An optical fusion gate for W-states

We introduce a simple optical gate to fuse arbitrary size polarization entangled W-states to prepare larger W-states. The gate requires a polarizing beam splitter (PBS), a half wave plate (HWP) and two photon detectors. We study numerically and analytically the necessary resource consumption for preparing larger W-states by fusing smaller ones with the proposed fusion gate. We show analytically that resource requirement scales at most sub-exponentially with the increasing size of the state to be prepared. We numerically determine the resource cost for fusion without recycling where W-states of arbitrary size can be optimally prepared. Moreover, we introduce another strategy which is based on recycling and outperforms the optimal strategy for non-recycling case.


Introduction
Quantum entanglement is at the heart of many quantum information processing (QIP) tasks such as quantum teleportation [1], quantum key distribution (QKD) [2] and quantum computation [3]. Among the multipartite entangled states, W, GHZ and cluster states form inequivalent classes in that they cannot be transformed into each other by local operations and classical communication (LOCC) [4]. Recent studies have shown that for specific type of QIP tasks, specially designed multi-partite entangled states are required: Cluster states have been proposed as a universal substrate for measurement based quantum computation [5], GHZ class has not only been shown to be useful for quantum teleportation [6], quantum secret sharing [7,8] and QKD [9] but also to be the only one for reaching consensus in distributed networks when no classical post-processing is allowed [10]. On the other hand, W-class is proposed as a resource for QKD [11] and for the optimal universal quantum cloning machine [12] as well as shown to be the only pure state to exactly solve the problem of leader election in anonymous quantum networks [10]. It is thus important to efficiently prepare states of different classes not only for practical applications but also for the fundamental study of quantum information.
It is known that starting with Bell pairs, probabilistic quantum parity checking gates, the so-called fusion gates, can be efficiently used to grow large scale cluster states [13]. A GHZ state can be prepared and expanded by one photon with each successful application of the same fusion gate [13,14]. So far a similar study on n-partite W-states, defined as | W n = | n − 1, 1 / √ n where | n − k, k is the sum of all states with n − k zeros (H-polarized photon) and k ones (V -polarized photons), have not been carried out. Given that W states are optimal in the amount of pairwise entanglement when n − 2 parties are discarded [4,15] and have persistency of n − 1 which is much larger than those of GHZ and cluster states [16], it becomes a necessity to probe the bounds on the efficiency of preparing large scale W-states and the resource requirements. It is also worth noting that | W n constitute an entangled web and fully interconnected quantum network. Therefore, such a study may shed light on the scalability of entangled webs of W-states [17].
In Ref. [31], we proposed an optical gate formed by a pair of 50:50 beamsplitters which accepts one photon from | W n to expand it into | W n+2 with a success probability of (n + 2)/16n using an ancillary state of two H-polarized photons | 2 H . The gate can be cascaded in which each successful gate operation increases the size of W-state by two photons: Cascading this gate k-times prepares the state W n+2k with a success probability of 2 −4k (1 + 2k/n). We experimentally demonstrated this optical gate and successfully generated three-photon and four-photon polarization-based W-states [33]. In Refs. [32,34], on the other hand, it was shown that | W n can be expanded to | W n+1 with a success probability of (n + 1)/5n using a polarization dependent beamsplitter and an ancillary state of H-polarized single photon | 1 H . The cascade operation of this gate expands a W-state by one photon at each successful step: Cascaded application of this gate k-times prepare the state W n+k with a success probability of 5 −k (1 + k/n). As it is clear, for both of these gates the success probability decreases exponentially as k increases. Recycling in case of failure is not possible; thus resource required to prepare a large W-state scales exponentially.
In this paper we present a theoretical proposal for preparing large scale W-state networks using a fusion mechanism. We have previously introduced the basic principles of this W-state fusion mechanism and the gate for its realization in Refs. [35,36,37]. Here, we compare resource requirements for preparing arbitrarily large W-states using this fusion mechanism under various scenarios. We show that resource for the proposed fusion mechanism scales subexponentially. Although we phrase our proposal for qubits encoded in horizontal (H) and vertical (V) photon polarization, the fusion mechanism is of wider applicability. The primary resource we will make use of is three photon polarization entangled W-states | W 3 . These can be prepared using the gate given in Ref. [31], with a success probability of 3/16 starting with a single photon | 1 V and two photons in Fock state | 2 H , and the gate in Ref. [32], with a success probabilities 3/10 with a single photon and a Bell pair. Alternatively, two polarization entangled Bell states can be used to create this initial W-state as shown in Ref. [26].

