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arXiv:2501.16834 · Physical Review A

Coherence-assisted entanglement activation during open evolution

Masaya Takahashi, Jarosław K. Korbicz

Published Aug 26, 2025• 0 citations
arXivHTMLPDF

Abstract

The fact that in the course of an open evolution there is, in a vast majority of cases, the creation of system-environment entanglement out of the initial coherence, has been known for many years. However, how much entanglement can be created under general assumptions has not been analyzed yet. In this work, we investigate system-environment entanglement generation under a broad and important class of interactions known as pure dephasing interactions and for arbitrary initial conditions. Our main results are the following bounds on the relative entropy of entanglement $E_r(σ_{SE})$ after evolution: $ C_r(ρ_S) - H(\mathcal{I}|\mathcal{M}) \leq E_r(σ_{SE}) \leq C_r(ρ_S) - C_r(σ_S) + H(\mathcal{I})$, connecting it on one side to the initial relative entropy of coherence, $C_r(ρ_S)$, and information extractable from the environment, $H(\mathcal{I}|\mathcal{M})$, and on the other side to the decoherence strength, $ C_r(ρ_S) - C_r(σ_S)$, making quantitatively precise the statement that larger entanglement leads to more decoherence. As an illustration, we apply our results to the spin-boson model, where a qubit interacts with a bosonic bath, showing time evolution of the entanglement and the Markovian/non-Markovian crossover. Our results help better understand and quantify the complicated correlations, produced during an open evolution, which will hopefully lead to better and new decoherence mitigation techniques.

