One, Two, Many...

[Hubener, Mari & Eisert, Wick’s Theorem for Matrix Product States, Phys. Rev. Lett. 110 040401 (2013)]

 

Physics is about knowing what to ignore. You know that you can’t follow every detail, but you hope that you can pick out what matters. This is what those beautiful firework-diagrams are for in the Large Hadron Collider, and what your computer does when it shrinks your photos down to kilobyte files. Identifying important particle tracks or using a transformation that makes an image very small are tricks that make it possible to deal efficiently with systems that would otherwise be too complex or too large to analyse. Nowhere is the need for this type of simplification more apparent than in the burgeoning field of quantum technologies, which seeks to produce new types of device with radically enhanced capabilities by harnessing the physics of large quantum systems. A major goal of this research is to develop quantum computers, which could process information exponentially faster than ordinary computers. This extraordinary possibillity arises because when quantum objects are combined, they can become entangled. Entanglement is a type of correlation between quantum particles which can only be captured by describing the system as a whole, so that it does not make sense to talk about the state of an individual particle without reference to the others. This requirement of a holistic description makes it generally difficult to analyse and predict the behaviour of large quantum systems, and this is precisely why they are useful as a resource for doing computations that are not ordinarily feasible. However many large quantum systems are not fully globally entangled: suppose a group of particles are placed next to each other (like atoms in a crystal). In that case, only neighbouring particles interact with each other, so that even though every particle is connected to the others, the connections are only local -- just between neighbours. This type of state arises naturally in many situations so it would be very useful to be able to analyse these types of states. In addition, they could be used as a resource for quantum computation, provided that some additional ingredients are available. Intuitively, the fact that these states have a simple structure, with only local connections between neighbouring particles, should make them easier to describe. Instead of requiring an exponentially large number of parameters, which are required for a globally entangled state, it should be possible to take advantage of the local structure to ignore many of these parameters and describe the system much more efficiently.

In fact, this has now been achieved by a group in Berlin, who showed in a recent paper how to describe these locally connected matrix product states using a simplified scheme [Hubener, Mari & Eisert, Wick’s Theorem for Matrix Product States, Phys. Rev. Lett. 110 040401 (2013)]. In particular, they analyzed correlation functions, which are the correlations between different positions in the group of particles. It turns out that the most complicated correlation in the state is a three-point correlation. Any other correlation, involving four of five or a hundred points, can always be broken down into three-point (and two-point) correlations. This discovery represents an exponential saving in the efficiency of characterising these states, and it will make it possible to analyse many naturally arising quantum systems, ranging from condensed matter to biology. The work is a perfect example of carefully working out what you need to know, and what you don’t need to know, in a physical system. By showing that only correlations between pairs or triplets of particles are necessary, the challenge of designing large quantum devices has been rendered possible. There is still much work to do, but the task ahead is now merely difficult, and no longer impossible!

This work was undertaken as part of the Q-ESSENCE project.

Project & realization: Pixels United.
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