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The concept of information plays a fundamental role in our everyday experience, but is conspicuously absent in framework of classical physics. Over the last century, quantum theory and a series of other developments in physics and related subjects have brought the concept of information and the interface between an agent and the physical world into increasing prominence. As a result, over the last few decades, there has arisen a growing belief amongst many physicists that the concept of information may have a critical role to play in our understanding of the workings of the physical world, both in more deeply understanding existing physical theories and in formulating of new theories. In this paper, I describe the origin of the informational view of physics, illustrate some of the work inspired by this view, and give some indication of its implications for the development of a new conception of physical reality.

The concept of information plays a fundamental role in our everyday experience. We generally view ourselves as embodied finite beings immersed in a physical world of which we possess limited knowledge. Each of us constantly directs our attention to aspects of the world about which we wish to gain more information, and that information is supplied by our sensory systems in a virtually endless stream and is dynamically shaped by our mental faculties into an ever-changing predictive model of the world. That our predictive model provides us with limited information about the future is evidenced by our constant experience of making predictions (consciously or unconsciously) which—even in the simplest situations which we habitually encounter—rarely precisely coincide with what actually happens. According to this view, the concept of information enters in two distinct ways: (1) At any moment, the information available to us through our sensory systems about the external world is limited; and (2) the information provided by our predictive model about the future is also limited.

Despite its ubiquity in everyday experience, the concept of information is absent in the mechanical conception of reality that underlies classical physics. This conception, which underpinned the development of physics for over two hundred and fifty years—from the seventeenth-century mechanics of Galileo and Newton to the nineteenth-century electrodynamics of Faraday and Maxwell, articulates a highly-idealized view of reality and our relationship to it. According to this view, in essence, the totality of all that exists in the physical world consists of matter and fields which sit on the fixed stage of space, and which evolve according to universal laws in step with a universal time, with no fundamental limits on how well an ideal agent can know the state of all that exists.

In this mechanical conception of reality, the concept of information is redundant in both of the senses identified above. First, an agent, in principle, has complete information about the state of reality, so there is no sense in maintaining a distinction between the state of reality and an ideal agent’s knowledge of that state. Second, in principle, such an agent who is also in possession of a description of the universal laws of motion can exactly predict any future (or past) event, and thus has complete information about the state of reality at all other times, past or future.

Over a period of over two centuries, the unprecedented success of the physical theories built within the classical framework—chiefly Newtonian mechanics and Maxwellian electrodynamics—naturally gave rise to the belief that the assumptions and idealizations underlying that framework were basically sound. In particular, it gave rise to the belief that the issues of limited access to information and limited predictive information which are characteristic of our everyday existence have no bearing on our quest to precisely formalize nature’s innermost workings, these issues simply being nuisances which we (as non-ideal agents) must live with. The assumptions and idealizations of the classical framework certainly did not go unchallenged, either within physics community itself or within the wider intellectual culture that had been profoundly affected and challenged by the mechanical world-view spawned by classical physics. Nevertheless, largely owing to the fecundity of the classical framework, these assumptions essentially held sway in the physics community until the early part of the twentieth century.

By the 1890s, classical physics was in a highly developed state. However, there were some clouds on the horizon, curious experimental facts—such as the spectrum of frequencies of light emitted by heated bodies—that stubbornly resisted explanation within the existing theories of classical physics. The explanation of these facts, initiated by Max Planck, would, over the next thirty years, bring about the development of quantum theory—an entirely new theoretical framework for physical theories that would replace the classical framework. Within a few years after its creation, it was clear that the quantum framework departed radically from the framework of classical physics, and was in many respects at odds with the assumptions and idealizations underlying the mechanical conception of reality. In particular, in the quantum framework, an agent has,

Since the creation of quantum theory, it has been widely debated whether these non-classical features, particularly the radically different status of the measurement-performing agent, are fundamental, reflecting the very structure of physical reality, and so ought to be regarded as generic features of physical theories, or are only apparent, emergent from some yet-to-be-discovered more fundamental theory closer in nature to the theories of classical physics which lacks most (if not all) of these features.

