![]() ![]() Note that the evolution from an initially entangled state may be accompanied by the arbitrary gain in entropy. Then, according to the Stinespring-Kraus dilation theorem 6 is the quantum channel. Let be the unitary operator describing the temporal evolution of the grand system. Indeed, let us consider joint evolution of the grand system, comprising a given quantum system and a reservoir initially prepared in a disentangled state,, where is the density matrix of the reservoir. To connect the general result (1) and the related mathematical H-theorem formulation to the realm of physics note that any quantum system interacting with the reservoir generates a quantum channel. Finally, it is noteworthy that there exist certain classes of states that evolve with even if the channel is not unital 11. 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. ![]()
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