Local expansion of photonic W state using a polarization dependent beamsplitter

We propose a simple probabilistic optical gate to expand polarization entangled W states. The gate uses one polarization-dependent beamsplitter and a horizontally polarized single photon as an ancilla. The gate post-selectively expands $N$-photon W states to $(N+1)$-photon W states. A feasibility analysis considering the realistic experimental conditions show that the scheme is within the reach of the current quantum optical technologies.


Introduction
Entanglement is the most important resource to realize quantum information processing tasks which surpass the efficiency of their classical correspondences as well as provide solutions to problems which are intractable with the classical resources [1] - [6]. The simplest form of this resource is the one between two parties, so called bipartite entanglement. Thanks to the theoretical and experimental efforts within the past two decades, we now have a clear understanding of the structure and characteristics of bipartite entangled states. It is well known that any bipartite state can be prepared from a maximally entangled bipartite state by local operation and classical communication (LOCC). It is generally accepted that the structure and dynamics of entanglement become more complex as the number of parties sharing entanglement increases. This sets a challenge in the theoretical and experimental studies of multipartite entanglement. Among many interesting features of multipartite entangled states, the widely-known one is the presence of inequivalent classes such as Greenberger-Horne-Zeilinger (GHZ), W and cluster states which cannot be converted into each other by stochastic local operations and classical communication (SLOCC) [7]. Experimental preparation and characterization of multipartite entangled states thus is not only essential for a better understanding of the quantum mechanics but also for the realization of state-specific information processing tasks. For example, W state has been shown to be the only pure state to exactly solve the problem of leader election in anonymous quantum networks whereas GHZ-state has been shown to be the only pure state to achieve consensus in distributed networks where no classical post-processing is allowed [8].
The photonic N-particle W state is represented by being the superposition of all possible permutations of N − k photons with horizontal (H) polarization and k photons with vertical (V) polarization, e.g., 3. W state has the peculiar property that every photon pair has the optimal amount of pairwise entanglement [9,10,11]. Such an entanglement structure forms a web-like system where every qubit has bonds with every other qubit, and the bipartite entanglement survives even if all the other (N − 2)-qubits are discarded. In recent years, there has been a number of theoretical proposals for the use of W states in multiparty protocols such as QKD [12], leader election in anonymous quantum networks [8] and teleportation, and for the preparation of W-states in different systems including optics [13] - [18], ion traps [19] and NMR [20]. Some of these proposals have been realized in experiments [6,19], [21] - [24].
In a recent study, we introduced an elementary optical gate to expand a state | W N to a state | W N +2 by local manipulation on a single site without accessing all the qubits of the initial W state [18]. While such a local expansion was known for GHZ and cluster states, it was a challenge for W states as (i) the marginal states of the remaining untouched N − 1 qubits is different for | W N and | W N +1 implying that the expansion process cannot be achieved unitarily, and (ii) the added new qubit should W K W K+1 Figure 1. Schematic description of local expansion of a W state using a gate with two inputs, one of which is an ancillary state and the other is provided from the W state.
form pairwise entanglement with each of the untouched N − 1 qubits of the original W state [see Fig. 1]. The gate proposed in Ref. [18] consists of two 50:50 beamsplitters and a half wave plate (HWP), and expands any W state by two qubits as the ancillary state used in the gate is an H-polarized two-photon Fock state. In this paper, on the other hand, we propose a simpler probabilistic optical gate, which is based on postselection, to expand a state | W N by only one qubit. This new gate, which is shown in Fig. 2, is composed of a polarization-dependent beamsplitter (PDBS), an H-polarized single photon as an ancilla and a HWP for phase compensation. The gate operates as a one-input two-output gate which can expand the state | W N to the state | W N +1 . This paper is organized as follows: In Sec. 2, we describe the principles of the gate operation. Sec. 3 includes a discussion of how this basic gate structure can be used to expand any polarization entangled W state. In Sec. 4, we give a scheme for the experimental realization of this gate and carry out a feasibility analysis under realistic conditions. Finally, in Sec. 5, we give a brief summary and conclusions.

