ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
208 12. Multiparty quantum communication with Gaussian resources To demonstrate the promiscuous sharing, one would then need to discard each one of the modes at a time, and perform standard two-user teleportation between the remaining pair of parties. The optimal fidelity for this two-user teleportation, which is achieved exactly for r1 = r2 [see Eq. (12.14)], is F opt 2:red = 3 3 + √ . (12.29) 3 + 6e−4¯r Again, one should implement the three possible configurations and take the average fidelity as figure of merit. As anticipated in Sec. 12.2.3.2, this fidelity cannot reach unity because the entanglement in the shared mixed resource remains finite, and in fact F opt 2:red saturates to 3/(3 + √ 3) ≈ 0.634 in the limit of infinite squeezing. Finding simultaneously both F opt 3 and F opt 2:red above the classical threshold F cl ≡ 1/2, Eq. (12.1), at fixed squeezing ¯r, would be a clear experimental fingerprint of the promiscuous sharing of tripartite CV entanglement. Theoretically, this is true for all ¯r > 0, as shown in Fig. 12.5. From an experimental point of view, the tripartite teleportation network has been recently implemented, and the genuine tripartite shared entanglement unambiguously demonstrated by obtaining a nonclassical teleportation fidelity (up to 0.64 ± 0.02) in all the three possible user configurations [277]. Nevertheless, a nonclassical fidelity F2:red in the teleportation exploiting any two-mode reduction was not observed. This fact can be consistently explained by taking into account experimental noise. In fact, even if the desired resource states were pure GHZ/W states, the unavoidable effects of decoherence and imperfections resulted in the experimental production of mixed states, namely of the noisy GHZ/W states discussed in Sec. 7.4.2. It is very likely that the noise was too high compared with the pumped squeezing, so that the actual produced states were still fully inseparable, but laid outside the region of promiscuous sharing (see Fig. 7.6), having no entanglement left in the two-mode reductions. However, increasing the degree of initial squeezing, and/or reducing the noise sources might be accomplished with the state-of-the-art equipment employed in the experiments of Ref. [277] (see also [226]). The conditions required for a proper test (to be followed by actual practical applications) of the promiscuous sharing of CV entanglement in symmetric three-mode Gaussian states, as detailed in Sec. 12.2.3.2, should be thus met shortly. As a final remark, let us observe that repeating the same experiment but employing T states, introduced in Sec. 7.3.2, as resources (engineerable as detailed in Sec. 10.1.2.3), would be another interesting option. In fact, in this case the expected optimal fidelity is strictly smaller than in the case of GHZ/W states, confirming the promiscuous structure in which the reduced bipartite entanglement enhances the value of the genuine tripartite one. With the same GHZ/W shared resources (but also with all symmetric and bisymmetric three-mode Gaussian states, including T states, noisy GHZ/W states and basset hound states, all introduced in Chapter 7), one may also test the power of the unitary localization of entanglement by local symplectic operations [GA5] (presented in Chapter 5), as opposed to the nonunitary localization of entanglement by measurements [248] (described in Fig. 12.4), needed for the teleportation network. Suppose that the three parties Alice, Bob and Claire share a GHZ/W state. If Bob and Claire are allowed to cooperate (non-locally), they can combine their respective modes at a 50:50 beam-splitter, as depicted in Fig. 5.1. The result is an entangled state shared by Alice and Bob, while Claire is left with an uncorrelated state. The optimal fidelity of standard teleportation from Alice to Bob with
