Positivity of the assignment map implies complete positivity of the reduced dynamics
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Positivity of the assignment map implies complete positivity of the reduced dynamics Iman Sargolzahi1,2 Received: 13 March 2020 / Accepted: 11 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Consider the set S = {ρ S E } of possible initial states of the system-environment. The map which assigns to each ρ S ∈ Tr E S a ρ S E ∈ 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 . Keywords Open quantum systems · Hermitian maps · Completely positive maps · Markov states · Relative entropy · Entanglement
1 Introduction In quantum information theory, it is common to assume that quantum operations are linear trace-preserving completely positive (CP) maps [1]: ρ = (ρ) =
E i ρ E i† ,
i
E i† E i = I ,
(1)
i
where ρ and ρ are the initial and final states (density operators) of the system, respectively. In addition, E i are linear operators, and I is the identity operator.
B
Iman Sargolzahi [email protected] ; [email protected]
1
Department of Physics, University of Neyshabur, Neyshabur, Iran
2
Research Department of Astronomy and Cosmology, University of Neyshabur, Neyshabur, Iran 0123456789().: V,-vol
123
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I. Sargolzahi
For example, consider a bipartite quantum system S = AB. In entanglement theory, it is assumed that the entanglement, between the two separated parts A and B, does not increase under local quantum operations [2]. Assuming quantum operations as CP maps, an entanglement measure (monotone) M is constructed as a non-increasing function, under local CP maps [2,3], i.e., M (ρ AB ) ≥ M ( A ⊗ B (ρ AB )) ,
(2)
where A and B are CP maps, as Eq. (1), on the parts A and B, respectively. Now, consider the case that the bipartite system S = AB is not closed and interacts with its environment E = E A E B , where E A and E B are the local environments of the parts A and B, respectively. In addition, assume that the time evolution of the whole system-environment is local as U S E = U AE A ⊗ U B E B , where U AE A and U B E B are unitary operators on AE A and B E B , respectively. Then, the reduced dynamics of the system S = AB may not be given as a local CP map A ⊗ B , in general, and so the entanglement may increase, during such a local evolution of the whole system-environment [4–10]. In entanglement theory, one usually consider only one initial state ρ AB
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