Fusion Gate for W-states
Let us consider the following scenario: Two spatially separated administrators, Alice and Bob, decide to merge their small scale entangled webs | W n A and | W m B into a larger entangled web | W γ A∪B with the help of a trusted third party Claire. In order to do this each transmits one qubit of their web to Claire who acts locally on the received two qubits with the fusion gate and inform them when the task is successful. The question is whether such a local manipulation is possible or not, and if possible then how does the scheme looks like. The polarization entangled W-states of Alice and Bob are where the photon in the mode 1 (2) are sent to Claire by Alice (Bob) and those in mode a (b) are kept at Alice's (Bob's) side. We see that the photon pairs Claire receives are and | 1 H 1 | 1 H 2 with the respective probabilities P VV = 1/nm, P HV = (n − 1)/nm, P VH = (m − 1)/nm, and P HH = (n − 1)(m − 1)/nm.
The output modes 3 and 4 of the first PBS are measured by the detectors D1 and D2 which are formed by a HWP, PBS and a pair of photon counters.
We start by describing the fusion mechanism which is a parity check operation (see Fig.1). The photons in two spatial modes are mixed on a PBS after exchanging the polarization of the photon in one of the modes by π/2. After the PBS, photons in the output modes are measured in The combined action of the HWP and PBS on the input photons is as follows: where the subscript numbers denote the spatial modes of the fusion gate. It is clear that a coincidence detection between the detectors D1 and D2 takes place when the photons in modes 1 and 2 have orthogonal polarizations, and no coincidence is observed when the photons in modes 1 and 2 have the same polarizations. Moreover, the detectors cannot discriminate between the two cases which lead to coincidence detection.
The case when D1 detects photons but not D2 implies that both of the initial Wstates have lost their V-polarized photons. Therefore, the remaining photons will all be H-polarized. Thus, we end up with a product state (networks are destroyed). Such events which we call as failure take place with a probability of P f (W n , W m ) = P VV = 1/nm. On the other hand, for the case where D2 detects photons but D1 does not implies that each of the initial W-states have lost one H-polarized photon. Thus, we will have two separate W-states with smaller number of qubits, | W n−1 and | W m−1 with probability P r (W n , W m ) = P HH = (n − 1)(m − 1)/nm. This shortened W-states can be recycled using the same fusion mechanism later. Now let us look at the cases which lead to a coincidence detection closely. When both D1 and D2 detect photons in the same state | D (or |D ), the state of the remaining photons becomes where we have used When one of the detectors detects a photon in | D and the other in |D , the state of the remaining photons will be the same as Eq.(3) but with a minus sign which can be corrected by applying a π-phase shift in one of the modes. Thus a coincidence detection signals the successful fusion operation and the preparation of a W-state with n + m − 2 photons with the success probability P s (W n , W m ) = (n + m − 2)/nm. Note that an attempt to fuse | W 2 with | W n will not expand the W-state; successful events will prepare only the state | W n . Expansion requires that both n and m are greater than or equal to 3.
We give an example of fusion operation in Fig.2, which show the success, failure or recycling processes. Recycling is performed until either a Bell pair | W 2 = (| HV + | V H )/ √ 2 or a product state is obtained.. In principle, the resultant Bell states can be further recycled to prepare a | W 3 using the gate introduced in Ref. [26]; however, in this study we do not consider such recycling.
As the above discussions show the fusion gate for W-states differ from that for Cluster states in two ways: (i) Fusion gates for cluster states operate with constant success probability of P s regardless of the size of the cluster states attempted to be fused. However, success probability of fusion gate for W-state depends on the size of the W-states: As n and m increases, P f decreases but this does not necessarily leads to an increase of the same order in P s , instead the probability P r of recyclable events increases. (ii) Failure events in cluster state fusion leads to two cluster states shortened by one qubit, similarly to the recyclable events in fusion of W-states. On the other hand, a failure in fusion for W-states leads to complete destruction of both W-states. These make the analysis of fusion and expansion of W-states much more difficult.

Cost of Preparing Arbitrary-Size W states
In this section, we compare the performances of various strategies using the proposed fusion gate for the preparation of arbitrary size W-states. We will answer the question "How does the required resource to prepare a W-state of N-photons | W N scale?" in various scenarios with and without recycling process.
In this section, we switch to a new index to represent the size of W states, which differs from the orignal by 2: The benefit of the lower-case notation is to make the result of successful fusion more intuitive: successful fusion of w m and w n simply produces w m+n . The recyclable outcome leaves the states in the same form as in the original notation, namely, w m−1 and w n−1 .
The probabilities associated with the three outcomes are now written as We use the notation R[w m ] for the resource cost (i.e., the number of | w 1 states required) of producing state w m , which is a (m + 2) qubit W state | W m+2 . In our analysis, we consider | W 3 ≡ | w 1 as the basic resource provided with unit cost, i.e., R[w 1 ] = R[W 3 ] = 1. For m ≥ 2, the value of R[w m ] will vary depending on the strategies.
When we do not use recycling and try to produce w m+n from w n and w m , the costs are simply related as which is frequently used in the later analysis. The cases with recycling are more complicated and will be treated separately below.
In the following, we treat this problem for linear and exponential growth strategies with and without recycling and derive analytical bounds for resource complexity. We provide the optimal strategy for fusion without recycling, and also introduce a strategy based on fusing W states of similar sizes which turns out to provide the best resource scaling among all the strategies considered here.

