Figure 1

(Color online) Demonstration of tripartite entanglement sudden death in GHZ states. Left: Expectation value of the entanglement witness $-\mathrm{Tr}\left({\mathcal{W}}_G\rho \right)$ as a function of the initial state parametrized by $q$, and the dephasing constant $\kappa t$. At sufficiently low values of $q$ or high values of $\kappa t$ the state can no longer be detected by a GHZ-type entanglement witness. Center: $-\mathrm{Tr}\left({\mathcal{W}}_{W2}\rho \right)$ as a function of the same parameters. This entanglement witness shows when the state can no longer be detected as one with tripartite entanglement. Right: The negativity with respect to any qubit partition (and thus in this case the negativity is equivalent to the tripartite negativity). When this is zero the state has no distillable entanglement. For $q=1$ (initial pure state) the negativity is equal to $-\mathrm{Tr}\left({\mathcal{W}}_{W2}\rho \right)$ meaning that the ${\mathcal{W}}_{W2}$ entanglement witness always detects the presence of the tripartite entanglement. When $q<1$ the state is mixed and the witness does not always detect the tripartite entanglement which is known to be present via the tripartite negativity measure.Reuse & Permissions

Figure 2

(Color online) Demonstration of bipartite and tripartite entanglement sudden death in W states. Each plot shows the expectation value of the entanglement witness $-\mathrm{Tr}\left({\mathcal{W}}_W{\rho }_W\right)$ (solid line), the bipartite concurrence (medium dashed line), and the negativity (large dashed line) as a function of the dephasing constant $\kappa t$. Left: $q=1$, the initial state is pure. In this case $-\mathrm{Tr}\left({\mathcal{W}}_W{\rho }_W\right)$ goes to zero in finite time, but the other entanglement measures do not. Center: $q=0.95$, for any $q<1$ the concurrence undergoes ESD before the negativity (which is equal to the tripartite negativity) goes to zero. Right: $q=0.75$.Reuse & Permissions

Figure 3

(Color online) Demonstration of entanglement sudden death in ${\rho }_{gb}$-type states. Each plot shows $C_{12}$ (solid line), $N$ (large dashed line), $N_3$ (medium dashed line), $-\mathrm{Tr}\left({\mathcal{W}}_{Ggb}{\rho }_{gb}\right)$ (small dashed line), and $-\mathrm{Tr}\left({\mathcal{W}}_{Wgb}{\rho }_{gb}\right)$ (dotted line) as a function of the dephasing constant $\kappa t$. Left: $q=1$, the initial state is pure. In this case the tripartite entanglement witnesses go to zero in finite time but the concurrence, negativity, and tripartite negativity do not undergo ESD. Center: $q=0.9$, for any sufficiently high value of $q$, where $q<1$ the concurrence undergoes ESD before the negativity but after the tripartite negativity. Right: $q=0.6$, the concurrence undergoes ESD before the tripartite negativity and even before the tripartite entanglement witness. Thus, there is detectable tripartite entanglement in the system though there is no concurrence remaining when one of the qubits is traced over.Reuse & Permissions

Figure 4

(Color online) Top left: Expectation value of entanglement witness $-\mathrm{Tr}\left({\mathcal{W}}_{WH}{\rho }_{e1}\right)$ for single qubit dephasing error strength parameter $p$, before syndrome measurement and recovery, as a function of $p$ and the initial state of the unencoded qubit parametrized by $\alpha$. Lack of detection by the entanglement witness is evident despite the fact that error correction always works. Top center: $-\mathrm{Tr}\left({\mathcal{W}}_{WH}{\rho }_e\right)$ with dephasing of strength $p$ on all three qubits. Lack of detection occurs at much lower values of $p$ than when dephasing affects only one of the qubits. Top right: Negativity (which is equal to $N_3$) of ${\rho }_e$ with dephasing of all qubits. ESD of negativity requires much higher values of $p$ than lack of detection by the tripartite entanglement witness. Bottom left: Purity $P$ of final single qubit state after error correction and decoding when the syndrome measurement does not detect an error. Bottom center: $P$ of final single qubit state after error correction and decoding when the syndrome measurement does detect an error. Bottom right: Comparison of $P$ and the expectation value of the entanglement witnesses as a function of the initial state, parametrized by $\alpha$, and the dephasing strength $p$. The contours are for $-\mathrm{Tr}\left({\mathcal{W}}_{WH}{\rho }_e\right)=0.001$ (solid line), $N=0.001$ (long dashed line), $P=0.99$ if an error was not detected (medium dashed line), and $P=0.99$ if an error was detected (small dashed line). The shape of the curves for the entanglement witness and $N$ are similar though the entanglement witness fails to detect the entanglement significantly before ESD of $N$. However, while areas of higher purity and more entanglement are at lower values of $p$ and values of $\alpha$ closer to $1∕\sqrt{2}$, there is no apparent correlation between the presence of entanglement and the success of the QEC code.Reuse & Permissions