Constructor Theory of Information and Its Application in Hybrid Quantum Systems
Beyond Quantum Computation: Constructor Theory
Table of Contents
01) Constructor Theory
02) Hume, Russel vs Popper Philosophy
03) Universal Constructor
04) Counterfactuals
05) Constructor theory of Information
06) DeWitt’s Totalitarian property
07) Information Media
08) Principles of Information Media
09) Superinformation Media
10) Non-classicality within Superinformation
11) Hybrid Quantum Systems
Introduction
Quantum Computation fundamentally relies on quantum theory. However, what if we could envision a ‘quantum theory of information without quantum theory’? Why? To unify both classical and quantum information to create a general theory of information. This intriguing question sets the stage for a deeper exploration of the foundations of quantum information.
Key Questions
➡ Q1: Can we have a ‘quantum theory of information without quantum theory’?
➡ Q2: Can this ‘can/cannot’ approach provide the foundation for a scale-independent, dynamics-independent extension of quantum information theory?
Contrary to the obvious perception, it is not a theory of ‘constructors’ or their constructions. Instead, it examines which transformations (from input state of substrate to output state of substrates) can or cannot be caused & why.
Hume, Russel, and Popper
Deutsch challenges the prevailing notion that causation is merely a useful fiction in fundamental science. Contrary to Hume’s argument on the unobservable nature of causation, Deutsch aligns with Popper’s view, treating scientific theories as conjectured explanations, not inferences from evidence. In this perspective, observation serves as a test for theories, not a means of validation, rendering Hume’s argument irrelevant within this framework.
Universal Computer vs Universal Constructor
A Universal Computer encompasses all physically possible computations but has limitations, such as constructing a copy of itself from raw materials (Turing, Deutsch). On the other hand, a Universal Constructor is a programmable machine capable of performing any physically allowed task (Neumann). The latter resembles a universal 3D printer, it can reliably create a replica of itself, overcoming the limitation of self-replication associated with the Universal Computer.
Counterfactuals
Counterfactual Physical Principles, such as the Conservation of Energy, 2nd Law of Thermodynamics, and Heisenberg’s uncertainty principle, establish what is possible or impossible in a broader sense. These principles, dealing with counterfactual scenarios, transcend specific laws of motion. For instance, the Conservation of Energy asserts the impossibility of a perpetual motion machine, the 2nd Law highlights the impossibility of converting all heat into useful work, etc. These principles provide overarching constraints on physical possibilities.
Deutsch Recent Effort: Proving Universal Constructor from within Constructor Theory
Deutsch is currently developing a theory of a universal within constructor theory. If a universal constructor is not proved, it implies that our apparent understanding of and control over the world might not be as comprehensive as we believe.If there exists something that cannot be programmed for construction by a machine, then that signifies there are limits to what can be built. Should there be something in existence that cannot be constructed, it suggests a finite and replenishable resource in nature, akin to a single magnetic monopole. Consequently, if any machine requires this magnetic monopole to function and no other machine can operate without it, then that specific machine possesses capabilities unique from all others.
DeWitt’s Totalitarian Property
In quantum theory, the concept of the ‘totalitarian’ property, articulated by DeWitt, posits that the quantization of a given system implies the quantization of any other system to which it can be coupled. This assertion, however, seems contingent on specific dynamics within a defined framework. The pivotal question arises: can we extend this notion to scenarios where specific dynamics are either uncertain or cannot be assumed?
Application of Constructor Theory in Hybrid Quantum Systems
One intriguing application of constructor theory lies in its ability to make predictions without relying on specific dynamical laws. This proves particularly valuable when dealing with complex systems where the underlying dynamics are either intractable to calculate or remain entirely unknown. Consider the case of hybrid systems, where a quantum system (Q) interacts with a seemingly classical one (S). The theory allows us to pose the crucial question: does such a hybrid quantum system, with its distinct quantum and classical sectors, even exist within the framework of constructor theory? Can it be a viable model for our reality? This application is significant because it pushes beyond the limitations of traditional approaches, which often depend heavily on a pre-established knowledge of dynamical laws.
The advent of hybrid quantum systems, where a quantum entity interacts with a fully classical counterpart, challenges the universal applicability of DeWitt’s ‘totalitarian’ property. Does quantum theory seamlessly accommodate such scenarios without presupposing identical dynamics across diverse systems? This prompts an exploration of a more inclusive framework that transcends the limitations of specific dynamics.
To refine DeWitt’s proposition, it becomes essential to move beyond its exclusive application to quantum theory. A more nuanced proposal should be applicable not only to quantum systems but also to other classical theories. To refine DeWitt’s proposition, it becomes essential to move beyond its exclusive application to quantum theory. A more nuanced proposal should be applicable not only to quantum systems but also to other classical theories.
