Quantum entanglement distillation with metamaterials

Md. Abdullah al Farooqui; Justin Breeland; Muhammad I. Aslam; Mehdi Sadatgol; Şahin K. Özdemir; Mark Tame; Lan Yang; Durdu Ö. Güney

doi:10.1364/OE.23.017941

Quantum entanglement distillation with metamaterials

Md. Abdullah al Farooqui, Justin Breeland, Muhammad I. Aslam, Mehdi Sadatgol, Şahin K. Özdemir, Mark Tame, Lan Yang, and Durdu Ö. Güney

Optics Express
Vol. 23,
Issue 14,
pp. 17941-17954
(2015)
•https://doi.org/10.1364/OE.23.017941

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Md. Abdullah al Farooqui, Justin Breeland, Muhammad I. Aslam, Mehdi Sadatgol, Şahin K. Özdemir, Mark Tame, Lan Yang, and Durdu Ö. Güney, "Quantum entanglement distillation with metamaterials," Opt. Express 23, 17941-17954 (2015)

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Table of Contents Category
- Metamaterials
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Metamaterials (160.3918)
- Plasmonics (250.5403)
- Quantum information and processing (270.5585)
- Quantum optics (270.0270)
- Multiphoton processes (270.4180)

About this Article
History
- Original Manuscript: April 10, 2015
- Manuscript Accepted: June 18, 2015
- Published: July 1, 2015

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Open Access

Abstract

We propose a scheme for the distillation of partially entangled two-photon Bell and three-photon W states using metamaterials. The distillation of partially entangled Bell states is achieved by using two metamaterials with polarization dependence, one of which is rotated by $π / 2$ around the direction of propagation of the photons. On the other hand, the distillation of three-photon W states is achieved by using one polarization dependent metamaterial and two polarization independent metamaterials. Upon transmission of the photons of the partially entangled states through the metamaterials the entanglement of the states increases and they become distilled. This work opens up new directions in quantum optical state engineering by showing how metamaterials can be used to carry out a quantum information processing task.

1. Introduction

Metamaterial structures, man-made and usually periodic with subwavelength feature sizes, enable a wide variety of exotic applications including invisibility cloaks [1], compact antennas [2,3] for mobile stations, quantum levitation [4], optical analogue simulators [5,6], solar photovoltaics [7], metaspacers [8], and many others. Furthermore, the interplay between metamaterials and surface plasmons in the optical region can lead to applications superior to conventional ones, such as ultra-high resolution imaging [9] and high-precision optical lithography [10].

Under sufficiently large wavelengths, many metamaterials can be approximated by a highly dispersive and lossy homogeneous medium with effective constitutive parameters [11–19]. Spontaneous and stimulated emission processes were studied using the quantization of the electromagnetic field in such dispersive dielectric materials [20] and negative index media [21,22], although the results were not generalized to inherently discrete metamaterial structures. When the wavelength of the external electromagnetic field is substantially smaller, the metamaterial structure cannot be approximated by an effective homogeneous medium anymore; rather it starts to behave as a photonic crystal [23]. The history of studying quantum processes in photonic crystals, which can be designed by low-loss materials unlike most metamaterials, is more extensive and richer compared to metamaterials [24–26].

On the other hand, in plasmonics, the effect of subwavelength hole arrays on the properties of polarization-entangled photons has been investigated [27]. Here, it was found experimentally that the two-particle entanglement survives transmission through such a medium, where the plasmonic arrays convert incident entangled photons first into localised surface plasmons and then back to free-space photons. The process was later described by classical scattering [28] and linear transformation theory [29]. The scattering theory approach was further used in Ref [28]. to determine the conditions on polarization-dependent transmission probabilities of photons in partially entangled Bell pairs for entanglement distillation. The quantum description of surface plasmons is now well established [30], and a quantum description of the photon-to-surface plasmon conversion process based on attenuated reflection [31,32] and the scattering of surface plasmon polaritons from a plasmonic beam splitter [33] have been developed, amongst many other schemes [30]. Most relevant to this work is the recent experimental demonstration of effective transduction of multimode quantum correlations achieved by employing localized surface plasmons in plasmonic subwavelength hole arrays [34].

