Sudden death of entanglement, rebirth of magic

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 $\hat{I}\pm |0^n⟩+{\hat{I}}^2|1^n⟩$ ($0<\hat{I}\pm <{\hat{I}}^2$) loses its magic at a lower damping strength ${\hat{I}}_{-}^3$ and regains it at a higher one ${\hat{I}}_{+}^3$, while entanglement is irreversibly lost at ${\hat{I}}_e^3$. This magic--entanglement complementarity, ${\hat{I}}_e^3+{\hat{I}}_{+}^3=1$ 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 $\hat{I}\pm$, 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 $\hat{I}{\pm }^2/2$. Local dissipation further divides pure stabilizer states into magic-generators and magic-insulators: at two qubits, the Bell state $|\hat{I}{¦}^{+}⟩$ generates magic immediately, while its Bell-state partner $|\hat{I}{¨}^{+}⟩$ remains stabilizer. Together, magic and entanglement reveal a symmetry invisible to either alone.

44 pages, 12 figures

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