Transfer of entanglement from nonlocal photon to non-Gaussian CV states

Abstract

Continuous variable (CV) entanglement refers to the type of entanglement of quantum wave-like systems that are described by continuous variables in an inherently infinite-dimensional space. It can become a crucial resource for quantum communication, sensing and computation. We propose the mechanism of transfer of quantum entanglement (TQE) from a nonlocal photon to two initially separate single-mode squeezed vacuum (SMSV) states. The nonlocal photon is the only original quantum resource from which entanglement is transferred to CV states of a certain parity without them directly interacting with each other in a deterministic manner. Measurement induced CV parity entanglement is tuned using initial squeezing and the beam splitter (BS) parameter allowing us to estimate the probability of transfer of maximum entanglement at sufficiently high brightness to be 0.2344 . If, instead of the original SMSV states, we use those from which one photon is initially subtracted, then the heralded technique can turn the probabilistic maximum entanglement transfer protocol into a nearly deterministic one, the probability of which is >0.98. Such a perfect TQE from the nonlocal photon to a maximally parity-entangled CV state can be considered the most suitable for applications, since it preserves the trade-off between the probability and brightness of the output non-Gaussian states.

Figures (5)

• Figure 1: (a-c). Comparison of (a) DV-DV ($A_1$-$A_2$ and $A_3$-$A_4$ nodes) and (b) CV/DV-CV ($A_1$-$A_2$ and $A_3$-$A_4$ nodes) swapping via either (a) DV BSM or (b) CV BSM and (c) TQE from DV quantum resource to initially separated CV states. The red vertical arrows correspond to the separate states in the nodes $A_1$ and $A_4$ which become connected at the top. The notations DV and CV indicate the initial DV and final CV-CV entanglement and PS means photon subtraction.
• Figure 2: A realistic scheme used for the TQE from a nonlocal photon to two input SMSV states that are distant from each other and do not directly interact with each other. Each of the SMSV states interacts with the nonlocal photon on two beam splitters, with subsequent registration of coinciding photons in measuring modes by PNR detectors. Perfect TQE with maximum output CV entanglement is only possible under certain conditions. To turn the perfect TQE protocol into a nearly deterministic one, it is necessary to use CV states in equation \ref{['eq:12']} instead of SMSV states.
• Figure 3: (a,b) (a) Graphical dependencies of the success probabilities $P_{k_1=k_2}$ for the perfect $TQE$ obtained by optimization by parameter $B=B_{k_1=k_2}$on initial squeezing $S\,\,(dB)$ both with (curve $P_{0-4}$) and without (curve $P_{1-4}$) no click contribution, where subscripts $0-4$ and $1-4$ means taking into account the results of measurements $k=0,1,2,3,4$ and $k=1,2,3,4$, respectively. The optimizing values in (b) both $B_{0-4}$ and $B_{1-4}$ appear as two parallel horizontal lines due to the fact that $HRBSs$ with $B\gg1$ are required when the vacuum contribution is taken into account. In reality, the $B_{1-4}$ has a form as shown in the insert, which indicates the use of the beam splitters with larger transmissivity $T>R$.
• Figure 4: (a-h). Contour graphs of the amplitude distortion factor $b_{ij}$ (on the left side, that is in a), c), e) and g)) and success probabilities $P_{ij}$ of generation of the $CV$ entanglement with help of a nonlocal photon (on the right side, that is, the graphs in (b), (d), (f) and (h)) in dependency on the squeezing S and BS parameter $B_{ij}$, where $i$ and $j$ are the number of measured photons. Visually, the absence of the amplitude distorting factor, i.e. $b_{ij}=1$ can be accurately determined by the shades of red: the darker the red, the closer $b_{ij}$ is to $1$. Corresponding probabilities $P_{ij}$ can also be estimated by the shades of red using a graduated color bar. The dark red curves correspond to the maximum values of $P_{ij}$. There is a correlation between the $b_{ij}=1$ curves and the maximum values of $P_{ij}$. The range of values of $S$ and $B$ is chosen to ensure the condition $b_{ij}\le 1$.
• Figure 5: (a,b) (a) Plots of the TQE probabilities $P_{CV}^{(1)}$, optimized over $B=B_{CV}^{(1)}$, versus the initial squeezing $S$ when using the input CV states in equation \ref{['eq:12']}. Two dependencies correspond to cases with inclusion of the vacuum contribution $P_{0-4}$ and without it $P_{1-4}$. Including the vacuum contribution only slightly increases the overall probability $P_{CV}^{(1)}$ of maximum output CV entanglement. The optimizing values $B_{0-4}$ and $B_{0-1}$ coincide in (b), i.e. $B_{CV}^{(1)} =B_{0-4}=B_{1-4}=276.6$, which is expressed by one horizontal line.