Positivity of the assignment map implies complete positivity of the reduced dynamics

2019/06/30 by Iman Sargolzahi · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Dynamics (music) #Markov chain #Quantum Information and Cryptography #Quantum many-body systems #Set (abstract data type) #State (computer science) #Unitary state #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1007/s11128-020-02810-6

published in Quantum Information Processing 19(9) (Springer Science+Business Media) · 8 pages, As an application, the case that the system is a qubit, is added

openalex created_date 2019/07/12 · openalex publication_date 2020/08/24 · arxiv created 2020/08/29 · arxiv updated 2020/09/01 · openalex updated_date 2026/08/05

Abstract

Consider the set S=\lbraceρ_SE\rbrace of possible initial states of the system-environment. The map which assigns to each ρ_S∈ Tr_ES a ρ_SE∈ S is called the assignment map. The assignment map is Hermitian, in general. In this paper, we restrict ourselves to the case that the assignment map is, in addition, positive and show that this implies that the so-called reference state is a Markov state. Markovianity of the reference state leads to existence of another assignment map which is completely positive. So, the reduced dynamics of the system is also completely positive. As a consequence, when the system S is a qubit, we show that if S includes entangled states, then either the reduced dynamics is not given by a map, for, at least, one unitary time evolution of the system-environment U, or the reduced dynamics is non-positive, for, at least, one U.

Citations

• Stochastic Dynamics of Quantum-Mechanical Systems
• Completely positive maps and classical correlations
• Completely positive maps and entropy inequalities
• Completely positive dynamical semigroups of N-level systems
• Reduced Dynamics Need Not Be Completely Positive
• Vanishing quantum discord is not necessary for completely positive maps
• Sudden Birth versus Sudden Death of Entanglement in Multipartite Systems
• Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality
• Completely positive linear maps on complex matrices
• Entanglement revival can occur only when the system–environment state is not a Markov state
• Quantum entanglement