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arXiv:2605.08923 · Brazilian Journal of Physics

Entanglement increase from local interactions which lead to non-positive local reduced dynamics

Iman Sargolzahi

quant-phSubmitted May 9, 2026• Announced May 12, 2026• 0 citations
v1 · May 9, 2026v2 · Oct 13, 2026
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Abstract

Consider a bipartite quantum system S=AB such that each part interacts only with its local environment. Under such circumstances, one expects that the entanglement between parts A and B does not exceed its initial value during the time evolution. In fact, this is the case if the reduced dynamics of the system is given by $\mathcal{E}_{A}\otimes \mathcal{E}_{B}$, where $\mathcal{E}_{A}$ and $\mathcal{E}_{B}$ are quantum channels, i.e., completely positive trace-preserving maps. But, the reduced dynamics of the system may be given by a map as $Ψ_{A}\otimes Ψ_{B}$, where $Ψ_{A}$ and $Ψ_{B}$ are local non-positive maps. Then, the entanglement between A and B can exceed its initial value, as was shown in the case studied by Jordan et al. [Phys. Rev. A 76, 022102 (2007)]. In this paper, we first explore the general circumstances under which one can find such cases as they found. Next, we introduce another general procedure which leads to local non-positive maps that cause entanglement exceeding.

Journal reference
Brazilian Journal of Physics (2026) 56:238
DOI
10.1007/s13538-026-02153-0
Comments
7 pages, 1 figure

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Consider a bipartite quantum system S=AB such that each part interacts only with its local environment. Under such circumstances, one expects that the entanglement between parts A and B does not exceed its initial value during the time evolution. In fact, this is the case if the reduced dynamics of the system is given by EA⊗EB\mathcal{E}_{A}\otimes \mathcal{E}_{B}EA​⊗EB​, where EA\mathcal{E}_{A}EA​ and EB\mathcal{E}_{B}EB​ are quantum channels, i.e., completely positive trace-preserving maps. But, the reduced dynamics of the system may be given by a map as ΨA⊗ΨBΨ_{A}\otimes Ψ_{B}ΨA​⊗ΨB​, where ΨAΨ_{A}ΨA​ and ΨBΨ_{B}ΨB​ are local non-positive maps. Then, the entanglement between A and B can exceed its initial value, as was shown in the case studied by Jordan et al. [Phys. Rev. A 76, 022102 (2007)]. In this paper, we first explore the general circumstances under which one can find such cases as they found. Next, we introduce another general procedure which leads to local non-positive maps that cause entanglement exceeding.
We consider a bipartite quantum system S=ABS=ABS=AB such that the part AAA is isolated from the environment EEE and only the part BBB interacts with EEE. Under such circumstances, entanglement of the system may experience decreases and increases, during the evolution of the system. Here, we show that the entanglement of the system can exceed its initial value, under such local interaction, even though, at the initial moment, there is no entanglement in the environment and the system and the environment are only classically correlated. The case which is studied in this paper possesses another interesting feature too: The reduced dynamics of the system can be modeled as a completely positive map. In addition, we introduce the concept of inaccessible entanglement to explain why entanglement can exceed its initial value, under local interactions, in open quantum systems.
We consider a bipartite quantum system [Formula: see text] (including parties [Formula: see text] and [Formula: see text]), interacting with an environment [Formula: see text] through a localized quantum dynamics [Formula: see text]. We call a quantum dynamics [Formula: see text] localized if, e.g. the party [Formula: see text] is isolated from the environment and only [Formula: see text] interacts with the environment: [Formula: see text], where [Formula: see text] is the identity map on the part [Formula: see text] and [Formula: see text] is a completely positive (CP) map on the both [Formula: see text] and [Formula: see text]. We will show that the reduced dynamics of the system is also localized as [Formula: see text], where [Formula: see text] is a CP map on [Formula: see text], if and only if the initial state of the system-environment is a Markov state. We then generalize this result to the two following cases: when both [Formula: see text] and [Formula: see text] interact with a same environment, and when each party interacts with its local environment.
We fully characterize bipartite entanglement-annihilating (EA) channels that destroy entanglement of any state shared by subsystems and, thus, should be avoided in any entanglement-enabled experiment. Our approach relies on extending the problem to EA positive maps, the cone of which remains invariant under concatenation with partially positive maps. Due to this invariancy, positive EA maps adopt a well characterization and their intersection with completely positive trace-preserving maps results in the set of EA channels. In addition to a general description, we also provide sufficient operational criteria revealing EA channels. They have a clear physical meaning since the processes involved contain stages of classical information transfer for subsystems. We demonstrate the applicability of derived criteria for local and global depolarizing noises, and specify corresponding noise levels beyond which any initial state becomes disentangled after passing the channel. The robustness of some entangled states is discussed.
Positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement. In the multipartite case, however, the direct application of a positive but not completely positive map cannot distinguish if a state is genuinely multipartite entangled or just entangled across some bipartition. We thus generalise this bipartite concept to the multipartite setting by introducing non-positive maps that are positive on the subset of biseparable states but can map to a non-positive element if applied to a genuine multipartite entangled state. We explicitly construct examples of multipartite non-positive maps, obtained from positive maps via a lifting procedure, that in this fashion can reveal genuine multipartite entanglement in a robust way.
We show that the dynamics of any open quantum system that is initially correlated with its environment can be described by a set of d2{d}^{2}d2 (or less) completely positive maps, where ddd is the dimension of the system. Only one such map is required for the special case of no initial correlations. The same maps describe the dynamics of any system-environment state obtained from the initial state by a local operation on the system. The reduction of the system dynamics to a set of completely positive maps allows known numerical and analytic tools for uncorrelated initial states to be applied to the general case of initially correlated states, which we exemplify by solving the qubit dephasing model for such states, and provides a natural approach to quantum Markovianity for this case. We show that this set of completely positive maps can be experimentally characterized using only local operations on the system, via a generalization of noise spectroscopy protocols. As further applications, we first consider the problem of retrodicting the dynamics of an open quantum system which is in an arbitrary state when it becomes accessible to the experimenter and explore the conditions under which retrodiction is possible. We also introduce a related one-sided or limited-access tomography protocol for determining an arbitrary bipartite state, evolving under a sufficiently rich Hamiltonian, via local operations and measurements on just one component. We simulate this protocol for a physical model of particular relevance to nitrogen-vacancy centers and in particular show how to reconstruct the density matrix of a set of three qubits, interacting via dipolar coupling and in the presence of local magnetic fields, by measuring and controlling only one of them.
Given an open quantum system SSS with environment EEE, we establish a one-to-one correspondence between completely positive trace-preserving (CPTP) maps from SSS to EEE and assignment maps initializing the composite system SESESE in a possibly correlated state prior to unitary evolution. We show that every possible joint state ρSEρ_{SE}ρSE​ can be realized by such assignment maps, which are inherently Hermiticity-preserving and trace-preserving (HPTP). While these maps sacrifice global positivity, we derive a simple necessary and sufficient condition for any local state of SSS to belong to their domain of validity. Furthermore, we prove that the geometry of this domain is governed by the joint state assigned to the maximally mixed input, ensuring that a non-zero volume of admissible local states can readily coexist with initial system-environment entanglement. Finally, we provide an operational protocol for witnessing such limits of validity.