Over the last few decades, bolstered by many developments in physics (such as black hole physics) and other disciplines (such as Shannon’s theory of information), the view has arisen amongst a growing number of physicists that the concept of information may indeed have a fundamental role to play in our understanding of quantum theory and, more generally, the physical world, a role as fundamental as that occupied by such concepts as space, time, matter and energy in classical physics. In recent years, this

The purpose of this paper is to describe the origin of the informational view of physics, to illustrate some of the recent work inspired by this view, and to give some indication of its broader implications for the development of a coherent new conception of physical reality that is capable of providing a viable alternative to the mechanical conception of classical physics and of guiding the future development of physics.

The remainder of this paper is organized as follows. In

In order to understand the role of information in quantum theory, it is helpful to first summarize the essential ideas underlying classical physics.

Classical physics can be usefully divided into three components, namely:

A conception of the physical world, namely the mechanical conception of reality;

A precise conceptual and mathematical framework, which formalizes this mechanical conception;

Classical physical theories—in particular Newtonian mechanics and Maxwellian electrodynamics—which are built within this framework.

As mentioned in the introduction, classical physics is underpinned by a mechanical conception of reality. According to this conception, the physical universe is a vast machine whose state at any time is precisely describable in every detail, and which evolves in time according to quantitatively precise laws. These laws operate in the same manner at all times and at all places within the universe, and yet themselves remain unchanged. Furthermore, according to this conception, ideal agents can make measurements to learn about the state of the universe as precisely as they wish without disturbing it, and can build up a knowledge of that state by aggregating knowledge gained about spatially-disjoint regions of the universe.

The core of the mechanical conception is formalized in the framework of classical physics, which provides the conceptual and mathematical framework for the theories of classical physics. In this framework, one speaks abstractly about a

Every measurement is assumed to yield an outcome which is determined by the state of the system, but without affecting the state of the system itself. Accordingly, such a measurement is represented by a map from state space to a space of possible outcomes. In principle, there exists an informationally-complete measurement from whose outcome it is possible to infer the state of the system. Such a measurement is represented by a bijective map, so that there is a one-to-one correspondence between states of the system and outcomes of the measurement. An ideal agent, who by definition can perform such an informationally-complete measurement, will have complete knowledge of the physical state of a system. On the basis of such a measurement, an ideal agent would therefore be able to assert that, with certainty, the system is in a definite state, also known as a

Finally, if a system is composed of sub-systems, the state of the system is determined by the states of its sub-systems. This means that agents can build up knowledge of a system by aggregating knowledge about its sub-systems.

A particular classical physical theory is built within this framework by specifying the state,

In the framework of quantum physics, just as in the classical framework, a model of a physical system consists of (a) a complete theoretical description (the mathematical

For a composite system composed of subsystems, the framework also specifies the pure state of the composite system when the subsystems are in known pure states. However, in quantum theory a composite system can be in a pure state even when the subsystems are not.

In the quantum framework, just as in the classical framework, the dynamical mapping is bijective, representing deterministic and reversible temporal evolution. However, the quantum and classical frameworks differ sharply in two ways—in their models of the measurement process and in their descriptions of composite systems.

As described above, in the classical framework, an agent can perfectly and completely learn the state of a physical system, and furthermore can do so without disturbing the system to any significant degree. In contrast, the quantum framework posits a model of the measurement process that has four distinct, non-classical features:

The first of these features, discreteness, is that, when certain measurements are performed on physical systems, the number of possible outcomes can be finite or countably infinite, where this discreteness is not due to boundary conditions but is intrinsic to the system. This stands in contrast with the classical assumption that all physical quantities (such as the position of a particle) can take a continuum of possible values. Hence, discreteness challenges the classical idea that the continua of space and time are the fundamental bedrock of physical reality.

