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arXiv:1302.2059 · Physical Review A

Hamiltonian composite dynamics can almost always lead to negative reduced dynamics

James M. McCracken

Published Aug 5, 2013• 5 citations
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Abstract

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.

DOI
10.1103/physreva.88.022103

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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.
The reduced dynamics of an open quantum system SSS, interacting with its environment EEE, is not completely positive, in general. In this paper, we demonstrate that if the two following conditions are satisfied, simultaneously, then the reduced dynamics is completely positive: (1) the reduced dynamics of the system is linear, for arbitrary system-environment unitary evolution UUU; and (2) the reduced dynamics of the system is linear, for arbitrary initial state of the system ${\ensuremath{\rho}}_{S}$.
The title may be a bit misleading. Perhaps, "On the Complete Positivity of Reduced Quantum Dynamics," would be a more fitting title. Determining whether or not completely positive (CP) maps are required to describe open system quantum dynamics is an extremely important issue concerning the fundamental mathematical foundations of QM, as well as many other areas of physics. it had been typically believed that only CP maps actually describe the dynamical evolution of open quantum systems, however there has been speculation as to whether this is a strict constraint on the mathematical and physical structure of stochastic quantum dynamical maps. The objective of this thesis is to demonstrate that given a particular unitary operator, an initial system state, a final system state, and the dimension of the environment state, there exists no CP map with a composite system-environment, product initial state that is compatible with the given constraints on the reduced quantum dynamics of the system under investigation.
We show that complete positivity is not only sufficient but also necessary for the validity of the quantum data-processing inequality. As a consequence, the reduced dynamics of a quantum system are completely positive, even in the presence of initial correlations with its surrounding environment, if and only if such correlations do not allow any anomalous backward flow of information from the environment to the system. Our approach provides an intuitive information-theoretic framework to unify and extend a number of previous results.
We show that complete positivity is not only sufficient but also necessary for the validity of the quantum data-processing inequality. As a consequence, the reduced dynamics of a quantum system are completely positive, even in the presence of initial correlations with its surrounding environment, if and only if such correlations do not allow any anomalous backward flow of information from the environment to the system. Our approach provides an intuitive information-theoretic framework to unify and extend a number of previous results.
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.