In this letter, we continue along the direction of subwavelength arrays and extend the entanglement distillation process in Ref [28]. to partially entangled multipartite systems, in particular, partially entangled $3$ -photon W states using plasmonic metamaterials [17,35]. We use the global entanglement measure defined in Ref [36]. to quantify the entanglement of the quantum states. This entanglement measure is scalable and can be applied to any number of two-level quantum particles.

Maximally entangled states are central resource for quantum information processing. However, due to decoherence and dissipation during their preparation, storage and distribution, entanglement between particles are degraded resulting in non-maximally entangled or partially mixed states. Entanglement protection, distillation, concentration or purification protocols are needed to extract highly entangled states from non-maximally entangled states. These protocols have been well-studied and experimentally demonstrated for bipartite entangled states [37–40]. As the number of particles forming entangled states increases, the entanglement structure becomes more complex and diverse, and inequivalent entanglement classes emerge. Among these W and GHZ states are the commonly studied multipartite entangled states. Recently, there has been several theoretical and experimental works on the efficient preparation, expansion and fusion of W states to build entanglement webs with large number of nodes [41–46]. Naturally, entanglement protection, distillation, purification and concentration protocols should be extended to such multipartite entangled states. Entanglement distillation schemes for partially entangled [47–49] and arbitrary [50–52] W states based on multiphoton [47–49,52] and single-photon multimode [50] entanglement concentration protocols have been theoretically proposed using linear [47,48,52] and nonlinear [49,50] optical elements, and coupled quantum dot and cavity systems [51]. While these protocols provide relatively more efficient distillation of less-entangled states, our approach provides a simple, fast and straightforward distillation of partially entangled Bell and $3$ -photon W states without requiring any sophisticated protocols and their optical implementation.

2. Background theory

2.1. Entanglement measure

A scalable multipartite entanglement measure for the quantum state $| ψ 〉$ consisting of $n$ two-level quantum particles in the Hilbert space ${(C^{2})}^{\otimes n}$ is defined as [36]

Q (| ψ 〉) = \frac{4}{n} \sum_{j = 1}^{n} D (μ_{j} (0) | ψ 〉, μ_{j} (1) | ψ 〉) .

Here,

D (| u 〉, | v 〉) = \sum_{x < y} {| u_{x} v_{y} - u_{y} v_{x} |}^{2}

is the norm-squared of the wedge product of the quantum states

| u 〉, | v 〉 \in {(C^{2})}^{\otimes n - 1}

, which can be written as

| u 〉 = \sum u_{x} | x 〉

and

| v 〉 = \sum v_{y} | y 〉

, where

0 \leq x, y < 2^{n - 1}

are

(n - 1)

-bit strings. Additionally, considering that the Hilbert space

{(C^{2})}^{\otimes n}

has

2^{n}

basis states

| b_{1} \dots b_{n} 〉

, where

b_{j} \in {0, 1}

, in Eq. (1), we have

μ_{j} (b) | b_{1} \dots b_{n} 〉 = δ_{b b_{j}} | b_{1} \dots \hat{b_{j}} \dots b_{n} 〉 .

In this definition,

\hat{}

denotes absence. Using Eqs. (1-3), it can be shown that for each

n \in ℤ_{\geq 2}, Q : {(C^{2})}^{\otimes n} \to ℝ

is an entanglement measure [34]. For example,

Q (| ψ 〉) = 0

for a product state

| ψ 〉

, and

Q (| ψ 〉) = 1

for the EPR state

| ψ 〉 = (| 00 〉 + | 11 〉) / \sqrt{2} .

2.2. Entanglement distillation

Figure 1 illustrates our entanglement distillation scheme based on plasmonic metamaterials. Incident partially entangled photons travel through a distillation system and emerge as maximally entangled photons at the output. The distillation system consists of a set of appropriately designed plasmonic metamaterials.

Fig. 1 (a) Distillation of partially entangled photons using (b) appropriately designed plasmonic metamaterials.

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Below, we will first show that it is possible to transmit generalized $n$ -photon W states through plasmonic metamaterials without reducing the quality of entanglement. This is the generalization of previously demonstrated plasmon-assisted transmission of maximally entangled Bell pairs [27,28] to the transmission of $n$ -photon W states. Then, we will show the distillation of partially entangled Bell pairs [28] and $3$ -photon W states using plasmonic metamaterials. Although the former has been shown theoretically [28] in the context of Ref [27], the latter has been shown here only.