The second of these features is that, in contradistinction to one of the basic tenets of the mechanical conception of reality, it is

The quantum formalism, as the classical framework, is concerned with idealized measurements which are repeatable—that is, measurements which, when immediately repeated, yield the same outcome with certainty. If one considers such measurements, then it follows as a direct consequence of the fact that measurements are probabilistic that, in general, they disturb the state of the system upon which they are performed. In fact, they disturb it almost completely—almost no trace of the pre-measurement state of the system is left in the post-measurement state.

The fourth feature, complementarity, can be precisely expressed in a number of different ways. Perhaps the simplest is to say that, unlike the situation in classical physics, one cannot perform a repeatable measurement on a system which yields information about all of the degrees of freedom of the state of the system.

Taken together, the probabilistic nature of measurement outcomes and complementarity severely constrain an agent who wishes to learn about the state of a physical system which has been prepared by an ideal agent. First, due to complementarity, an observer cannot rely upon one type of measurement, but must use more than one. Second, due to the probabilistic nature of measurement outcomes, the agent must perform many measurements on identical copies of the system. Yet, after a finite number of measurements, the agent’s knowledge about the state will still be imperfect. It is only in the unattainable, idealized limit of an infinite number of measurements that the agent’s knowledge of the state becomes complete. Therefore, unless an agent has directly prepared a system herself, or knows precisely how a system was prepared, there is always an informational gap between the theoretical description of the underlying reality—the quantum state of the system—and her knowledge of that state.

In order to make the above remarks more concrete, consider Stern-Gerlach measurements performed upon a particle with a magnetic moment. Classically, one can represent the magnetic moment (or

Quantum mechanically, the situation is radically different. First, the state of the spin is represented as a two-dimensional complex vector,

The Stern-Gerlach measurement will yield

Now, the Stern-Gerlach measurement oriented in the

Taking stock of the above, we can identify two natural ways in which the agent is informationally constrained:

(1)

Given the state of the system and the measurement to be performed, the experimenter lacks information about the outcome that will be obtained. In the above example, prior to performing a Stern-Gerlach measurement in the

Quantitatively, prior to performing the measurement, the experimenter has uncertainty

(2)

If an experimenter is presented with a system in an unknown state and wishes to learn what that state is, the quantum framework imposes two kinds of fundamental limits. First, due to the probabilistic and disturbance features of measurements, the outcome of a single measurement performed on the system provides scant information about the state of the system. In practice, in order to build up any useful knowledge of the state, the experimenter must perform a large number of measurements on identically-prepared copies of the system. Furthermore, due to complementarity, a single type of measurement only provides access to one-half of the degrees of freedom of the state of the system, so that the experimenter must perform other types of measurement in order to build up information about all of the degrees of freedom in the state.

In the electron spin example, the experimenter wishes to learn about the unknown state,

If the experimenter obtains

The amount of information the data thus provides the experimenter about

which is finite for finite

Furthermore, the data string,

One of the basic premises of classical physics is that one can conceptually

In essence, in the classical framework, a description of the whole is a simple aggregate of the description of its parts. As a consequence, an agent who in practice is necessarily restricted to studying spatial regions of the physical universe of limited extent at any one time can nonetheless aggregate information gained about these regions to form a description of larger regions. According to the classical framework, by proceeding in this manner, an agent can aspire to an arbitrary precise description of an arbitrary large part of the physical universe.

In contrast, the quantum formalism asserts that physical reality is

More generally, quantum theory implies that a physical system becomes entangled with other systems as it interacts with them. For example, if a proton and an electron interact with one another locally, they become entangled with one another. Furthermore, this entanglement persists if they are subsequently separated from one another, no matter how widely. Consequently, a typical physical system as it is found in nature is generically entangled with many other physical systems, including those at great distance from it. However, an agent studying that system has no way of determining the precise nature of these entanglements by studying the system alone. Furthermore, when an agent chooses to perform measurement on any given system, the agent breaks any entanglement between that system and other systems elsewhere. Thus, an agent who singles out a particular system for study gains limited information about its entanglements to other systems, and furthermore disturbs the system by irrevocably breaking those entanglements.

