Three laser beams intertwine into a triangle. Each is linked to the other two: this is the structure of the new scheme that physicists call "fully connected three-mode squeezed vacuum." This is not just three independent light sources: inside a special crystal, triples of entangled photons are born, and the connection between them resembles a complete graph of three points, where all possible lines are drawn.
The main surprise: pairwise control of these beams is impossible. If you take any two of the three, the quantum influence of one on the other (the so-called "steering") is zero. This is like three people holding hands in a circle: individually, no one leads anyone, but together they can move in harmony. However, you can direct one beam to control the remaining two simultaneously: this is collective steering, and it is stronger the brighter the light.
The physicists also tested how robust this connection is to losses. Light passes through fogged glass: at 50% losses, collective steering of one beam over two disappears when the glass transmits only 58% of the light. In contrast, the reverse steering, two beams over one, holds up to 50.2% transparency. The quantum entanglement itself between particles does not vanish as long as any light passes through the glass. Finally, the authors added another trick: if you "subtract" one photon not from a single beam but from a mixture of two beams, a special quantum ripple (negativity of the Wigner function) appears in the remaining light, even with losses up to 33%.
as written by the authors arXiv:2607.17629
We investigate a fully-connected three-mode squeezed vacuum (FC-C3MSV) state, where all three modes are pairwise coupled through nonlinear interactions in a triangle ($$K_3$$) topology. Using the integration-within-ordered-product technique, we derive the normal product form of the squeezing operator and obtain the covariance matrix directly from the Bogoliubov transformation. Under symmetric coupling, the physical state is genuinely tripartite entangled for any nonzero squeezing, while the three Armstrong-type witnesses provide a finite-window sufficient experimental test; in the chain-type C3MSV only one of these witnesses is violated. We find that, despite two-mode entanglement, the fully-connected topology admits \emph{no} two-mode Gaussian steering ($$\mathcal{G}^{i\to j}=0$$) between any pair of physical modes; the steering resource is instead collective one-mode-versus-two steering $$\mathcal{G}^{i\to jk}$$, which is $$\theta$$-independent and grows with $$r$$. We analyze independent vacuum losses and obtain critical transmittances for steering survival: under full symmetric loss at $$r=0.5$$, one-to-two collective steering disappears at $$\eta\approx0.58$$, whereas reverse two-to-one collective steering survives down to $$\eta\approx0.502$$ and the underlying two-mode entanglement persists for all $$\eta>0$$. Finally, we revisit photon subtraction using a normalized phase-space derivation. A photon subtraction on a single physical mode does not generate Wigner negativity on another single mode, consistent with the absence of two-mode steering. Wigner negativity can instead be generated when Bob subtracts from the collective mode $$(b+c)/\sqrt{2}$$, with a loss threshold $$\eta_c\approx0.667$$ at $$r=0.5$$. These results distinguish pairwise and collective nonclassical resources in the FC-C3MSV and clarify the operational role of the complete-graph topology.
In the chain scheme, where beams are connected sequentially (like train cars), only one of the three Armstrong tests shows quantum correlation. The triangular scheme, however, lights up all three: it's like the difference between a garland where bulbs are connected in a chain and a triangle where each bulb is connected to the other two.