The classical-quantum divergence of complexity in modelling spin chains

Whei Yeap Suen1, Jayne Thompson1, Andrew J. P. Garner1, Vlatko Vedral1,2,3, and Mile Gu1,4,5

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore
2Atomic and Laser Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom
3Department of Physics, National University of Singapore, 3 Science Drive 2, Singapore 117543
4School of Mathematical and Physical Scieces, Nanyang Technological University, Singapore
5Complexity Institute, Nanyang Technological University, Singapore

The minimal memory required to model a given stochastic process - known as the statistical complexity - is a widely adopted quantifier of structure in complexity science. Here, we ask if quantum mechanics can fundamentally change the qualitative behaviour of this measure. We study this question in the context of the classical Ising spin chain. In this system, the statistical complexity is known to grow monotonically with temperature. We evaluate the spin chain's quantum mechanical statistical complexity by explicitly constructing its provably simplest quantum model, and demonstrate that this measure exhibits drastically different behaviour: it rises to a maximum at some finite temperature then tends back towards zero for higher temperatures. This demonstrates how complexity, as captured by the amount of memory required to model a process, can exhibit radically different behaviour when quantum processing is allowed.

Can quantum information fundamentally change the way we perceive what is complex? Statistical complexity quantifies the minimal information we must store about a process to simulate its future behaviour using classical information processing. Here, we construct a quantum variant of this measure, which also allows for simulation using quantum mechanical systems. The resulting complexity measure – quantum statistical complexity – exhibits drastically different qualitative behaviour than its classical counterpart.

We apply this measure to the Ising spin chain – a series of magnetically interacting spins that can each be aligned in one of two directions. The classical statistical complexity of an Ising spin chain only ever increases with temperature. On the other hand, the quantum statistical complexity rises to a maximum at some finite temperature then tends back towards zero for higher temperatures. This difference in the qualitative behaviour of complexity measures shows us that when we also consider quantum perspectives, our conclusions about “What is complex?” may be drastically changed.

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