## Spekkens Toy Theory

There is about a week until I'll attend "The New Directions" conference in Washington DC, and after the conference I'll have plenty of fresh new ideas to present. In the meantime I'll devote this and next post to discuss the so-called Psi-Epistemic point of view of Quantum Mechanics.

The basic intuition originates from phase space where particles have a well define position and momenta, and a probability distribution in phase space corresponds to genuine lack of knowledge. In the realist epistemic point of view, the wavefunction corresponds to knowledge about an underlying ontic reality. This ontic reality is left unspecified: it could be classical physics with hidden variable, it could be the wavefunction itself, or it could be something completely new and undiscovered.

The key question is this: can the ontic state exist in more than one epistemic state? If yes, then a measurement in quantum mechanics simply reveals the pre-existing reality. There are a lot of roadblocks to construct such an epistemic model, but the point of view taken by Spekkens was different: let's not construct a model which recovers completely quantum mechanics predictions, but let's construct a simple epistemic toy theory and see what unintuitive quantum phenomena get a simple explanation.

The basic idea is that of simulating spin 1/2 particle measurements on 3 axis: x, y, z. Here is a picture from the excellent review paper by Matt Leifer: http://arxiv.org/pdf/1409.1570v2.pdf

In Spekkens toy model there are 2 x states: + and - and 2 y states: + and -. The system is at any point in one of the 4 possible states: ++, +-, -+, --, but here is the catch: you cannot measure both at the same time.  Moreover, during measurement the particle makes a jump from the unmeasured state to the other.

The spin x measurement corresponds to measuring the x coordinate, the spin y measurement correspond to measuring the y coordinate, while the measurement of spin z corresponds to measuring the "sameness of x and y" coordinates.

Repeatable measurements always yields the same outcome, just as in the quantum case, and measurement of "spin x" followed by a measurement of "spin y" perturbs the system (remember the jumping of the unmeasured coordinate during measurement) and a third measurement of "spin x" gives a random outcome.

Now here are the successes of the toy model:
• nonorthogonal pure states cannot be perfectly distinguished by a single measurement
• no-cloning
• non-uniqueness of decomposition of mixed states
Given such impressive successes a lot of people in the quantum foundations fell in love with the realistic epistemic point of view. No full blown realistic epistemic model for quantum mechanics was ever developed, and the PBR theorem which I'll talk about next time crushed any hopes for it (or so is my opinion).

Of course there is the other possibility of having a non-realistic epistemic interpretation, like Copenhagen and neo-Copenhagen and this possibility is alive and well.