Entanglement enhancement through multirail noise reduction for continuous-variable measurement-based quantum-information processing

We study theoretically the teleportation of a controlled-phase (cz) gate through measurement-based quantum-information processing for continuous-variable systems. We examine the degree of entanglement in the output modes of the teleported cz-gate for two classes of resource states: the canonical cluster states that are constructed via direct implementations of two-mode squeezing operations and the linear-optical version of cluster states which are built from linear-optical networks of beam splitters and phase shifters. In order to reduce the excess noise arising from finite-squeezed resource states, teleportation through resource states with different multirail designs will be considered and the enhancement of entanglement in the teleported cz gates will be analyzed. For multirail cluster with an arbitrary number of rails, we obtain analytical expressions for the entanglement in the output modes and analyze in detail the results for both classes of resource states. At the same time, we also show that for uniformly squeezed clusters the multirail noise reduction can be optimized when the excess noise is allocated uniformly to the rails. To facilitate the analysis, we develop a trick with manipulations of quadrature operators that can reveal rather efficiently the measurement sequence and corrective operations needed for the measurement-based gate teleportation, which will also be explained in detail.

Figure 1

Excess-noise reduction for measurement-based teleportation through multirail designs for CV cluster states proposed by van Loock et al. [20] with (a) single-rail, (b) two-rail, and (c) general multirail structures. Here each circle represents a qumode, with the one labeled $\alpha$ the input mode and those unlabeled the cluster modes. The solid lines connecting the cluster nodes represent two-mode correlations which can be established through two-mode squeezing operations or linear-optical networks (see the text). In each figure, the coupling between the input mode $\alpha$ and the cluster is represented by a dashed line. To teleport the state of $\alpha$ toward the right of the cluster, a sequence of quadrature measurements must be applied to the cluster modes. Upon feeding forward the measurement outcomes and applying the corresponding corrective operations to the rightmost node of the cluster, as indicated by the arrow, one would get the output mode $\mu$ which holds the teleported state.

Figure 2

Teleportation of a cz gate through a linear four-mode cluster. Here and in all figures below, the input modes $\alpha ,\beta$ are coupled to the cluster nodes via either a QND coupling or a beam-splitter coupling (see the text), which is represented by a dashed line here. To the right of the cluster, the output modes $\mu$ and $\nu$ are generated from nodes 2 and 3, respectively, where the arrows indicate appropriate corrective operations in accordance with measurement outcomes in the measurement sequence. For an ideal cluster state, the teleported state registered by $\mu ,\nu$ would be identical to that of the input modes $\alpha ,\beta$ operated with a cz gate. Depending on the context, the cluster state here (and also in other figures of this paper) can be a canonical one or a linear-optical one.

Figure 3

cz-gate teleportation using a linear four-mode cluster with a node arrangement different from that of Fig. 2.

Figure 4

cz-gate teleportation using clusters with multirail designs of (a) single-rail, (b) two-rail, (c) three-rail, and (d) $N$-rail structures. Namely, the “teleporting arms” (i.e., the segments of nodes 3–2–1 and 4–5–6) of the single-rail cluster in panel (a) are replaced with multirail structures in panels (b)–(d) according to the designs of Fig. 1. Here the dashed lines and the arrows have the same meanings as those in Figs. 2 and 3.

Figure 5

The log-negativity ${\tilde{E}}_N$ for the output modes $\mu ,\nu$ of the teleported cz gate normalized with respect to the ideal (infinite squeezing) value plotted as a function of the squeezing parameter $r$. Here the resource states are (a) canonical and (b) linear-optical multirail clusters with the number of rails $N=1$ (dashed lines), 2 (dotted lines), 3 (dot-dashed lines), and 100 (solid lines). In each panel, the inset shows the vicinity of the squeezing level at which the log-negativity reaches $50\%$ of the ideal value.

Figure 6

Panels (a) and (b): $W_g$ as a function of the signal gain $g$ at $-3.91$ dB ($r=0.45$) squeezing for teleportations with (a) canonical and (b) linear-optical clusters with multirails $N=$ 1 (dashed lines), 3 (dot-dashed lines), and 100 (solid lines). Panels (c) and (d): $W_g$ plotted vs the squeezing parameter $r$ at fixed gain values (c) $g=1.0$ and (d) $g=0.5$ for teleportations with canonical (solid lines) and linear-optical (dashed lines) clusters with the number of rails $N=100$. In each panel, the dotted line depicts the corresponding entanglement bound.

Figure 7

A network of QND gates for the single-mode teleportation through an $N$-rail canonical cluster shown in the lower left corner of this figure. Here “S” are single-mode squeezers that generate momentum-squeezed vacuum modes, “QND” are QND gates that entangle the pairs of intersecting modes, and “D” are the homodyne detectors for the qumodes. Each qumode is denoted by a numbered pulse for the corresponding cluster node, and similarly for the input mode $\alpha$ and the output mode $\mu$. The teleportation is divided into three stages (see text), which are indicated by vertical dotted lines. The gate “correc. op's” in stage (iii) represents corrective operations for the output mode, where the double wires on the sides stand for signals fed forward from measurement outcomes of other qumodes.

Figure 8

A snapshot for the single-QND counterpart for stage (ii) of the multirail teleportation depicted in Fig. 7. Here the midrail modes $\{2,4,\cdots ,(N+2\left)\right\}$ are produced sequentially from the squeezer at a fixed interval, so that they can meet qumodes 1 and 3 at appropriate times at the QND gate. Note that for the optical paths the path lengths illustrated here are not in actual proportions and the loop for qumodes 1 and 3 is in fact part of a helical path; see text and also Fig. 9.

Figure 9

Single-QND realization for the transition from stage (i) [panel (a)] to the initial steps of stage (ii) [panels (b), (c)] for the teleportation shown in Fig. 7. For clarity, we show only part of the helical path for modes 1 and 3.