Tripartite entanglement

Despite the great importance of entanglement, a necessary and sufficient condition for the entanglement of a given state is only known for two-qubit systems [46]. The determina-tion of entanglement for multipartite states, however, is an open quesdetermina-tion. For three qubits it was shown in Ref. [47] that two different kinds of genuine multipartite entanglement (i.e. each party is entangled with each other party) can occur. Namely, each tripartite entangled state can (with nonzero probability) be converted² by LOCC [48] to either one of two standard forms, the GHZ state, |GHZi = ^√1

2(|↑↑↑i+|↓↓↓i) [49], or the W state,

|Wi = ^√1

3(|↑↑↓i+|↑↓↑i+|↓↑↑i), which are mutually unrelated under LOCC. For GHZ-like entanglement, a measure was invented [50], the3-tangle τ. It allows for a reliable distinction between the two classes of entanglement, as it is zero for all W-type states (and all separable states, of course), whereas it is greater than zero for all states in the GHZ class. An expression forτ in terms of the coefficients of the state in the standard basis is given in appendix D.1.

A tool for detection of any kind of genuine tripartite entanglement for arbitrary states is not at hand; however, if some knowledge about the state under investigation is provided, entanglement witnesses (EW) can be used [51, 52]. These are observables with a positive ex-pectation value for all (bi-)separable states (in generaln−1 partite entangled states), whereas a negative expectation value indicates the presence of tripartite (n-partite) entanglement³. The common way to construct an EW for a state|ψi is

W =α1l− |ψihψ|, (5.1)

whereα is the maximal squared overlap of |ψiwith any biseparable or fully separable state.

Determination of α is in general complicated⁴, but we can use the proximity of the states under investigation (as described in 3.3.2 and 3.4) to W and GHZ states, respectively, to make use of known values forα. In order to measure EWs, they must be decomposed into a sum of local measurements. This as well is a demanding task and we will again refer to previous work done on this topic [55, 56, 57].

5.1.1 Entanglement of state |E₃i

As pointed out in 3.4.1,|E₃i is for zero energy bias close to|GHZi= ^√1

2(|↑↑↑i+|↓↓↓i). For constructing a GHZ witness adapted to|E₃i, the maximal squared overlap of|E₃i with non-GHZ entangled states is required, though not known. We therefore choose an EW suitable for detecting the state|GHZifor whichαis known (α= 3/4) and thus make use of the proximity of|E₃i to|GHZi. However, instead of using the EW [58]

Wf_GHZ = 3

41l− |GHZihGHZ|, (5.2)

2No operational criterion for the existence of such a transformation between two given states is known.

3The object of study here is tripartite entanglement. However, the concept of entanglement witnesses applies to multipartite entanglement as well.

4Since one has to minimize over all product states, i.e. over a convex hull of states, numerical calculations involving the theory of convex optimization are commonly used [52, 53, 54].

5.1Tripartite entanglement 41

Figure 5.1: 3-tangle and expectation value of the GHZ witness W_GHZ (explicit form in the text) for the state |E₃i. As can be seen, both quantities indicate a strong GHZ-like entanglement around²= 0.

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

Figure 5.2: 3-tangle and expectation value of the Bell operator ˆM_GHZ (explicit form in the text) for the state |E₃i. The high 3-tangle coincides with a significant violation of the Bell inequality.

we apply an EW introduced in Ref. [57], which can be measured with two collective local measurement settings,

The two local settings here are the© ˆ these, all correlators appearing in (5.3) can be computed. In contrast, the EWWf_GHZ requires four measurement settings (see table 5.1).

The 3-tangle τ(|E₃i) and the expectation value hE₃|W_GHZ|E₃i for varying energy bias are shown in figure 5.1. Both quantities indicate a strong (cp. limiting case: τ_max=τ(|GHZi) = 1,hW_GHZi_min =hGHZ|W_GHZ|GHZi=−3/4) tripartite entanglement of GHZ type in a narrow range around zero energy bias (here, only|E₃iis plotted,|E₄ishows the same behavior). The range of negative expectation value forW_GHZis even a little smaller than the range of non-zero 3-tangle. This reflects the fact that EWs need to be adapted to the state under investigation and can take positive expectation value even if the associated entanglement is present. In section 5.2, we will comment on the violation of Bell inequality as displayed in figure 5.2.