Gate operation for expanding W state
The details of the proposed gate are shown in Fig. 2. The key component in this gate is the PDBS whose reflection and transmission coefficients depend on the polarization of the input light. The action of a PDBS for H-polarized photons and V-polarized photons can be written aŝ whereâ † jH (â † jV ) denotes the creation operator of H (V)-polarized photon in the j-th mode of PDBS, and µ (ν) is the transmission coefficient for H (V)-polarization. The gate uses an H-polarized photon as the ancilla in mode 2, and a photon in mode 1 with an arbitrary polarization as the input. The successful operation of the gate is signalled by a coincidence detection which occurs when there is one photon in each of the output modes 3 and 4. In order to understand the working principle of this gate for W-state preparation and expansion, it is enough to consider its action on two possible cases: where | vac stands for the vacuum state. Using the relations given in Eqs. (1) and (2) for the PDBS, we find that these input states are transformed into and In the above equations, only the underlined terms lead to successful gate operation and we will focus only on those terms. It is seen that when the input photon is in V-polarization, the coincidence detection will postselect the state parameters are chosen such that µ + ν = 1. It means that this gate works as an "entangling gate". The probability of this event is then 2µν.
Next, we see that if the input photon is from a Bell state a triple coincidence at modes 0, 3 and 4 will postselect the state If the weights of the components of this superposition state in Eq. (5) are made equal, then Eq. (5) will be of the form | W 3 except a π-phase shift which can be compensated using a HWP in mode 4. The equalization of the weights occurs when Second equality in Eq. 6 imposes the condition µ + ν = 1 which is the same condition obtained above for Bell state preparation. Solving the remaining equalities under the condition µ+ν = 1, we find that one should choose µ = (5− √ 5)/10 and ν = (5+ √ 5)/10. Inserting these values of µ and ν into Eqs. (3) -(4), and imposing the coincidence detection, we find that the successful gate operation is characterized by the following transformations where we have included the effect of the HWP in mode 4. Putting all together, we conclude that this gate can prepare the Bell state | W 2 with a probability of 2/5 starting with a V-polarized photon in mode 1, and the | W 3 state with a probability of 3/10 starting with the Bell state | W 2 in modes 0 and 1. This success probability for | W 3 state preparation is a significant improvement over other linear optics schemes existing in the literature. Among the already proposed schemes, the one in Ref. [18] has the highest success probability given as 3/16 which is less than that of the present scheme.

Expansion of polarization entangled W states
Here, we show that the same gate can be used to prepare and expand arbitrary W states. In the following, we will represent an N- N where the subscript 1 denotes the spatial mode of the photon that is input to the gate and 1 denotes the remaining N − 1 modes of | W N . Using this notation, the transformation in Eq. (7) can be represented as | 1 H 1 | 1 H 2 → 1/5| 2 H , 0 V and | 1 V 1 | 1 H 2 → 1/5| 1 H , 1 V . Thus, we find that upon the selection of the successful events, the action of the gate is given as Using these relations, it is straightforward to show that the successful gate operations performs the following transformation on an initial | W N : Thus we conclude that the gate expands a given W-state | W N to | W N +1 by one photon with a success probability of (N + 1)/5N. The success probability will approach the constant 1/5 when N becomes very large. This analysis shows clearly that the proposed gate can be used in two different ways: (i) a given arbitrary-size W-state | W N can be expanded by one at each successful operation of the gate which takes place with the probability (N + 1)/5N, e.g., a probability of 4/15 for the expansion of | W 3 to | W 4 and (ii) starting from a V-polarized input photon, an arbitrary-size W-state can be prepared by cascade application of the gate. For example, cascading k of this gate will prepare the state | W k+1 with a probability of (k + 1)5 −k .

Practical considerations for an experimental implementation
In this section, we introduce an experimental scheme for the implementation of this gate to expand the Bell state | W 2 to | W 3 , and discuss the effects of realistic conditions on the performance of the gate. We will focus on the effects of imperfections in (a) the preparation of the | W 2 and the ancillary state, | 1 H , (b) the detection of the successful events, and (c) the deviations of the parameters of PDBS from its optimal values.

Basic scheme
We propose the scheme given in Fig. 3 for the practical implementation of the proposed gate. In this scheme, the output of a pulsed laser (PL) with angular frequency ω 0 in the visible range of the spectrum is frequency doubled in a nonlinear crystal to produce pulses of ultraviolet (UV) light of angular frequency 2ω 0 . These UV pulses are then used to pump twice in forward and backward directions a pair of nonlinear crystals, which are stacked together such that their optical axes are orthogonal to each other [25]. The crystals are for Type-I spontaneous parametric down conversion (SPDC) to produce photon pairs in two modes (idler and signal) with the same polarization and at half the frequency of the pump beam. In the forward pumping direction, the polarization of the UV beam is set to vertical so that an H-polarized photon pair in modes 2 and 2 ′ are generated from which the required ancillary state | 1 H in mode 2 can be prepared. The remaining (non-down-converted) portion of the UV beam first passes through a quarter wave plate (QWP) which changes its polarization into an ellipsoidal polarization. A mirror placed after the QWP back-reflects this beam and sends it through the QWP again which further changes the polarization of the beam into diagonal polarization. This diagonally polarized beam pumps the crystals in the backward direction creating the entangled photon pair ( Changing the polarization of the photon in the mode 0 (idler) of the SPDC output will prepare the | W 2 in the spatial modes 0 and 1. Then the ancillary photon in mode 2 and the photon in mode 1 of | W 2 are mixed at the PDBS. The successful events are selected by a four-fold coincidence detection by ON/OFF detectors placed at the modes 0, 2 ′ , 3 and 4 as seen in Fig. 3.

