For an infinite-dimensional quantum system the entropy is not continuous 10, and this situation requires special consideration. Thus, therefore, and the channel is unital. It then follows that for the chaotic state that already has the maximal entropy,, the entropy cannot grow. Indeed, let us assume that for any initial state of a system with N-dimensional Hilbert space, the entropy gain in a channel Φ is non-negative. Moreover, for a quantum system endowed with the finite N-dimensional Hilbert space, the unitality condition becomes not only a sufficient, but also the necessary condition for non-diminishing entropy. Then within the framework of the QIT one can formulate the quantum H-theorem as follows: the entropy gain during evolution is nonnegative if the system evolution can be described by the unital channel. (1) vanishes,, so that the entropy gain is non-negative. There exists a wide class of channels, the so-called unital channels, defined by the relation, for which the right hand side of Eq. This formula was derived from the monotonicity property 9 of the relative entropy under the quantum channel Φ :, where. A remarkable general result of the QIT states that the entropy gain in a channel is 8 To describe quantum dynamics of an open system, the quantum information theory introduces the so-called quantum channel (QC) defined as a trace-preserving completely positive map,, of a density matrix 6. In this communication we show how the results of QIT apply to physical quantum systems and phenomena establishing thus non-diminishing von Neumann’s entropy in physics and formulate the conditions under which the evolution accompanied by non-diminishing entropy arises within pure quantum mechanical framework. At the same time there have been a remarkable progress in quantum information theory (QIT), which formulated several rigorous mathematical theorems about the conditions for a non-negative entropy gain 6 ,7. As this proof yet invoked concepts going beyond pure quantum mechanical treatment, the nonstop tireless search for the quantum mechanical foundation of the H-theorem have been continuing ever since, see ref. He defined entropy through quantum mechanical density matrix as, and offered a proof of non-decreasing entropy resting on the final procedure of macroscopic measurement. Striving to bypass molecular chaos hypothesis, unjustified within the classical mechanics, John von Neumann proposed 4 pure quantum mechanical origin of the entropy growth. Boltzmann’s kinetic equation rests on the molecular chaos hypothesis which assumes that velocities of colliding particles are uncorrelated and independent of position. The H-theorem states that if f( x v τ) is the distribution density of molecules of the ideal gas at the time τ, position x and velocity v, which satisfies the kinetic equation, then entropy defined as is non-diminishing, i.e. In the 1870-s, Ludwig Boltzmann published his celebrated kinetic equation and the H-theorem 1 ,2 that gave the statistical foundation of the second law of thermodynamics 3. We further demonstrate that the typical evolution of energy-isolated quantum systems occurs with non-diminishing entropy. We discuss the manifestation of the second law of thermodynamics in quantum physics and uncover special situations where the second law can be violated. Here we build on the mathematical formalism provided by QIT to formulate the quantum H-theorem in terms of physical observables. However, relation of these results formulated in terms of entropy gain in quantum channels to temporal evolution of real physical systems is not thoroughly understood. Remarkable progress of quantum information theory (QIT) allowed to formulate mathematical theorems for conditions that data-transmitting or data-processing occurs with a non-negative entropy gain.
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