Applications

A pervasive issue in contemporary physics : hybrid systems.

In the traditional conception of physics, we usually rely on dynamical laws to make predictions: specific dynamical constraints that tell us what happens in the universe in space and time. But there are domains where the current laws are expected to break down.

Commentary to the illustration

A parrot and an atom: both are information media. One macroscopic system that can talk and learn, one sub-microscopic system following an evolution along clearly stated and accurate physical laws. While one system emerges from the other, the laws governing the microscopic and macroscopic realms seem irreconcilably different. Yet, we can identify what evolutions can and cannot be made to happen under a variety of conditions: their counterfactual properties. For example, identifying two different physically possible flying positions of a parrot and two different energetic states of an atom allow for us to encode binary logic in either system, irrespective of their detailed dynamics. Thus, one is able to transfer information between both systems, despite all their differences and our incomplete knowledge of their dynamical evolution. This also allows us to probe one system with the other to explore other principles and boundaries, whilst leaving space for new properties of each system to be discovered upon refinement of our observational means and explanatory theories.

An important case is  that of hybrid systems: where a quantum system (for instance, a qubit) interacts with another system that may not fully obey quantum theory, or whose dynamical trajectories are so complicated that they cannot be computed exactly. A notorious example of the former case is gravity: we have several inequivalent formulations of a quantum theory of gravity, but the best theory of gravity that we possess at present is still Einstein’s general relativity, which is not quantum at all. General relativity says that spacetime curves according to where a mass is; but if a mass is quantum, it can be in two locations at the same time, and this breaks the structure of general relativity. This is why we are uncertain about how to treat a hybrid system made of a quantum mass in a superposition of different locations with a univocal dynamical law. And constructor theory can help with this problem.

Another example (of the second kind of hybrid systems) is a macroscopic system with lots of degrees of freedom (e.g. a living cell or even a cat!) interacting with a small qubit. Such systems have been key to thought experiments like Schrödinger cat, but their dynamics may be intractable.

So how do we make predictions in these cases?

Theories of principles, such as constructor theory, come to the rescue. This is because they do not rely on specific laws of motion, but they are more general than them: they constrain laws of motion, those that are known and those that are yet to be known.

Hybrid systems have been occasionally treated in the past with a principle-driven approach: if you are in for some physics details, you can look up Wheeler and Bondi’s derivation of gravitational red-shift from the equivalence of mass and energy; DeWitt’s and Feynman’s arguments for quantising gravity; Einstein’s well-known derivations in the theory of relativity; Hawking’s and Bekenstein’s thermodynamic analysis of black holes.

In fact, the logic of principles underlies the branch of physics called thermodynamics. In existing thermodynamics, the principles only hold at a macroscopic scale. The laws of thermodynamics can be derived from microscopic dynamics using statistical mechanics, but this leads to approximate, probabilistic laws that remain inadequate to describe microscopic systems. Constructor theory can solve this problem as its principles are scale-independent: they hold at all scales, not just at the microscopic or at the macroscopic scale. These principles can describe the thermodynamic laws of qubits, macroscopic objects, and systems where the two interact. We therefore expect constructor-theoretic principles to be formidable tools to deal with hybrid systems.

Principles in constructor theory

Interoperability of information

Two systems that each can behave like an information medium with a given capacity (say, each is a bit), and when considered jointly still behave like an information medium, of larger capacity (two bits worth of information).

Important feature: to formulate the principle you don’t need to assume any specific dynamical framework; or a specific scale. The principle is scale– and dynamics-independent. Compare this with the case of the no-cloning theorem in quantum information, where you need to assume quantum theory’s laws of motion in order to prove it; or for example the second law of thermodynamics in one of the traditional formulations, where it is necessary to assume a macroscopic scale for it to be applicable.

Commentary to the illustration

A perpetuum mobile sustains periodic dynamics without an external source of energy. This object is forbidden by the laws of conservation of energy, which we can visualise as requiring all paths that the energy can take to be circular. This requirement of energy-recirculation is scale-independent: It needs to be fulfilled by the simplest resonator in the center of this illustration as much as the most intricate macroscopic structures the resonator may couple to. The interoperability of information works with the same logic: like energy, information can be defined in a scale-independent way, and the principle of interoperability of information says it can be copied from one information medium to any other, irrespective of their physical details.

Principle of locality (or no action at a distance)

Commentary to the illustration

Two separate entities float above an abstract communication plane, a “sea of light”. Information cannot travel faster than the speed at which this plane transmits signals, the speed of light. The two objects are reading out and encoding information e.g. through objects that intersect with that plane, sensing or causing ripples on its surface. This happens with a rhythm that is partially dictated by the inner workings of the objects and partially influenced by environmental interactions. The entities’ inner workings, with their tilted gearwheels and hidden parts, hint at a degree of imperfection and uncertainty concerning the composition of each object and the computational pathways that read out- and create stimuli.

Given a partition into subsystems A and B, a transformation that acts on A only cannot change B.

Important note: The principle of locality is a pillar of both of our fundamental theories of physics: general relativity and quantum theory. Note the misleading, unfortunate terminology related to “Bell-type non-locality”, which is used to indicate particular kinds of correlations that arise in entangled systems. Despite prevalent misconceptions, these quantum correlations perfectly satisfy the principle of locality!

A remarkable application of these two principles is a recently proposed witness of non-classicality to check if a hybrid system (which may or may not obey quantum theory) has itself some quantum features1. It is based on the fact that if the principles of locality and interoperability hold, then one can conclude the following:  if a system M can entangle two quantum systems Q and Q’, then M must be itself non-classical. This means it must have one of the basic features of quantum systems. Namely, the Heisenberg uncertainty principle must be satisfied by two of its variables, which must behave just like position and energy for an electron. If you can determine one with certainty, you cannot determine the other, and vice-versa. This is an exciting application of constructor theory, which underlies a recently proposed experiment to test quantum effects in gravity. This test also indirectly tests constructor theory’s interoperability principle, because it is based on it. For example, if you could create entanglement despite having a classical object as a mediator, it would mean that either the interoperability law or locality are falsified.

1. For further information about the tests to witness non-classicality in gravity, see: Marletto and Vedral (2017), Bose et al. (2017)
Illustrations by Angelina Frank (2020)

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