5.1.2 Entanglement of state |E2i

More GHZ-like states can be found among the eigenstates. In figure 5.3, the 3-tangleτ(|E₂i) and the expectation values for two EWshE₂|W⁽¹⁾

GHZ|E₂iandhE₂|W⁽²⁾

GHZ|E₂ifor varying energy bias are displayed. We obtain a strong (limiting case: hW⁽¹⁾

GHZi_min = hW⁽²⁾

GHZi_min = −1/4) entanglement for a finite energy bias ² ≈ ±2.6∆ and a high residual entanglement in the range between these two maxima.

42 5 Entanglement properties

Figure 5.3: 3-tangle and expectation value of the GHZ witnessesW_GHZ⁽¹⁾ and W_GHZ⁽²⁾ for the state|E₂i. For finite energy bias ²≈ ±2.6∆, we find a peaking 3-tangle as well as nega-tive expectation value for the two GHZ wit-nesses, indicating entanglement of the GHZ class. Moreover, the entanglement is more robust to detuning of the energy bias com-pared to the situation for|E₃i.

-8 -6 -4 -2 0 2 4 6 8

Figure 5.4: 3-tangle and expectation value of the Bell operator ˆM_GHZfor the state|E₂i.

A significant violation of the corresponding Bell inequality over a relatively large range of ²can be observed.

Left maximum at ²≈ −2.6∆

The explicit form of the state constituting the left maximum is

|E⁽¹⁾₂ i=−α©

Thus, this state is close to the state

|GHZi= 1

onto the state|GHZi has the form Uˆ_E⁽¹⁾₂ =e^iπ/4

nRˆ_x(−π/2) ˆR_z(−π) ˆR_y(π/2) o_⊗3

. (5.7)

The EW is chosen such as to detect tripartite entanglement in the proximity of|GHZi[56], W⁽¹⁾

GHZ = 3

41l− |GHZihGHZ|= 3

41l−Uˆ_E⁽¹⁾₂ ^†|GHZihGHZ|Uˆ_E⁽¹⁾₂ . (5.8) For an optimal decomposition ofW⁽¹⁾

GHZ refer to table 5.1.

5.1Tripartite entanglement 43 Right maximum at ²≈2.6∆

For the state in the right maximum we find

|E₂⁽²⁾i=α©

|↓↓↑i+|↓↑↓i+|↑↓↓i − |↑↑↑iª

−β©

|↑↑↓i+|↑↓↑i+|↓↑↑i − |↓↓↓iª

(5.9) withα andβ as above. |E₂⁽²⁾i is the totally flipped counterpart to|E₂⁽¹⁾i and therefore shows the same behavior after being flipped back by ˆR_x(−π)^⊗3,

Uˆ_E⁽²⁾₂ |E₂⁽²⁾i ≈ |GHZi with Uˆ_E⁽²⁾₂ = ˆU_E⁽¹⁾₂ Rˆ_x(−π)^⊗3 . (5.10) The corresponding EW looks like

W⁽²⁾

GHZ= ˆR_x(π)^⊗3W⁽¹⁾

GHZRˆ_x(−π)^⊗3. (5.11)

The optimal decomposition ofW_GHZ⁽²⁾ can again be looked up in table 5.1.

Whereas the behavior of the highest excited states|E₃iand |E₄ifor vanishing energy bias as described above was expected, we make the surprising observation –referring to figure 5.3–

of the existence of GHZ entangled states also in the parameter regime of finite energy bias.

Moreover, the entanglement shown by|E₂i is more stable to deviations from the optimal en-ergy bias than the one of|E₃i, overcoming a major drawback caused by the antiferromagnetic nature of the coupling.

5.1.3 Entanglement in the degenerate subspaces

-8 -6 -4 -2 0 2 4 6 8

Figure 5.5: 3-tangle and expectation value of the W witness W_W (explicit form in the text) for the state |ψ_max^L i. The vanishing 3-tangle excludes entanglement of the GHZ type, whereas the negative expectation value of the W witness indicates a W type entan-glement.

Figure 5.6: 3-tangle and expectation value of the Bell operator ˆM_W (explicit form in the text) for the state |ψ_max^L i. The maxi-mal violation of the Bell inequality for the W-equivalent state is not as high as for the GHZ-equivalent states above, however, the violation persists over a large range of ².

In figure 5.5, the 3-tangle and the expectation value of the EWhψ_max^L |W_W|ψ_max^L ifor the max-imally entangled superposition|ψ^L_maxiin the low-energy subspace is displayed (corresponding behavior for|ψ^H_maxi—not shown separately).

