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arXiv:2605.22603

Sudden death of entanglement, rebirth of magic

Chenfeng Cao

quant-phSubmitted May 21, 2026• Announced May 22, 2026• 0 citations• Revised Jul 15, 2026 (v2)
v1 · May 21, 2026v2 · Jul 15, 2026
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Abstract

Local Markovian noise cannot bring entanglement back, but it can bring magic back. Unlike separability, stabilizer membership is not preserved by local channels, allowing dissipation to push states out of the stabilizer polytope as well as in. Under local amplitude damping, the $n$-qubit GHZ family $α|0^n\rangle+β|1^n\rangle$ ($0<α<β$) loses its magic at a lower damping strength $γ_-$ and regains it at a higher one $γ_+$, while entanglement is irreversibly lost at $γ_e$. This magic--entanglement complementarity, $γ_e+γ_+=1$ for every $n$, reflects a system--environment duality of amplitude damping. Within real phase-covariant Markovian semigroups the phenomenon is mapped out in full: zero-temperature rebirth occurs if and only if $T_2>T_1$, unital dynamics produce no rebirth, and sufficiently weak thermal excitation confines rebirth to a finite magic island ending in a second sudden death. For small $α$, the reborn magic resides in a fully separable state with all proper marginals stabilizer, yet parity-syndrome extraction concentrates it onto a single qubit for magic-state distillation, without loss of expected robustness and with optimal per-register yield $α^2/2$. Local dissipation further divides pure stabilizer states into magic-generators and magic-insulators: at two qubits, the Bell state $|Φ^+\rangle$ generates magic immediately, while its Bell-state partner $|Ψ^+\rangle$ remains stabilizer. Together, magic and entanglement reveal a symmetry invisible to either alone.

Comments
44 pages, 12 figures

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Local Markovian noise cannot bring entanglement back, but it can bring magic back. Unlike separability, stabilizer membership is not preserved by local channels, allowing dissipation to push states out of the stabilizer polytope as well as in. Under local amplitude damping, the nnn-qubit GHZ family α∣0n⟩+β∣1n⟩α|0^n\rangle+β|1^n\rangleα∣0n⟩+β∣1n⟩ (0<α<β0<α<β0<α<β) loses its magic at a lower damping strength γ−γ_-γ−​ and regains it at a higher one γ+γ_+γ+​, while entanglement is irreversibly lost at γeγ_eγe​. This magic--entanglement complementarity, γe+γ+=1γ_e+γ_+=1γe​+γ+​=1 for every nnn, reflects a system--environment duality of amplitude damping. Within real phase-covariant Markovian semigroups the phenomenon is mapped out in full: zero-temperature rebirth occurs if and only if T2>T1T_2>T_1T2​>T1​, unital dynamics produce no rebirth, and sufficiently weak thermal excitation confines rebirth to a finite magic island ending in a second sudden death. For small ααα, the reborn magic resides in a fully separable state with all proper marginals stabilizer, yet parity-syndrome extraction concentrates it onto a single qubit for magic-state distillation, without loss of expected robustness and with optimal per-register yield α2/2α^2/2α2/2. Local dissipation further divides pure stabilizer states into magic-generators and magic-insulators: at two qubits, the Bell state ∣Φ+⟩|Φ^+\rangle∣Φ+⟩ generates magic immediately, while its Bell-state partner ∣Ψ+⟩|Ψ^+\rangle∣Ψ+⟩ remains stabilizer. Together, magic and entanglement reveal a symmetry invisible to either alone.
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Decoherence due to the unwanted interaction between a quantum system and environment leads to the degradation of quantum coherence. In particular, for an entangled state, decoherence makes a loss of entanglement and Bell nonlocality known as entanglement sudden death (ESD), and Bell nonlocality sudden death (BNSD). Here, we theoretically investigate the entanglement and Bell nonlocality of a bipartite entangled state under three types of decoherence, amplitude damping, phase damping, and depolarizing. Our result provides the bound of decoherence strength that does not lose the entanglement and Bell nonlocality. In addition, we find two interesting features. One is that the entanglement can survive even though one of the entangled qubits is affected by a large strength of decoherence if the other qubit is affected by a small enough strength of decoherence except for the depolarizing. The second one is that when a specific form of entangled state is under amplitude damping, the Bell nonlocality shows an asymmetric behavior respect to the decoherence strengths on each qubit. Our work provides comprehensive information on ESD and BNSD for the bipartite entangled state which will be useful to implement quantum information processing in the presence of decoherence.
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We systematically characterize quantum entanglement in all sixteen combinations of two-qubit Bell states across three operational regimes: pure superposition, noiseless mixing, and asymmetric mixing of one noisy and one pure Bell state. For pure superpositions, we calculate the von Neumann entropy and show that entanglement depends critically on the relative phase phi between superposed states. For mixed states, we use concurrence C under three physically motivated decoherence channels. The Depolarizing (DP) and Phase Damping (PD) channels are modeled as global, collective operations on the two-qubit Hilbert space, while Amplitude Damping (AD) is treated as independent local decay on each qubit, reflecting the asymmetric, energy-dissipating nature of spontaneous emission. A central and counterintuitive result is that increasing noise can raise concurrence in specific parameter regimes across all three channels. We further map the boundary between mathematical entanglement (C > 0) and operational quantum non-locality using the Horodecki criterion, finding this boundary to be strongly channel-dependent. Under Phase Damping, the Bell-local entanglement regime vanishes entirely for identical and same-bases mixtures, meaning any surviving entanglement guarantees a CHSH violation. Under Depolarizing noise, a large Bell-local region requires mixing probability r > 1/sqrt(2) to recover non-locality at maximum noise. For Amplitude Damping, certain mixtures exhibit a non-monotonic Horodecki parameter M(T): concurrence decreases monotonically with noise, yet the capacity for CHSH violation is lost at intermediate noise levels and restored at high noise, as maximal damping repurifies the noisy branch. These results provide a unified analytical reference for entanglement and non-locality management across Bell-state combinations and decoherence channels relevant to NISQ architectures.
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