Linear Growth Strategies
Here, we analyze the resource requirements for linear growth strategies with and without recycling. The strategy is based on repeated fusion of a fixed-size W-state to an already existing W-state (see Fig.3). Let us assume that we want to expand w m by fusing it with w n repeatedly. If fusing is successful, bring another w n and fuse it with the state prepared in the previous level. In this way after the k-th level of successively successful gate operations, the state w m+kn is prepared with the probability (m+kn+2)/(m+2)(n+2) k . At each level of successful operation, the size of the state increases by n, i.e., {m, n} → {m + n, n} → {m + 2n, n} . . . → {m + (k − 1)n, n} → {m + kn}.
3.1.2. Linear growth with recycling As shown in Sec. 2, recyclable failure of the fusion gate, which takes place with probability P r (w n , w m ) = (n + 1)(m + 1)/(n + 2)(m + 2), leads to a reduction in the size of the initial W states by one qubit, e.g., {m, n} → {m − 1, n − 1}. Therefore, the remaining W states can be recycled. Here we include this recycling into the linear-growth strategy and see how the cost R[w m ] changes. The averaged cost R[w m+1 ] can be written as R[w m+1 ] = R[w m ] + ∆ m+1 , where ∆ m+1 is the averaged cost of creating w m+1 when a shorter W state w m is given. Let us calculate ∆ m+1 as follows. Suppose that state w m is given, and we apply a fusion gate to it with w 1 , by paying a unit cost. At probability p m ≡ P s (w m , w 1 ), the gate succeeds, and no additional cost is required. At probability q m ≡ P r (w m , w 1 ), we are left with state w m−1 , and it takes additional cost of ∆ m + ∆ m+1 to obtain w m+1 . Finally, at probability 1 − p m − q m , the gate fails completely and we need full cost R[w m+1 ] to produce w m+1 . These observations lead to and thus we have This recursive formula can be numerically solved with R[w 1 ] = 1 and R[w 2 ] = 9/2, which is depicted in Fig. 5 (red colored dotted box). When m is large, p m ∼ 1/3 and . We thus conclude that although recycling reduces the required resources from O(3 m ) to O(2 m ), it does not change the resource scaling law: Regardless of whether recycling is performed or not, required amount of resource to prepare a desired state using linear-growth strategies scales exponentially.

Optimal Strategy without Recycling
We have seen that strategies to grow W states by a constant amount at each step is not so efficient, even if recycling is introduced. We should thus turn to other strategies  for efficiency. In this subsection, we numerically determine the optimal cost over all the strategies without recycling. Let us start from simple examples. In order to produce w 2 , the only way is to fuse two w 1 states, since w 2 cannot be produced when larger W states are fused under the assumption of no recyclinig. The optimal cost R[w 2 ] opt is thus given by P s (w 1 , w 1 ) −1 (1 + 1) = 9/2. Similarly, there is only one way to produce w 3 , leading to R[w 3 ] opt = P s (w 2 , w 1 ) −1 (1 + 9/2) = 66/5. In the case of w 4 , on the other hand, there are two possible ways, {w 1 , w 3 } and {w 2 , w 2 } with the respective successful fusion probabilities of 2/5 and 3/8. Since we know the optimal costs for preparing w 2 and w 3 , we calculate the cost of preparing w 4 from {w 1 , w 3 } as (5/2)(1 + 66/5) = 71/2, whereas that from {w 2 , w 2 } as (8/3)(9/2 + 9/2) = 24. The latter strategy is better, and hence R[w 4 ] opt = 24. In this way, the optimal cost of any state can be numerically calculated using the recursive formula The calculated values of {R[w N ] opt } are presented in Fig.5 (green colored dotted circle), which suggests a sub-exponential resource scaling.

