# Is linearity of the quantum state space a necessary postulate in the reconstruction of quantum theory?

This question is about quantum reconstruction. I am new to this topic, and I decided to read some papers on it.

I selected some works which follow an "information-focused" approach. The authors of these works try to formulate QM through an alternate set of axioms, which avoids the standard quantum state-space structure (in terms of Hermitian operators/linear spaces) and is centered on the idea that the physical state can carry only a limited amount of information.

Some of these authors, after stating the postulates of the theory, use the standard QM formalism to get their results. This can be done because their “central” postulate about information includes some extra hypothesis about the linearity of the state space. Perhaps in some authors this extra hypothesis is somehow hidden, for example in [1] and [3], see below (but this is just my opinion).

Of course, since their works are on the reconstruction of QM, their aim is to obtain the same properties of standard QM, and linearity is among these properties.

My questions are:
1) When I say that linearity is somehow hidden, is my interpretation right?

2) Are there other theoretical works on quantum reconstruction which follow this idea of “limited information” systems but avoid any hypothesis about linearity? (but, anyway linearity will be a consequence of the axioms of that theories)

Here are some references:
[1] Caslav Brukner and Anton Zeilinger: http://quantmag.ppole.ru/Articles/Quo_Vadis_Quantum_Mechanics.pdf#page=60

Carlo Rovelli:
[2] "Relational quantum mechanics" https://arxiv.org/abs/quant-ph/9609002

Borivoje Dakic and Caslav Brukner:
[3] Quantum theory and beyond: Is entanglement special? https://arxiv.org/abs/0911.0695

Even in Hardy's most cited work ([4] https://arxiv.org/abs/quant-ph/0101012), which doesn't follow an information-focused approach, there is a very strong "Simplicity" axiom which involves mathematical properties of the state space, closely related to linearity and Hilbert structure.

• What do you mean? Quantum mechanics is inherently linear! – Norbert Schuch Oct 18 at 15:06
• Of course linearity is among the basic hypotheses of the standard formulations of QM. I am talking about different set of axioms that are equivalent to those of standard QM. In the works I cited, linearity seems to be considered as a sort of background hypothesis and never mentioned clearly. – PFerro Oct 18 at 15:19
• We can formulate QM starting with an algebra $\mathcal{A}$ generated by observables -- a C*-algebra -- and define a state to be a map $\rho:\mathcal{A}\to\mathbb{C}$ satisfying certain properties, as described in my answer here. Then the GNS theorem implies the existence of a representation of $\mathcal{A}$ as linear operators on a Hilbert space. In this approach, we don't assume a linear structure on the state space (we derive it instead), but we do still assume that each individual state is a linear map $\rho$. – Chiral Anomaly Oct 18 at 22:21
• This post might be of interest to you: physics.stackexchange.com/questions/285635/… – D. Halsey Oct 19 at 13:37
• Thanks for editing the question, this is much better. You probably want to look at Gisin's theorem: Gisin, "Weinberg's non-linear quantum mechanics and supraluminal communications," dx.doi.org/10.1016/0375-9601(90)90786-N , Physics Letters A 143(1-2):1-2 – Ben Crowell Oct 20 at 14:37