## Saturday, August 06, 2005

### Is Hilbert space discrete?

New paper, to appear on arxiv.org on Tuesday.

Paper: hep-th/0508039
From: Stephen D. H. Hsu
Date: Sat, 6 Aug 2005 03:15:50 GMT (21kb)

Title: Is Hilbert space discrete?
Authors: R. Buniy, S. Hsu and A. Zee
Comments: 4 pages, revtex, 1 figure

We show that discretization of spacetime naturally suggests discretization of Hilbert space itself. Specifically, in a universe with a minimal length (for example, due to quantum gravity), no experiment can exclude the possibility that Hilbert space is discrete. We give some simple examples involving qubits and the Schrodinger wavefunction, and discuss implications for quantum information and quantum gravity.

Figure 1: A possible discretization of the Bloch sphere (qubit Hilbert space). Points on each disc (of size $\epsilon$) are identified. Points between discs can be assigned to the nearest disc.

Anonymous said...

Steve,

Very interesting article.

BTW: when you write "no experiment can exclude the possibility that Hilbert space is discrete", do you mean no experiment yet has excluded the possibility? Because later on you write about experimental deviations from linear superposition property... Hmm maybe you mean if the discreteness is really small...

Could you someday write about the latest approaches on building a quantum computer? I recently read that topological considerations should lead to stable quantum states (say in FQHE), making quantum computation closer to reality (even if at low temperatures...

MFA

steve said...

MFA,

I should clarify. Because of minimum length (e.g., due to quantum gravity), there seems to be a fundamental limit on our ability to measure discreteness of H (Hilbert space). So, if the size of the discreteness \epsilon is sufficiently small, it will escape our notice no matter how clever are our experiments (machines will collapse into black holes before they can detect the smallest \epsilons).

However, it is possible that discreteness in H might be larger than this threshold value which is concealed by minimum length. In that case, experimentalists might see a signal before, e.g., gravity becomes an important effect in their measurement.

Hope that makes sense :-)

I'm not up to date on the latest proposals for quantum computing devices (only so much time in the day ;-). I suggest you check out Dave Bacon's blog as he is an active researcher in quantum information...

Anonymous said...

Excellent explanations, Steve. I now understand the point. Very interesting.

Many thanks!

MFA