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arXiv:2605.18639

Open quantum dynamics without Complete Positivity: a criticism

Fabio Benatti, Dariusz Chruściński, Saverio Pascazio

quant-phmath-phSubmitted May 18, 2026• Announced May 19, 2026• 0 citations
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Abstract

The requirement of complete positivity is very often regarded as a fundamental consistency condition for the description of open quantum dynamics. We critically examine this requirement and discuss both its physical motivations and its limitations. We analyze proposals based on restricting the domain of non-completely positive maps to subsets of compatible initial states. Using isotropic states as a concrete example, we show that such domain restrictions become increasingly severe with growing system dimension, revealing an intrinsic weakness of the compatibility-based approach.

Journal reference
Open Systems & Information Dynamics 2026
Comments
10 pages

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The requirement of complete positivity is very often regarded as a fundamental consistency condition for the description of open quantum dynamics. We critically examine this requirement and discuss both its physical motivations and its limitations. We analyze proposals based on restricting the domain of non-completely positive maps to subsets of compatible initial states. Using isotropic states as a concrete example, we show that such domain restrictions become increasingly severe with growing system dimension, revealing an intrinsic weakness of the compatibility-based approach.
The requirement of complete positivity is very often regarded as a fundamental consistency condition for the description of open quantum dynamics. We critically examine this requirement and discuss both its physical motivations and its limitations. We analyse proposals based on restricting the domain of non-completely positive maps to subsets of compatible initial states. Using isotropic states as a concrete example, we show that such domain restrictions become increasingly severe with growing system dimension, revealing an intrinsic weakness of the compatibility-based approach.
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.
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It is now increasingly realized in the study of open system dynamics that initial correlations do not pose a conceptual difficulty as traditionally believed. A similar methodology as used to describe initial product states can be adopted, with the only difference being that the reduced dynamics is possibly not completely positive, entailing that only a restricted set of initial reduced states lead to a physically valid dynamics. Here we study the interplay between the initial correlations, especially in the form of highly entangled states, and the system–environment unitary. In particular, for almost any initial entangled state, one can furnish infinitely many joint unitaries that generate CP dynamics on the system. Restricting to the case of initial, pure entangled states, we obtain the scaling of the dimension of the set of these unitaries and show that it is of zero measure in the set of all possible interaction unitaries.
Maps that are not completely positive (CP) are often useful to describe the dynamics of open systems. An apparent violation of complete positivity can occur because there are prior correlations of the principal system with the environment, or if the applied transformation is correlated with the state of the system. We provide a physically motivated definition of accessible non-CP maps and derive two necessary conditions for a map to be accessible. We also show that entanglement between the system and the environment is not necessary to generate a non-CP dynamics. We describe two simple approximations that may be sufficient for some problems in process tomography, and then outline what these methods may be able to tell us in other situations where non-CP dynamics naturally arise.
The description of the dynamics of a system that may be correlated with its environment is only meaningful within the context of a specific framework. Different frameworks rely upon different assumptions about the initial system-environment state. We reexamine the connections between complete positivity and quantum discord within two different sets of assumptions about the relevant family of initial states. We present an example of a system-environment state with nonvanishing quantum discord that leads to a completely positive map. This invalidates an earlier claim about the necessity of vanishing quantum discord for completely positive maps. In our final remarks, we discuss the physical validity of each approach.