ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
98 5. Multimode entanglement under symmetry 1 2 3 1 2 3 2' 3' Figure 5.1. “If You Cut The Head Of A Basset Hound, It Will Grow Again” (by F. Illuminati, 2001; see also [207], Chapter 1). Graphical depiction of the process of unitary localization (concentration) and delocalization (distribution) of entanglement in three-mode bisymmetric Gaussian states [GA5] (or “basset hound” states), described in the text. Initially, mode 1 is entangled (entanglement is depicted as a waving string) with both modes 2 and 3. It exists a local (with respect to the 1|(23) bipartition) symplectic operation, realized e.g. via a beam-splitter (denoted by a black thick dash), such that all the entanglement is concentrated between mode 1 and the transformed mode 2 ′ , while the other transformed mode 3 ′ decouples from the rest of the system (unitary localization). Therefore, the head of the basset hound (mode 3 ′ ) has been cut off. However, being realized through a symplectic operation (i.e. unitary on the density matrix), the process is reversible: operating on modes 2 ′ and 3 ′ with the inverse symplectic transformation, yields the original modes 2 and 3 entangled again with mode 1, without any loss of quantum correlations (unitary delocalization): the head of the basset hound is back again. once more, that such an entanglement switch is endowed with maximum (100%) efficiency, as no entanglement is lost in the conversions. This fact may have a remarkable impact in the context of quantum repeaters [41] for communications with continuous variables.
5.2. Quantification and scaling of entanglement in fully symmetric states 99 5.1.3.1. The case of the basset hound. To give an example, we can consider a bisymmetric 1 × 2 three-mode Gaussian state, 12 where the CM of the last two modes (constituting subsystem SB) is assumed in standard form, Eq. (2.54). Because of the symmetry, the local symplectic transformation responsible for entanglement concentration in this simple case is the identity on the first mode (constituting subsystem SA) and just a 50:50 beam-splitter transformation B2,3(1/2), Eq. (2.26), on the last two modes [268] (see also Sec. 9.2.1). The entire procedure of unitary localization and delocalization of entanglement [GA5] is depicted in Fig. 5.1. Interestingly, it may be referred to as “cut-off and regrowth of the head of a basset hound”, where in our example the basset hound pictorially represents a bisymmetric three-mode state. However, the breed of the dog reflects the fact that the unitary localizability is a property that extends to all 1×N [GA4] and M ×N [GA5] bisymmetric Gaussian states (in which case, the basset hound’s body would be longer and longer with increasing N). We can therefore address bisymmetric Gaussian states as basset hound states, if desired. In this canine analogy, let us take the freedom to remark that fully symmetric states of the form Eq. (2.60), as a special case, are of course bisymmetric under any bipartition of the modes; this, in brief, means that any conceivable multimode, bipartite entanglement is locally equivalent to the minimal two-mode, bipartite entanglement (consequences of this will be deeply investigated in the following). Pictorially, remaining in the context of three-mode Gaussian states, this special type of basset hound state resembles a Cerberus state, in which any one of the three heads can be cut and can be reversibly regrown. 5.2. Quantification and scaling of entanglement in fully symmetric states In this Section we will explicitly compute the block entanglement (i.e. the entanglement between different blocks of modes) for some instances of multimode Gaussian states. We will study its scaling behavior as a function of the number of modes and explore in deeper detail the localizability of the multimode entanglement. We focus our attention on fully symmetric L-mode Gaussian states (the number of modes is denoted by L in general to avoid confusion), endowed with complete permutation invariance under mode exchange, and described by a 2L × 2L CM σ β L given by Eq. (2.60). These states are trivially bisymmetric under any bipartition of the modes, so that their block entanglement is always localizable by means of local symplectic operations. Let us recall that concerning the covariances in normal forms of fully symmetric states (see Sec. 2.4.3), pure L-mode states are characterized by ν − β = ν+ β L = 1 in Eq. (2.61), which yields z1 = (L − 2)(b2 − 1) + (b 2 − 1) [L ((b 2 − 1) L + 4) − 4] 2b(L − 1) z2 = (L − 2)(b2 − 1) − (b 2 − 1) [L ((b 2 − 1) L + 4) − 4] 2b(L − 1) , . (5.13) 12 The bipartite and genuinely tripartite entanglement structure of three-mode Gaussian states will be extensively investigated in Chapter 7, based on Ref. [GA11]. The bisymmetric three-mode Gaussian states will be also reconsidered as efficient resources for 1 → 2 telecloning of coherent states in Sec. 12.3, based on Ref. [GA16].
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98 5. Multimode entanglement under symmetry<br />
1 2 3<br />
1 2 3<br />
2' 3'<br />
Figure 5.1. “If You Cut The Head Of A Basset Hound, It Will Grow<br />
Again” (by F. Illuminati, 2001; see also [207], Chapter 1). Graphical depiction<br />
of the process of unitary localization (concentration) and delocalization (distribution)<br />
of entanglement in three-mode bisymmetric Gaussian states [GA5]<br />
(or “basset hound” states), described in the text. Initially, mode 1 is entangled<br />
(entanglement is depicted as a waving string) with both modes 2 and 3.<br />
It exists a local (with respect to the 1|(23) bipartition) symplectic operation,<br />
realized e.g. via a beam-splitter (denoted by a black thick dash), such that all<br />
the entanglement is concentrated between mode 1 and the transformed mode<br />
2 ′ , while the other transformed mode 3 ′ decouples from the rest of the system<br />
(unitary localization). Therefore, the head of the basset hound (mode 3 ′ ) has<br />
been cut off. However, being realized through a symplectic operation (i.e. unitary<br />
on the density matrix), the process is reversible: operating on modes 2 ′<br />
and 3 ′ with the inverse symplectic transformation, yields the original modes 2<br />
and 3 entangled again with mode 1, without any loss of quantum correlations<br />
(unitary delocalization): the head of the basset hound is back again.<br />
once more, that such an entanglement switch is endowed with maximum (100%)<br />
efficiency, as no entanglement is lost in the conversions. This fact may have a remarkable<br />
impact in the context of quantum repeaters [41] for communications with<br />
continuous variables.
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