ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

198 12. Multiparty quantum communication with Gaussian resources
mom-sqz r 1
noise n 1
BS
50:50
pos-sqz r 2
noise n 2
Figure 12.1. Optical generation of two-mode symmetric mixed Gaussian
states, by superimposing two independently squeezed single-mode noisy beams
at a 50:50 beam-splitter. The output states can be employed as resources
for CV teleportation of unknown coherent states. For r1 = r2 ≡ r and
n1 = n2 = 1 (meaning absence of noise), the output states reduce to those of
Fig. 9.1.
12.2.1. Optimal fidelity of two-party teleportation and bipartite entanglement
The two-user CV teleportation protocol [39] would require, to achieve unit fidelity,
the sharing of an ideal (unnormalizable) EPR resource state [73], i.e. the simultaneous
eigenstate of relative position and total momentum of a two-mode radiation
field. An arbitrarily good approximation of the EPR state, as we know, is
represented by two-mode squeezed Gaussian states of Eq. (2.22) with squeezing
parameter r → ∞.
As remarked in Sec. 9.2, a two-mode squeezed state can be, in principle, produced
by mixing a momentum-squeezed state and a position-squeezed state, with
squeezing parameters r1 and r2 respectively, through a 50:50 ideal (lossless) beamsplitter.
In practice, due to experimental imperfections and unavoidable thermal
noise the two initial squeezed states will be mixed. To perform a realistic analysis,
we must then consider two thermal squeezed single-mode states, 30 described by the
following quadrature operators in Heisenberg picture
ˆq sq
1 = √ n1e r1 ˆq 0 1 , ˆp sq
1 = √ n1e −r1 ˆp 0 1 , (12.5)
ˆq sq
2 = √ n2e −r2 0
ˆq 2 , ˆp sq
2 = √ n2e r2 0
ˆp 2 , (12.6)
where the suffix “0” refers to the vacuum. The action of an ideal (phase-free) beamsplitter
operation ˆ Bi,j(θ) on a pair of modes i and j, corresponding to a phase-space
rotation of an angle θ, is defined by Eq. (2.25).
When applied to the two modes of Eqs. (12.5,12.6), the beam-splitter entangling
operation (θ = π/4) produces a symmetric mixed state [131], depending on the
squeezings r1,2 and on the thermal noises n1,2, as depicted in Fig. 12.1. The CM
30 Any losses due to imperfect optical elements and/or to the fibre or open-air propagation
of the beams can be embedded into the initial single-mode noise factors.