Figure 4.3: Depiction of the entanglement distribution protocol. Solid lines: The first two steps of the protocol considered here. Dotted lines: The final step of the protocol, described in [153], that localises the entanglement to two-mode entanglement. Initially, modesAand C are orthogonally squeezed states and modeBis the vacuum, then correlated displacements are added to the modes. Modes A and C are mixed on a beamsplitter resulting in a three-mode state that is entangled across the A|BC splitting but separable otherwise. For the full protocol, modes B and C are mixed on a beamsplitter to localise the entanglement between modesA andB.
4.2
Entanglement distribution by separable states
The protocol described in the previous section is based on a similar principle as the protocol for entanglement distribution by separable states described in [151, 153], and implemented in [167]. Here, I briefly describe the first two steps of the protocol in order to draw comparisons with the scheme in the previous section and further highlight the entangling power of a beamsplitter. Entanglement distribution by separable states was first proposed for qubits in [51] and for CV in [151, 153]. It was experimentally demonstrated simultaneously for qubits [72] and for CV [167, 212].
4.2.1 Theoretical Scheme
Whereas in the previous scheme the initial state consisted of two modes, in this protocol the initial state is a three-mode state. The protocol is depicted in Fig. 4.3. Initially, mode
A is in a position-squeezed state, mode C is a momentum-squeezed state and mode B is the vacuum. This means 2hxˆ2ai= 2hpˆ2Ci= exp(−2r) and 2hpˆ2ai= 2hxˆ2Ci= exp(2r) where
r >0 is the squeezing parameter. These modes are then displaced by ˆ xA→xˆA+ ¯x, pˆC →pˆC−p,¯ xˆB→xˆB+ √ 2¯x, pˆB→pˆB+ √ 2¯p, (4.20) where ¯x and ¯p are uncorrelated classical displacements following Gaussian distributions with zero mean and varianceshx¯2i=hp¯2i=σ2≡(e2r−1)/2. Note that the displacements add enough noise to destroy the initial squeezing present in modesAandC, meaning that each mode is individually classical. Next, modes A and C are mixed on a balanced
beamsplitter to create a new correlated three-mode state. Since modes A and C are classical and uncorrelated, the two output modes must be separable. The covariance matrix of the state after the beamsplitter is
γABC = (cosh(2r) +σ2)1 2σ2σz (sinh(2r)−σ2)σz 2σ2σ z (1 + 4σ2)1 −2σ21 (sinh(2r)−σ2)σz −2σ21 (cosh(2r) +σ2)1 . (4.21)
The separability properties of this state can be checked by the PPT criterion. The state is clearly separable across theB|AC bipartition since it was created by LOCC across this splitting. The state is entangled across the A|BC bipartition for r >0 and σ2 ≥0. The state is separable across theC|AB bipartition forr >0 as long as 2σ2 ≥e2r−1, which is why it was chosen as 2σ2≡(e2r−1) in this case. Therefore this scheme has demonstrated
the effect where two uncorrelated classical states mixed together on a beamsplitter result in entanglement, as long as they are suitably correlated to a third mode. This is a similar effect to that discussed in the previous section and further demonstrates the entangling power of a beamsplitter.
4.2.2 Experimental Implementation
The full entanglement distribution scheme in this form was implemented experimentally in [167], and the first two steps were given greater focus in [50]. A schematic of the protocol implemented in [50] is shown by the blue circles and ellipses of Fig. 4.2. The steps of the experiment were carried out as described in the previous section. Measurement of the Stokes operators gave the covariance matrix of the final state as
γABC = 20.90 1.10 5.17 −8.59 −7.80 −1.68 1.10 25.31 −5.04 −6.76 1.00 14.64 5.17 −5.04 11.87 −0.45 4.95 4.49 −8.59 −6.76 −0.45 18.88 −8.61 6.04 −7.80 1.00 4.95 −8.61 20.68 0.80 −1.68 14.64 4.49 6.04 0.80 24.65 . (4.22)
The required separability properties can be checked using the PPT criteria. The relevant eigenvalues needed to find the three-mode separability properties are shown in Table 4.2. This shows that after the beamsplitter, the state is entangled across theA|BC bipar- tition but separable across the C|AB and B|AC bipartitions, as predicted by the theory. Therefore the experiment has shown that entanglement can be generated by mixing two uncorrelated classical states on a beamsplitter, as long as they are suitably correlated to a third mode. Note that since there is only entanglement across one bipartition, it is impossible for there to be two-mode entanglement. For example if modes A and C were entangled, there would have to be entanglement across both the A|BC and C|AB split- tings. This shows that the generated entanglement is genuine three-mode entanglement not caused by a nonclassical state entering one of the ports of the beamsplitter. Similarly to the previous protocol, there must be some global nonclassicality to allow entanglement to be created. In this case it is global squeezing, which is quantified by the eigenvalue min[(eig(γABC)] = 0.609±0.003<1.
Due to the specific entanglement properties of the state with covariance matrix (4.22), the state can be used for entanglement distribution by separable states. As was shown in