A simple proof of monogamy of entanglement

Zhi‐Xiang Jin, Shao-Ming Fei

Limin Gao, Fengli Yan, Ting Gao

Guillaume Aubrun, Alexander Müller‐Hermes, Martin Plávala

Zackary Jorquera, Alexandra Kolla, Steven Kordonowy +2 more

We prove new monogamy of entanglement bounds for two-local qudit Hamiltonians of rank-one projectors without one-local terms. In particular, we certify the maximum energy in terms of the maximum matching of the underlying interaction graph via low-degree sum-of-squares proofs. Algorithmically, we show that a simple matching-based algorithm approximates the maximum energy to at least 1 / d for general graphs and to at least 1 / d + Θ ( 1 / D ) for graphs with bounded degree, D . This outperforms random assignment, which, in expectation, achieves energy of only 1 / d 2 of the maximum energy for general graphs. Notably, on D -regular graphs with degree, D ≤ 5 , and for any local dimension, d , we show that this simple matching-based algorithm has an approximation guarantee of 1 / 2 . Lastly, when d = 2 , we present an algorithm achieving an approximation guarantee of 0.595 , beating that of \cite{parekh_optimal_2022}, which gave an approximation ratio of 1 / 2 .

S. Shelly Sharma, Naresh Kumar Sharma

Xi-Jun Ren, Yanni Shi

In this paper, we prove a stronger entanglement monogamy inequality in a 2⊗2⊗3 system's pure state |Ψ⟩ABC. Specifically, we show that the linear entropy of ρA, which is the entanglement between A and BC, is always larger than the sum of the square of concurrence between A and B and the square of concurrence of assistance between A and C. Our proof is based on direct generalizations of the qubit system's results. Our inequality is stronger than the known monogamy inequality of concurrence and shows that the entanglement of assistance always comes from the existing entanglement. However, our inequality also shows that unlike the three-qubit case, in higher dimensional systems the entanglement between A and BC cannot be completely transformed into bipartite entanglement with assistance. Through our proof, we also give some cases when the inequality reduces to an equality.