Here are some comments by Robert Spekkens on my essay, divided into two posts for length reasons:
Dear Tobias,
Thank you for directing me to your paper. I largely agree withyour perspective. I am also of the opinion that it is best *not* toconsider the toy theory as a probabilistic theory. I have often saidthat all that is required for this theory is modal logic, that is, thelogic of possibility and necessity. Bob Coecke and his students havebeen working on category-theoretic presentations of the toy theory andwithin that framework, one can be a bit more formal about this. I attachan article which I am writing with them and which is available on Bob'swebsite (http://www.comlab.ox.ac.uk/people/bob.coecke/) but not yetpublished. The phrase "possibilistic theory" is actually used anddiscussed therein (see remark 4.5). See also arXiv:0808.1037v1[quant-ph].
Despite the naturalness of thinking of the toy theory as apossibilistic theory, most people's intuitions go the other way and I amconstantly asked why I didn't choose to include arbitrary convexcombinations of the epistemic states. I take it that you are askingyourself the same question. Here is my response.
First, note that allowing such convex sums of epistemic states would, strictly speaking, violate the knowledge-balance principle because for such states one could not say that one has the answer to half the questions in a canonical set. I therefore disagree with your statementon p. 4 that "the knowledge balance principle advocated in [Sp] would still be satisfied".
Second, one cannot close the set of epistemic states under convex combination while simultaneously keeping a restricted notion of coherent combination without doing violence to the quality of the analogy with quantum theory. Here is what I say in section VII.C of my paper.
[quote] We have seen that there are two types of binary operations defined for epistemic states in the toy theory, analogous to convex combinations and coherent superpositions of quantum states. However, these operations are partial; they are not defined for every pair of epistemic states. It might therefore seem desirable to close the set of epistemic states in the toy theory under convex combination with arbitrary probability distributions. In this case, the set of allowed epistemic states for a single elementary system would have the shape of an octahedron in the Bloch sphere picture. Hardy's toy theory, for instance, has this feature [35]. Such a variant of our toy theory has also been consideredby Halvorson [61]. However, there is an important sense in which such a theory is less analogous to quantum theory than the one presented in this paper. The toy theory shares with quantum theory the feature that every mixed state has multiple convex decompositions into pure states, whereas in this modified version, there are many mixed states that have unique decompositions. Similarly, in the toy theory, as in quantum theory, every mixed state has a "purification"-a correlated state between the system of interest and another of equal size such that the marginal over the system of interest is equal to the mixed state in question-whereas in the modifiedversion, there are many mixed states that have only a single purification.
[quote] The problem with the modified theory is that although convex combination has been extended to a full binary operation rather than a partial one, the coherent binary operations have not been so extended. Moreover, although one has allowed arbitrary weights in the convex combinations, one has not allowed the analogue of arbitrary amplitudes and phases for the coherent binary operations. It is likely that a better analogy with quantum theory can be obtained only if both operations are generalized. Unfortunately, it is unclear how to do so in a conceptually well-motivated way. [end quote]