Enter the Constructor Theory of Information — a framework that holds promise in addressing this challenge. By leveraging Constructor Theory principles, it becomes possible to formulate a set of assumptions that extends beyond specific dynamics. This not only enhances the universality of the ‘totalitarian’ property but also contributes to a more comprehensive understanding of the interaction between quantum and classical systems.
Information Media
The Constructor Theory of Information aims to establish a framework that treats both classical and quantum information equivalently, providing a generalization of quantum conditions without relying on quantum theory. The primary objective is to define a class of systems that extends beyond classical bits, encompassing a broader notion of information media. An information medium is conceptualized as a physical system or substrate with specific inherent tasks. These tasks involve permutations on a designated permutation set and a copy task.
In this context, an information medium is characterized by a set X of disjoint attributes where the following tasks are feasible: 1) all permutation tasks within the specified permutation set, and 2) the copy task.
To illustrate, consider an example involving tasks such as ‘permutation’ and ‘copy,’ where the value of the first slot is copied to the second slot. The defined set X with these distinctive properties is termed an ‘information variable,’ serving as a generalized concept that goes beyond traditional notions of classical information, aligning with the goals of the Constructor Theory of Information.
Principles About Information Media
Interoperability Principle
The combination of two information media with information variable
and
is an information medium with information variable
(Informally: ‘Information variables can be copied from any information medium to any other information medium of at least the same capacity’)
Superinformation Media
The concept of a ‘Superinformation medium’ within the Constructor Theory of Information introduces an intriguing hierarchy that extends beyond traditional information media. Let’s quickly go over some definition first:
Superinformation medium: An information medium with at least two variables X and Y whose union is not an information variable.
Example: a qubit with the information variables X and Y
Consider a qubit as an example, where the information variables X and Y correspond to non-commuting observable, and their eigenstates constitute information variables that are individually permutable and copyable.
What does it mean for information variables too be individually permutable and copyable? Let’s break it down.
Non-Commuting Observable: In quantum mechanics, observables are properties of a system that can be measured. Non-commuting observables are properties that cannot be measured simultaneously with arbitrary precision. For example, you cannot precisely measure both the position and momentum of a particle at the same time.
Individually Permutable and Copyable: If we look at the eigenstates of X and Y individually, we can rearrange them (permute) and make copies of them. This is a characteristic of information variables — we can manipulate and duplicate the information associated with X or Y on its own.
In simple terms, think of X and Y as different characteristics of a quantum system, like its spin in different directions. The eigenstates are the possible ways these characteristics can manifest. This means we can mess around with these characteristics individually, rearrange them, and make copies. It’s a bit like saying if we know a property of the qubit, we can change it or duplicate it. However, the intriguing part comes when we try to deal with both properties simultaneously.
However, when we examine the setwise union of X and Y, this combined set becomes non-copyable. The qubit, therefore, exhibits a unique property — its union of information variables is not consistently an information variable. This distinct characteristic classifies the qubit as a Superinformation medium.
Let’s prove this logically? Let’s assume that X and Y are individual information variables, meaning that each of them is permutable and copyable. This implies that for any set of eigenstates associated with X or Y, permutation and copy tasks are feasible. Now, consider the setwise union of X and Y, denoted as X ∪ Y. If X and Y were both information variables, then their union would also be an information variable. However, the definition of a Superinformation medium posits that the union of X and Y does not form an information variable. If X ∪ Y were copyable, it would contradict the definition of Superinformation medium because a copy task would be possible on the combined set. (Hence proven)
Notably, the hierarchy established in this framework reveals that Superinformation media form a subclass of information media, imposing additional restrictions. Moreover, quantum systems emerge as a subset of Superinformation media, showcasing that Superinformation media encapsulate all the qualitative properties of quantum systems. Remarkably, this framework achieves a comprehensive understanding of the information-theoretic structure of quantum systems without directly invoking the formalism of quantum theory.
Significantly, within this framework, a hierarchical structure becomes evident, elucidating that Superinformation media constitute a specific category within information media, thereby imposing supplementary constraints. Furthermore, it becomes apparent that quantum systems manifest as a distinct subset of Superinformation media, illustrating that Superinformation media encompass all the inherent qualitative attributes of quantum systems. Noteworthy is the achievement of a thorough comprehension of the information-theoretic architecture of quantum systems through this framework, without the direct invocation of quantum theory’s formalism.
‘Non-classicality’ within the superinformation framework
A system is ‘non-classical’ if it has at least two incompatible variables X and Z, one of which is an information variable.
‘Incompatible’ means that it is impossible that X and Z are copied simultaneously to perfect accuracy. This generalises the idea of non-commutativity to the superinformation framework. Note that it does not necessarily mean that both of these variables are information variables. It’s a slightly weaker condition than being a superinformation medium. A medium with non-classicality is not necessarily a superinformation medium but it has features that are not purely classical.
