Classical and quantum error correction are presented in the form of Maxwell's demon and their efficiency analysed from the thermodynamic point of view. We explain how Landauer's principle of information erasure applies to both cases. By then extending this principle to entanglement manipulations we rederive upper bounds on purification procedures, thereby linking the ‘no local increase of entanglement’ principle to the second law of thermodynamics.
Classical and quantum error correction are presented in the form of Maxwell's demon and their efficiency analysed from the thermodynamic point of view. We explain how Landauer's principle of information erasure applies to both cases. By then extending this principle to entanglement manipulations we rederive upper bounds on purification procedures, thereby linking the ‘no local increase of entanglement’ principle to the second law of thermodynamics.
We review a general theory of thermodynamics of information processing. The background of this topic is the recently-developed nonequilibrium statistical mechanics and quantum (and classical) information theory. These theories are closely related to the modern technologies to manipulate and observe small systems; for example, macromolecules and colloidal particles in the classical regime, and quantum-optical systems and quantum dots in the quantum regime. First, we review a generalization of the second law of thermodynamics to the situations in which small thermodynamic systems are subject to quantum feedback control. The generalized second law is expressed in terms of an inequality that includes the term of information obtained by the measurement, as well as the thermodynamic quantities such as the free energy. This inequality leads to the fundamental upper bound of the work that can be extracted by a “Maxwell's demon”, which can be regarded as a feedback controller with a memory that stores measurement outcomes. Second, we review generalizations of the second law of thermodynamics to the measurement and information erasure processes of the memory of the demon that is a quantum system. The generalized second laws consist of inequalities that identify the lower bounds of the energy costs that are needed for the measurement and the information erasure. The inequality for the erasure leads to the celebrated Landauer's principle for a special case. Moreover, these inequalities enable us to reconcile Maxwell's demon with the second law of thermodynamics. In these inequalities, thermodynamic quantities and information contents are treated on an equal footing. In fact, the inequalities are model-independent, so that they can be applied to a broad class of information processing. Therefore, these inequalities can be called the second law of “information thermodynamics”.
Abstract In addition to treating quantum communication, entanglement, error correction, and algorithms in great depth, this book also addresses a number of interesting miscellaneous topics, such as Maxwell's demon, Landauer's erasure, the Bekenstein bound, and Caratheodory's treatment of the second law of thermodynamics. All mathematical derivations are based on clear physical pictures which make even the most involved results — such as the Holevo bound — look comprehensible and transparent. Quantum information is a fascinating topic precisely because it shows that the laws of information processing are actually dependent on the laws of physics. However, it is also very interesting to see that information theory has something to teach us about physics. Both of these directions are discussed throughout the book. Other topics covered in the book are quantum mechanics, measures of quantum entanglement, general conditions of quantum error correction, pure state entanglement and Pauli matrices, pure states and Bell's inequalities, and computational complexity of quantum algorithms.
The aim of quantum error correction (QEC) is to counteract the decoherence of qubits,which arises from their unavoidable interaction with the environment. In this work, westudy the lower bound of entropy production in quantum error correction, asking the cen-tral question: Is it possible to fight decoherence in a reversible way?A previous work established that ideal quantum error correction can, in principle,be performed reversibly. In this work, however, we consider fault-tolerant quantum errorcorrection, which explicitly accounts for faults occurring during the correction processitself. We show that when noise affects the circuit during the error-correcting operations,quantum error correction can no longer be reversible, because of unused correlations thatmust be dissipated at each correction cycle.This result can be better understood by drawing an analogy with Maxwell’s demon.In quantum error correction, the goal is to remove entropy from the data qubits. Ancillaryqubits are used to record the occurrence of errors (syndrome diagnosis), and this infor-mation is then employed to correct them (recovery operation). At the end of each cycle,the ancillary qubits are reset in order to start a new cycle. This is closely analogous to aMaxwell demon, which uses a memory to store information about a system, exploits thatinformation to reduce the system’s entropy, and finally resets its memory, dissipating heatinto the environment in accordance with Landauer’s principle.In this analogy, the demon’s memory corresponds to the ancillary qubits, while thesystem corresponds to the data qubits. During syndrome diagnosis, information aboutthe errors is recorded in the ancillary qubits, creating some correlations between dataand ancillary qubits. In the recovery operation, these correlations are used to correctthe errors on the data qubits and thus reduce their entropy. At the end of the cycle, theancillary qubits are reset to |0⟩, which generates a heat flow to the environment as dictatedby Landauer’s principle.For ideal quantum error correction, this process can, in principle, be performed re-versibly, thereby reaching equality in Landauer’s bound.However, when errors occur on the ancillary qubits during the QEC process (in the lan-guage of Maxwell’s demon, when faults affect the demon’s memory before the feedbackstep) these errors propagate to the data qubits, creating unwanted correlations betweendata and ancillas. Such correlations generate an additional finite heat flow to the bath,that prevents reaching equality in Landauer’s principle.
Adiabatic measurements, followed by feedback and erasure protocols, have often been considered as a model to embody Maxwell’s Demon paradox and to study the interplay between thermodynamics and information processing. Such studies have led to the conclusion, now widely accepted in the community, that Maxwell’s Demon and the second law of thermodynamics can peacefully coexist because any gain provided by the demon must be offset by the cost of performing the measurement and resetting the demon’s memory to its initial state. Statements of this kind are collectively referred to as second laws of information thermodynamics and have recently been extended to include quantum theoretical scenarios. However, previous studies in this direction have made several assumptions, particularly about the feedback process and the demon’s memory readout, and thus arrived at statements that are not universally applicable and whose range of validity is not clear. In this work, we fill this gap by precisely characterizing the full range of quantum feedback control and erasure protocols that are overall consistent with the second law of thermodynamics. This leads us to conclude that the second law of information thermodynamics is indeed universal : it must hold for any quantum feedback control and erasure protocol, regardless of the measurement process involved, as long as the protocol is overall compatible with thermodynamics. Our comprehensive analysis not only encompasses new scenarios but also retrieves previous ones, doing so with fewer assumptions. This simplification contributes to a clearer understanding of the theory.
This article discusses the concept of information and its intimate relationship with physics. After an introduction of all the necessary quantum mechanical and information theoretical concepts we analyse Landauer's principle which states that the erasure of information is inevitably accompanied by the generation of heat. We employ this principle to rederive a number of results in classical and quantum information theory whose rigorous mathematical derivations are difficult. This demonstrates the usefulness of Landauer's principle and provides an introduction to the physical theory of information.