Distillation of photon entanglement using a plasmonic metamaterial

Plasmonics is a rapidly emerging platform for quantum state engineering with the potential for building ultra-compact and hybrid optoelectronic devices. Recent experiments have shown that despite the presence of decoherence and loss, photon statistics and entanglement can be preserved in single plasmonic systems. This preserving ability should carry over to plasmonic metamaterials, whose properties are the result of many individual plasmonic systems acting collectively, and can be used to engineer optical states of light. Here, we report an experimental demonstration of quantum state filtering, also known as entanglement distillation, using a metamaterial. We show that the metamaterial can be used to distill highly entangled states from less entangled states. As the metamaterial can be integrated with other optical components this work opens up the intriguing possibility of incorporating plasmonic metamaterials in on-chip quantum state engineering tasks.

the spectral position of the resonance. Each array has a footprint of 100 µm × 100 µm and contains antennas with a horizontal spacing of 200 nm and nominal lengths between 95 nm and 110 nm. After exposure, the samples were developed in a 1:3 solution of methyl isobuthyl ketone (MIBK) and isopropanol. This process dissolves the long-chained molecules of the photoresist that have been broken up during the exposure process and thus creates a mask. Onto this mask, 30 nm of gold were deposited by high-vacuum electron-beam evaporation. To lift off the PMMA mask and the excess gold, the samples were exposed to a bath of Allresist remover AR 300-70 at 50° C until the lift-off was completed.

Quantum process tomography
The general form of a quantum channel corresponding to a completely positive map on the state ρ is given by ρ → , where the elements of the χ matrix are which gives Tr χ = (1 + T ! )/2. This channel is trace preserving (and unitary) only for T ! = 1. In order to obtain the elements of an experimental χ matrix for a given single-qubit channel ε, one can probe it with the four states H , V , D and R , which allow the reconstruction of the action of ε on the different elements of an arbitrary input state: , . From this it is straightforward to extract out the χ matrix elements [1]. To obtain the different probe state outputs ε i i , we prepare each of the probe states and send them into the metamaterial. The output states are then obtained from quantum state tomography. Note, however, that the channel is expected to be non-trace preserving. Thus we must weight the different output states by their relative probability of being produced, given a probe state was input to the channel. For instance, the probe state V is only expected to be transmitted through the metamaterial with probability T ! in the ideal case, thus the output state ε V V = V V would be produced with probability T ! and any channel reconstruction would need to weight ε V V by the factor T ! . More generally, for a fixed time period we count the number of output states transmitted by a given input probe state when there is no metamaterial present (glass substrate only). This is obtained by measuring the total number of counts for measurements in the H / V basis, providing a reference value, N ! ! , for each probe state i when there is no metamaterial (corresponding to the identity operation). In the presence of the metamaterial we again count the number of output states transmitted by a given probe state using the H / V basis, which provides the value N ! . The relative probability of an output state being produced by the metamaterial given a probe state was input is then given by This weighting of the probe state outputs ε i i leads to a non-trace preserving χ matrix. In Fig. S1, we show the reconstructed χ matrices for the seven different metamaterials studied in our experiment.
The process fidelity F ! = Tr χχ !" χ ! / Tr(χ)Tr(χ !" ) of each χ matrix with respect to the ideal partial polarizer χ !" is maximized over the variable T ! , leading to the ideal χ !" matrices shown to the right of the corresponding experimentally reconstructed ones. The process fidelities are given in the caption along with the maximized T ! values.

Density matrices of distilled states for pure input states
In Figure S2 we show the density matrices of the distilled states from each of the seven metamaterials.

Density matrices of distilled states for non-maximally entangled partially mixed input states
To prepare non-maximally entangled partially mixed states in our experiment, we implement a phase damping channel by using a quartz plate sandwiched between two HWPs inserted into the path of one of the photons. The quartz plate induces phase damping in the polarization basis by introducing a delay between photons with horizontal polarization and those with vertical polarization. This delay is comparable to the coherence time of the two terms in the nonmaximally entangled input state but much shorter than the coincidence window and therefore produces an effective phase damping effect. The HWPs enable the amount of phase damping to be controlled by rotating the polarization basis in which the phase damping occurs. The Kraus operators corresponding to this optical configuration are given by where U !"# (θ) = cos θ sin θ sin θ − cos θ and I is the identity operation. Here, each matrix is written in the H /|V〉 basis, λ represents the degree of phase damping corresponding to the difference of group velocity between each polarization (due to the quartz plate). Using the Kraus representation, the partially mixed state produced by acting on the non-maximally where |Φ ! 〉 corresponds to the pure state defined in Eq. (1), and each element is exactly calculated as the following: We can obtain a simple approximate form of the density matrix for the non-maximally entangled partially mixed state by omitting terms that are higher than the second order of λ and θ, and cross terms to first order, The final state is then approximately given by where K ! represents the Kraus operator of the metamaterial. We obtained the experimental parameters λ !"# and ε !"# summarized in Table 1 using the following relations given by the approximate form of the density matrices where ρ !"# represents the experimentally reconstructed density matrix in the distillation of partially mixed states. A similar derivation to the above can be performed for the nonmaximally entangled partially mixed initial state for Ψ ! . Figure S3 shows all the density matrices of the initial partially mixed states and final distilled states that are summarized in Table 1