Do toy models of quantum mechanics help us better understand "regular" quantum mechanics?

My answer is a resounding "yes!" First of all, let me say that I agree with Steve that it is better to talk about toy theories than toy models. In fact, I favor the terminology “foil theory” for this sort of thing. Essentially, if we want an answer to the question “why quantum theory?” we need to study the alternatives, the ways the world might have been – the foils to quantum theory. It is only against a landscape of possible theories that one can identify what is special about quantum theory.

One can certainly devise foil theories by a modification of certain features of the formalism of quantum theory. However a more fruitful approach, in my view, is to define novel foil theories by deriving their formalism from a set of physical principles within an operational framework. Recently, there have been several approaches of this sort. For instance, there has been a lot of work in the last few years on foil theories that allow violations of Bell inequalities that are stronger than those allowed in quantum theory, but which still do not allow superluminal signalling.

There have been several workshops on this topic. The first was in Cambridge in 2007. Matt Leifer provides a synopsis here: http://www.fqxi.org/community/forum/topic/86. The next two happened at the ETH in Zurich in 2008 and 2010: http://www.qit.ethz.ch/workshops/IPLN2008. Finally, there will be a conference this year at Perimeter Institute: http://www.perimeterinstitute.ca/Events/Conceptual_Foundations_and_Foils_for_QIP. Looking up the participants of these workshops on the arxiv will yield lots of relevant papers.

My own work in this area starts with a classical statistical theory, and posits a principle that restricts how much information an agent can have about the classical state. The article “In defense of the epistemic view of quantum states: a toy theory”, http://arxiv.org/abs/quant-ph/0401052 describes one such foil theory, and the following talk of mine from 2008 summarizes the broader research program at a nontechnical level: http://pirsa.org/08020051/. The foil theories one obtains in this way (epistemically-restricted classical statistical theories) reproduce a wide variety of quantum phenomena, such as the noncommutativity of measurements, interference, many features of entanglement, no cloning, teleportation, mutually unbiased bases, and many others. What they suggest about the interpretation of quantum theory is that quantum states are states of incomplete knowledge rather than states of reality.

Do toy models of quantum mechanics help us better understand "regular" quantum mechanics?

My answer is a resounding "yes!" First of all, let me say that I agree with Steve that it is better to talk about toy theories than toy models. In fact, I favor the terminology “foil theory” for this sort of thing. Essentially, if we want an answer to the question “why quantum theory?” we need to study the alternatives, the ways the world might have been – the foils to quantum theory. It is only against a landscape of possible theories that one can identify what is special about quantum theory.

One can certainly devise foil theories by a modification of certain features of the formalism of quantum theory. However a more fruitful approach, in my view, is to define novel foil theories by deriving their formalism from a set of physical principles within an operational framework. Recently, there have been several approaches of this sort. For instance, there has been a lot of work in the last few years on foil theories that allow violations of Bell inequalities that are stronger than those allowed in quantum theory, but which still do not allow superluminal signalling.

There have been several workshops on this topic. The first was in Cambridge in 2007. Matt Leifer provides a synopsis here: http://www.fqxi.org/community/forum/topic/86. The next two happened at the ETH in Zurich in 2008 and 2010: https://web.archive.org/web/20130801040747/http://www.qit.ethz.ch/workshops/IPLN2008. Finally, there will be a conference this year at Perimeter Institute: https://web.archive.org/web/20110524065750/http://www.perimeterinstitute.ca/Events/Conceptual_Foundations_and_Foils_for_Quantum_Information_Processing/Abstracts/. Looking up the participants of these workshops on the arxiv will yield lots of relevant papers.

My own work in this area starts with a classical statistical theory, and posits a principle that restricts how much information an agent can have about the classical state. The article “In defense of the epistemic view of quantum states: a toy theory”, http://arxiv.org/abs/quant-ph/0401052 describes one such foil theory, and the following talk of mine from 2008 summarizes the broader research program at a nontechnical level: http://pirsa.org/08020051/. The foil theories one obtains in this way (epistemically-restricted classical statistical theories) reproduce a wide variety of quantum phenomena, such as the noncommutativity of measurements, interference, many features of entanglement, no cloning, teleportation, mutually unbiased bases, and many others. What they suggest about the interpretation of quantum theory is that quantum states are states of incomplete knowledge rather than states of reality.