12.2. Equivalence between entanglement and optimal teleportation fidelity 209 opt 1 0.9 0.8 0.7 0.6 0.5 0 5 10 15 20 r dB Figure 12.5. Expected success for an experimental test of the promiscuous sharing of CV tripartite entanglement in finite-squeezing GHZ/W states. Referring to the check-list in Sec. 12.2.3.2: the solid curve realizes point (i), being the optimal fidelity F opt 3 of a three-party teleportation network; the dotted curve realizes point (ii), being the optimal fidelity F opt 2:uni of two-party teleportation exploiting the two-mode pure resource obtained from a unitary localization applied on two of the modes; the dashed curve realizes point (iii), being the optimal fidelity F opt 2:red of two-party teleportation exploiting the twomode mixed resource obtained discarding a mode. All of them lie above the classical threshold F cl ≡ 0.5, providing a direct evidence of the promiscuity of entanglement sharing in the employed resources. the unitarily localized resource, reads F opt 2:uni = 1 4 cosh(4¯r) + 5 − 2 cosh(4¯r) − 1 + 1 3 −1 . (12.30) Notice that F2:uni is larger than F opt 3 , as shown in Fig. 12.5. This is true for any number N of modes, and the difference between the two fidelities — the optimal teleportation fidelity employing the unitarily-localized two-mode resource, minus the N-party optimal teleportation-network fidelity, corresponding to a two-party teleportation with nonunitarily-localized resources, Eq. (12.25) — at fixed squeezing increases with N. This confirms that the unitarily localizable entanglement of Chapter 5 is strictly stronger than the (measurement-based) localizable entanglement [248] of Fig. 12.4, as discussed in Sec. 5.1.3. This is of course not surprising, as the unitary localization generally requires a high degree of non-local control on the two subset of modes, while the localizable entanglement of Ref. [248] is defined in terms of LOCC alone. 12.2.4. Degradation of teleportation efficiency under quantum noise In Sec. 7.4.1 we have addressed the decay of three-partite entanglement (as quantified by the residual Gaussian contangle) of three-mode states in the presence of losses and thermal noise. We aim now at relating such an ‘abstract’ analysis to precise operational statements, by investigating the decay of the optimal teleportation
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12.2. Equivalence between entanglement and optimal teleportation fidelity 209<br />
opt<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0 5 10 15 20<br />
r dB<br />
Figure 12.5. Expected success for an experimental test of the promiscuous<br />
sharing of CV tripartite entanglement in finite-squeezing GHZ/W states.<br />
Referring to the check-list in Sec. 12.2.3.2: the solid curve realizes point (i),<br />
being the optimal fidelity F opt<br />
3 of a three-party teleportation network; the<br />
dotted curve realizes point (ii), being the optimal fidelity F opt<br />
2:uni of two-party<br />
teleportation exploiting the two-mode pure resource obtained from a unitary<br />
localization applied on two of the modes; the dashed curve realizes point (iii),<br />
being the optimal fidelity F opt<br />
2:red of two-party teleportation exploiting the twomode<br />
mixed resource obtained discarding a mode. All of them lie above the<br />
classical threshold F cl ≡ 0.5, providing a direct evidence of the promiscuity of<br />
entanglement sharing in the employed resources.<br />
the unitarily localized resource, reads<br />
F opt<br />
2:uni =<br />
<br />
1<br />
<br />
4 cosh(4¯r) + 5 − 2 cosh(4¯r) − 1 + 1<br />
3<br />
−1 . (12.30)<br />
Notice that F2:uni is larger than F opt<br />
3 , as shown in Fig. 12.5. This is true for any<br />
number N of modes, and the difference between the two fidelities — the optimal<br />
teleportation fidelity employing the unitarily-localized two-mode resource, minus<br />
the N-party optimal teleportation-network fidelity, corresponding to a two-party<br />
teleportation with nonunitarily-localized resources, Eq. (12.25) — at fixed squeezing<br />
increases with N. This confirms that the unitarily localizable entanglement of<br />
Chapter 5 is strictly stronger than the (measurement-based) localizable entanglement<br />
[248] of Fig. 12.4, as discussed in Sec. 5.1.3. This is of course not surprising,<br />
as the unitary localization generally requires a high degree of non-local control on<br />
the two subset of modes, while the localizable entanglement of Ref. [248] is defined<br />
in terms of LOCC alone.<br />
12.2.4. Degradation of teleportation efficiency under quantum noise<br />
In Sec. 7.4.1 we have addressed the decay of three-partite entanglement (as quantified<br />
by the residual Gaussian contangle) of three-mode states in the presence of<br />
losses and thermal noise. We aim now at relating such an ‘abstract’ analysis to precise<br />
operational statements, by investigating the decay of the optimal teleportation
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