Quantum and Classical Ergotropy from Relative Entropies

The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. Given that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state, we define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both quantum as well as classical scenarios is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem.     

Quantum Mixtures in the Early Universe

In Relativistic Schrödinger Theory for spinning matter, there exist mixtures which have vanishing spin density (S 0). Such a fermionic, but spinless quantum fluid is studied in detail. Its energy-momentum density can develop negative pressure which is able to accelerate the expansion of the universe and to create matter energy through continuous bouncing at minimal radius (cosmic pumping).     

Early Universe Quantum Processes in BEC Collapse Experiments

We show that in the collapse of a Bose–Einstein condensate (BEC)⁴certain processes involved and mechanisms at work share a common origin with corresponding quantum field processes in the early universe such as particle creation, structure formation, and spinodal instability. Phenomena associated with the controlled BEC collapse observed in the experiment of Donley et al. (Donley, E., et al. (2001), Nature 412, 295; Claussen, N. (2003), PhD Thesis, University of Colorado; Claussen, N., et al. (2003), Physical Review A 67, 060701(R))(they call it “Bose–Nova,” see also Chin, J., Vogels, J., and Ketterle, W. (2003), Physical Review Letters 90, 160405) such as the appearance of bursts and jets can be explained as a consequence of the squeezing and amplification of quantum fluctuations above the condensate by the dynamics of the condensate. Using the physical insight gained in depicting these cosmological processes, our analysis of the changing amplitude and particle contents of quantum excitations in these BEC dynamics provides excellent quantitative fits with the experimental data on the scaling behavior of the collapse time and the amount of particles emitted in the jets. Because of the coherence properties of BEC and the high degree of control and measurement precision in atomic and optical systems, we see great potential in the design of tabletop experiments for testing out general ideas and specific (quantum field) processes in the early universe, thus opening up the possibility for implementing “laboratory cosmology.” This essay has the same content as v2 of Calzetta and Hu (2002), with a few references updated. For more details, see Calzetta and Hu (2002). ⁴For an excellent introduction to BEC theory, see Pethick and Smith (2002).     

Quantum Creation of Highly Massive Particles in the Very Early Universe

Quantum creation of very massive particles in the gravitational background of anisotropically perturbed Minkowski space-time is discussed. In this framework of semiclassical gravity the quantum mechanically produced heavy particles which made the initial space-time unstable and ushered into the FRW expansion phase at the Planck order epoch of the universe can account for the energy density at that epoch. Also, both the conformal and nonconformal particle-creations in the FRW era of the early universe after the Planck order epoch are investigated. In this consideration the total particle number of the observable universe as well as the present value of photon-to-baryon ratio are obtained in agreement with their accepted values from the observational facts. The existence of very massive particles at the very early period of the universe is also discussed here with the suggestion of an observational test.     

Relating Entropies of Quantum Channels

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi–Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.     

The Classical Regime of a Quantum Universe Obtained Through a Functional Method

The functional method, introduced to deal withsystems endowed with a continuous spectrum, is used tostudy the problem of decoherence and correlations in asimple cosmological model.     

The Universe and the Quantum Computer

It is first pointed out that there is a common mathematical model for the universe and the quantum computer. The former is called the histories approach to quantum mechanics and the latter is called measurement-based quantum computation. Although a rigorous concrete model for the universe has not been completed, a quantum measure and integration theory has been developed which may be useful for future progress. In this work we show that the quantum integral is the unique functional satisfying certain basic physical and mathematical principles. Since the set of paths (or trajectories) for a quantum computer is finite, this theory is easier to treat and more developed. We observe that the sum of the quantum measures of the paths is unity and the total interference vanishes. Thus, constructive interference is always balanced by an equal amount of destructive interference. As an example we consider a simplified two-slit experiment.     

Quantum Origin of the Universe

The quantum origin of the Universe due to a positive cosmological constant is studied.     

Generalized Conditional Entropies of Partitions on Quantum Logic

For partitions on quantum logic, the R閚yi and Tsallis conditional entropies are introduced. Several relations between the conditional entropies of such partitions are derived.     

Note on Entropies of Quantum Dynamical Systems

We review some techniques and notions for quantum information theory. It is shown that the dynamical entropies is discussed and some numerical computations of these entropies are carried for several states.     

A Family of Monotone Quantum Relative Entropies

We study here the elementary properties of the relative entropy ${\mathcal{H}_\varphi(A, B) = {\rm Tr}[\varphi(A) - \varphi(B) - \varphi'(B)(A - B)]}$ for φ a convex function and A, B bounded self-adjoint operators. In particular, we prove that this relative entropy is monotone if and only if φ′ is operator monotone. We use this to appropriately define ${\mathcal{H}_\varphi(A, B)}$ in infinite dimension.     

Quantum Birth of the Rotating Universe

Quantum birth of the Universe of the Bianchi type IX filled by a rotating anisotropic liquid is studied. The Wheeler–De Witt equation is derived for the model under consideration. The tunneling wave function as a solution to the equation is found using the WKB method. The tunneling coefficient of the Universe is calculated. The probabilities of quantum birth of the Universe with and without rotation are compared for different formulations of the problem.     

Magnetic Fields in the Early Universe

We discuss magnetic fields in the early universe— their origin, their possible detection, andlimits and values today and at early times.     

Generalized Conditional Entropies of Partitions on Quantum Logic

For partitions on quantum logic, the Rényi and Tsallis conditional entropies are introduced. Several relations between the conditional entropies of such partitions are derived.     

Evolution of the Early Universe

The evolution of the very early Universe from a vacuum-like state to the epoch of leptoquark decay is investigated.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 33–35, March, 2005.