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Register a new userThe Definition of Entropy for Quantum Unstable Systems: A View-Point Based on the Properties of Gamow States
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In this paper, we review the concept of entropy in connection with the description of quantum unstable systems. We revise the conventional definition of entropy due to Boltzmann and extend it so as to include the presence of complex-energy states. After introducing a generalized basis of states which includes resonances, and working with amplitudes instead of probabilities, we~found an expression for the entropy which exhibits real and imaginary components. We discuss the meaning of the imaginary part of the entropy on the basis of the similarities existing between thermal and time evolutions.
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The Copenhagen interpretation of quantum mechanics, which first took shape in Bohrs landmark 1928 paper on complementarity, remains an enigma. Although many physicists are skeptical about the necessity of Bohrs philosophical conclusions, his pragmatic message about the importance of the whole experimental arrangement is widely accepted. It is, however, generally also agreed that the Copenhagen interpretation has no direct consequences for the mathematical structure of quantum mechanics. Here I show that the application of Bohrs main concepts of complementarity to the subsystems of a closed system requires a change in the definition of the quantum state. The appropriate definition is as an equivalence class similar to that used by von Neumann to describe macroscopic subsystems. He showed that such equivalence classes are necessary in order to maximize information entropy and achieve agreement with experimental entropy. However, the significance of these results for the quantum theory of measurement has been overlooked. Current formulations of measurement theory are therefore manifestly in conflict with experiment. This conflict is resolved by the definition of the quantum state proposed here.
The estimation of the density matrix of a $k$-level quantum system is studied when the parametrization is given by the real and imaginary part of the entries and they are estimated by independent measurements. It is established that the properties of the estimation procedure depend very much on the invertibility of the true state. In particular, in case of a pure state the estimation is less efficient. Moreover, several estimation schemes are compared for the unknown state of a qubit when one copy is measured at a time. It is shown that the average mean quadratic error matrix is the smallest if the applied observables are complementary. The results are illustrated by computer simulations.
Three paradigms commonly used in classical, pre-quantum physics to describe particles (that is: the material point, the test-particle and the diluted particle (droplet model)) can be identified as limit-cases of a quantum regime in which pairs of particles interact without getting entangled with each other. This entanglement-free regime also provides a simplified model of what is called in the decoherence approach islands of classicality, that is, preferred bases that would be selected through evolution by a Darwinist mechanism that aims at optimising information. We show how, under very general conditions, coherent states are natural candidates for classical pointer states. This occurs essentially because, when a (supposedly bosonic) system coherently exchanges only one quantum at a time with the (supposedly bosonic) environment, coherent states of the system do not get entangled with the environment, due to the bosonic symmetry.
For the XXZ subclass of symmetric two-qubit X states, we study the behavior of quantum conditional entropy S_{cond} as a function of measurement angle thetain[0,pi/2]. Numerical calculations show that the function S_{cond}(theta) for X states can have at most one local extremum in the open interval from zero to pi/2 (unimodality property). If the extremum is a minimum the quantum discord displays region with variable (state-dependent) optimal measurement angle theta^*. Such theta-regions (phases, fractions) are very tiny in the space of X state parameters. We also discover the cases when the conditional entropy has a local maximum inside the interval (0,pi/2). It is remarkable that the maxima exist in surprisingly wide regions and the boundaries for such regions are defined by the same bifurcation conditions as for those with a minimum. Moreover, the found maxima can exceed the conditional entropy values at the ends of interval [0,pi/2] more than by 1%. This instils hope in the possibility to detect such maxima in experiment.
We study properties of moving relativistic quantum unstable systems. We show that in contrast to the properties of classical particles and quantum stable objects the velocity of moving freely relativistic quantum unstable systems can not be constant in time. We show that this new quantum effect results from the fundamental principles of the quantum theory and physics: It is a consequence of the principle of conservation of energy and of the fact that the mass of the quantum unstable system is not defined. This effect can affect the form of the decay law of moving relativistic quantum unstable systems.
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