What does classical mean here? In the realm of classical systems, a singular information variable encapsulates all of its attributes, thereby allowing for the representation of its entirety through information variables. This characteristic implies the feasibility of copying and permuting said information variable in all conceivable manners.
Why not define information as Shannon would?
A important inquiry arises as to why we refrain from delineating information in a manner akin to Shannon information. This decision stems from the recognition that our objective is not to articulate the physical attributes requisite for systems to embody what we denote as information. Shannon posited that distinguishable states exist; however, his focus did not extend to interrogating the fundamental attributes that render a system capable of containing what he termed ‘information.’ Instead, he contended that the discernibility of distinct states was paramount, without delving into the physical underpinnings of this discernibility. Consequently, we adopt an implicit definition of information, conceptualizing it as an entity transmissible by systems capable of performing designated tasks, denoted as information media.
An information-theoretic argument for the totalitarian property of QT
Assume three general principles:
- Locality (no action at a distance)
- Interoperability of information
- 1:1 dynamics (dynamical law is logically reversible, general dynamical transformation that doesn’t use group theory)
An information-theoretic argument for the totalitarian property of QT
Theorem 1: (Generalization of DeWitt’s theorem): If it is possible to couple a superinformation medium Q with an information medium S via a copy-like interaction, then S must be non-classical.
The totalitarian property suggests a robust witness of non-classicality for the system S.
Use two superinformation media to extract S’s non-classical features!
Assume:
- Locality
- Interoperability of information
Theorem 2 (Witness of non-classicality): If S can locally mediate entanglement between two superinformation media Q and Q’, then S is non-classical.
In Chiara Marletto’s constructor theory of information framework, locality isn’t about spooky action at a distance, but rather about preserving the integrity of information across isolated systems. Unlike the Bell non-locality, where entangled particles defy classical notions of locality, constructor theory maintains that when information processing occurs in one system, its variables shouldn’t instantaneously and mysteriously affect another. Think of it like this: imagine you’re baking a cake in one kitchen, while your sibling simultaneously paints in another. No matter how much delicious chaos unfolds in your culinary domain, the paintbrushes across the hallway won’t suddenly start dancing on their own. Constructor theory’s locality encapsulates this idea.
This principle plays a crucial role in defining information itself. The theory postulates that information resides in the “attributes” of variables, like the cake’s temperature or the color of your sibling’s brushstrokes. Locality ensures that tasks performed on one set of variables, within their isolated “kitchen,” can’t magically alter the attributes of variables in another unrelated “room.” This effectively prevents information, whether about cake batter or paint splatters, from teleporting without any tangible interaction.
The connection between locality and quantum mechanics also becomes apparent. In the quantum world, observables, like a particle’s spin, commute if they pertain to independent systems. When you measure the spin of a particle in one system, it doesn’t magically flip the spin of its entangled partner in another, even if their fates are mysteriously intertwined. Constructor theory’s locality mirrors this behavior, with unitaries and CP maps representing tasks that, like measuring spin, can influence attributes within a system without having any ghostly repercussions on variables elsewhere.
The significance of locality extends beyond mere philosophical tidiness. It underpins how constructor theory describes experiments. By preventing information from instantaneously zipping between isolated systems, the theory can explain measurement processes and information acquisition rigorously, without resorting to non-local explanations. It’s like having a well-defined recipe for baking knowledge, where ingredients flow only through designated channels, never magically appearing from thin air across the kitchen counter.
Conclusion and Applications
Constructor Theory is a candidate to expand on the theory of quantum computation, and ultimately to deliver the theory of the universal constructor. It provides novel physical principles to understand systems that go beyond current dynamical laws — e.g. by unifying quantum and classical information. Constructor theory focuses on fundamental physical constraints and information transformations, offering a potentially more general and robust way to predict and understand the behavior of even poorly understood systems. It’s like building a bridge between what we know (the quantum and classical worlds) and what we don’t, paving the way for exploring uncharted territory in physics and beyond.
References
[1] Deutsch, D., & Marletto, C. (2015, February). “Constructor theory of information.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2174), 20140540. Retrieved from http://dx.doi.org/10.1098/rspa.2014.0540.
[2] Marletto, C., & Vedral, V. (2020, October). “Witnessing nonclassicality beyond quantum theory.” Physical Review D, 102(8). Retrieved from http://dx.doi.org/10.1103/PhysRevD.102.086012
[3] Marletto, C., & Vedral, V. (2017, December). “Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity.” Physical Review Letters, 119(24), 240402. Retrieved from https://link.aps.org/doi/10.1103/PhysRevLett.119.240402.
[4] Deutsch, D. (2013). “Constructor Theory.” arXiv preprint, arXiv:1210.7439. Retrieved from https://arxiv.org/abs/1210.7439.
