Demonstration of a fully hineable entangling gate for continuous-variable cluster computation

The one-way model of measurement-based quantum computation [1] is a fascinating alternative to the standard unitary-gate-based circuit model, for qubits as well as for continuous-variable (CV) encodings. Such one-way schemes are realized through single-qubit or single-mode measurements together with outcome-dependent feedforward operations on a preprepared multi-partite entangled state, the so-called "cluster state". By choosing an appropriate set of measurement bases on a sufficiently large cluster state, an arbitrary unitary operation can be implemented for the corresponding encoding.To date, various cluster states have been experimentally generated. Further, a fixed-strength entangling gate has been demonstrated by using an optical CV cluster state as a resource [2]. In the case of a fixed-strength entangling gate, unwanted noise might propagate along it during computations. Therefore, it is necessary for more efficient computation to have full tuneability of the entanglement strength. We demonstrate a fully tuneable entangling gate (TZ) by propagating two arbitrary, independent inputs through an optical graph state comprised of three quantum modes in a cluster state configuration. Its entangling strength is perfectly controlled by the selection of measurement on the cluster state. The schematic of our experiment is shown in Fig. 1(a). The three mode linear cluster state is represented, in the ideal case, as the zero eigenstate δ3 = p3 - x2. Here, x of linear combinations of the canonical operators: δ1 = p1 - x2, δ2 = p2 - x1 - x3, and andp are canonical position and momentum operators, respectively, satisfying the commutation relation [x, p] = i/2 (h = 1/2). Two input states (labeled by α and β) enter the cluster state via half beam splitters (HBSs). By measuring half of their outputs and mode 2, and then implementing appropriate corrections, we can teleport entanglement onto modes α and β, which are relabeled as μ and ν in the output of Fig. 1(a). The input-output relation is written as ξμν = S+K K ξ αβ + δ, where ξjk = (xj, pj, xk, pk)T , S = V 0 0 2 0 V , and K = V . K S+K 0 1/ 2 g/ 2 0 Here, the matrix S+K K corresponds to the TZ gate, whose gain g describes the entanglement strength. In K S+K our experiment, this entangling gain g = tanθ is perfectly controlled by the homodyne measurement angle θ for V mode 2. δ= 0,( δ1 + g δ2)/ V 2,0,( δ3 + g δ2)/ 2T represents the excess noises of our TZ gate, which vanishes in the limit of infinite squeezing. In order to verify the entangling ability of our TZ gate, we investigate the Partial Transposition(PT)-symplectic eigenvalues of the output states. They are identical to the logarithmic negativities in the case of Gaussian states [3]. A state is entangled if its PT-symplectic eigenvalue is below 0.25, and the smaller PT-symplectic eigenvalue shows stronger entanglement. The dependence of the PT-symplectic eigenvalues of the output states on the TZ gain g is shown in Fig. 1(b). When the entangling gain increases, our measurement results of them monotonically decrease and cross from separable region to inseparable region, which corresponds to monotonic increasing of entanglement strength. Furthermore, they agree with theoretical predictions, showing the successful implementation of the fully tuneable entangling gate.