To make these comments more concrete, let us consider a bipartite system consisting of two particles, each with spin. In the simplest case, the spin component of each particle can be represented by a two-dimensional complex vector, and one possible state of the composite system that represents the spin components is

Now, if an agent performs a Stern-Gerlach measurement in the

We also note that, after the above-mentioned Stern-Gerlach measurement is performed on the system in the state

It is not obvious by inspection of the above example (nor by casual inspection of the quantum formalism itself) that, when the measurement on the first particle is made and the state of the composite system is thereby changed (becoming disentangled), there is an

Bell’s argument concerns the correlations between the outcomes of measurements performed on different sub-systems of a composite system. The quantum framework predicts that if these sub-systems are entangled, then these measurement outcomes are correlated. For instance, in the above example, as we have already mentioned, if two experimentalists were to perform separate Stern-Gerlach measurements in the

Now, correlations such as these are also possible in the classical framework. For example, if Alice and Bob are each given a classical spin and told that the spins are both pointing up or both pointing down, then, when they subsequently make a measurement in the

Bell’s theorem implies that there is

Remarkably, although this non-local connection exists, the quantum framework implies that this connection cannot be used by two agents to send

In view of its extraordinary implications, the interpretation of Bell’s theorem remains controversial. For example, various authors have claimed that, either explicitly or implicitly, Bell’s theorem makes one or more additional assumptions (such as assuming the validity of counterfactual reasoning) beyond those stated, and argue that it is more reasonable to deny these additional assumptions than to accept them and the concomitant conclusion that the validity of quantum predictions implies the violation of locality (for recent examples, see the papers by Blaylock [

As described in the Introduction, over the last few decades the view has arisen amongst a growing number of physicists that the concept of information may have a fundamental role to play in our understanding of quantum theory and in our search for new physical theories. Quantum theory, with its non-classical features (described in the previous section) and unprecedented empirical success over the span of close to a century, has played a key role in the emergence of the informational view. However, one can identify several key stages in which other developments in physics and related subjects have also played a crucial role. In order to put the informational view in an appropriately broad context, and, reciprocally, to be able to appreciate the possible broader impact of this view, the key developments will now be briefly described in approximate chronological order.

As mentioned in the Introduction, in the two centuries following its articulation, the assumptions and idealizations underlying mechanical conception of classical physics were challenged from many directions. One early important source of challenge came from physicist-philosopher Ernst Mach (1838-1916). Mach developed the notion that a physical theory should not be regarded so much a description of how the world

“The goal which it (physical science) has set itself is the simplest and most economical abstract expression of facts.”

In expressing such an attitude, Mach was seeking to undermine the emphasis hitherto placed on the conceptual framework of a physical theory (the mechanical framework of classical physics in particular), arguing that experimental facts, not concepts, ought to be regarded as the primary reality, while the concepts which we use to frame these facts ought to be regarded with a certain suspicion. He also asserted that, in order to place concepts on the firmest possible foundation, ideally every concept in the conceptual framework of a physical theory ought to be definable in terms of definite experimental operations, a view that came to be known as

Mach also argued that abstract physical theories, with the uniformity that their so-called universal laws promise, are in fact

“In mentally separating a body from the changeable environment in which it moves, what we really do is to extricate a group of sensations on which our thoughts are fastened and which is of relatively greater stability than the others, from the stream of all our sensations. Suppose we were to attribute to nature the property of producing like effects in like circumstances; just these like circumstances we should not know how to find. Nature exists once only. Our schematic mental imitation alone produces like events.”