Consider the $n$ -photon W state,

\begin{array}{l} | W_{n} 〉 = \frac{1}{\sqrt{n}} ({| 1 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} + {| 0 〉}_{1} {| 1 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} \\ + {| 0 〉}_{1} {| 0 〉}_{2} {| 1 〉}_{3} ... {| 0 〉}_{n} + ... + {| 0 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 1 〉}_{n}) . \end{array}

Assume that each photon in the

n

-photon W state described by Eq. (4) is sent to different metamaterial slabs. Then, the final output state after the photons exit the metamaterial slabs becomes,

\begin{array}{l} | {W^{'}}_{n} 〉 = \frac{1}{\sqrt{n}} (t^{(1)} {| 1 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} + t^{(2)} {| 0 〉}_{1} {| 1 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} \\ ​ ​ ​ ​ ​ ​ ​ ​ ​ + t^{(3)} {| 0 〉}_{1} {| 0 〉}_{2} {| 1 〉}_{3} ... {| 0 〉}_{n} + ... ​ + t^{(n)} {| 0 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 1 〉}_{n}), \end{array}

where

{| 0 〉}_{i}

and

{| 1 〉}_{i}

represent horizontal and vertical polarization states for the

i th

photon, respectively, and

t^{(i)} = \frac{t_{01} t_{02} \dots t_{1 i} \dots t_{0 n}}{Z} .

1 / Z

is the normalization factor for the output state

| {W^{'}}_{n} 〉

t_{σ i}

is the probability amplitude for either the horizontally

(σ = 0)

(σ = 1)

vertically polarized ith photon being transmitted into its same initial polarization state after exiting the respective metamaterial slab. This means that Eqs. (5) and (6) cannot be used in their current form for bianisotropic metamaterials [53,54], since we assume no cross-coupling between different polarizations. Although bianisotropic metamaterials may provide additional degree of freedom for further manipulation of incident quantum states, we do not need such a level of complexity here to demonstrate multipartite entanglement distillation. Moreover, metamaterials without or with negligible bianisotropy are readily available [55].

Using the entanglement measure in Eq. (1), the entanglement in the output state $| {W^{'}}_{n} 〉$ in Eq. (5) can be found as

Q (| {W^{'}}_{n} 〉) = \frac{8 \sum_{i < j} {| t^{[i]} |}^{2} {| t^{[j]} |}^{2}}{n {(\sum_{i = 1}^{n} {| t^{[i]} |}^{2})}^{2}},

where we define

t^{[i]} = t^{(i)} Z = t_{01} t_{02} \dots t_{1 i} \dots t_{0 n}

using Eq. (6). For example, we can easily show for

3

-photon W state (i.e.,

n = 3

) that entanglement can be preserved without reducing its quality after the photons are transmitted through plasmonic metamaterial slabs, because for

n = 3

, Eq. (7) can be written as

Q (| {W^{'}}_{3} 〉) = \frac{8}{3} \frac{({| t^{[1]} |}^{2} {| t^{[2]} |}^{2} + {| t^{[1]} |}^{2} {| t^{[3]} |}^{2} + {| t^{[2]} |}^{2} {| t^{[3]} |}^{2})}{{({| t^{[1]} |}^{2} + {| t^{[2]} |}^{2} + {| t^{[3]} |}^{2})}^{2}}

if we assume

t^{[i]} = t^{[j]}

, we obtain

Q (| {W^{'}}_{3} 〉) = Q (| W_{3} 〉) = 8 / 9.

Hence, provided that photons are transmitted through identically designed metamaterial slabs, the original entanglement is not degraded despite photon losses in the metamaterial slabs. This condition can be easily satisfied by polarization independent metamaterials [55]. However, we should mention that even though the quality of the entanglement can be sustained at arbitrarily low transmittances, the efficiency of the process decreases due to losses in the metamaterial slabs.

In fact, Eq. (7) is not only consistent with Refs [27,28], where the plasmon assisted transmission of maximally entangled Bell pairs were shown to be possible without reducing the quality of entanglement, but it also demonstrates that plasmon assisted transmission is valid for generalized $n$ -photon W states. That is

Q (| {W^{'}}_{n} 〉) = Q (| W_{n} 〉) = \frac{4 (n - 1)}{n^{2}} .

