Friday, September 23, 2011

Quantum computational matter

Stephen Bartlett gave a colloquium on this topic today.
The ground states of quantum antiferromagnets contain substantial amounts of entanglement (although how much is hard to quantify). Hence, one might hope they are a resource that might be used in quantum computation. Stephen described some recent work [see this PRL and this PRL ] which considers how this might be done. The focus seems to be on gapped systems which have hidden symmetries and can be described as tensor network states. The prime example is the Haldane spin-1 antiferromagnetic Heisenberg chain which is adiabatically connected to the AKLT model.

I was reminded of some work I was involved in a few years ago (described in this PRL) which showed how one could take any spin singlet state [not just the ground state!] from a spin-1/2 chain and perform projective Bell measurements (on pairs of spins) along the chain and teleport quantum states with perfect fidelity along the chain. I was wondering what the relationship (similarities and differences) of our work was with this more recent work.

1 comment:

  1. Hi Ross,

    One of the central defining features of the measurement-based model for quantum computation has been the restriction to 'single particle' measurements. This restriction places stringent requirements on the resource states, in a way that has proven to be quite useful for proving all sorts of results in quantum computation. If you relax this requirement, and allow for Bell measurements (as you do in your work), then it is difficult to distinguish between entanglement initially present in the state, and entanglement generated by the measurements themselves. For example, I could use Bell measurements to teleport along *any* spin chain as follows: first, I perform Bell measurements between even-odd pairs of particles to generate a dimerised state of Bell pairs (assuming my measurements obey von Neumann's projection postulate), and then follow with Bell measurements between odd-even particles to teleport along the resulting dimerised state. While your use of the entangling power of these Bell measurements appears benign (i.e., you don't subsequently use the Bell states generated by your measurement), it is in general quite difficult to maintain this distinction.

    Restricting to single-particle measurements, as in the AKLT example you describe in your paper, removes this ambiguity: all entanglement in the systems is necessarily resulting from the initial resource state, because single-particle measurements cannot generate entanglement. This perspective lead to the very useful definition of 'localizable entanglement' by Frank Verstraete and collaborators, and a lot of subsequent work to present sufficient conditions for a state to have the long-ranged localizable entanglement needed to teleport along a spin chain. Building on this idea, a variety of string order parameters can be used to characterise this property, and some of these relate to the order parameter that you propose in your work. (See for example L. Campos Venuti and M. Roncaglia, Phys. Rev. Lett. 94, 207207 (2005).) I'm very excited about the recent results by Wen and collaborators (see Phys. Rev. B 83, 035107 (2011)), that I highlighted in my talk, as they can be used to provide *necessary* and sufficient conditions for this property of teleporting down a spin chain, in terms of the symmetries and their representations on a tensor network description.

    I should add, though, that the notion of single-particle measurements has itself been tricky to define in general, especially if you want to use ideas from renormalization to redefine what you mean by a single 'particle'. Perhaps someone will come up with a better way to capture this notion in a fully general sense. In the meantime, the continued success of the measurement-based model for quantum computation suggests that this notion is basically on the right track.