Extracting an entangled photon pair from collectively decohered pairs at a telecommunication wavelength

We experimentally demonstrated entanglement extraction scheme by using photons at the telecommunication band for optical-fiber-based quantum communications. We generated two pairs of non-degenerate polarization entangled photons at 780~nm and 1551~nm by spontaneous parametric down-conversion and distributed the two photons at 1551~nm through a collective phase damping channel which gives the same amount of random phase shift on the two photons. Through local operation and classical communication, we extracted an entangled photon pair from two phase-disturbed photon pairs. An observed fidelity of the extracted photon pair to a maximally entangled photon pair was 0.73 $\pm$ 0.07 which clearly shows the recovery of entanglement.


Introduction
Faithful distribution of photonic entangled states between two distant parties is one of the significant issues in the field of quantum information processing such as quantum teleportation [1], superdense coding [2] and entanglement-based quantum key distribution [3]. In order to perform these quantum information processing, schemes for robust distribution of entanglement are required. Entanglement distillation [4][5][6] is one of the protocols for this purpose and several schemes have been proposed and demonstrated experimentally [7][8][9][10][11]. In previous proof-ofprinciple experiments, photons at visible wavelength were used. However, for practical opticalfiber-based quantum communication over a long distance, photons in telecommunication bands should be used to benefit from a low photon loss in an optical fiber.
In this paper, we report an experimental demonstration of the entanglement extraction based on linear optics and a post-selection [7] by using two non-degenerated photon pairs at a visible wavelength of 780 nm and a telecom wavelength of 1551 nm. The sender Alice keeps the two visible photons and sends the two telecom photons to the receiver Bob through a collective phase damping channel (CPC) which adds the same phase shifts to the two photons. Such a collective noise channel appears in many practical situations [12][13][14] and discussed in many quantum communication protocols, such as reference-frame-free quantum communication [15][16][17][18], decoherence-free quantum communication [19][20][21][22][23] and computation [24,25]. Through the transmission through the CPC, the entanglement of each photon pair is lost. However, after performing the quantum operation on two photons of 780 nm at Alice's side and a projective measurement on one of the two photons at Bob's side, Alice and Bob can extract an entangled photon pair. An observed fidelity of the extracted photon pairs to a maximally entangled state is 0.73 ± 0.07, which shows extraction of the entanglement from two phase-disturbed photon pairs shared between Alice and Bob.

The theory of the entanglement extraction
We first introduce the demonstrated entanglement extraction scheme. The purpose of two parties, Alice and Bob, is to share a maximally entangled photon pair denoted by |φ + ≡ (|HH + |VV )/ √ 2 , where |H and |V represent horizontal (H) and vertical (V) polarization states of a photon, respectively. As shown in Fig. 1, Alice generatesρ 1234 =φ + 12 ⊗φ + 34 , whereφ + ≡ |φ + φ + |. The subscripts represent the spatial modes of the photons. She then sends the photons in modes 2 and 4 to Bob through a CPC. By denotingẐ ≡ |H H| − |V V| andẐ(θ ) ≡ exp(−iθẐ/2), the CPC acting on the two photons in modes 2 and 4 transformŝ ρ 1234 tô Because the density operator of each photon pair Tr 34 (12) [ρ ′ 1234 ] = (|HH 34(12) HH| 34(12) + |VV 34(12) VV| 34 (12) )/2 has no entanglement, Alice and Bob do not share any entangled photon pairs as long as they treat the photon pairs separately. However, they can extract a maximally entangled state of a photon pair fromρ ′ 1234 of the whole system by local operation and classical communication in the following way.
Alice performs the quantum parity check (QPC) [26] on photons in modes 1 and 3. Kraus operators of the QPC in Fig. 1 By performing a phase flip operation when the photon in mode 4 is projected onto |− , |φ − is corrected to |φ + . As a result, Alice and Bob share |φ + with a success probability Tr [F †Fρ QPC ]=1/4. In our experiment, we do not perform the phase flip operation, and the success probability becomes 1/8. We notice that in the scheme, the effect of the phase disturbance on the photons in modes 2 and 4 is compensated by the QPC on the photons 1 and 3 at Alice's side. Such a non-local cancellation of the phase fluctuation is the result of the use of a significant property of the maximally entangled states, which is described by the relation (Î 1(3) ⊗Ẑ 2(4) )|φ + 12(34) = (Ẑ 1(3) ⊗Î 2(4) )|φ + 12(34) [22,23].