Having shown that plasmonic metamaterial does not deteriorate the quality of the entanglement in the W states, now we consider the distillation of partially entangled $n$ -photon W-class states,

\begin{array}{l} | Φ_{n} 〉 = α {| 1 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} \\ + β ({| 0 〉}_{1} {| 1 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} + {| 0 〉}_{1} {| 0 〉}_{2} {| 1 〉}_{3} ... {| 0 〉}_{n} + ... + {| 0 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 1 〉}_{n}), \end{array}

where

{| α |}^{2} + (n - 1) {| β |}^{2} = 1.

After the photons exit the metamaterial slabs, their final state reaches,

\begin{array}{l} | {Φ^{'}}_{n} 〉 = α t^{(1)} {| 1 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} \\ + β (t^{(2)} {| 0 〉}_{1} {| 1 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} + t^{(3)} {| 0 〉}_{1} {| 0 〉}_{2} {| 1 〉}_{3} ... {| 0 〉}_{n} \\ + ... + t^{(n)} {| 0 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 1 〉}_{n}) . \end{array}

Then, the entanglement in the final output state

| {Φ^{'}}_{n} 〉

using Eq. (1) can be found as

Q (| {Φ^{'}}_{n} 〉) = \frac{8}{n} ({| α β t^{(1)} |}^{2} \sum_{i = 2}^{n} | t^{(i)} |^{2} + {| β |}^{4} \sum_{i = 2}^{n} \sum_{i < j} {| t^{(i)} t^{(j)} |}^{2}) .

2.2.1. Distillation of partially entangled Bell states

By setting $n = 2$ in Eq. (11) and using Eq. (14), one can demonstrate the distillation of the resultant partially entangled Bell states $| Φ_{2} 〉$ using plasmonic metamaterials. To achieve this, we first rewrite Eq. (14) for $n = 2$ as

Q (| {Φ^{'}}_{2} 〉) = \frac{4 {| α β |}^{2} τ_{1}}{{({| α |}^{2} + {| β |}^{2} τ_{1})}^{2}},

where

τ_{1} = {| t^{[2]} |}^{2} / {| t^{[1]} |}^{2} .

If the maximal entanglement condition

Q (| {Φ^{'}}_{2} 〉) = 1

as any physical solution, then the partially entangled Bell states

| Φ_{2} 〉

can be distilled by plasmonic metamaterials. Indeed, this condition has a mathematical solution at

τ_{1} = {| α |}^{2} / {| β |}^{2},

which can be satisfied physically, for example, by choosing

| t_{01} | = | t_{12} | = \sqrt{| α |},

| t_{11} | = | t_{02} | = \sqrt{| β |} .

The required probability amplitudes can be achieved by a polarization-dependent plasmonic metamaterial design.

2.2.2. Distillation of partially entangled three-photon states

We can extend the above approach to the distillation of partially entangled $3$ -photon W states. After the partially entangled $3$ -photon W states exit from the plasmonic metamaterial slabs, the degree of entanglement in the final state $| {Φ^{'}}_{3} 〉$ , using Eq. (14), becomes

Q (| {Φ^{'}}_{3} 〉) = \frac{8}{3} \frac{{| α β |}^{2} τ_{2} + {| β |}^{4} τ_{3}^{2}}{{({| α |}^{2} + {| β |}^{2} τ_{2})}^{2}},

where

τ_{2} = \frac{({| t^{[2]} |}^{2} + {| t^{[3]} |}^{2})}{{| t^{[1]} |}^{2}},

τ_{3} = \frac{| t^{[2]} | | t^{[3]} |}{{| t^{[1]} |}^{2}} .

Changing the variables we can write Eqs. (19) and (20), respectively, as

τ_{2} = u^{2} + v^{2},

τ_{3} = u v,

where

u = \frac{| t^{[2]} |}{| t^{[1]} |},

v = \frac{| t^{[3]} |}{| t^{[1]} |} .

Noticing that Eq. (21) is the equation for a circle with radius

\sqrt{τ_{2}}

we can further write variables

u

and

v

in terms of

θ,

such that

u = \sqrt{τ_{2}} \cos θ,

v = \sqrt{τ_{2}} \sin θ,

where

0 \leq θ \leq π / 2 .