Mach’s emphasis on the primacy of experimental data and the concomitant demotion of the metaphysical assumptions and idealizations of physical theory to a secondary status had a strong impact on the development of physics in the early twentieth century. The theory of relativity—both special and general—developed by Einstein bears testimony to the freedom of thought (from undue attachment to the assumptions underlying classical physics) fostered by Mach’s point of view. For example, Einstein’s key insight leading to the special theory of relativity was that the apparent conflict between Maxwell’s equations and Galilean invariance could be resolved by giving up the Newtonian notion of absolute time (which had also been previously criticized by Mach as not operationally well-grounded) in favor of the notion that two observers in relative motion do not, in general, agree on the amount of time that passes between two events.

Heisenberg, one of primary contributors to quantum theory, was—via Einstein—also influenced by Mach’s ideas. While trying to understand the regularities in the frequency spectra of light emitted by excited atoms, he recognized that the assumption hitherto made that an electron orbits an atomic nucleus in the same way that a planet orbits a star was empirically not well-founded and so set it aside, instead seeking to directly capture the numerical regularities seen in the observed frequency spectra.

Although Mach did not speak about information

Thermodynamics is concerned with the transfer of heat energy between physical systems, and the interconversion of heat energy and mechanical energy (“work”). In thermodynamics, a physical system is described in terms of such large-scale variables as volume, temperature, and pressure. This mode of description is intentionally coarse—thermodynamics is, by design, tailored to the practical concerns and constraints of human agents interacting with physical systems. Although it intentionally does not adopt a fundamental mode of description, thermodynamics is nevertheless a quantitative, precise science with conceptually elegant and mathematically precise laws that concern the definition of temperature (the zeroth law of thermodynamics), the mechanical equivalent of heat energy (the first law), and the transfer of heat between bodies at different temperatures (the second law).

Statistical mechanics, as developed in the last few decades of the ninetieth century, sought to build a bridge capable of connecting the thermodynamic macro-description of a physical system with its micro-description supplied by classical physics. Its major accomplishment was the recognition that the thermodynamic state variable known as

In one of its simplest statements, the second law of thermodynamics asserts that heat does not spontaneously flow from a cooler to a hotter body. As a statement of our everyday experience, this statement is unobjectionable. However, it is at odds with a classical mechanical description of the situation, which (as described in

In the process of articulating his point of view, Maxwell formulated an ingenious thought-experiment. Maxwell asks the reader to consider a microscopic creature who is a “very observant and neat-fingered being”, capable of following the positions and velocities of all of the molecules in a box of gas. Such a being, Maxwell argued, would be able to sort the molecules to create a macroscopic temperature difference between two sides of the box without the expenditure of any work, thereby generating a violation of the second law of thermodynamics.

In his argument, Maxwell tacitly assumed, as per the idealizations of classical physics, that it is possible in principle for an agent to obtain arbitrarily precise information about the state of a physical system without in any way affecting the physical world. In 1929, Leo Szilard proposed an ingenious solution to Maxwell’s demon (as Maxwell’s microscopic creature came to be known) that brought this assumption into question [

Taken together, thermodynamics and statistical mechanics played an important role in the evolution of physicists’ thinking in at least three distinct respects. First, thermodynamics demonstrated that it was possible to construct a rigorous, mathematical theory to describe the physical world from the point of view of non-ideal agents, so that the construction of a physical theory did not

In 1948, the electrical engineer Claude Shannon introduced a measure of the information gained when one learns of the outcome of a probabilistic process [

Shannon showed that it was possible to formalize these and a few other intuitive ideas in the form of mathematical postulates, and derive from these postulates a definite expression for the information gain applicable to any probabilistic process. More precisely, if a probabilistic process has

The establishment of a precise, quantitative measure of information associated with learning the outcome of a probabilistic process raised the question of whether this measure could provide new understanding into the formalism of existing theories of physics that involve probabilistic processes. In 1957, Jaynes showed that, indeed, it was possible to reinterpret the derivation of the formalism of statistical mechanics due to Gibbs at the turn of the century as an application of a new principle of inference, namely the Principle of Maximum Entropy, which states [

