title: Information Erasure and Thermodynamics
Creators: name: Croucher, Toshio J
Publisher: name: Griffith University
Year: 2023
Subjects: subject: erasure
subject: thermodynamics
subject: statistical
subject: mechanics

description: This thesis contains the results of an extensive study motivated by Vaccaro and Barnett's work on information erasure via entropy maximisation constrained by multiple conserved quantities, and the exciting technologies that may arise from their research. The study focuses on three main topics. The first investigates fluctuations in the cost of performing information erasure using a spin reservoir, rather than a conventional thermal reservoir. This allows the erasure process to be completed via the conservation of spin angular momentum. To my knowledge this is the first instance where fluctuation theorem has been analysied in this manner. Here, I introduce the first law of spin thermodynamics in terms of conservation of spin angular momentum, where the spin equivalent of work is called spinlabor, and the spin equivalent of heat, spintherm. I determine the spinlabor probability distribution for the erasure protocol. It is of particular interest to investigate the probability for the spinlabor to violate the inequality ⟨Ls⟩ ≥ γ−1 ln 2, where ⟨Ls⟩ is the spinlabor cost of erasure and γ is a Lagrange multiplier equivalent to inverse spin temperature. I derived three bounds on the suppression of the probability violations; more specifically, the higher ϵ is, the less the probability of violating the bound, where ϵ is the departure from the bound γ−1 ln 2. These bounds advance the field of fluctuation theorem and statistics by placing a limit on the minimum cost to erase one bit of information, and also it, shows that the suppression is greater than previously thought. In addition, I derived a Jarzynski-like equality for the erasure protocol. In the second topic, I extend Vaccaro and Barnett's erasure scheme to allow for variations. Here, I address the issue on how the average spinlabor cost can violate the inequality ⟨Ls⟩ ≥ γ−1 ln 2. During the process of solving this problem, I derived three separate bounds for the spinlabor erasure cost using three different methods. The first was derived by lower-bounding the summation of the expected erasure spinlabor cost. The second was derived by applying Jensen's inequality to the derived Jarzynski-like equality for the extended erasure scheme. Finally, the third bound was derived from the integral fluctuation theorem formalism that was analysied for this erasure protocol. I determined which bound provided the tightest bound on the spinlabor erasure cost. By incorporating the first law of spin angular momentum, my findings show that although the total spinlabor cost can violate the bound ⟨Ls⟩ ≥ γ−1 ln 2, the total spintherm cost always adheres to the bound. In the research discussed above, the reservoir utilised for erasure was assumed to be infinitely large. This assumption was used to ensure that spin temperature (spin equivalent to temperature, T) of the spin reservoir remains constant throughout the erasure process. The assumption of constant reservoir temperature is commonly made when erasing via thermal reservoirs. In this third topic, I relaxed the infinitely-large reservoir assumption to provide a more pragmatic analysis of the erasure protocol. Here, I answer the question of what finite reservoir size is necessary for the erasure statistics to be comparable to the infinite counterpart, within some tolerance? I present the mathematical formalism required to capture the dynamics of the erasure process for finite sized reservoirs and the probability distribution of the spinlabor cost. I use the Jensen-Shannon Divergence to compare the spinlabor probability distribution of the infinite and finite reservoirs. The Jensen-Shannon Divergence is a symmetric method of comparing how closely related two probability distribution are. I show that perfect erasure is not possible for a finite reservoir, which is consistent with the third law of thermodynamics. Ancilla spins are used during the erasure process and are initially in the state |↓⟩⟨↓| corresponding to the logical 0 state. For an infinite reservoir the ancillas are automatically returned to their initial state, since perfect erasure occurs. However, I find as a result of imperfect erasure, when using a finite reservoir there is an additional cost of returning the ancillas to their initial state. This cost, in a worse case scenario, can almost be the same as the spinlabor cost required to actively erase the contents of the memory.