Then we can rewrite Eq. (14) as

Q (| {Φ^{'}}_{3} 〉) = \frac{8}{3} \frac{{| α β |}^{2} τ_{2} + {| β |}^{4} {(\frac{τ_{2}}{2} \sin (2 θ))}^{2}}{{({| α |}^{2} + {| β |}^{2} τ_{2})}^{2}} .

In comparison with Eq. (18), which contains two unbounded free parameters (i.e., $τ_{2}$ and $τ_{3}$ ), Eq. (27) contains only one unbounded (i.e., $0 \leq τ_{2} < \infty$ ) and one bounded free parameter (i.e., $0 \leq θ \leq π / 2$ ). This simplifies the analytical determination of what transmittance values are required through each plasmonic metamaterial slab.

The distillation of the incident partially entangled $3$ -photon W states requires $Q (| {Φ^{'}}_{3} 〉) = 8 / 9$ as can be calculated from Eq. (10). Indeed, one can show that this is exactly the maximum value that $Q (| {Φ^{'}}_{3} 〉)$ can take and is achieved when $τ_{2} = τ_{2 \max} = 2 {| α |}^{2} / {| β |}^{2}$ and $θ = θ_{\max} = π / 4 .$ Substituting $τ_{2 \max}$ and $θ_{\max}$ into Eqs. (25) and (26) gives

u = v = \frac{| α |}{| β |} .

Using Eqs. (8), (23), (24), and (28), and choosing

| t_{02} | = | t_{03} | = | t_{12} | = | t_{13} |,

| t_{01} | = | α |,

| t_{11} | = | β |,

we can obtain

Q (| {Φ^{'}}_{3} 〉) = 8 / 9 .

Although this set of transmission probabilities is not the only solution set for the distillation of partially entangled

3

-photon W states, it can be easily satisfied by one polarization-dependent and one polarization-independent plasmonic metamaterial design, as we will now show.

3. Proof-of-principle metamaterials and protocols

3.1. Metamaterial designs for entanglement distillation

The sketches in Fig. 2 show two different views (i.e., front and back) of the unit cell of a metamaterial structure which can be tailored as either a polarization-independent or polarization-dependent plasmonic metamaterial by choosing the strip widths $w_{0}$ and $w_{1}$ either the same or slightly different, respectively. The two-possible incident field configurations are also illustrated. $| 0 〉$ and $| 1 〉$ denote horizontal and vertical polarizations, respectively, and k denotes the direction of propagation. This structure is the two-dimensional (i.e., functional for two orthogonal polarizations) version of the surface plasmon driven negative index metamaterial studied in detail in Ref [17].

Fig. 2 Two different views of the unit cell of a plasmonic metamaterial structure. The unit cell consists of a gold thin film in the middle and two gold nano-patterned structures on both sides of the thin film. The nano-patterned structures are the same on both sides except that they are diagonally shifted by $a / \sqrt{2}$ in their planes with respect to each other where $a$ is the unit cell size for the square lattice. The metamaterial is designed to be functional under normally incident light indicated by wave vector k and polarizations $| 0 〉$ and $| 1 〉 .$ The metamaterial can be designed as polarization-independent (or polarization dependent) by choosing the strip widths $w_{0}$ and $w_{1}$ equal (or slightly different).

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3.1.1. Polarization independent metamaterial

Figure 3(a) shows the transmission, reflection, and absorption spectra of an example polarization-independent design under normally incident light. All the simulations are performed by using finite element based COMSOL software package. Gold layers shown in Fig. 2 are described by Drude model with the bulk plasma frequency of $f_{p} = 2175 THz$ and the collision frequency $f_{c} = 6.5 THz$ [56]. Polyimide has the relative permittivity $ε_{r} = 3.5.$ All the geometric parameters used in the simulation are given in the caption. This metamaterial structure shows three different extraordinary transmission windows near the plasmonic resonances around 375THz, 450THz, and 500THz. The retrieved [11–19] effective index results in Fig. 3(b) show that the first two lower frequency resonances provide positive index bands while the high frequency resonance provides a negative index band with a low figure of merit. The Lorentzian-like resonances observed in the retrieved effective permittivity and permeability displayed in Fig. 3(c) show that the first two lower frequency resonances have electric type while the high frequency resonance is of a magnetic type.