DOI
10.1103/8vc7-64fk

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The fact that in the course of an open evolution there is, in a vast majority of cases, the creation of system-environment entanglement out of the initial coherence, has been known for many years. However, how much entanglement can be created under general assumptions has not been analyzed yet. In this work, we investigate system-environment entanglement generation under a broad and important class of interactions known as pure dephasing interactions and for arbitrary initial conditions. Our main results are the following bounds on the relative entropy of entanglement Er(σSE)E_r(σ_{SE})Er​(σSE​) after evolution: Cr(ρS)−H(I∣M)≤Er(σSE)≤Cr(ρS)−Cr(σS)+H(I) C_r(ρ_S) - H(\mathcal{I}|\mathcal{M}) \leq E_r(σ_{SE}) \leq C_r(ρ_S) - C_r(σ_S) + H(\mathcal{I})Cr​(ρS​)−H(I∣M)≤Er​(σSE​)≤Cr​(ρS​)−Cr​(σS​)+H(I), connecting it on one side to the initial relative entropy of coherence, Cr(ρS)C_r(ρ_S)Cr​(ρS​), and information extractable from the environment, H(I∣M)H(\mathcal{I}|\mathcal{M})H(I∣M), and on the other side to the decoherence strength, Cr(ρS)−Cr(σS) C_r(ρ_S) - C_r(σ_S)Cr​(ρS​)−Cr​(σS​), making quantitatively precise the statement that larger entanglement leads to more decoherence. As an illustration, we apply our results to the spin-boson model, where a qubit interacts with a bosonic bath, showing time evolution of the entanglement and the Markovian/non-Markovian crossover. Our results help better understand and quantify the complicated correlations, produced during an open evolution, which will hopefully lead to better and new decoherence mitigation techniques.
Decoherence of qubits is one main limitation in quantum information processing, where the qubits themselves can be mixed because of decoherence and additional decoherence may generated through interactions with a bath. Here, we present a complete study of qubit-environment entanglement in the case of interactions that lead to pure dephasing of the qubit by allowing the initial state of the qubit to be mixed, exceeding previous restrictions. We find that the criteria for separability that hold for pure initial states of the qubit are reduced to witness status. Yet, the evolution of the witnesses reflects the array of features that entanglement can exhibit in the mixed case but are forbidden in the pure-qubit case. Most notably, this includes sudden birth of entanglement as the initial phase of the evolution is typically separable. Moreover, there is a stark numerical advantage to the study of entanglement for large environments when using the witness rather than the traditional measures available. We include the study of entanglement generated between a spin qubit and a sparse spin environment to illustrate the peculiarities of the evolution of entanglement in this case, as well as to emphasize the level of reliability of the entanglement witness.
We study entanglement generated between a charge qubit and a bosonic bath due to their joint evolution which leads to pure dephasing of the qubit. We tune the parameters of the interaction, so that the decoherence is quantitatively independent of the number of bosonic modes taken into account and investigate how the entanglement generated depends on the size of the environment. A second parameter of interest is the mixedness of the initial state of the environment which is controlled by temperature. We show analytically that for a pure initial state of the environment, entanglement does not depend on environment size. For mixed initial states of the environment, the generated entanglement decreases with the increase of environment size. This effect is stronger for larger temperatures, when the environment is initially more mixed, but in the limit of an infinitely large environment, no entanglement is created at any finite temperature.
We study entanglement generated between a qubit and a super-ohmic bosonic bath due to their joint evolution which leads to pure dephasing of the qubit. We tune the parameters of the interaction, so that the decoherence is quantitatively independent of the number of bosonic modes taken into account (and corresponds to realistic decoherence seen in small excitonic quantum dots coupled to an environment of phonons) and investigate, how the entanglement generated depends on the size of the environment. A second parameter of interest is the mixedness of the initial state of the environment which is controlled by temperature. We show analytically that for a pure initial state of the environment (zero temperature here), entanglement does not depend on environment size. At finite temperatures (mixed initial state of the environment), the generated entanglement decreases with the increase of environment size. This effect is stronger for larger temperatures, when the environment is initially more mixed, but in the limit of an infinitely large environment, no entanglement is created at any finite temperature.
An evolution between a system and its environment which leads to pure dephasing of the system may either be a result of entanglement building up between the system and the environment or not (the second option is only possible for initially mixed environmental states). We find a way of distinguishing between entangling and nonentangling evolutions for systems which are larger than a single qubit. The generalization of the single-qubit separability criterion to larger systems is not sufficient to make this distinction (it constitutes a necessary condition of separability). A set of additional conditions for the operators describing the evolution of the environment depending on the state of the system is required. We find that the commutation of these environmental operators with the initial state of the environment does not guarantee separability; products of the operators need to commute among themselves for a pure-dephasing evolution not to be accompanied by system-environment entanglement generation. This is a qualitative difference with respect to the single-qubit case, since it allows for a system to entangle with an initially completely mixed environment.
We find that when a qubit initialized in a pure state experiences pure dephasing due to interaction with an environment, separable qubit-environment states generated during the evolution also have zero quantum discord with respect to the environment. What follows is that the set of separable states which can be reached during the evolution has zero volume, and hence, such effects as sudden death of qubit-environment entanglement are very unlikely. In the case of the discord with respect to the qubit, a vast majority of qubit-environment separable states is discordant, but in specific situations zero-discord states are possible. This is conceptually important since there is a connection between the discordance with respect to a given subsystem and the possibility of describing the evolution of this subsystem using completely positive maps. Finally, we use the formalism to find an exemplary evolution of an entangled state of two qubits that is completely positive, and occurs solely due to interaction of only one of the qubits with its environment (so one could guess that it corresponds to a local operation, since it is local in a physical sense), but which nevertheless causes the enhancement of entanglement between the qubits. While this simply means that the considered evolution is completely positive, but does not belong to local operations and classical communication, it shows how much caution has to be exercised when identifying evolution channels that belong to that class.
We determine the total state dynamics of a dephasing open quantum system using the standard environment of harmonic oscillators. Of particular interest are random unitary approaches to the same reduced dynamics and system-environment correlations in the full model. Concentrating on a model with an at times negative dephasing rate, the issue of non-Markovianity will also be addressed. Crucially, given the quantum environment, the appearance of non-Markovian dynamics turns out to be accompanied by a loss of system-environment correlations. Depending on the initial purity of the qubit state, these system-environment correlations may be purely classical over the whole relevant time scale, or there may be intervals of genuine system-environment entanglement. In the latter case, we see no obvious relation between the build-up or decay of these quantum correlations and Non-Markovianity.