ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
230 13. Entanglement in Gaussian valence bond states γγγγ 0000 1 γγγγ 0000 2 γγγγ 0000 3 (a) (b) r -r r 1,3 50:50 r 1,2 Figure 13.7. Optical production of bisymmetric three-mode Gaussian states, used as building blocks for the valence bond construction. (a) Three initial vacuum modes are entangled through two sequential twin-beam boxes, the first (parametrized by a squeezing degree r1,3) acting on modes 1 and 3, and the second (parametrized by a squeezing degree r1,2) acting on the transformed mode 1 and mode 2. The output is a pure three-mode Gaussian state whose CM is equivalent, up to local unitary operations, to the standard form given in Eq. (13.1). (b) Detail of the entangling twin-beam transformation. One input mode is squeezed in a quadrature, say momentum, of a degree r (this transformation is denoted by stretching arrows →| |←); the other input mode is squeezed in the orthogonal quadrature, say position, of the same amount (this anti-squeezing transformation is denoted by the corresponding rotated symbol). Then the two squeezed modes are combined at a 50:50 beam-splitter. If the input modes are both in the vacuum state, the output is a pure two-mode squeezed Gaussian state (twin-beam state), with entanglement proportional to the degree of squeezing r, as in Fig. 9.1. (see also Ref. [238]), to realize bisymmetric three-mode Gaussian states of the form Eq. (13.1). As shown in Fig. 13.7, one can start from three vacuum modes and first apply a two-mode squeezing operation (twin-beam box) to modes 1 and 3, characterized by a squeezing r1,3, then apply another twin-beam operation to modes 1 and 2, parametrized by r1,2. The symplectic operation Ti,j(ri,j) describing the twin-beam transformation (two-mode squeezing plus balanced beam-splitter) acting on modes i and j is given by Eq. (2.28) and pictorially represented in Fig. 13.7(b). The output of this optical network is a pure, bisymmetric, three-mode Gaussian state with a CM BS γ B = T1,2(r1,2)T1,3(r1,3)T T 1,3(r1,3)T T 1,2(r1,2) , γγγγ s γγγγ s γγγγ x
of the form Eq. (13.1), with 13.3. Optical implementation of Gaussian valence bond states 231 γs = 1 diag 2 e−2r1,2 e 4r1,2 cosh (2r1,3) + 1 , 1 2 e−2r1,2 4r1,2 cosh (2r1,3) + e , γx εss = = diag {cosh (2r1,3) , cosh (2r1,3)}, 1 diag 2 e−2r1,2 e 4r1,2 cosh (2r1,3) − 1 , 1 2 e−2r1,2 εsx = 4r1,2 cosh (2r1,3) − e , √ r1,2 diag 2e cosh (r1,3) sinh (r1,3) , − √ 2e −r1,2 cosh (r1,3) sinh (r1,3) . (13.7) By means of local symplectic operations (unitary on the Hilbert space), like additional single-mode squeezings, the CM γ B can be brought in the standard form of Eq. (13.2), from which one has √ x + 1 r1,3 = arccos √ , 2 √−x 3 + 2x2 + 4s2x − x r1,2 = arccos 4x + 1 2 . (13.8) For a given r1,3 (i.e. at fixed x), the quantity r1,2 is a monotonically increasing function of the standard form covariance s, so this squeezing parameter which enters in the production of the building block (see Fig. 13.7) directly regulates the entanglement distribution in the target GVBS, as discussed in Sec. 13.2. The only unfeasible part of the scheme seems constituted by the ancillary EPR pairs. But are infinitely entangled bonds truly necessary? In Ref. [GA13] we have considered the possibility of using a Γ in given by the direct sum of two-mode squeezed states of Eq. (2.22), but with finite r. Repeating the analysis of Sec. 13.2 to investigate the entanglement properties of the resulting GVBS with finitely entangled bonds, it is found that, at fixed (x, s), the entanglement in the various partitions is degraded as r decreases, as somehow expected. Crucially, this does not affect the connection between input entanglement and output correlation length. Numerical investigations show that, while the thresholds sk for the onset of entanglement between distant pairs are quantitatively modified — namely, a bigger s is required at a given x to compensate the less entangled bonds — the overall structure stays untouched. As an example, Fig. 13.5(b) depicts the entanglement distribution in six-mode GVBS obtained from finitely entangled bonds with r = 1.1, corresponding to ≈ 6.6 dB of squeezing (an achievable value [224]). This ensures that the possibility of engineering the entanglement structure in GVBS via the properties of the building block is robust against imperfect resources, definitely meaning that the presented scheme is feasible. Alternatively, one could from the beginning observe that the triples consisting of two projective measurements and one EPR pair can be replaced by a single projection onto the EPR state, applied at each site i between the input mode 2 of the building block and the consecutive input mode 1 of the building block of site i + 1 [202]. The output of all the homodyne measurements would conditionally realize the target GVBS.
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230 13. Entanglement in Gaussian valence bond states<br />
γγγγ 0000 1<br />
γγγγ 0000 2<br />
γγγγ 0000 3<br />
(a)<br />
(b)<br />
r<br />
-r<br />
r 1,3<br />
50:50<br />
r 1,2<br />
Figure 13.7. Optical production of bisymmetric three-mode Gaussian states,<br />
used as building blocks for the valence bond construction. (a) Three initial<br />
vacuum modes are entangled through two sequential twin-beam boxes, the first<br />
(parametrized by a squeezing degree r1,3) acting on modes 1 and 3, and the<br />
second (parametrized by a squeezing degree r1,2) acting on the transformed<br />
mode 1 and mode 2. The output is a pure three-mode Gaussian state whose<br />
CM is equivalent, up to local unitary operations, to the standard form given<br />
in Eq. (13.1). (b) Detail of the entangling twin-beam transformation. One<br />
input mode is squeezed in a quadrature, say momentum, of a degree r (this<br />
transformation is denoted by stretching arrows →| |←); the other input mode<br />
is squeezed in the orthogonal quadrature, say position, of the same amount<br />
(this anti-squeezing transformation is denoted by the corresponding rotated<br />
symbol). Then the two squeezed modes are combined at a 50:50 beam-splitter.<br />
If the input modes are both in the vacuum state, the output is a pure two-mode<br />
squeezed Gaussian state (twin-beam state), with entanglement proportional to<br />
the degree of squeezing r, as in Fig. 9.1.<br />
(see also Ref. [238]), to realize bisymmetric three-mode Gaussian states of the<br />
form Eq. (13.1). As shown in Fig. 13.7, one can start from three vacuum modes<br />
and first apply a two-mode squeezing operation (twin-beam box) to modes 1 and 3,<br />
characterized by a squeezing r1,3, then apply another twin-beam operation to modes<br />
1 and 2, parametrized by r1,2. The symplectic operation Ti,j(ri,j) describing the<br />
twin-beam transformation (two-mode squeezing plus balanced beam-splitter) acting<br />
on modes i and j is given by Eq. (2.28) and pictorially represented in Fig. 13.7(b).<br />
The output of this optical network is a pure, bisymmetric, three-mode Gaussian<br />
state with a CM<br />
BS<br />
γ B = T1,2(r1,2)T1,3(r1,3)T T 1,3(r1,3)T T 1,2(r1,2) ,<br />
γγγγ s<br />
γγγγ s<br />
γγγγ x
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