Fig. 3 (a) Reflectance (R), transmittance (T), and absorbance (A) of a polarization independent plasmonic metamaterial. Retrieved effective (b) refractive index, (c) relative electrical permittivity ( $ε_{r} = {ε^{'}}_{r} + i {ε^{″}}_{r}$ ) and relative magnetic permeability ( $μ_{r} = {μ^{'}}_{r} + i {μ^{″}}_{r}$ ). The strip widths $w_{0} = w_{1} = 40 nm .$ The lattice constant $a = 80 nm .$ The thicknesses of the thin film and the strips are 5nm and 11nm, respectively. The strips are separated from the thin film in the middle by 8nm. The thickness of the unit cell along the direction of propagation is 100nm. The dashed green line in (b) indicates the first Brillouin zone edge.

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3.1.2. Polarization dependent metamaterial

By choosing $w_{1}$ slightly different than $w_{0},$ we can lift the degeneracy between the horizontal and vertical polarizations. The transmission spectra for the polarization-dependent plasmonic metamaterial, choosing $w_{0} = 39 nm$ and $w_{1} = 45 nm$ and keeping the remaining parameters fixed, is displayed in Fig. 4. The green and blue curves correspond to transmittance through the metamaterial structure for horizontal and vertical polarizations, respectively.

Fig. 4 Transmittance for horizontally and vertically polarized light. $w_{0} = 39 nm,$ $w_{1} = 45 nm .$ Other parameters are the same as in Fig. 3.

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3.2. Entanglement distillation protocol

Having shown the metamaterial designs above, we describe below how to use these plasmonic metamaterials for the distillation of partially entangled (i) Bell states $| Φ_{2} 〉$ and (ii) $3$ -photon W states $| Φ_{3} 〉 .$

3.2.1. Distillation protocol for partially entangled Bell states

The sketch in Fig. 5 describes the distillation of partially entangled Bell states $| Φ_{2} 〉 .$ Using Eqs. (16) and (17), we notice that we only need two polarization-dependent plasmonic metamaterials with the same design. However, one of the metamaterial structures has to be rotated by $π / 2$ (denoted by * in Fig. 5) around the normal axis. The incident partially entangled Bell state $| Φ_{2} 〉 = α {| 1 〉}_{1} {| 0 〉}_{2} + β {| 0 〉}_{1} {| 1 〉}_{2}$ is then distilled when the Photon 1 is transmitted through the metamaterial of Design I and the Photon 2 is transmitted through the metamaterial of Design I*, such that the transmittances through the metamaterial of Design I for the horizontal and vertical polarizations are tuned to $| α |$ and $| β |,$ (i.e., modulus of probability amplitudes) respectively, and Design I* is the orthogonal replica of Design I. We have already designed such a metamaterial structure with the transmittance spectra shown in Fig. 4. The condition for Design I is satisfied at frequency 396THz where the transmittance for the horizontal and vertical polarizations are $| α | = 0.6$ and $| β | = 0.8,$ respectively. That is why this metamaterial structure, together with its orthogonal replica, can be readily used for the distillation of partially entangled Bell states $| Φ_{2} 〉 .$

Fig. 5 Distillation of partially entangled Bell states $| Φ_{2} 〉 .$ The first photon (i.e., Photon 1) in the partially entangled state travels through the metamaterial of Design I and the second photon (i.e., Photon 2) travels through the metamaterial of Design I*. The transmittance of the metamaterial with Design I is $| α |$ for the horizontal polarization $| 0 〉$ and $| β |$ for the vertical polarization $| 1 〉 .$ The metamaterial of Design I* is obtained by rotating the Design I around k by $π / 2 .$ For example, for $| α | = 0.6$ and $| β | = 0.8,$ Design I refers to the metamaterial structure operating at 396THz, considered in Fig. 4.

