# Entanglement distribution and distillation

In the beginning of the protocol, Alice possesses two separable modes A and C, while Bob possesses mode B, which is separable from Alice’s modes. In the subsequent step I, Alice sends the ancillary mode C, which is neither entangled with mode A nor with mode B, to Bob. To obtain two-mode entanglement (step II), Bob mixes his modes B and C. One output mode is then discarded, while the other one turns out to be entangled with A.

Our setup for entanglement distribution by separable states: The initial three-mode Gaussian state is prepared by an independent source and is distributed between Alice and Bob. The preparation starts with a squeezed state, which interferes with a vacuum state at a balanced beam splitter. The beam splitter output A is sent directly to Alice, while the other output is superimposed with a thermal state at a second balanced beam splitter. After the state preparation, Alice possesses modes A and C, while Bob holds mode B. The separability properties of this three-mode state (ABC) are checked by a tomographic reconstruction of the full three-mode covariance matrix with the balanced homodyne detectors BHDA, BHDB, and BHDC.

For investigating the separability properties of the three-mode state (ABC), we apply the positive partial transposition criterion (PPT) to the measured state. This criterion is equivalent to finding the symplectic eigenvalues of the covariance matrix of the partially transposed state. If the smallest symplectic eigenvalue µk, in the following called a PPT value, is below 1, the state is inseparable with respect to the k|ij splitting. We named the PPT values for the different splittings after the single mode: PPTA for the (A|BC) splitting, PPTB for (B|AC), and PPTC for (C|AB). Our protocol thus requires PPTA < 1 (=inseparable) and PPTB, PPTC >1 (=separable) to verify the appropriate three-mode state for distributing entanglement by separable states.

As input states, we used a squeezed state with -1.8 and 5.1 dB noise reduction or amplification in the amplitude and phase quadrature, respectively, and an elliptical thermal state (“hot squeezed state”) with 9.6 and 10.2 dB noise amplification. The resulting three-mode covariance matrix directly leads to the PPT values PPTA= 0.89, PPTB= 1.1, and PPTC=1.07.

Thus, the measured state fulfilled the requirements for distributing entanglement via separable states. After the prepared three-mode state had been checked for its separability properties, ancilla mode C, which was separable from modes A and B, was sent to Bob. Twomode entanglement between Alice and Bob was generated by superimposing modes C and B at the balanced beam splitter BS3 with the appropriate phase, which was controlled manually. The criterion by Duan et al. resulted in 3.4 (< 4), which proved that entanglement was successfully distributed by separable states.

This work was published in Physical Review Letters.

Contact: Daniela Schulze

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