Effects of SPDC and imperfect detection
Imperfections in the photon detectors affect the gate in two ways: (i) Recording some of the successful events as the failure due to non-unit quantum efficiency, and (ii) reporting some of the failures as the successful due to dark counts and/or due to the fact that detectors cannot resolve the photon number. In the following, without loss of generality, we neglect the errors due to dark counts. This is acceptable as the dark counts rates of current detectors are very low [26]. Moreover, the requirement of four-fold coincidence detection in our scheme significantly reduces the probability of false events due to dark counts. Neglecting the dark counts, the positive operator valued measure (POVM) elements for ON/OFF photondetectors become where Π 0 and Π 1 are, respectively, elements for no click (OFF) and for a click (ON) [27]. Returning back to our gate, we see that if there is only one photon in each of the modes 1 and 2, then the success probability of having one photon in each of modes 3 and 4 becomes 3η 4 /10. Note that the error due to (ii) occurs when there are more than one photon in either or both of the modes 3 and 4. This takes place when either or both of the backward and forward SPDC processes prepare two or more photon pairs. In practical settings, SPDC suffers from the non-deterministic nature of the process: The output of the SPDC contains vacuum with high probability and the probability of a photon pair generation is low. Moreover, although the probability is much lower, there are cases when multiple pairs of photons are generated. The generated state in the forward direction becomes where g = 1 − γ 2 and γe iφp is proportional to the complex amplitude of the pump field.
Assuming that the losses in the forward and backward pumping are negligible, the state in the backward direction can be written as where Combining the above expressions, we find that four-fold coincidence detection postselects the state, where we have focused on the terms up to γ 3 by considering that in practice, γ 2 ∼ O(10 −4 ) is very small. The PDBS transforms modes 1 and 2 of | Ψ 0122 ′ according to the relations given in Eqs. (1) and (2). Let | Ψ ′ 0342 ′ be the state after the transformation. Using POVM given in Eq. (11), the four-fold coincidence detection probability p c can be calculated as where Π j 1 is the POVM for "click" events at the detection in mode j. p t and p f respectively corresponds to probability of true and false coincidences. We see in Eq. (15) two contributions, p t and p f . The true coincidences (p t ) are due to the | W 2 01 | 1 H 2 | 1 H 2 ′ term in | Ψ 0122 ′ and the false coincidences (p f ) originates from multiple pairs of photons. Plugging the value µ = (5 − √ 5)/10 in these terms, we find that the ratio of the true coincidences to the total coincidence events becomes It is clearly seen that almost all the four-fold coincidence detections are true coincidences within the range of realistic values of η and γ.

Effect of deviation in the PDBS parameter
In this section, we consider the effect of deviations in the parameters of PDBS from its ideal values of µ = (5 − √ 5)/10 and ν = (5 + √ 5)/10 on the probability and the fidelity of expanding | W 2 into | W 3 . Let us assume that the reflection coefficients of PDBS for H-and V-polarized photons are deviated from the ideal values by δ and ∆, respectively. Then the action of the imperfect PDBS on H-polarized light and Vpolarized light becomeŝ Using these expressions, we calculated the probability of coincidence detection and the fidelity of the output state to the desired one. We omit the analytic expressions since they are rather lengthy and complicated. Instead, we depict the constant fidelity and constant probability contours as a function of δ and ∆ in Fig. 4. We see that the effect of δ on the fidelity is much larger than that of ∆. We can thus tolerate larger deviations from the ideal value for ∆.

Conclusion
In this paper, we have proposed a simple probabilistic optical gate for expanding polarization entangled W states and analyzed its feasibility taking into account the imperfections encountered in practice. The proposed gate is based on post-selection process to expand | W N by one qubit into | W N +1 by locally acting on one of its qubits. A remarkable feature of this gate is that starting with a Bell state, it can prepare tripartite entangled W-state with a success probability of 3/10 which is the highest among all the proposed schemes so far. Moreover, the gate does not need stabilization of optical paths and does not employ sub-wavelength adjustments. Our feasibility analysis shows that the proposed gate can be implemented by the current experimental technologies. Thus this gate will provide a simple and useful tool to probe interesting features of multipartite W states.