Remember the film ‘The Dark Knight’ in 2008 where the Joker tells Harvey Dent that he’s an agent of chaos? Would you have ever imagined that an understanding of chaos helps in numerous fields such as in weather forecasting, the study of the behaviours of ecosystems, computer science, economics, etc.?
Classical physics, governed by Newton’s laws of motion, is deterministic. Chaos Theory is a branch of mathematics that deals with the question of how systems governed by deterministic laws of motion become unpredictable and exhibit seemingly random behaviour. Chaos is also popularly known as “The Butterfly Effect”, due to which systems become extremely sensitive to small perturbations. A butterfly flapping its wings in Brazil might cause a hurricane in Japan. Another manifestation of chaos is “mixing”. A drop of ink in a glass of water completely spreads in the cup, giving it a light blue colour.
But how does chaos work at the atomic level? Physics and behaviours at the atomic level are not governed by Newton’s laws. They are governed by quantum mechanics. In this paper, the authors Mr. Abinash Sahu, Mr. Naga Dileep Varikuti, Mr. Bishal Kumar Das, and Prof. Vaibhav Madhok from the Department of Physics, Indian Institute of Technology Madras, Chennai, India, have studied chaos in the quantum world.
Is there an analogy of chaos as a drop of ink spreading in a glass of water in quantum mechanics? Here an operator can be considered as a drop of ink, and the water as the Hilbert space, or a Krylov subspace. In quantum mechanics, an operator is a mathematical entity that represents a physical quantity. Operators are used to describe the quantum systems and measurement of physical properties. The Hilbert space is the mathematical space where all possible states of a quantum system can be described. A subspace of Hilbert space is called the Krylov space.
The question that the authors of this paper ask regarding chaos in the quantum realm is – is unpredictability a source of useful information? This question is at odds with itself, as unpredictability and information are at opposite ends of the spectrum. How can you get information from unpredictability? And how can information arise from unpredictability?
However, if everything is known and predictable, we have no new information to gain. We only gain information after the outcome of an event when its occurrence is unpredictable. In other words, we learn from the occurrence of the hurricane whether or not the butterfly flapped its wings. Information is gained from unpredictable outcomes by what is known as quantum tomography. Quantum tomography is like detective work to find out more about the state of a quantum system.
To conclude, the authors have studied operator spreading in many-body quantum systems. This paper gives an operational interpretation for operator spreading in quantum chaos. The protocol of this paper can be experimentally realized in experiments involving atoms and laser light with state-of-the-art technology. This study is of use in quantum computing.
Prof. M. S. Santhanam, from the Indian Institute of Science Education and Research, Pune, India, noted the importance of the study done by the authors with the following comments: “Classical chaos implies unpredictability beyond a short time horizon due to exponential spread of near-by classical trajectories. In quantum domain, it is not easy to characterise this idea since the notion of trajectories does not exist. But there are other ways of doing it. One of them is to characterise the spreading of a mathematical object called the operator in quantum regime. This is a mathematical jugglery and a crucial question is “how does one do it in practice?” Vaibhav Madhok’s work shows a remarkable result that operator spreading can be inferred from rate of information gain in quantum tomography which can be measured in experiments. Thus, it can be measured and quantified in experiments. This is particularly useful in emerging areas of quantum computing to understand the effects of quantum chaos in many-body systems. I see this as an interesting contribution since it connects many different ideas from the classical notion of entropy to information gain in chaotic quantum systems.”
Article by Akshay Anantharaman
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