Sequentially generated entanglement, macroscopicity, and squeezing in a spin chain

We study quantum states generated by a sequence of nearest neighbor bipartite entangling operations along a one-dimensional chain of spin qubits. After a single sweep of such a set of operations, the system is effectively described by a matrix product state (MPS) with the same virtual dimension as the spin qubits. We employ the explicit form of the MPS to calculate expectation values and two-site correlation functions of local observables, and we use the results to study fluctuations of collective observables. Through the so-called macroscopicity and the squeezing properties of the collective spin variables they witness the quantum correlations and multiparticle entanglement within the chain. Macroscopicity only occurs over the entire chain if the nearest neighbor interaction is maximally entangling, while a finite, sequential interaction between nearest neighbor particles leads to squeezing of the collective spin.

Figure 1

A two-qubit operation $U$ is sequentially applied between the nearest neighbor qubits along a one-dimensional chain. We assume that the first qubit initially occupies a state $|\varphi \rangle =c_0|0\rangle +c_1|1\rangle$, while all the other qubits occupy the state $|0\rangle$.

Figure 2

The controlled-$U^0$ operators, acting on the target qubit if and only if the control qubit is $|1\rangle$, are sequentially applied between nearest neighbor qubits. We choose the first qubit in $|\varphi \rangle =\frac{|0\rangle +|1\rangle }{\sqrt{2}}$.

Figure 3

The collective variance of $A_z$ as a function of $N$. The curves from top to bottom correspond to rotation by $a=\pi ,\pi -0.1,\pi -0.2,\pi -0.3,\pi -0.4$. For all angles, we observe a quadratic behavior for small values of $N$, but as we gradually increase the value of $N$, the scaling becomes linear except in the $\pi$ rotation, which leads to controlled-not operation.

Figure 4

Maximal collective squeezing in the limit of large $N$. The dashed red curve represents accompanying values of the mean spin $\langle J_z\rangle$ and the minimum transverse spin variance ${(△J_{\frac{\pi }{4}})}^2$, while the solid black (dotted blue) curve shows the minimum transverse variance ${(△J_{\theta })}^2$ allowed for given $\langle J_z\rangle$ if the spins are separable (at most pairwise entangled) [11]. For large $J_z$ the dashed red curve lies below the dotted blue one, so the spin squeezed state generated by sequential pairwise interaction contains at least three spin entangled components.