### Description

Vacuum is one of the most interesting phenomena of the real world. A two-level atom, frequently treated as a qubit, when unavoidably touched by vacuum, is unstable on its excited level, even if it does not interact with any other system. All states except the ground state suffer from decay due to interaction with vacuum radiation modes. The atom falls to the ground state in asymptotic time and a photon is emitted, a process known as spontaneous decay.

The situation becomes more interesting when multiple atoms are present and placed far apart, such that they can be considered as sitting in distinguishable vacuum states. As in the single-atom case, the excited elements of the multi-atom density matrix decay exponentially to zero. However, previous results of Yu and Eberly [Science **323**, 598 (2009)] have shown that two-atom entanglement (also called nonseparability) has a transition to zero in such systems that is temporally abrupt. Such dynamics of entanglement is non-analytic, and this bizarre behaviour is usually termed entanglement sudden death (ESD).

It is then natural to ask how entanglement among more than two qubits behaves when the qubits decay independently in their respective cavities. Are there also sudden transitions in these more complicated systems? Specifically, what is the requirement for ESD in these systems? These open questions are made difficult by the following two obstacles. First, existing entanglement measures cannot quantify the degree of genuine multipartite entanglement (GME), the measure of the entanglement resource that plays a central role in quantum information and communication tasks such as the three-qubit quantum teleportation. Second, even if a GME measure is used, it is an extremely difficult task to evaluate it for general mixed states, which are ubiquitous for vacuum decay.

In this work, we provide solutions to the above two problems. For the GME measure, we use our recently developed pure-state measure *Concurrence Fill*, which was proven to detect genuine tripartite entanglement [see Xie and Eberly, Phys. Rev. Lett. **127**, 040403 (2021)]. For the mixed-state issue, we avoid deriving an analytic formula, which is currently impossible. Instead, we apply a previous result to convert the evaluation of mixed-state entanglement into a practical numerical task [Eisert, Brandão, and Audenaert, New J. Phys. **9**, 46 (2007)].

With these methods, we study the entanglement dynamics for three-qubit systems, wherein each qubit interacts with its own vacuum environment via the amplitude-damping channel. We numerically examine the dynamics of the Concurrence Fill measure of genuine tripartite entanglement among the three qubits, when different initial three-party states are selected.

Specifically, when the initial state is the generalized GHZ state, $|\text{GHZ}\rangle=\cos\theta|eee\rangle+\sin\theta|ggg\rangle$, where $|e\rangle$ and $|g\rangle$ are the excited and ground states for each atom, ESD occurs when the amplitude of the all-excited state $|eee\rangle$ exceeds a threshold value. Furthermore, we show that ESD does not occur if the initial state is the singly-excited $W$ state, $|W\rangle=(|egg\rangle+|geg\rangle+|gge\rangle)/\sqrt{3}$, or the doubly-excited $\overline W$ state, $|\overline W\rangle=(|gee\rangle+|ege\rangle+|eeg\rangle)/\sqrt{3}$.

These observations imply that when the three-qubit initial state is pure, the presence of the all-excited $|eee\rangle$ state is a necessary requirement for ESD to occur, despite the fact that $|eee\rangle$ is a product state which cannot exhibit ESD alone. To test this novel conjecture, we consider as the initial state the superposition of $|eee\rangle$ with the no-ESD $|W\rangle$ state, $|S\rangle=\cos\theta|eee\rangle+\sin\theta|W\rangle$. In agreement with our conjecture, ESD occurs when the amplitude of $|eee\rangle$ exceeds a threshold value. Our results bring new insights toward a more complete understanding of entanglement sudden death.

Presenter name | Songbo Xie |
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How will you attend ICAP-27? | I am planning on in-person attendance |