Exponential Growth Strategy without Recycling
Although the discussion in the previous subsection enables us to numerically calculate the optimal cost in the case of no recycling, it does not tell us how the optimal strategy look like or how the cost scales in the limit of large target size. Here we consider a specific strategy without recycling, based on fusing two states of the same size to double it. We show that this strategy works under the optimal cost, and derive an analytical expression of the cost in order to see the scaling over the target size.
Here we only consider production of a state w N whose size is written as N = 2 k . In what we call an exponential-growth strategy, the state w N is produced by fusion of two W states w N/2 with equal size. The state w N/2 is in turn generated from fusion of state w N/4 . When no recycling is done, Eq. (8) leads to a simple relation among the costs in this strategy: If we define a l ≡ (1 + 2 l−1 )R[w 2 l ], we have a l+1 = 2 l+1 (1 + 2 1−l )a l and thus which is plotted in Fig.5. We see that the cost coincides with the optimal cost for no recycling derived in the previous section, indicating that the strategy of fusing two state of the same size is very cost-effective. Since the coefficient γ k is finite, namely, γ k ≤ lim k→∞ γ k = 21.458 . . ., the scaling of the cost in the limit of large k is O(2 k(k+1)/2 ), or equivalently, in the limit of large N, which is sub-exponential in N.

Fusing States of Similar Sizes with Recycling
We have seen that fusing two W states of the same size is advantageous in the case of no recycling. Here we propose a strategy with recycling, which tries to fuse W states of similar sizes. The performance of the strategy is then evaluated through Monte Carlo simulations. Let us classify generated W states into sets {S l } according to their sizes, such that state w m belongs to set S l when m ∈ (2 l−1 , 2 l ]. The idea is to perform fusion operation between two states belonging to the same set. It is easy to see that fusion of two states w m and w n belonging to the same set S l will produce state w m+n in S l+1 upon successful operation. If the operation is complete failure, the states w n and w m are discarded. In case of recyclable outcome, the resultant states w m−1 and w n−1 belong to either S l or S l−1 . More precisely, our strategy to prepare a state belonging to set S k+1 is described in the following algorithm, which dictates the order in which the recycled W states should be used. In the description below, µ l represents the number (0,1, or 2) of states already generated in set S l , R is the cost (the number of consumed states w 1 ), and ξ is a pointer to the 'current' working set.
(2) Depending on the values of ξ and µ ξ , do one of the following procedure.
(3) S ξ should have two states, which we denote w n and w m , and we apply a fusion gate on them. Empty S ξ and set µ ξ → 0. Do one of the following procedures depending on the result of the gate operation. (Complete failure): Go back to step 2.
The above strategy creates a W-state of (actual) size N = 2 k + 3 or larger. We have done Monte-Carlo simulations for k = 0, . . . , 6, and Fig. 5 (black square) shows the cost averaged over 1000 runs for each value of k. It is clearly seen that this strategy, based on fusing the states of similar sizes with recycling, outperforms all the other strategies considered in previous sections.
Linear growth without recycling Linear growth with recycling Exponential growth without recycling Optimal Strategy without recycling Fusing states of similar sizes with recycling Figure 5. Comparison of the expected amount of resources R[N] to prepare a W N for strategies introduced in the text: Linear growth by one, i.e., {n, 1} → {n + 1}, with and without recycling, Exponential growth, i.e., {n, n} → {2n} , without recycling, Fusing W states of similar sizes, and optimal strategy without recycling. Each point corresponds to the average of 1000 trials. Note that sizes given here are the actual sizes of the states.

Conclusion
In this paper, we introduced an optical fusion gate to fuse W-states to prepare a W-state of larger sizes and discussed the scaling laws for the required resources for preparing a W-state using the proposed fusion gate.
We introduced four different strategies with different resource requirements depending on whether recycling is allowed or not. We first demonstrated both analytically and numerically that resource requirement for linear growth strategies, which consider repeated fusion of a fixed size W-state with the already existing Wstate. These strategies scale exponentially regardless of whether re-cycling is performed or not, although recycling allows reduction in the required resources. Then we calculated the optimal cost for fusion without re-cycling. Next, we considered exponential growth strategies in which fusion gate is always applied to two states of the same size. We derived analytical expressions showing that the required resources scale subexponentially. Interestingly, non-recycling exponential growth strategy appears to have the same resource scaling as the optimal strategy with no-recycling, implying that the former is the optimal solution for fusion without recycling. Finally, through numerical simulations we demonstrated a strategy, in which states with the closest sizes are fused, provides the best resource scaling for the proposed fusion gate among the strategies investigated in this study.
The proposed fusion gate and discussed fusion strategies outperform the previously proposed W-state preparation and expansion gates in terms of required resources when the size of the state to be prepared is large. Our study does not exclude the possibility of the presence of a better strategy or a strategy with a polynomial scaling for the proposed fusion gate. We hope that this study will initiate further work and continuing discussions on the optimal ways of a fusing/preparing/expanding W-states and understanding the structure of larger multipartite entangled states.