Anima Ex Machina

Originally uploaded by hzenilc.

Models and Simulations 2
11 – 13 October 2007
Tilburg University, The Netherlands

I attended this conference one month ago. Among several interesting talks, one in particular caught my attention. It was given by Michael Seevinck from the Institute for History and Foundations of Science at Utrecht, The Netherlands. His talk was about the foundations of Quantum Mechanics, and there were many NKS related topics that it brought  to mind. He talked about reconstructing Quantum Mechanics (QM) from scratch by exploring several restricted models in order to solve the so-called measurement problem, to deal with the nonlocality of quantum correlations, and with its alleged non-classicality, there being  no consensus on  the meaning of Quantum Mechanics  (Niels Bohr said once: “If you think you have understood quantum mechanics, then you have not understood quantum mechanics.”—More quotes of this sort on QM here).  The restrictons chosen in order to reconstruct the theory must be physical principles and not  theoretical assumptions. In other words, one approaches the problem contrariwise than is traditional, taking the least possible restrictions and exploring the theories that can be built thereon. The speaker characterized  this approach  as the “study [of]  a system from the outside” in order to ”reconstruct the model”. It is basically a pure NKS approach: “Start from a general class of possible models and try to constrain it using some physical principles so as to arrive at the model in question (in this case QM).”

One can then proceed to ask such questions as how one might identify QM uniquely, what it is that makes QM quantum, what set of axioms in the model is to be used, and which of them are necessary and sufficient? The question of meaning, previously asked of the formalism, is removed, and bears, if at all, only on the selection and justification of  first principles. Seevinck came up with the following interesting statement: “The partially ordered set of all questions in QM is isomorphic to the partially ordered set of all closed subspaces of a separable Hilbert space” (one of Mackey’s axioms in his axiomatisation of 1957). He added: “They (the principles)have solely an epistemic status. The personal motives for adopting certain first principles should be bracketed. One should be ontologically agnostic. The principles should be free of ontological commitment.” And further: “…axioms are neutral towards philosophical positions: they can be adopted by a realist, instrumentalist, or subjectivist.” He cited Clifton, Bub and Halverson who provided the following quantum information constraints used to derive quantum theory:

1. No superluminal information transfer via measurement.

2. No broadcasting

3. No secure bit commitment

Seevinck’s methodology in further detail is: Start with a general reconstruction model with a very weak formalism. Gradually see what (quantum) features are consequences of what added physical principles, and also see which features are connected and which features are a consequence of adding which principle. One thereby learns which principle is responsible for which element in the (quantum) theoretical structure.

One can generate further foundational questions over the whole space of restricted models, e.g.  how many of them:

- forbid superluminal signalling?

- allow nonlocality, and to what extent?

- solve NP-complete problems in polynomial time?

An important question which arises concerns whether intrinsic randomness would be of a different nature in different models or whether all of them would yield to deterministic randomness.

His talk slides are available online. Highly recommended.

Among other interesting people I met was Rafaela Hillebrand, of  the Institute for The Human Future at Oxford University. The Institute’s director, Nick Bostrom, has proposed an interesting theory concerning the likelihood that our reality is actually  a computer simulation. I have myself approached the  question in my work on experimental algorithmic complexity, in particular in my work on  the testability and the skepticism content of the simulation hypothesis. I will post on that subject later. The subject of thought experiments–in which I have an interest– was one that came up frequently.

In an exchange of emails, Seth Lloyd and I discussed the topic I wrote about some posts ago. Here is some of it.

According to Lloyd, there is a perfectly good definition of a quantum Turing machine (basically, a Turing machine with qubits and extra instructions to put those qubits in superposition, as above). A universal quantum computer is a physical system that can be programmed (i.e., whose state can be prepared) to simulate any quantum Turing machine. The laws of physics support universal quantum computation in a straightforward way, which is why my colleagues and I can build quantum computers. So the universe is at least as powerful as a universal quantum computer. Conversely, he says, a number of years ago he proved that quantum computers could simulate any quantum system precisely, including one such as the universe that abides by the standard model. Accordingly, the universe is no more computationally powerful than a quantum computer.

The chain of reasoning, to jump to the quantum computer universe view, seems to be 1 and 2 implies 3 where 1, 2 premises and the conclusion 3 are:

1 the universe is completely describable by quantum mechanics
2 standard quantum computing completely captures quantum mechanics
3 therefore the universe is a quantum computer.

Seth Lloyd claims to have proved the connection between 1 and 2, which probably puts the standard (or some standard) theory of quantum mechanics and the standard quantum computing model in an isomorphic relation with each other.

Lloyd’s thesis adds to the conception of the Universe as a Turing computer an important and remarkable claim (albeit one that depends on the conception of the quantum computer), viz. that the Universe is not only Turing computable, but because it is constituted by quantum particles which behave according to quantum mechanics, it is a quantum computer.

In the end, the rigid definition of qubit together with the versatility of possible interpretations of quantum mechanics allows, makes difficult to establish the boundaries of the claim that the universe is a quantum computer. If one does assume that it is a standard quantum computer in the sense of the definition of a qubit then a description of the universe in these terms assumes that quantum particles encode only a finite amount of information as it does the qubit, and that the qubit can be used for a full description of the world.

Quantum computation may have, however, another property that may make it more powerful than Turing machines as Cristian Calude et al. have suggested. That is the production of indeterministic randomness for free. Nevertheless, no interpretation of quantum mechanics rules out the possibility of deterministic randomness even at the quantum level. Some colleagues, however, have some interesting results establishing that hidden variables theories may require many more resources in memory to keep up with known quantum phenomena. In other words hidden variable theories are more expensive to assume, and memory needed to simulate what happens in the quantum world grows as bad as it could be for certain deterministic machines. But still, that does not rule out other possibilities, not even the hidden variables theories, even if not efficient in traditional terms.

This is important because this means one does not actually need ‘true’ randomness, the kind of randomness assumed in quantum mechanics. So one does not really need quantum mechanics to explain the complexity of the world or to underly reality to explain it, one does require, however, computation, at least in this informational worldview. Unlike Lloyd and Deutsch, it is information that we think may explain some quantum phenomena and not quantum mechanics what explains computation (neither the structures in the world and how it seems to algorithmically unfold), so we put computation at the lowest level underlying physical reality.

Lloyd’s thesis adds to the conception of the Universe as a Turing computer an important and remarkable claim (albeit one that depends on the conception of the quantum computer), viz.  that the Universe is not only Turing computable, but because it is constituted by quantum particles which behave according to quantum mechanics, it is a quantum computer computing its future state from its current one. The better we understand and master such theories, the better prepared we would be to hack the universe in order to perform the kind of computations–quantum computations–we would like to perform.

I would agree with Rudy Rucker too as to why Seth Lloyd assigns such an important role to quantum mechanics in this story. Rudy Rucker basically says that being a subscriber to quantum mechanics, Lloyd doesn’t give enough consideration to the possibility of deterministic computations. Lloyd writes, “Without the laws of quantum mechanics, the universe would still be featureless and bare.” However, though I am one among many (including Stephen Wolfram) who agree  that it is unlikely that the universe is a cellular automaton, simply because cellular automata are unable to reproduce quantum behavior from empirical data (but note that Petri and Wolfram himself attempt explanations of quantum processes based on nets), there’s  absolutely no need to rush headlong into quantum mechanics. If you look at computer simulations of physical systems, they don’t use quantum mechanics as a randomizer, and they seem to be able to produce enough variations to feed a computational universe. Non-deterministic randomness is not neccesary; pseudorandomness or unpredictable computation seem to be enough.