ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

222 13. Entanglement in Gaussian valence bond states
(1)
(2)
(3)
(4)
Alice
0
1 2
Bob
Alice
0
1 2
Bob
Alice
0
1 2
Bob
Alice
0
1 2
Bob
(9)
Figure 13.2. How a Gaussian valence bond state is created via continuousvariable
entanglement swapping. At each step, Alice attempts to teleport her
mode 0 (half of an EPR bond, depicted in yellow) to Bob, exploiting as an
entangled resource two of the three modes of the building block (denoted at
each step by 1 and 2). The curly bracket denotes homodyne detection, which
together with classical communication and conditional displacement at Bob’s
side achieves teleportation. The state will be approximately recovered in mode
2, owned by Bob. Since mode 0, at each step, is entangled with the respective
half of an EPR bond, the process swaps entanglement from the ancillary chain
of the EPR bonds to the modes in the building block. The picture has to be
followed column-wise. For ease of clarity, we depict the process as constituted
by two sequences: in the first sequence [frames (1) to (4)] modes 1 and 2 are
the two input modes of the building block (depicted in blue); in the second
sequence [frames (5) to (8)] modes 1 and 2 are respectively an input and an
output mode of the building block. As a result of the multiple entanglement
swapping [frame (9)] the chain of the output modes (depicted in red), initially
in a product state, is transformed into a translationally invariant Gaussian
valence bond state, possessing in general multipartite entanglement among all
the modes (depicted in magenta).
to the physical one, will affect the structure and entanglement properties of the
target GVBS. This link is explored in the following Section.
We note here that the Gaussian states generally constructed according to the
above procedure are ground states of harmonic Hamiltonians (a property of all
GVBS [202]). This follows as no mutual correlations are ever created between the
operators ˆqi and ˆpj for any i, j = 1, . . . , N, due to the fact that both EPR bonds
and building blocks are chosen from the beginning in standard form. The final CM
2
Bob
Alice
0
1
2
Bob
Alice
0
1
2
Bob
Alice
0
1
2
Bob
Alice
0
1
(5)
(6)
(7)
(8)