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arXiv:2610.36936

Qubit-environment entanglement in pure dephasing dynamics for mixed initial states

Małgorzata Strzałka, Nico Peters, Laura Ares, Jan Sperling, Katarzyna Roszak

quant-phSubmitted Oct 27, 2026• Announced Oct 28, 2026• 0 citations
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Abstract

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.

AI Summary

The paper studies when pure-dephasing interactions generate entanglement between an initially mixed qubit and an arbitrarily sized, initially mixed environment, assuming an initially uncorrelated joint state. It extends earlier results for pure initial qubit states, where equality of the two conditional environmental states, $R_{00}(t)=R_{11}(t)$, is necessary and sufficient for separability. For mixed qubits, this equality remains sufficient for separability, as shown by an explicit product-state decomposition. Entanglement detection is developed using a family of principal minors of the partially transposed joint state. Negative minors certify entanglement, but nonnegative minors no longer establish separability. A Schur-complement transformation reduces the partial-transpose positivity test from a $2N\times2N$ matrix to an $N\times N$ matrix for an environment of dimension $N$. Its negative eigenvalues detect entanglement missed by the selected minors. An analytically tractable single-spin environment and numerical examples of an NV-center qubit coupled to one, two, three, or five noninteracting nuclear spins demonstrate finite initial intervals without entanglement, subsequent sudden birth, and repeated death and revival. Interactions that entangle initially pure qubits can also remain nonentangling for sufficiently mixed initial qubits. The selected-minor witness can miss entire entangled intervals, particularly when entanglement is weak. The analysis primarily addresses negative-partial-transpose entanglement; it does not provide a complete separability criterion for mixed initial qubits or resolve possible bound entanglement. The NV examples assume initially uncorrelated environmental spins and neglect interactions between them.

No sign of AI-generated textIncrementalOverclaimsPossible error

Central claim may not hold: The claimed complete study leaves possible PPT entanglement unresolved, so its mixed-state results do not constitute a complete separability characterization. Equations (Mi) and (D2) fail for allowed rank-deficient environmental states, including a pure environmental qubit for which they miss an explicitly negative partial-transpose eigenvalue. The eigenvalues of the Schur-complement matrix D do not quantify negativity, because the transformation preserves inertia rather than eigenvalue magnitudes.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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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.
The paper studies when pure-dephasing interactions generate entanglement between an initially mixed qubit and an arbitrarily sized, initially mixed environment, assuming an initially uncorrelated joint state. It extends earlier results for pure initial qubit states, where equality of the two conditional environmental states, R00(t)=R11(t)R_{00}(t)=R_{11}(t)R00​(t)=R11​(t), is necessary and sufficient for separability.
For mixed qubits, this equality remains sufficient for separability, as shown by an explicit product-state decomposition. Entanglement detection is developed using a family of principal minors of the partially transposed joint state. Negative minors certify entanglement, but nonnegative minors no longer establish separability. A Schur-complement transformation reduces the partial-transpose positivity test from a 2N×2N2N\times2N2N×2N matrix to an N×NN\times NN×N matrix for an environment of dimension NNN. Its negative eigenvalues detect entanglement missed by the selected minors.
An analytically tractable single-spin environment and numerical examples of an NV-center qubit coupled to one, two, three, or five noninteracting nuclear spins demonstrate finite initial intervals without entanglement, subsequent sudden birth, and repeated death and revival. Interactions that entangle initially pure qubits can also remain nonentangling for sufficiently mixed initial qubits. The selected-minor witness can miss entire entangled intervals, particularly when entanglement is weak.
The analysis primarily addresses negative-partial-transpose entanglement; it does not provide a complete separability criterion for mixed initial qubits or resolve possible bound entanglement. The NV examples assume initially uncorrelated environmental spins and neglect interactions between them.
We propose a qubit-environment entanglement measure which is tailored for evolutions that lead to pure dephasing of the qubit, such as are abundant in solid-state scenarios.The measure can be calculated directly form the density matrix without minimization of any kind.In fact, it does not require knowledge of the full density matrix and it is enough to know the initial qubit state and the states of the environment conditional on qubit pointer states.This yields a computational advantage over standard entanglement measures, which becomes large when there are no correlations between environmental components in the conditional environmental states.In contrast to all other measures of mixed-state entanglement, the measure has a straightforward physical interpretation directly linking the amount of information about the qubit state which is contained in the environment to the amount of qubit-environment entanglement.This allows for a direct extension of the pure-state interpretation of entanglement generated during pure dephasing to mixed states, even though pure-state conclusions about qubit decoherence are not transferable.
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.
Abstract We propose a scheme for the detection of qubit–environment entanglement which requires only operations and measurements on the qubit, all within reach of current experimental state-of-the-art. The scheme works for any type of interaction which leads to pure dephasing of the qubit as long as the initial qubit state is pure. The scheme is direct in the sense that it allows the detection of entanglement present in the system at time τ\tau τ τ after the initialization of the qubit in a superposition state. It requires a measurement on the qubit at time τ\tau τ τ and a comparison of the post-measurement evolution to the evolution obtained by a modified scheme. It becomes particularly simple when one of the qubit states is neutral with respect to the environment, such as in case of the most common choice of the NV center spin qubit or for excitonic charge qubits, when the environment is initially at thermal equilibrium. In this case, the post-measurement evolution needs to be compared only to the standard decoherence which is obtained without any qubit manipulation after the preparation of the initial state.
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 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.
Decoherence is an unavoidable effect of the quantum system coupled to an environment, which leads to the system evolving from a pure state to a mixed one. A critical issue here is that the reduction of purity is quantum or classical, which corresponds to whether or not entanglement is generated between the system and environment. We report a proof-of-principle experimental investigation of such entanglement in a simple but paradigmatic case, in which a qubit system interacts with a pure dephasing channel. The pure dephasing process is realized in a linear photonic system, and the experiment is a validation in a very controlled setting. By adopting previous methods to detect qubit-environment entanglement, we prove that such entanglement can be detected by performing local measurements on only one subsystem, even when the other subsystem in the pair cannot be detected. Our experiment paves the way for investigating features of open quantum system dynamics.