Experimental setup
The experimental setup is shown in Fig. 2 (a). Two entangled photon pairs are generated by spontaneous parametric down conversion (SPDC) at source A and B. The setup of the photon pair sources is shown in Fig. 2 (b). The pump laser is based on a fiber laser (wavelength: 1037 nm, pulse width: 381 fs, repetition rate: 80 MHz), which is frequency doubled such that the center wavelength is 519 nm with the power of 1.32 W. The polarization of the pump beam is set to be diagonal to the axes of two adjacent phase-matched 1-mm-thick β -barium borate (BBO) crystals by the half wave plate (R45). In this experiment, we select the nondegenerate photon pair at a visible wavelength of 780 nm and a telecom wavelength of 1551 nm among the photon pairs generated from the BBOs. After the BBO crystal, the pump beam is removed by using a pair of dichroic mirrors (DM1) whose reflectance for the photons at 780 nm and 1551 nm is over 99 % and transmittance for the photons at 519 nm is ∼ 97 %. After DM1, the visible photons and telecom photons are separated into different spatial modes by DM2. For both photons, the group delays between |H and |V are compensated by 6.17 mm and 7.44 mmthick quartz crystals (Quartz1) in the paths of the visible and the telecom photons, respectively. The relative phase shift between |HH and |VV is compensated by tilting 0.6 mm-thick quartz crystals (Quartz2). As shown in Fig. 2 (a), two visible photons from photon source A and B are injected into a polarizing beamsplitter (PBS) simultaneously by adjusting a moving mirror (M). The spectra of the two output photons from the PBS are filtered by interference filters (IFs) with a bandwidth of 3 nm, and then coupled to single-mode fibers followed by silicon avalanche photodetectors (quantum efficiency: 60 %) D1 and D3. On the other hand, two telecom photons pass through a liquid crystal retarder (LCR). The LCR adds eight phase shifts nπ/4 (n = 0, 1, 2, · · · , 7) between |H and |V of the two photons by switching the applied voltage. The operations of the LCR on the two photons are nominally described by the Kraus operators {Ẑ nπ/4 2 ⊗Ẑ nπ/4 4 } n=0,1,2,···,7 . Because this transformation is exactly the same completely positive and trace preserving map as the one given in Eq. (1), the LCR simulates the CPC. The telecom photons are sent through the interference filters (IFs) with a bandwidth of 10 nm and then they are coupled to single-mode fibers followed by two InGaAs avalanche photodetectors (quantum efficiency: 25 %) D2 and D4 which are gated by the electric signal from D1. The electric signal from D3 is connected to a time-to-digital converter (TDC) as a start signal, and electric signals from D2 and D4 are used as stop signals of the TDC. We postselect the events where two stop signals are clicked within the time difference of 2 ns, which guarantees the four-fold coincidence among D1, D2, D3 and D4.

Experimental results
We first characterized the initial two photon pairs from photon pair sources A (ρ 12 ) and B (ρ 34 ). By performing the quantum state tomography [27] with diluted maximum-likelihood algorithm [28], we reconstructed the density operators of the photon pairs as shown in Figs. 3 (a) and (b). Observed fidelities ofρ 12 andρ 34 to |φ + were 0.92 ± 0.01 and 0.94 ± 0.01, respectively, which implies that the two photon pairs were highly entangled. The detection rates ofρ 12 andρ 34 were 920 Hz and 620 Hz, respectively. We estimated that the photon pair generation rate is of the order of 10 −2 , which indicates that the multi-photon emission from the photon sources is negligibly small.
Next we reconstructed the density operators when we applied phase fluctuations to each pho- ton pair by using the LCR. Figures. 3 (c) and (d) show the reconstructed density matrices of ρ ′ 12 andρ ′ 34 with phase fluctuations. Off-diagonal elements of the density matrices disappeared, which indicates that the LCR effectively simulates the phase-damping channel. Observed fidelities ofρ 12 andρ 34 to |φ + were 0.50 ± 0.01 and 0.46 ± 0.02, respectively.
Before demonstrating the entanglement extraction scheme, we performed the Hong-Ou-Mandel (HOM) interference [29] of the photons in modes 1 and 3 which are heralded by the photon detection at D4 and D2, respectively. The experimental result is shown in Fig. 4. We clearly observed the HOM dip around zero delay point. The observed visibility at zero delay was 0.80 ± 0.05. The full width at the half maximum (FWHM) was calculated as ∼ 204 µm by fitting the experimental data with the Gaussian.
Finally, we reconstructed the density operatorρ final when we performed the entanglement extraction scheme on the two photon pairs. The experiment was performed at the zero delay point of the HOM dip in Fig. 4. The reconstructed density matrix is shown in Fig. 5. An observed fidelity ofρ final to |φ + was 0.73 ± 0.07, clearly exceeding the threshold value of 0.5 to show that the extracted pair of photons are entangled. This means that entanglement was successfully distributed at the telecom regime in the presence of corrective phase fluctuations.

Discussion
In the following, we consider the reason for the degradation of the observed values of the visibility of the HOM interference between the photons in modes 1 and 3 and the fidelity of ρ final after the entanglement extraction scheme. Because the probability of the multi-photon pair emission is sufficiently low, the two-photon state obtained by SPDC after IFs is expressed as where |vac is the vacuum state, andâ † k (ω) is a creation operator at the angular frequency of ω in spatial mode k(= 1, 2, 3, 4). Φ(ω, ω ′ ) is a product of the spectral amplitude of the two-photon state generated from the SPDC and the transmission coefficients of the IFs for the visible and telecom photons. Because the detectors distinguish neither angular frequencies nor exact arriving times of the photons, the four-fold coincidence probability P 1234 among the detectors D1, D2, D3 and D4 is given by the sum of all the frequency contributions, which is proportional to dω 1 dω 2 dω 3 dω 4 | vac|â 1 (ω 1 )â 3 (ω 3 )Û|φ (ω 2 ) 1 |φ (ω 4 ) 3 | 2 . Here |φ (ω 2 ) 1 ≡ vac|â 2 (ω 2 )|Ψ 12 = dωΦ(ω, ω 2 )â †