For example, the canonical distribution of statistical physics can be derived by imposing the constraint that the expected energy of a system,

In 1916, Schwarzchild published an exact solution of Einstein’s field equations of general relativity for a point mass of mass

In the early 1970s, it was established that, when quantum mechanical effects are taken into account, a black hole can be treated within the framework of thermodynamics—it has an entropy (for example,

However, according to general relativity alone, the internal structure of the black hole involves a non-denumerably infinite number of degrees of freedom, so that one would expect this lack of information to be infinite. The fact that it is actually

Black hole physics has been of relevance to the emergence of the information view of physics in a number of ways. First, it provided the first example of a physical system whose precise physical state is, according to general relativity,

The elementary unit of information is a bit (short for binary digit), which takes the value zero or one. Abstractly, a computation can be regarded a deterministic map from one string of bits to another string of bits. In 1936, Turing proposed a minimal abstract computing device—a

Turing’s characterization of a computing machine presupposes the essential idealizations underlying classical physics, and can indeed be regarded as a formalization of the notion of a mechanical process that underlies classical physics. In particular, direct analogs to the the classical notions of state, deterministic dynamics, and ideal measurement are all clearly visible in Turing’s construction.

In proposing a concrete realization of the notion of a mechanical computing machine, Turing took a crucial step in bringing physics and computation together: If a computation is to actually be carried out step-by-step by a machine of our imagination that we could potentially construct, then the design of that machine must conform to the known laws of physics, and its capacities will necessarily be constrained by those laws. Any change in those laws, or indeed any disagreement about the content of those laws, has the potential to impact any putative computing machine. There have been two important developments in this direction.

First, as mentioned in

Second, Turing’s computing machine presumes the framework of classical physics. It is natural to wonder whether information processing fundamentally differs when embodied in the framework of quantum theory and, in particular, whether quantum theory imposes new constraints on how the information encoded in quantum systems can be manipulated or whether new things are possible when the full richness of quantum reality (such as the phenomenon of entanglement described in

Let us first consider how information is encoded in, and read from, a quantum system. The elementary unit of quantum information is not the bit but the

Now, one might imagine that there might at least exist a process which is capable of

But quantum theory does not only entail restrictions. Remarkably, over the last thirty years, it has been found that, when information is embodied in quantum systems, it is possible to carry out information processing tasks which are impossible or very difficult using classical information processing. For example, one can in principle build a

Most of the innovations allowed by quantum information processing depend crucially upon the use of

Over the last few decades, there has arisen a growing belief amongst many physicists that the concept of information may have a critical role to play in our understanding of workings of the physical world. This view has been perhaps best articulated by John A. Wheeler under the slogan

“ ‘

and

“What we call reality consists of a few iron posts of observation between which we fill an elaborate papier-mâché of imagination and theory.”

As described in

As described in the previous section, the quantum framework brings about a relationship between an agent and the physical world which differs in a number of respects from that which exists in the classical framework, one in which the concept of information naturally plays a key explanatory role. This fact, together with the insights into the nature of quantum reality gained through an exploration of quantum information processing, naturally gives rise to the question of whether specific predictions of quantum theory and the mathematics of quantum theory—whose exact physical origin has long been regarded as obscure—can be derived from information-theoretic principles. As I will elaborate below, work carried out over the last thirty years has shown that this indeed appears to be the case.

