given the specification of the input attribute and the subsidiary theory).[24]

Thus, whether other observables than the ones of axiom 2 may be sharp is irrelevant to the automaton's output. Otherwise, those observables would have to be mentioned in the program to condition the automaton's output (thus specifying a player for a different game). This fact shall be repeatedly used in §7c.

In classical decision theory, any monotonic rescaling of the utility function causes no change in choices; so, without losing generality, I shall assume that, whenever the observable is predicted by the automaton to be sharp on the pay-off substrate with value x, the automaton outputs a substrate holding a sharp .

Properties of the value function

As I promised, axiom 1 and 2 imply crucial properties, which were construed as independent decision-theoretic axioms in earlier treatments, but here follow from the other axioms and the principles of constructor theory, under decision-supporting superinformation theories. Namely:

(P1) 's preferences must be constant in time. Otherwise, the observable ‘elapsed time’ would have to condition the program, violating axiom A2.[25]

(P2) Substitutability of games [8,38]. The value of a game in isolation must be the same as when composed with another game . Otherwise, again, the automaton's program would have to be conditioned on the observable ‘what games is composed with’, violating axiom A2.

(P3) Additivity of composition. Setting and , P2 implies:

(P4) Measurement neutrality [10]. Games where the X-adder (under the given labellings) has physically different implementations have the same value—otherwise the observable ‘which physical implementation’ would have to be included in the automaton's program, violating axiom A2. Measurement neutrality, in turn, implies the following additional properties:

a. The game —where the computation (for any permutation over ) acts on S immediately before the adder (figure 6)—is a -adder, where denotes the re-labelling of given by . Hence, it can be regarded as a particular physical implementation of the game . Therefore, measurement neutrality implies:

Figure 6.

The game (left) is a particular physical implementation of the game (right).

b. As is a physical implementation of (figure 5), by measurement neutrality the values of games on composite substrates must equal that of composite games: (The last step follows from (7.2).)

c. Let and be the X-intrinsic parts and of the attributes and (§4), where is the attribute of prepared by measuring on in the attribute , with as a target. By prepending a measurer of to the X-adder in the game , so that the measurer's source is the input of the X-adder (figure 7, bottom right), one still has the same X-adder. Thus, the X-game (figure 7, top) has the same physical implementation as the X-game (figure 7, bottom left). Similarly, by concatenating a measurer of to an ‘X’-adder so that the former's target is the game substrate of the latter, one obtains an X-adder overall, whereby the ‘X’-game has the same physical implementation as the X-game . The same applies if one considers , for any permutation over X. Measurement neutrality implies:

Figure 7.

Equivalence of games: the X-game (top) has the same physical implementation as the game obtained by prepending a measurer of to the game G() (bottom left); the latter, in turn, is equivalent to G()(bottom right).

d. Shift rule [8]. Define the uniform shift as the permutation , for real k. The game (figure 8, right) is a particular implementation of the game (figure 8, left), where the -adder is realized by measuring on in the attribute , for a fixed . By measurement neutrality and (7.4):

Figure 8.

The game (right) is a particular implementation of the game (left).

(P5) The ‘equal value’ property. In quantum theory, a superposition or mixture of orthogonal quantum states, such that X-games played with them have the same value v, has value v [8]. A generalization of this property holds under decision-supporting superinformation theories. A crucial difference from the argument in [8] is that here there is no need to invoke observed outcomes or relative states to prove this property. Let be an information variable of a game substrate , such that X-games played with or have the same value v: . Consider the information attribute , having an X-partition of unity, with the property that is a generalized mixture of —whereby . I shall now prove that the X-game with game attribute also has value v: . In the trivial case that or , . Consider the case where, instead, is a non-trivial generalized mixture of : . In quantum theory, is the attribute of being in a superposition or mixture of distinguishable quantum states —e.g. the state for complex . I shall show that is equivalent to another game with value v, obtained by applying the following procedure on the game substrate .

First, measure on holding the attribute , with an ancillary system as a target of the -measurer. This delivers the substrate in the attribute . By the properties of the intrinsic part (§3), still is a generalized mixture of the attributes in . Thus, , because both and have the same specification qua X-game attributes. Next, apply a constructor for the following task (a control operation in quantum theory): if ‘’ is sharp with value on , nothing happens on ; whereas if ‘’ is sharp with value on , then on is replaced by . Applying does not change the value of the game on , for the attributes on the game substrate before (i.e. ) and after (i.e. ) the action of have the same specification: they are both generalized mixtures of attributes in . Thus, , as promised.

(P6) The reflection rule. Under decision-supporting superinformation theories, the property that I shall call the reflection rule follows. It is the superinformation version of Deutsch's [8] ‘zero-sum rule’, but, again, my derivation does not depend on relative states or observed outcomes in universes. Consider the attributes and , satisfying the conditions (R1)–(R4) (§6). By (7.5) and (R3) it follows that: for any over X. Hence, by the additivity of composition (property (P3)), it also follows that: for any permutations , over X. Consider the game where satisfies conditions R4 (§6) and the reflection R is the permutation over X defined by . The adder in the game is a measurer of , which, in turn, is an X-comparer (as defined in (3.4)) on the substrates : it measures the observable «whether the two substrates hold the same value x», where the output corresponds to «». By the consistency of successive measurements of on the same substrate with attribute (§3), the output variable of that measurer is sharp with value 0 when presented with an attribute which, like , has the property that —where . In quantum theory, as I said, the attribute is that of being in the quantum state , which is a 0-eigenstate of the observable . Thus, by definition of value, the value of must be 0.

