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arXiv:quant-ph/9708029 · Physical Review A

Complete positivity of nonlinear evolution: A case study

Marek Czachor, Maciej Kuna

Published Jul 1, 1998• 31 citations
arXivHTMLPDF

Abstract

Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown in a concrete example.

DOI
10.1103/physreva.58.128

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Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown in a concrete example.
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.
Quantum systems which interact with an unknown environment cannot be described in terms of a unitary evolution on the system alone. For such evolution one can use a map from one density operator to another and use any other known information to model the system. Such maps are required to be positive (at least on their domain) — they take positive density operators to positive density operators — so as to be physically reasonable. A map [Formula: see text] which is positive but not completely positive (CP) is one which takes positive operators to positive operators, but when extended by the identity [Formula: see text], i.e. [Formula: see text] does not give a positive map for some [Formula: see text]. The map is CP if and only if the extension is positive for all [Formula: see text]. Recently some effort has been put forth to try to understand if, and/or, under what circumstances one might utilize a non-CP map. Here, after a tutorial-type introduction to maps from one density operator to another, some examples which do not fit the standard prescription (SP) for deriving a CP map are given. In cases where the physical system is constrained by some knowledge of the interaction, it is possible that the SP cannot be used directly. Our example is robust to changes in the initial and final conditions.
Many situations in quantum theory and other areas of physics lead to quasiprobabilities which seem to be physically useful but can be negative. The interpretation of such objects is not at all clear. In this paper, we show that quasiprobabilities naturally fall into two qualitatively different types, according to whether their non-negative marginals can or cannot be matched to a non-negative probability. The former type of quasiprobabilities, which we call viable, are qualitatively similar to true probabilities, but the latter type, which we call nonviable, may not have a sensible interpretation. Determining the existence of a probability matching given marginals is a nontrivial question in general. In simple examples, Fine's theorem indicates that inequalities of the Bell and Clauser-Horne-Shimony-Holt (CHSH) types provide criteria for its existence, and these examples are considered in detail. Our results have consequences for the linear positivity condition of Goldstein and Page in the context of the histories approach to quantum theory. Although it is a very weak condition for the assignment of probabilities, it fails in some important cases where our results indicate that probabilities clearly exist. We speculate that our method, of matching probabilities to a given set of marginals, provides a general method of assigning probabilities to histories and we show that it passes the Di\'osi test for the statistical independence of subsystems.
Complete positivity is a ubiquitous assumption in the study of quantum systems interacting with the environment. It will be shown that Hamiltonian evolution of a quantum system and its environment can be negative (i.e., not completely positive) in the energy basis, by showing that such evolution is almost always negative for given initial conditions. As such, ignoring or ``correcting'' experimental data that is not completely positive may cause the loss of important information regarding system-environment correlations and coupling. Complete positivity assumptions are an important part of many quantum information theories and it is important to understand how (and why) it appears to be possible to violate the complete positivity requirement in the examples presented here. A relationship between the negativity of an evolution and the eigenvalues of the Hamiltonian will be shown and experimental verification of negative reduced dynamics will be proposed.