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I know that the title of the question is rather vague, so let me clarify what I mean.

For a quantum system, the set of states has infinitely (even continuously) many extreme points, i.e. there are infinitely many pure states. By the set of states, I mean the convex set of density operators $\mathcal{S} \subseteq \text{Herm}(\mathcal{H})$ on a Hilbert space $\mathcal{H}$ rather than the set of unit vectors in the Hilbert space $\mathcal{H}$. So I am talking about a convex subset of the vector space of Hermitian operators on $\mathcal{H}$ (for my puroposes, we can restrict to finite dimensions).

For example, the set of states for a spin-1/2 particle is given by the Bloch ball. Every point on the boundary of the Bloch ball is an extreme point of the Bloch ball and therefore a pure state. This boundary is a sphere, containing continuously many states.

I am wondering whether there are physicists who have come up with reasons why the set of states of a (quantum) system should be discretized instead of continuous in the sense that it should only have finitely many extreme points / pure states. To see geometrically what I mean, see the following figure:


More precisely, my question is: What are possible reasons or motivations why the set of states of a (quantum) system should be the convex hull of finitely many points (i.e. a polytope), instead of having infinitely many pure states?

Looking for such reasons, I have come across papers like this one:

Discreteness and the origin of probability in quantum mechanics

They say that quantum-gravitational considerations might suggest some kind of discretization of the states of a system. First of all, I don't really understand their argumentation yet why there should be discreteness. Moreover, I'm not sure whether the kind of discreteness they propose is the same as the one I'm asking for (e.g. I'm not sure whether their state spaces are convex).

If anyone can help me with this kind of argumentation (by explaining it or pointing to additional sources) or if anyone has come across other arguments for discreteness, I would be glad to see them.

I'm aware of the fact classical systems can be described by a simplex, therefore bearing this kind of discreteness. However, I'm interested in reasons why quantum state spaces should be discretized.


Thanks for the answers so far. Perhaps I should be clear about a few things: I do not think that quantum state spaces are discrete, and I do not really try to justify discreteness of quantum state spaces. I also see reasons why a quantum state space cannot be discrete. I am just curious whether other physicists have come up with justifications or reasons for discreteness. Why? To see what ideas are ruled out by arguments against discreteness.

The answers so far are not uninteresting, but they provide reasons why a quantum state space should not be discrete. I see that my question somehow asks for justification of something that cannot be, but I am still interested in what possible motivations for discreteness there could be.

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There is no known reason. I explained why this idea is difficult, but you are asking to justify it, when there is no known persuasive justification. –  Ron Maimon Jul 5 '12 at 14:41
A practical answer may lie in the fact discrete quantum spaces are easier to code up for computational simulation... –  Alex Nelson Jul 5 '12 at 16:00
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2 Answers

This idea violates the superposition principle of quantum mechanics, and picks out a preferred basis. As such, it wrecks the property of QM that the state space is invariant under arbitrary unitary transformations, only certain dicrete unitaries will preserve the phase space structure. Such an idea is a new thing, and you need to check that it makes sense, because on its face, such a thing will lead to irreversible decoherence in any finite state space.

For example, suppose I place a spin 1/2 particle in a magnetic field which I control. By using a weak field for long times, I can make the pure state point in any direction I want in the Block sphere. Saying that only a discrete set of states are allowed to be pure will break the rotation symmetry into something else, some directions of spin orientation will be allowed to be pure, while others will not. This can only happen if the effective pure states are extremely dense on the Block sphere.

If you insist that the discrete transformations of the Block sphere form a discrete subgroup of SU(N) (where N is the number of basis quantum states, and SU(N) is the full unitary group on these), then I think you are in trouble. The discrete subgroups of SU(N) are highly constrained, for SU(2) you essentially get the symmetry group of the Platonic solids, while for SU(N) it's open, but you would need that the discrete groups approximate SU(N) arbitrarily well at large N, which doesn't look likely to me. Anyway, with the Platonic solid groups, you would not be able to reproduce rotational invariance at all, since the electron would only be pure in a set of directions forming a dodecahedron (at best) and this is not a very good approximation to a sphere.

If you drop the insistence that the symmetry of the state space forms a group, then you might be ok, but then it is not clear how you make a mathematical structure which realizes this idea. If you say that only the full quantum state space of the whole universe is discrete, then you need to check that if you isolate a single electron, you can make an approximate block sphere for the single electron. These are difficult problems, and without a resolution to these, I think the idea must remain in the realm of the very speculative.

The discreteness ideas from quantum gravity come from Banks's bound on the universe's information content from the cosmological horizon entropy. If you take this seriously, the effective dimension of the Hilbert space that describes our causal patch (the universe) is finite and growing. This is inconsistent with unitary evolution. One way around this is to declare that the true string theory Hilbert space is defined in the asymptotic future, when the universe decays into a zero temperature vacuum. This seems hard to say, since we need to know what vacuum it will pick, and this is certainly not determined from our current observations.

My own speculation is that you need to think of the cosmological state as a classsical probability distribution over the different possible string theory end-states, with no coherence between the different possible end-states. This is extremely speculative, because there is no direct matching between density matrices and cosmological evolution known. Then the finite information content of the universe would not be so mysterious, it would just say, given that we know so little about the end state, we can't fit too much information into the universe, since each bit we have inside the cosmological horizon is a bit we must know about the endstate already today, since it is encoded in the asymptotic future. This is just me musing about an unsolved problem, I don't know the answer, but I think that this sort of thing is more firmly grounded in established physics than any sort of naive discretization of Hilbert space.

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I see at least two possible motivations to answer your questions.

A theoretical answer

A possible answer is given by the recent works on deducing quantum physics' Hilbert space from a few information theoretical axioms:

  • Chiribella et al. (2010). (I haven't read this paper)
  • Müller, Masanes. You'll be interested by section V.A Why bits are balls, where the deduce the strict convexity of the phase-space (excluding polytope) with their axiom 2, stating that all d-dimensional systems are equivalent.

An experimental answer

If you want your discretized Bloch ball to be symmetric, the only (3D) possibilities are the 5 platonic solids. Therefore, you do not have regular polyhedra coming arbitrarily close to the sphere, so it is easy to check that physics does not correspond to any of these situation. One you have more than 20 possible (non-orthogonal) pure qubit states, you now that the only solutions are :

  • the phase space is the Bloch ball,
  • the phase space is an irregular polyhedron, which leads to some problems when you "rotate" the ball,
  • the phase space has more than 3 (real) dimensions.
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