44 5 Entanglement properties

Because we know that|ψ^L_maxi(=|E₅i) is equivalent under LOCC to the W state (see 3.3.2), we can construct an EW making use of the known maximal overlap (α = 2/3 [58]) between the W state (and therefore the state|ψ_max^L i) and biseparable states,

W_W = 2

31l− |ψ^L_maxihψ_max^L |. (5.12) Its expectation value is positive on biseparable and fully separable states. It thus detects genuine tripartite entanglement in general, without distinguishing between entanglement of the W and the GHZ class. However, in connection with the 3-tangle, a distinction can be achieved, stating an entanglement of the W type in a large range for².

Reviewing the results of this section, we were able to find states with either kind of genuine tripartite entanglement, GHZ type as well as W type.

EW Local decomposition Wf_GHZ = ¹₈£

5·1l^⊗3−2 ˆσ_x^⊗3−σˆ_zσˆ_z1l−σˆ_z1l ˆσ_z−1l ˆσ_zσˆ_z+¹₂(ˆσ_x+ ˆσ_y)^⊗3+¹₂(ˆσ_x−ˆσ_y)^⊗3¤ W_GHZ = ⁷₄1l−σˆ^⊗3_x −¹₂£

ˆ

σ_zσˆ_z1l + ˆσ_z1l ˆσ_z+ 1l ˆσ_zσˆ_z¤ W⁽¹⁾

GHZ = ₁₆¹ £

10·1l^⊗3+ 4 ˆσ^⊗3_z −2(ˆσ_yˆσ_y1l + ˆσ_y1l ˆσ_y+ 1l ˆσ_yσˆ_y)−(ˆσ_z+ ˆσ_x)^⊗3−(ˆσ_z−σˆ_x)^⊗3¤ W⁽²⁾

GHZ = ₁₆¹ £

10·1l^⊗3−4 ˆσ^⊗3_z −2(ˆσ_yˆσ_y1l + ˆσ_y1l ˆσ_y+ 1l ˆσ_yσˆ_y) + (ˆσ_z−σˆ_x)^⊗3+ (ˆσ_z+ ˆσ_x)^⊗3¤ W_W = ₂₄¹ £

17·1l^⊗3−7 ˆσ^⊗3_x −3(ˆσ_x1l 1l + 1l ˆσ_x1l + 1l 1l ˆσ_x) + 5(ˆσ_xσˆ_x1l + ˆσ_x1l ˆσ_x+ 1l ˆσ_xˆσ_x)−

−(1l−σˆ_x+ ˆσ_z)⊗(1l−σˆ_x−^√₂³σˆ_y−¹₂σˆ_z)⊗(1l−σˆ_x+^√₂³σˆ_y− ¹₂σˆ_z)−

−(1l−σˆ_x−σˆ_z)⊗(1l−σˆ_x+^√₂³σˆ_y+¹₂σˆ_z)⊗(1l−σˆ_x−^√₂³σˆ_y+ ¹₂σˆ_z)−

−(1l−σˆ_x+ ˆσ_y)⊗(1l−ˆσ_x− ¹₂σˆ_y+^√₂³σˆ_z)⊗(1l−σˆ_x−¹₂ˆσ_y−^√₂³σˆ_z)−

−(1l−σˆ_x−σˆ_y)⊗(1l−ˆσ_x+ ¹₂σˆ_y−^√₂³σˆ_z)⊗(1l−σˆ_x+¹₂ˆσ_y+^√₂³σˆ_z)¤ Table 5.1: Local decomposition of the entanglement witnesses used above. The decom-position for Wf_GHZ (shown to be optimal in [58]) requires four collective measurement settings in contrast to the two settings needed forW_GHZ [57]. The optimal decomposi-tion forW_GHZ⁽¹⁾ can be found in Ref. [56] (five settings), the one forW_GHZ⁽²⁾ (five settings) was computed by rotating the individual Pauli operators occurring in W_GHZ⁽¹⁾ accord-ing to ˆR_x(π), W_GHZ⁽²⁾ = ˆR_x(π)^⊗3W_GHZ⁽¹⁾ Rˆ_x(−π)^⊗3. The decomposition for W_W (five settings) was obtained similarly from the optimized decompositionW_W⁽¹⁾ (five settings) derived in Ref. [55],W_W = ˆU^L†W_W⁽¹⁾Uˆ^L.

Im Dokument Efficient creation of multipartite entanglement for superconducting quantum computers (Seite 46-50)