On the other hand, as , where , and, by (7.7), both and have the same value, by the ‘equal-value’ property (e), one has: . Hence: where the last step follows by (7.4). Finally, one obtains, as promised: It follows that if some attribute is invariant under reflection () then , because:

The decision-theoretic argument under decision-supporting superinformation theories

I shall now show that if an attribute has X-partition of unity then thereby showing that a program satisfying the axioms in §7.1 is possible, under decision-supporting superinformation theories. The automaton with that program must value the game using the as if they were the probabilities of outcomes, without assuming (or concluding!) that they are. It places bets in the same way as a corresponding automaton would, if programmed with the same axioms of classical decision theory plus additional, ad hoc probabilistic axioms connecting probabilities with decisions (e.g. that the ‘prudentially best option’ maximizes the expected value of the gain [33]) and some stochastic theory with the as probabilities of outcomes x (e.g. quantum theory with the Born Rule).

My argument broadly follows the logic of Deutsch's or Wallace's formulation; however, individual steps rest on different constructor-theoretic conditions and do not use concepts specific to quantum theory or the Everett interpretation—e.g. subspaces, observed outcomes, relative states, universes and instances of the player in ‘universes’ or ‘branches of the multiverse’.

Consider the attribute , satisfying R1–R4. Recall that it is symmetric under swap: , so . This special case of ‘equal weights’ provides, as usual, the pivotal step in the decision-theory argument. The shift rule (7.6) with gives:

Now, can be implemented by composing the swap and the reflection R. By measurement neutrality (P4), the reflection rule (7.8) and the symmetry :

By comparison with (7.11): which is a special case of (7.10) when the game attribute has an X-partition of unity whose non-null elements are . This, the central result proved by Deutsch and Wallace, is now proved from the physical, constructor-theoretic properties of measurements. The general, ‘unequal weights’, case follows by an argument analogous to [8], with some conceptual differences (see appendix A).

What elements of reality the elements of a partition of unity in (7.10) represent depends on the subsidiary theory in question: it is not up to constructor theory (nor the decision-theory argument) to explain that. The decision-theoretic argument only shows that decisions can be made, when informed by a decision-supporting superinformation theory, under a non-probabilistic subset of the classical axioms of rationality, as if it were a stochastic theory with probabilities given by the elements in the partition of unity. In particular, in unitary quantum theory that argument is not (as it is often described) a ‘derivation of the Born Rule’, but an explanation of why, without the Born Rule, in the situations where it would apply, one must use the moduli squared of the amplitudes (and only those) to inform decisions. I have shown that the equivalent holds for any constructor-theory-compliant subsidiary theory. Thus, such theories can, like unitary quantum theory, account for the appearance of stochastic behaviour without appealing to any stochastic law.

Conclusion

I have reformulated the problem of reconciling unitary quantum theory, unpredictability and the appearance of stochasticity in quantum systems, within the constructor theory of information, where unitary quantum theory is a particular superinformation theory—a non-probabilistic theory obeying constructor-theoretic principles. I have provided an exact criterion for unpredictability, and have shown that superinformation theories (including unitary quantum theory) satisfy it, and that unpredictability follows from the impossibility of cloning certain sets of states, and is compatible with deterministic laws. This distinguishes it from randomness.

I have exhibited conditions under which superinformation theories can inform decisions in games of chance, as if they were stochastic theories—by giving conditions for decision-supporting superinformation theories. To this end, I have generalized to constructor theory the Deutsch–Wallace decision-theoretic approach, which shows how unitary quantum theory can inform decisions in those games as if the Born Rule were assumed.

My approach improves upon that one in that (i) its axioms, formulated in constructor-information-theoretic terms only, make no use of concepts specific to Everettian quantum theory, thus broadening the domain of applicability of the approach to decision-supporting superinformation theories; (ii) it shows that some assumptions that were previously considered as purely decision-theoretic, and thus criticized for being ‘subjective’ (namely, measurement neutrality, diachronic consistency, the zero-sum rule), follow from physical properties of superinformation media, measurers and adders, as required by the principles of constructor theory.

So the axioms of the decision-theory approach turn out not to be particular, ad hoc axioms necessary for quantum theory only, as it was previously thought; instead, they are either physical, information-theoretic requirements (such as the principle of consistency of measurement, interoperability and the conditions for decision-supporting superinformation theories) or general methodological rules of scientific methodology (such as transitivity of preferences) required by general theory testing. Deutsch [18] has shown how all decision-supporting superinformation theories (as defined in this paper) are testable in regard to their statements about repeated unpredictable measurements.

This paper and Deutsch's, taken together, imply that it is possible to regard the set of decision-supporting superinformation theories as a set of theoretical possibilities for a local, non-probabilistic generalization of quantum theory (alternative to, for example, ‘generalized probabilistic theories’ [37]), thus providing a new framework where the successor of quantum theory may be sought.