Here's the abstract, with the important bit highlighted by me:
At a fundamental level, the classical picture of the world is dead, and has been dead now for almost a century. Pinning down exactly which quantum phenomena are responsible for this has proved to be a tricky and controversial question, but a lot of progress has been made in the past few decades. We now have a range of precise statements showing that whatever the ultimate laws of Nature are, they cannot be classical. In this article, we review results on the fundamental phenomena of quantum theory that cannot be understood in classical terms. We proceed by first granting quite a broad notion of classicality, describe a range of quantum phenomena (such as randomness, discreteness, the indistinguishability of states, measurement-uncertainty, measurement-disturbance, complementarity, noncommutativity, interference, the no-cloning theorem, and the collapse of the wave-packet) that do fall under its liberal scope, and then finally describe some aspects of quantum physics that can never admit a classical understanding -- the intrinsically quantum mechanical aspects of Nature. The most famous of these is Bell's theorem, but we also review two more recent results in this area. Firstly, Hardy's theorem shows that even a finite dimensional quantum system must contain an infinite amount of information, and secondly, the Pusey--Barrett--Rudolph theorem shows that the wave-function must be an objective property of an individual quantum system. Besides being of foundational interest, results of this sort now find surprising practical applications in areas such as quantum information science and the simulation of quantum systems.Here's some more detail from within the paper:
At the heart of classical information theory is the idea of a classical bit – the information revealed by a single yes-no question. Our ability to quantify, encode and transform information has revolutionised the world in countless ways (telecommunications, the internet, computers, etc.), and its study has shed light on the foundations of physics. Central to this is the idea that information does not care how we choose to encode it – we can encode information on paper, in electronic pulses or carve it into stone. For almost all of history our encoding of information has been into classical degrees of freedom. However, Nature is quantum-mechanical and, in recent years, we have begun to use quantum degrees of freedom to encode information. A central question therefore arises: does information in quantum mechanics have the same properties as in classical mechanics?
Now, the state of even the simplest quantum system – a qubit – is specified by continuous parameters. This means that it requires an infinite amount of information to specify the state exactly. For example, the amplitude α of |0i in the superposition α|0i + β|1i could encode the decimal expansion of π. Thus, at first glance, it seems that that quantum systems can carry vastly more information than classical systems. However, Holevo [22, 42, 43] showed only a single bit of classical information can ever be extracted from a qubit system via measurement. Further, in spite having a continuous infinity of pure states, quantum computation do not suffer from the the problems that rule out analog classical computers . Powerful theorems on the discretization of errors  tell us that we do not need to correct a continuum of errors, but only particular discrete types. These surprising characteristics present a basic conundrum: how is it that qubits behave as if they are discrete systems when their state space forms a continuum?
As already discussed, in classical statistical mechanics we can consider the allowed macrostates: the set of probability distributions over some state space Λ of microstates. It is easy to see that these distributions also form a continuum – even if there is only a discrete finite set of microstates. As an example, consider the case of DNA bases, which can be in one of 4 microstates A, T, C or G. The macrostate for a single base is therefore a probability distribution p = (pA, pT , pC, pG), obeying Pj pj = 1 and 0 ≤ pj ≤ 1 for all j = A, T, C, G. The set of such distributions therefore forms a solid tetrahedron (a simplex) in 3-dimensional space, and there is a continuum of macrostates (see Figure 7).
The fact that qubits behave in many ways like discrete, finite systems would be easily explained if perhaps there were only a finite number of more fundamental states – like the finite number of DNA bases – and if the continuum of quantum states only represented our uncertainty about which one of them is occupied – like the continuum of DNA macrostates. Surprisingly, in spite of Holevo’s bound and the discretization of errors, this cannot be the case: any future physical
theory that reproduces the physics of finite-dimensional quantum systems must have an infinite number of fundamental states.
The paper then goes onto to explain Hardy's proof of this. It's math-y, and the interested reader (hello?) can go read it in the paper itself.
"Hardy" is Lucien Hardy, who seems to have made quite a name for himself in quantum theory, and is said to have devised a pretty simple proof back in 1992 that quantum physics must be non local.
But the "theorem" referred to about infinite information seems to come from a 2004 paper, which does not seem to be on arXiv.
But there is a 2010 paper by someone (from where, I do not see - another paper just gives a hotmail address for him!) disputing that Hardy is right on this. He argues that "infinite excess baggage also occurs in classical theories".
Well, what to make of this?
Am I wrong, or I am right, in suspecting that the idea of infinite information being necessary in a quantum universe to be pretty significant for a philosophical understanding of the nature of the universe?
It seems that Godel's Incompleteness Theorem gets all the attention from a philosophical implication point of view, but perhaps there is another theorem here that deserves similar thought.
Certainly, for the religious, the idea of infinite information tends to be associated with God, so if Hardy is right, does it suggest more of a Spinoza view of God rather than the Catholic view?
Bit deep for a Friday, hey?
Update: see, when the question is asked at Quora "is there an infinite amount of information in the Universe", most people answer "no".