Identifying genuine entanglement of lossy noisy very large scale continuous variable Greenberger-Horne-Zeilinger state

Abstract

Genuine entanglement identification of large scale systems is crucial for quantum computation, quantum communication and quantum learning advantage. In contrast to experiments, where noisy intermediate-scale programmable photonic quantum processors have been developed, theoretically very limited results have been achieved for detecting genuine entanglement of continuous variable multipartite systems. We propose a quite general and efficient entanglement detection framework for all kinds of multipartite entanglement of continuous variable systems based on uncertainty relations and the sign matrix technique. Matrix criteria are demonstrated and can be applied to various entanglement depth and k-separability problems of multimode systems. We illustrate the genuine entanglement conditions of continuous variable Greenberger-Horne-Zeilinger states of more than a hundred million modes in a photon loss and noise environment.

Figures (4)

• Figure 1: Genuine entanglement detection of lossy noisy CV-GHZ states, where $v=N/(N+1)$. The longitudinal straight lines in the subfigures are due to the additional condition. (a) $n=10^2$ modes, with $m=6, n_0=1$. (b) $n=10^4$ modes, with $m=13, n_0=1$. (c) $n=10^6$ modes, with $m=20, n_0=500000$. (d) $n=10^8$ modes, with $m=26, n_0=1$.
• Figure 2: (a) (b) $\kappa$ in Eqs. (\ref{['wee11a']}) and (\ref{['wee11b']}) as a function of $n_{0}$ for $n=10^2$ and $n=10^5$ mode systems.
• Figure 3: (a)(b) $\kappa$ in the triseparable case as a function of $n_0$ and $n_1$ for $n=100$ mode system with $m=6$ and $n=600$ system with $m=9$, respectively. There are $n_0,n_1,n-n_0-n_1$ modes in the three parts of the tripartite partition.
• Figure 4: (a).Various necessary conditions of separabilities and producibilities for the four mode CV-GHZ state, $v=0.0001$. (b).$n=20$, lines from from left to right are the biseparable, and verious producible conditions from 19-producibility to 10-producibility, $v=0.0001$. (c).$n=50$, lines from from left to right are the biseparable, and verious producible conditions from 49-producibility to 25-producibility, $v=0.0001$. (d).$n=105$, lines from from left to right are the biseparable, and various producible conditions from 104-producibility to 53-producibility, $v=0.01$.