Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1702.03702 · Open Systems & Information Dynamics

When the Assignment Map Is Completely Positive

Iman Sargolzahi, Sayyed Yahya Mirafzali

Published Sep 1, 2018• 6 citations
arXivHTMLPDF

Abstract

Finding the general set of system-environment states {ρSE} for which the reduced dynamics of the system is completely positive (CP) is the subject of some recent works. An advance in this context appeared in [7], where the problem was solved for the case of CP assignment map. Here, we restate this result using the framework introduced in [8]. This, we think, clarifies the mentioned result better and so leads to a generalization of it, straightforwardly.

DOI
10.1142/s1230161218500129

Related papers

Dynamics beyond completely positive maps: Some properties and applications

Hilary A. Carteret, Daniel R. Terno, Karol Życzkowski

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.

arXiv:quant-ph/0512167•Physical Review A•Apr 18, 2008•118 citations

Positivity in the presence of initial system-environment correlation

Kavan Kishore Modi, César A. Rodríguez-Rosario, Alán Aspuru‐Guzik

The constraints imposed by the initial system-environment correlation can lead to nonpositive dynamical maps. We find the conditions for positivity and complete positivity of such dynamical maps by using the concept of an assignment map. Any initial system-environment correlations make the assignment map nonpositive, while the positivity of the dynamical map depends on the interplay between the assignment map and the system-environment coupling. We show how this interplay can reveal or hide the nonpositivity of the assignment map. We discuss the role of this interplay in Markovian models.

arXiv:1203.5209•Physical Review A•Dec 12, 2012•36 citations

Some remarks on assignment maps

Fabio Masillo, G. Scolarici, Luigi Solombrino

We study the properties of general linear assignment maps, showing that positivity axiom can be suitably relaxed, and propose a new class of dynamical maps (generalized dynamics). A puzzling result, arising in such a context in quantum information theory, is also discussed.

Journal of Mathematical Physics•Jan 1, 2011•15 citations

LINEARITY VERSUS COMPLETE POSITIVITY OF THE EVOLUTION OF OPEN QUANTUM SYSTEMS

Russell R. Ceballos

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.

•Jan 1, 2014•0 citations

Open system quantum evolution and the assumption of complete positivity (A Tutorial)

Mark Byrd, Russell R. Ceballos, Eric Chitambar

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.

International Journal of Quantum Information•Aug 18, 2016•2 citations

Completely positive reduced dynamics with non-Markovian initial states

Nazifkar, Samira, Sargolzahi, Iman

Consider an open quantum system S, interacting with its environment E, and also an ancillary Hilbert space R. The reduced dynamics of the system S is given by a completely positive map, if the set S={ρSE} , of possible initial states of the system-environment, can be written as a steered set from a tripartite Markov state ƮRSE. In this paper, we call such steered states as Markovian states ρSE, and study a physical case, in which the reduced dynamics of the system can be completely positive, even when the initial states of the system-environment ρSE are non-Markovian.

Iranian Journal of Physics Research•Mar 1, 2023•0 citations
Finding the general set of system-environment states {ρSE} for which the reduced dynamics of the system is completely positive (CP) is the subject of some recent works. An advance in this context appeared in [7], where the problem was solved for the case of CP assignment map. Here, we restate this result using the framework introduced in [8]. This, we think, clarifies the mentioned result better and so leads to a generalization of it, straightforwardly.
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 constraints imposed by the initial system-environment correlation can lead to nonpositive dynamical maps. We find the conditions for positivity and complete positivity of such dynamical maps by using the concept of an assignment map. Any initial system-environment correlations make the assignment map nonpositive, while the positivity of the dynamical map depends on the interplay between the assignment map and the system-environment coupling. We show how this interplay can reveal or hide the nonpositivity of the assignment map. We discuss the role of this interplay in Markovian models.
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.
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.
Consider an open quantum system S, interacting with its environment E, and also an ancillary Hilbert space R. The reduced dynamics of the system S is given by a completely positive map, if the set S={ρSE} , of possible initial states of the system-environment, can be written as a steered set from a tripartite Markov state ƮRSE. In this paper, we call such steered states as Markovian states ρSE, and study a physical case, in which the reduced dynamics of the system can be completely positive, even when the initial states of the system-environment ρSE are non-Markovian.