Open quantum systems are those which interact in some uncontrolled way with other degrees of freedom which we refer to as their ‘environment’. Essentially all cognitive systems are ‘open’, since the brain is highly interconnected and thoughts and beliefs about one event can rarely be isolated from those about another.

In classical physics the distinction between closed and open systems is not of any profound significance, however in quantum theory open and closed systems obey different types of evolution laws and can have different qualitative behaviours. In particular interactions between a system and its environment can cause the system to lose its quantum properties and become ‘classical’.

Relatively little work has been done on modelling open quantum cognitive systems (a notable exception is the work of Fuss and Navarro, 2013), however this is likely to be critical in understanding the conditions under which we should and should not expect quantum behaviour in cognition.

I wrote a tutorial recently, currently under review at JMP, which includes an introduction to open quantum systems. The tutorial also includes other useful ideas, such as how to model imperfect measurements via POVMs, and how to model quantum states that include classical uncertainty. The tutorial was based on some earlier notes that I wrote for a workshop at CogSci which can be found here (pdf). You can also find the slides from my presentation here (pdf).

I was also recently asked to explain the idea of a `partial trace’ which occurs in discussions on open systems (and more generally in any composite system.) Some brief notes can be found here (pdf).

For those interested a very worthwhile introduction to the ideas of decoherence and emergent classicality is the paper by Halliwell. A classic review is given by Zurek.