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Workshop on Physics and Computation (PhysComp)

Gibbs Paradox And Equivalence Relation Between Quantum Information And Work.

Lev B. Levitin

Published Aug 24, 2005• 6 citations
Publisher

Abstract

The physical meaning of the entropy defect and information in quantum systems is analyzed in connection with Gibbs paradox and the M(uimLun work obtainable by use of a non-equilibrium system. The question is studied of the maximum amount of heat that can be converted into mechanical work in the process of isothermic mixing of two gases, whose molecules differ only in their quantum states, having non-commuting density matrices. It is shown that schemes for producing work that involve direct discrimination of nonorthogonal states (such as the method of semipermeable partitions) yield an amount of work that is determined not by the entropy ddect. but by the information in the system about its state, which reflects the fundamental irreversibility of the measurement process. An equivalence relation is suggested between quantum information and work.

DOI
10.1109/phycmp.1992.615545

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IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2012•0 citations
The physical meaning of the entropy defect and information in quantum systems is analyzed in connection with Gibbs paradox and the M(uimLun work obtainable by use of a non-equilibrium system. The question is studied of the maximum amount of heat that can be converted into mechanical work in the process of isothermic mixing of two gases, whose molecules differ only in their quantum states, having non-commuting density matrices. It is shown that schemes for producing work that involve direct discrimination of nonorthogonal states (such as the method of semipermeable partitions) yield an amount of work that is determined not by the entropy ddect. but by the information in the system about its state, which reflects the fundamental irreversibility of the measurement process. An equivalence relation is suggested between quantum information and work.
The issue of the Gibbs paradox is that when considering the mixing of two gases within classical thermodynamics, the entropy of mixing is a discontinuous function of the difference between the gases: it is finite for no matter how small difference, but vanishes for identical gases. The resolution offered in literature with help of quantum mixing entropy was later on shown to be unsatisfactory precisely where it sought to resolve the paradox. Within quantum thermodynamics entropy looses its dominant place. It is argued that the target of the paradox is shifted to the decrease of the available work upon mixing (mixing work), which in contrast to entropy is an unambiguous quantity. For almost identical gases the mixing work continuously goes to zero, thus resolving the paradox. In this approach the concept of ``difference between the gases'' attains a clear operational meaning related to possibilities of controlling the involved quantum states. The difficulties which prevent the resolution of the paradox in its entropic formulation do not arise here. The mixing work can increase when less precise operations are allowed. In contrast to the classical situation, in the quantum case the amount of mixing work can also increase when decreasing the degree of mixing between the gases, or when decreasing their distinguishability. These points go against a direct quantitative association of physical irreversibility with lack of information.
Our way to deal with the given topic is a connection of both the mathematical definitions of information entropies and their mutual relations within a system of stochastic quantities especially with thermodynamic entropies defined on an isolated system in which a realization of our (repeatable) observation is performed [it is a (cyclic) transformation of heat energy of an observed, measured system].We use the information description to analyze Gibbs paradox reasoning it as a property of such observation, measuring of an (equilibrium) thermodynamic system.We state a logical proof of the II. P.T. as a derivation of relations among the entropies of a system of stochastic variables, realized physically, and, the Equivalence Principle of the I., II. and III. Principle of Thermodynamics is formulated.
The issue of the Gibbs paradox is that when considering mixing of two gases within classical thermodynamics, the entropy of mixing appears to be a discontinuous function of the difference between the gases: it is finite for whatever small difference, but vanishes for identical gases. The resolution offered in the literature, with help of quantum mixing entropy, was later shown to be unsatisfactory precisely where it sought to resolve the paradox. Macroscopic thermodynamics, classical or quantum, is unsuitable for explaining the paradox, since it does not deal explicitly with the difference between the gases. The proper approach employs quantum thermodynamics, which deals with finite quantum systems coupled to a large bath and a macroscopic work source. Within quantum thermodynamics, entropy generally loses its dominant place and the target of the paradox is naturally shifted to a decrease of the maximally available work before and after mixing (mixing ergotropy). In contrast to entropy this is an unambiguous quantity. For almost identical gases the mixing ergotropy continuously goes to zero, thus resolving the paradox. In this approach the concept of "difference between the gases" gets a clear operational meaning related to the possibilities of controlling the involved quantum states. Difficulties which prevent resolutions of the paradox in its entropic formulation do not arise here. The mixing ergotropy has several counterintuitive features. It can increase when less precise operations are allowed. In the quantum situation (in contrast to the classical one) the mixing ergotropy can also increase when decreasing the degree of mixing between the gases or when decreasing their distinguishability. These points go against a direct association of physical irreversibility with lack of information.
The mixing of two different gases is one of the most common natural phenomena, with applications ranging from CO2_22​ capture to water purification. Traditionally, mixing is analyzed in the context of local thermal equilibrium, where systems exchange energy with a heat bath. Here, we study mixing in an isolated system with potentially non-equilibrium initial states, characterized solely by macroscopic observables. We identify the entropy of mixing as a special case of observational entropy within an observer-dependent framework, where both entropy and extractable work depend on the resolution of measurement. This approach naturally resolves the Gibbs mixing paradox in quantum systems: while an observer experiences a discontinuous increase in entropy upon learning of the existence of two particle types, this knowledge does not provide an advantage in work extraction if the types of particles remain operationally indistinguishable in their measurements. Finally, we derive a Landauer-like bound on the difference in energy extracted by two observers, where an "observational temperature" emerges, determined by the accessible information. These results provide a foundation for rigorously determining the energy required to unmix in non-equilibrium settings and extend beyond quantum systems, offering insights into the thermodynamics of isolated classical gases.
It is shown how information contained in the pair-wise correlations between atoms of a gas can be used to convert completely heat taken from a thermostat into mechanical work in a process of relaxation of the system to its thermal equilibrium state. Both classical correlations and quantum correlations (entanglement) are considered. The amount of heat converted into work is proportional to the entropy defect of the initial state of the system. The equivalence relation between information and work is explicitly demonstrated for the case of two-particle correlations. The amount of work obtained per particle is twice as large in the case of entanglement as in the case of classical correlations.
It is shown how information contained in the pair-wise correlations between atoms of a gas can be used to convert completely heat taken from a thermostat into mechanical work in a process of relaxation of the system to its thermal equilibrium state. Both classical correlations and quantum correlations (entanglement) are considered. The amount of heat converted into work is proportional to the entropy defect of the initial state of the system. The equivalence relation between information and work is explicitly demonstrated for the case of two-particle correlations. The amount of work obtained per particle is twice as large in the case of entanglement as in the case of classical correlations.