In order to better describe the difficulty involved in understanding quantum theory, it is helpful to first consider its origin. As described in

The first quantum theory was non-relativistic quantum mechanics, developed in distinct forms by Schroedinger and Heisenberg in 1925-1926. Its key achievement was accounting in quantitative detail for the light spectrum of hydrogen in a variety of different circumstances. Over the course of the subsequent few years, a general and rather beautiful mathematical formalism—the

Owing to the indirect process by which the quantum formalism was created, the physical origin of many of the mathematical aspects of the formalism was not clear. For example, the states of quantum systems are represented by complex vectors, but it was not clear precisely why complex numbers were necessary. As described in

The first view, held by many of the founders of quantum theory, such as Bohr, Heisenberg and Pauli, was that the non-classical features of the formalism, such as the statistical nature of its predictions, reflected the very structure of physical reality, and that the classical mechanical view of physical reality therefore had to be replaced by something fundamentally new.

In order to illuminate these non-classical features of the quantum formalism, some of these physicists attempted to identify related concepts in the existing philosophical literature or to develop new concepts of which these features could be regarded as particular instantiations, thereby placing these non-classical features in a broader philosophical light. For example, Bohr developed the concept of complementarity, which he expressed as meaning that the process of coming to know anything about some aspect of reality is an active process that unavoidably has the effect of bringing into existence some aspect of it at the expense of simultaneously rendering inaccessible some other aspect of it [

The second view, held by Einstein and some other physicists, maintained that notwithstanding the manifestly non-classical features of quantum theory, the classical mechanical view of reality did not require the revision of its fundamental tenets. Einstein, for instance, argued that the statistical nature of quantum predictions was simply an indication that the quantum description of reality was incomplete, and spent a significant part of his later life searching for a classical field theory that was capable of underpinning quantum theory. De Broglie, in particular, supported this point of view by showing that the quantum theory of an ensemble of particles could be re-written in a form closely akin to classical mechanics, albeit with some curious non-classical features.

In the intervening eighty years, various attempts have been made to provide a physically intelligible interpretation of the quantum formalism. However, although these interpretations—such as the Copenhagen interpretation, the de Broglie-Bohm interpretation and the Many Worlds interpretation—paint extraordinarily diverse pictures of the quantum world, there appears to be little possibility of objectively choosing between them as each interpretation appears both internally consistent and consistent with the known experimental facts.

Over the last thirty years, it has become increasingly recognized that interpretations by themselves do not provide an adequate understanding of quantum theory [

Attempts to reconstruct the mathematics of quantum theory have a long history and can, in fact, be traced back to the early 1930s. Indeed, Heisenberg, one of the creators of quantum theory, recognized that it would be highly desirable if the quantum formalism could be reconstructed using his uncertainty principle as a key axiom. Nonetheless, broadly speaking, reconstructive attempts prior to the 1980s tended towards highly abstract, intricate systems of axioms, and consequently made little impact [

Making some additional assumptions which will be described below, Wootters is able to show that, if the outcome probabilities,

In outline, the argument runs as follows. Consider an experimental arrangement consisting of a preparation stage and an analysis stage. The preparation stage consists of a Stern-Gerlach apparatus oriented at

As already mentioned in

Upon each analysis, the experimenter obtains one of two possible outcomes. After

The amount of information obtained about

Wootters then attempts to extend this principle to an

The postulates that have been employed are diverse. One class of postulates concerns bipartite systems. For example, one early investigation [

Another example of a postulate of this type is due to Barrett [

Another class of postulates concern the behavior of individual systems. For example, in recent work, I have shown that it is possible to construct the core of the quantum formalism (namely Feynman’s rules of quantum theory) by suitably formalizing the notion of complementarity and by postulating that certain measurements that yield no useful information about a physical system also do not disturb its state in any detectable way [

As described in

In contrast, the informational view of physics places the agent center stage, confers non-trivial properties (such as indeterminacy and complementarity) on the interface between the agent and the physical world in which the agent is immersed, and suggests that such a basic construct as space is, in fact, an approximation that neglects both the pervasive entanglement of widely-separated physical bodies and the postulate that only a finite amount of information is required to completely describe any region of space. As outlined above, all of these ideas are now underpinned by a considerable body of theoretical and experimental evidence. At one level, one can regard the contents of the informational view simply as summarizations of experimental observations. But, at a deeper level, one would like to place these separate ideas into a coherent conception of physical reality within whose framework these separate ideas appear natural or even expected and thus related. The formulation of such a conception is important not only to unify our present understanding for its own sake, but also to provide a reliable guide to the future development of physics.

