Four-concurrence in the transverse $XY$ spin-1/2 chain

We analyze the entanglement measure $C_4$ for specific mixed states in general and for the ground state of the transverse $XY$ spin-1/2 chain. We find that its factorizing property for pure states does not easily extend to mixed states. For cases where the density matrix is a tensor product, $C_4$ is definitely upper bounded by the product of the corresponding concurrences. In transverse $XY$ chains, we find that for large distances this condition goes conform with the working hypotheses of a factorizing property of density matrices in this limit. Additionally, we find that $C_4$ together with the genuine multipartite negativity makes it impossible to decide—at the present state of knowledge—which type of entanglement prevails in the system. In particular, this is true for all entanglement measures that detect SL-invariant genuine $n$-partite entanglement for different $n$. Further measures of SL-invariant genuine multipartite entanglement have to be considered here. $C_4$ is, however, of the same order of magnitude as the genuine multipartite negativity in Phys. Rev. B 89, 134101 (2014) and shows the same functional behavior, which we read as a hint towards the Greenberger-Horne-Zeilinger (GHZ) type of entanglement. Furthermore, we observe an interesting feature in the $C_4$ values that resembles a destructive interference with the underlying concurrence.

Figure 1

Here, the value of $C_4$ for $\rho =p|{\text{GHZ}}_4\rangle \langle {\text{GHZ}}_4|+(1-p)|\mathrm{Bell}\otimes \mathrm{Bell}\rangle \langle \mathrm{Bell}\otimes \mathrm{Bell}|$ is shown for $|{\text{GHZ}}_4\rangle =\left(\right|1111\rangle +|0000\rangle )/\sqrt{2}$ and $|\mathrm{Bell}\rangle$ is one of the two states $\left(\right|11\rangle +|00\rangle )/\sqrt{2}$ (red dash-dotted curve) and $\left(\right|11\rangle +i|00\rangle )/\sqrt{2}$ (black solid line).

Figure 2

$C_4$ and $C_{2;1,2}{C}_{2;3,4}$ are shown for $\rho \left(p\right)=p|{\mathrm{\Phi }}^{+}\otimes {\mathrm{\Phi }}^{+}\rangle \langle {\mathrm{\Phi }}^{+}\otimes {\mathrm{\Phi }}^{+}|+(1-p)|{\mathrm{W}}_4\rangle \langle {\mathrm{W}}_4|$. Here, $C_{2;i,j}$ is the concurrence of the reduced density matrix of the sites $i$ and $j$. It is clearly seen that the factorizing property of $C_4$ into the concurrences for pure states does not mean that it factorizes also for mixed states. Whereas $C_4$ linearly decreases, $C_{2;1,2}{C}_{2;3,4}$ has two distinct zeros at $p_1\sim 0.1716$ and $p_2=1/3$. Even if the square root is taken from the concurrences, this would mean only to replace the red dashed curves by a corresponding piecewise linear curve.

Figure 3

$C_4$ and $C_{2;1,2}{C}_{2;3,4}$ are shown for $\rho \left(p\right)={\rho }_2\left(p\right)\otimes {\rho }_2\left(p\right)$, where ${\rho }_2\left(p\right)=p|{\mathrm{Bell}}_1\rangle \langle {\mathrm{Bell}}_1|+(1-p)|{\mathrm{Bell}}_2\rangle \langle {\mathrm{Bell}}_2|$. Here, $C_{2;i,j}$ is the concurrence of the reduced density matrix of the sites $i$ and $j$ and $|{\mathrm{Bell}}_i\rangle$ can be arbitrary Bell states for $i=1,2$. It is clearly seen that the factorizing property of $C_4$ into the concurrences for pure states for tensor products of states transports into a $C_4$ being upper bounded by $C_{2;1,2}{C}_{2;3,4}$.

Figure 4

Here it is seen how the admixture of an additional Bell state influences the result; $C_4$ becomes precisely zero. The density matrix is taken to be $\rho =p|{\mathrm{GHZ}}_4\rangle \langle {\mathrm{GHZ}}_4|+(1-p)\left(q\right|{\mathrm{\Phi }}^{-}\otimes {\mathrm{\Phi }}^{-}\rangle \langle {\mathrm{\Phi }}^{-}\otimes {\mathrm{\Phi }}^{-}|+(1-q)|{\mathrm{\Psi }}^{-}\otimes {\mathrm{\Psi }}^{-}\rangle \langle {\mathrm{\Psi }}^{-}\otimes {\mathrm{\Psi }}^{-}\left|\right)$, with $|{\mathrm{\Phi }}^{\pm }\rangle =\left(\right|11\rangle \pm |00\rangle )/\sqrt{2}$, and $|{\mathrm{\Psi }}^{\pm }\rangle =\left(\right|10\rangle \pm |01\rangle )/\sqrt{2}$.

Figure 5

Here the value of $C_4$ is shown for the rank-3 density matrix $\rho =p|{\mathrm{GHZ}}_4^{\prime }\rangle \langle {\mathrm{GHZ}}_4^{\prime }|+(1-p)\left(q\right|\mathrm{Bell}\otimes \mathrm{Bell}\rangle \langle \mathrm{Bell}\otimes \mathrm{Bell}|+(1-q)|{\mathrm{W}}_4\rangle \langle {\mathrm{W}}_4\left|\right)$ with $|{\mathrm{GHZ}}_4^{\prime }\rangle =\left(\right|0100\rangle +|1011\rangle )/\sqrt{2}$, $|\mathrm{Bell}\otimes \mathrm{Bell}\rangle ={\mathrm{\Phi }}^{+}\otimes {\mathrm{\Psi }}^{-}$ with ${\mathrm{\Phi }}^{+}$ and ${\mathrm{\Psi }}^{-}$ as defined in Fig. 4, and $|{\mathrm{W}}_4\rangle =\left(\right|1000+|0100\rangle +|0010\rangle +|0001\rangle )/2\rangle$.

