In the above-mentioned paper by Dunkel and Trigger [Phys. Rev. A 71, 052102 (2005)] a hypothesis has been pursued that the loss of information associated with the quantum evolution of pure states, quantified in terms of an increase in the time of so-called Leipnik's joint entropy, could be a rather general property shared by many quantum systems. This behavior has been confirmed for the unconfined model systems and properly tuned initial data (maximally classical states). We provide two particular examples which indicate a complexity of the quantum evolution. In the presence of a confining (harmonic) potential Leipnik's entropy there may be nonincreasing for maximally classical initial data. Another choice of initial data implies periodicity in the time of the Leipnik entropy.
In the above-mentioned paper by Dunkel and Trigger [Phys. Rev. A 71, 052102 (2005)] a hypothesis has been pursued that the loss of information associated with the quantum evolution of pure states, quantified in terms of an increase in the time of so-called Leipnik's joint entropy, could be a rather general property shared by many quantum systems. This behavior has been confirmed for the unconfined model systems and properly tuned initial data (maximally classical states). We provide two particular examples which indicate a complexity of the quantum evolution. In the presence of a confining (harmonic) potential Leipnik's entropy there may be nonincreasing for maximally classical initial data. Another choice of initial data implies periodicity in the time of the Leipnik entropy.
Dunkel and Trigger [Phys. Rev. A {71}, 052102 (2005)] show that the Leipnik's joint entropy monotonously increases for the initially maximally classical Gaussian wave packet for a free particle. After expressing the joint entropy of the general Gaussian wave packets for quadratic Hamiltonians as S(t)=ln(e/2)+ln(2Deltax(t)Deltap(t)/hbar), we show that a class of general Gaussian wave packets does not warrant the monotonous increase of the joint entropy. We propose that the random-phase entropy with respect to the squeeze angle always monotonously increases even for non-maximally classical states.
Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of its observables; it trivially vanishes for pure quantum states. We propose a quantum entropy that quantifies the randomness of a pure quantum state via a conjugate pair of observables/operators forming the quantum phase space. The entropy is dimensionless, it is a relativistic scalar, it is invariant under canonical transformations and under CPT transformations, and its minimum has been established by the entropic uncertainty principle. We expand the entropy to also include mixed states. We show that the entropy is monotonically increasing during a time evolution of coherent states under a Dirac Hamiltonian. However, in a mathematical scenario, when two fermions come closer to each other, each evolving as a coherent state, the total system’s entropy oscillates due to the increasing spatial entanglement. We hypothesize an entropy law governing physical systems whereby the entropy of a closed system never decreases, implying a time arrow for particle physics. We then explore the possibility that as the oscillations of the entropy must by the law be barred in quantum physics, potential entropy oscillations trigger annihilation and creation of particles.
For a quantum state undergoing unitary Schrödinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator [R. S. Ingarden, Quantum information theory, Rep. Math. Phys. 10, 43 (1976)RMHPBE0034-487710.1016/0034-4877(76)90005-7]. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system composed of a set of highly interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.
Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of its observables; it trivially vanishes for pure quantum states. We propose a quantum entropy that quantifies the randomness of a pure quantum state via a conjugate pair of observables/operators forming the quantum phase space. The entropy is dimensionless, it is a relativistic scalar, it is invariant under canonical transformations and under CPT transformations, and its minimum has been established by the entropic uncertainty principle. We expand the entropy to also include mixed states. We show that the entropy is monotonically increasing during a time evolution of coherent states under a Dirac Hamiltonian. However, in a mathematical scenario, when two fermions come closer to each other, each evolving as a coherent state, the total system's entropy oscillates due to the increasing spatial entanglement. We hypothesize an entropy law governing physical systems whereby the entropy of a closed system never decreases, implying a time arrow for particle physics. We then explore the possibility that as the oscillations of the entropy must by the law be barred in quantum physics, potential entropy oscillations trigger annihilation and creation of particles.
For a quantum state undergoing unitary Schrodinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system comprised of a set of highly-interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.
The quantum entropy is usually defined using von Neumann's formula, which measures lack of information and vanishes for pure states. In contrast, we obtain a formula for the entropy of a pure state as it is measured from thermodynamic experiments, solely from the self-entanglement of the wave function, and find strong numerical evidence that the two are in agreement for nonintegrable systems, both for energy eigenstates and for states that are obtained at long times under the evolution of more general initial conditions. This is an extension of Boltzmann's hypothesis for classical systems, relating microscopic motion to thermodynamics.