To illustrate the kind of understanding that is sought, consider the probabilistic nature of measurement outcomes postulated by quantum theory. It is one thing to accept that measurements are indeterministic as an operational principle (that is, as a summary of what we find in our experiments), but quite another to accept at a philosophical level the idea that events

In order to make progress, it is helpful to recognize (as alluded to above) at the outset that the mechanical conception of reality is itself somewhat removed from our actual everyday experience of the world. As brought to light by the penetrating analyses of Hume, Mach, and others, the manifold abstractions upon which the classical framework is based are not as strongly rooted in our experience as we might sometimes suppose. For example, as has been long observed, the deterministic and reversible dynamics postulated by classical physics leads to model of reality in which there is no longer any fundamental distinction between past, present, and future. Such a model is in principle incapable of accounting for basic aspects of our experience, for example why we do not experience all times at once and instead experience reality unfolding gradually, and why we experience the past as fixed and the future as open and malleable [

If, however, one accepts the probabilistic nature of measurement as fundamental, then one cannot conceive of the future as objectively existing, and the sharp contrast between the past and future is naturally restored. Furthermore, as described in [

One of the most pressing challenges is the interpretation of the notion of measurement itself. Since the process of measurement as described by the quantum formalism is an active process which generates actual physical change in the system under observation, the formalism implies that there are in fact two distinct physical processes that a system can undergo: deterministic evolution and measurement. The question then arises: When is a physical process to be regarded as a measurement? In the classical framework, where a measurement passively registers the state of a system, this question is moot. However, in the quantum framework, due to the active nature of measurement, the question cannot be avoided. The quantum formalism itself provides no ready answer to this question. Thus far, it has been possible to apply the quantum formalism on the assumption that the process the formalism refers to as “measurement” only occurs when an agent performs a real measurement on a system. However, the question naturally arises whether, say, an electron interacting with a large protein molecule might undergo a physical process we call a measurement. At present, although there is no definitive answer to this question, several so-called spontaneous (or objective) collapse models have been proposed which imply that measurement-like processes occur spontaneously in nature, quite independently of probings initiated by macroscopic agents [

If one accepts that physical systems can undergo two distinct types of processes—deterministic evolution and a measurement-like probabilistic process—is there some way this can be understood at a deeper level? As mentioned earlier, Heisenberg and Pauli both suggested that there is a close parallel to the Aristotelian notions of potentiality and actuality, an idea that has been developed by various authors such as Whitehead [

Finally, if entanglement is indeed generic, then the status of space is fundamentally altered: Space is no longer a fundamental entity which mediates all interactions between material bodies, but is rather a useful approximate construct that neglects the fact the bodies also interact via an ever-changing web of inherently non-spatial connections. If this is so, a number of questions arise: Precisely

At present, then, it is far from clear how to combine the various facets of the emerging informational view into a coherent conception of physical reality. Nonetheless, what seems quite clear is that these various facets are now sufficiently well underpinned by theoretical and experimental work that they ought to be taken seriously as descriptions of how nature works, and the creation of such a conception is a vital next step.

More generally, the knowledge that an agent (be the agent ideal or non-ideal) possesses about the state of a system can be represented by a probability distribution over the state space of the system. This distribution itself is often also referred to as “the state” of the system. If the distribution picks out a single state, as would it be in the case of an ideal agent, it is said to be

This characterization holds true for an _{i}is the

If one is willing to sacrifice repeatability, then it is possible to perform measurements—known as

It is, however, possible for the two agents to distinguish between these two entangled states if they allowed to perform a sufficient number of