Figure 6

The value of $C_4$ for $\rho =p|{\mathrm{W}}_4\rangle \langle {\mathrm{W}}_4|+(1-p)\left(q\right|{\mathrm{Bell}}_1\otimes {\mathrm{Bell}}_1\rangle \langle {\mathrm{Bell}}_1\otimes {\mathrm{Bell}}_1|+(1-q)|{\mathrm{Bell}}_2\otimes {\mathrm{Bell}}_2\rangle \langle {\mathrm{Bell}}_2\otimes {\mathrm{Bell}}_2\left|\right)$ is shown for $|{\mathrm{W}}_4\rangle =\left(\right|1000\rangle +|0100\rangle +|0010\rangle +|0001\rangle )/\sqrt{2}$ and $|{\mathrm{Bell}}_i{\rangle =\left(\right|11\rangle +(-1)}^i|00\rangle )/\sqrt{2}$. It is zero only on the centerline between both Bell states and for the ${\mathrm{W}}_4$ state at $p=1$.

Figure 7

The product of two concurrences are shown for the same mixed state as in Fig. 6. The product of the two concurrences is zero on a whole region, which has a chocklike form.

Figure 8

The upper solid black line is the 1-tangle, $4det{\rho }_1$, and is an upper bound to ${\sum }_n{C}_2^2\left(n\right)$ (red dashed line). The difference of both is the “residual tangle” in the chain, which consists of multipartite entanglement beyond two sites.

Figure 9

We show various graphs of $C_4(1,n,1)$ for $n=1$ to 9. It is seen that a gap occurs for $n\ge 7$ around the critical point ${\lambda }_c=1$. In this region of vanishing $C_4$, the optimal decomposition states must be made of states from the null cone of $C_4$, including W, cluster, and states of $X$-type [4, 6], which all contribute to the PPT criterion [10]. As a comparison we also print $C_2^2\left(1\right)$, which would be an upper bound to $C_4$, if the state would be a tensor product. For $n\ge 3$, our results are at least compatible with that hypothesis.

Figure 10

The figure shows $C_4(1,1,2)$ together with $C_2\left(1\right)C_2\left(2\right)$. It is the only nonvanishing $C_4(1,n,2)$ that exists.

Figure 11

Various curves are plotted for $C_4(1,n,1)$ at the value $\gamma =0.5:C_4(1,1,1)$ (full orange curve) with the highest maximum down to the lowest maximum for $C_4(1,8,1)$ (full black curve) are shown together with $C_2^2\left(1\right)$ (black dashed curve). It drops down to zero around the critical point ${\lambda }_c=1$ as for the transverse Ising model. This remarkable feature is not seen around the factorizing point ${\lambda }_f={(1-{\gamma }^2)}^{-\frac{1}{2}}$, where the ground state is an exact site-wise tensor product. Here, $C_4(1,n,1)$ has to approach zero at least as quickly as $C_2{\left(1\right)}^2$, if the state is to a good approximation a tensor product. It, however, tends to lie a bit above $C_2{\left(1\right)}^2$ for $\lambda \lesssim {\lambda }_f$. It is seen that it is, however, upper bounded by $C_2^2\left(1\right)$ up to a value of $\lambda =1.125$ for $n\ge 5$. For $n\ge 4,C_4(1,n,1)$ is upper bounded by $C_2{\left(1\right)}^2$ above the factorizing field. This is seen in the inset. For $\lambda =2$ and $n\ge 2$, they all have values of about 0.0033.

Figure 12

$C_4(2,n,2)$ is compared with $C_2^2\left(2\right)$.

Figure 13

The behavior of $C_4(1,1,2)$ is shown (blue solid curve) and compared with $C_2\left(1\right)C_2\left(2\right)$ (black dashed curve). The curves are for $\gamma =0.55$. A similar behavior is observed as for $\gamma =0.58$ and $\gamma =0.59$ (see Figs. 15 and 17, respectively).

Figure 14

$\gamma =0.55:C4(1,1,3)$ already sets in considerably earlier than $C_2\left(1\right)C_2\left(3\right)$. The value of $C_4(1,1,3)$, where $C_2\left(1\right)C_2\left(3\right)$ sets in is considerably above 0.0012, which is more than $50\%$ of its maximum value.

Figure 15

$C_4(1,1,2)$ together with $C_2\left(1\right)C_2\left(2\right)$ is shown for $\gamma =0.58$. The plot is basically as in Fig. 13, except that the factorizing point has moved as ${\lambda }_f={(1-{\gamma }^2)}^{-\frac{1}{2}}$.

Figure 16

$C_4(1,1,3)$ together with $C_2\left(1\right)C_2\left(3\right)$ is shown for $\gamma =0.58$, whereas $C_4(1,1,3)$ has squeezed apparently against the critical point ${\lambda }_c$ with a high slope. It appears to destructively interfere with something that has approximately the same height as $C_2\left(1\right)C_2\left(3\right)$.

Figure 17

$C_4(1,1,2)$ together with $C_2\left(1\right)C_2\left(2\right)$ is shown for $\gamma =0.59$. The plot is basically as in Fig. 13, except that the factorizing point has moved as ${\lambda }_f={(1-{\gamma }^2)}^{-\frac{1}{2}}$.

Figure 18

$C_4(1,1,3)$ together with $C_2\left(1\right)C_2\left(3\right)$ is shown for $\gamma =0.59$. It has “overtaken” $C_2\left(1\right)C_2\left(3\right)$, which is still pinned at ${\lambda }_c$.