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3.2.2. Distillation protocol for partially entangled three-photon W states

Similarly, Fig. 6 describes the distillation of partially entangled $3$ -photon W states $| Φ_{3} 〉 .$ In this case, using Eqs. (29)-(31), we notice that in addition to a polarization-dependent metamaterial of Design I², we also need two polarization-independent metamaterials of Design II. The distillation of the partially entangled $3$ -photon W states $| Φ_{3} 〉$ is achieved by sending Photon 1 through the polarization-dependent metamaterial of Design I², Photons 2 and 3 through the polarization-independent metamaterials of Design II. The condition for Design I² can be satisfied in Fig. 4 at frequency 418THz where the transmittance for horizontal and vertical polarizations are ${| α |}^{2} = 0.8$ and ${| β |}^{2} = 0.2,$ respectively. On the other hand, the condition for Design II can be satisfied by using the polarization-independent metamaterial structure in Fig. 3.

Fig. 6 Distillation of partially entangled $3$ -photon W states $| Φ_{3} 〉 .$ The first photon (i.e., Photon 1) in the partially entangled state travels through the metamaterial of Design I², the second (i.e., Photon 2) and third photons (i.e., Photon 3) travel through polarization-independent metamaterials of Design II. The transmittance of the metamaterial with Design I² is ${| α |}^{2}$ for the horizontal polarization $| 0 〉$ and ${| β |}^{2}$ for the vertical polarization $| 1 〉$ (i.e., compare the required transmittances for Design I and Design I² for the naming). The metamaterial with Design II has polarization independent transmittance ${| γ |}^{2} .$ For example, for ${| α |}^{2} = 0.8$ and ${| β |}^{2} = 0.2,$ Design I² refers to the metamaterial structure operating at 418THz, considered in Fig. 4, while Design II refers to the polarization independent metamaterial structure considered in Fig. 3, operating at the same frequency.

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4. Conclusions

In summary, we have presented a scheme for the distillation of partially entangled Bell states and $3$ -photon W states using plasmonic metamaterials. Our technique extends the previous theoretical plasmon assisted distillation of partially entangled Bell states [28] to W states. In comparison with other W state distillation schemes [47–52], which have also been theoretical, we present a fast and straightforward method for the distillation of partially entangled $3$ -photon W states without requiring any sophisticated protocols and their associated optical implementations. Although we have used our own surface plasmon driven metamaterial designs to introduce the concept above, the realization of the proposed scheme is feasible by already fabricated plasmonic metamaterial structures. For example, the well studied fishnet metamaterial structures [55,57,58] would be ideal experimental platforms for such entanglement distillation processes. Our approach for entanglement distillation can be scaled or generalized to partially entangled or arbitrary $n$ -photon W states. Arbitrary manipulation of multipartite quantum states may be possible by appropriately designed plasmonic metamaterials. This capability for quantum manipulation of light may be further enhanced by tunable [59–67] and/or bianisotropic metamaterials [53,54] for optical quantum information processing applications. We should also mention that the new application of metamaterials presented here is another interesting aspect of our work and may lead to new directions by merging quantum information processing and metamaterials.

Acknowledgment

Work at the University of KwaZulu-Natal was supported by South African National Research Foundation and the National Institute for Theoretical Physics. The work at Michigan Technological University was supported in part by the National Science Foundation under grant Award No. ECCS-1202443.

References and links

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2. I. Bulu, H. Caglayan, K. Aydin, and E. Ozbay, “Compact size highly directive antennas based on the SRR metamaterial medium,” New J. Phys. 7, 223 (2005). [CrossRef]

3. H. Odabasi, F. Teixeira, and D. O. Güney, “Electrically small, complementary electric-field-coupled resonator antennas,” J. Appl. Phys. 113(8), 084903 (2013). [CrossRef]

4. U. Leonhardt and T. G. Philbin, “Quantum levitation by left-handed metamaterials,” New J. Phys. 9(8), 254 (2007). [CrossRef]

5. D. O. Güney and D. A. Meyer, “Negative refraction gives rise to the Klein paradox,” Phys. Rev. A 79(6), 063834 (2009). [CrossRef]

6. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]

7. A. Vora, J. Gwamuri, N. Pala, A. Kulkarni, J. M. Pearce, and D. O. Güney, “Exchanging Ohmic losses in metamaterial absorbers with useful optical absorption for photovoltaics,” Sci Rep 4, 4901 (2014). [CrossRef] [PubMed]

8. M. I. Aslam and D. O. Güney, “On negative index metamaterial spacers and their unusual optical properties,” Prog. Electromagn. Res. B 47, 203–217 (2013). [CrossRef]

9. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

10. T. Xu, Y. Zhao, J. Ma, C. Wang, J. Cui, C. Du, and X. Luo, “Sub-diffraction-limited interference photolithography with metamaterials,” Opt. Express 16(18), 13579–13584 (2008). [CrossRef] [PubMed]

11. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002). [CrossRef]

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23. F. Dominec, C. Kadlec, H. Němec, P. Kužel, and F. Kadlec, “Transition between metamaterial and photonic-crystal behavior in arrays of dielectric rods,” Opt. Express 22(25), 30492–30503 (2014). [CrossRef] [PubMed]

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45. F. Ozaydin, S. Bugu, C. Yesilyurt, A. A. Altintas, M. Tame, and Ş. K. Özdemir, “Fusing multiple W states simultaneously with a Fredkin gate,” Phys. Rev. A 89(4), 042311 (2014). [CrossRef]

46. T. Tashima, T. Kitano, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Demonstration of local expansion toward large-scale entangled webs,” Phys. Rev. Lett. 105(21), 210503 (2010). [CrossRef] [PubMed]

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References

View by:
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|
Year
|
Author
|
Publication

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]
I. Bulu, H. Caglayan, K. Aydin, and E. Ozbay, “Compact size highly directive antennas based on the SRR metamaterial medium,” New J. Phys. 7, 223 (2005).
[Crossref]
H. Odabasi, F. Teixeira, and D. O. Güney, “Electrically small, complementary electric-field-coupled resonator antennas,” J. Appl. Phys. 113(8), 084903 (2013).
[Crossref]
U. Leonhardt and T. G. Philbin, “Quantum levitation by left-handed metamaterials,” New J. Phys. 9(8), 254 (2007).
[Crossref]
D. O. Güney and D. A. Meyer, “Negative refraction gives rise to the Klein paradox,” Phys. Rev. A 79(6), 063834 (2009).
[Crossref]
D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]
A. Vora, J. Gwamuri, N. Pala, A. Kulkarni, J. M. Pearce, and D. O. Güney, “Exchanging Ohmic losses in metamaterial absorbers with useful optical absorption for photovoltaics,” Sci Rep 4, 4901 (2014).
[Crossref] [PubMed]
M. I. Aslam and D. O. Güney, “On negative index metamaterial spacers and their unusual optical properties,” Prog. Electromagn. Res. B 47, 203–217 (2013).
[Crossref]
J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]
T. Xu, Y. Zhao, J. Ma, C. Wang, J. Cui, C. Du, and X. Luo, “Sub-diffraction-limited interference photolithography with metamaterials,” Opt. Express 16(18), 13579–13584 (2008).
[Crossref] [PubMed]
D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]
X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).
[Crossref] [PubMed]
D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(3), 036617 (2005).
[Crossref] [PubMed]
J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B 80(3), 035109 (2009).
[Crossref]
D. O. Güney, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Connected bulk negative index photonic metamaterials,” Opt. Lett. 34(4), 506–508 (2009).
[Crossref] [PubMed]
D. O. Güney, T. Koschny, and C. M. Soukoulis, “Intra-connected three-dimensionally isotropic bulk negative index photonic metamaterial,” Opt. Express 18(12), 12348–12353 (2010).
[Crossref] [PubMed]
M. I. Aslam and D. O. Güney, “Surface plasmon driven scalable low-loss negative-index metamaterial in the visible spectrum,” Phys. Rev. B 84(19), 195465 (2011).
[Crossref]
M. I. Aslam and D. O. Güney, “Dual band double-negative polarization independent metamaterial for the visible spectrum,” J. Opt. Soc. Am. B 29(10), 2839–2847 (2012).
[Crossref]
S. Arslanagic, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antenn. Propag. M. 55(2), 91–106 (2013).
[Crossref]
P. W. Miloni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Opt. 42(10), 1991–2004 (1995).
[Crossref]
P. W. Milonni and G. J. Maclay, “Quantized-field description of light in negative-index media,” Opt. Commun. 228(1-3), 161–165 (2003).
[Crossref]
M. Ligare, “Propagation of quantized fields through negative-index media,” J. Mod. Opt. 58(17), 1551–1559 (2011).
[Crossref]
F. Dominec, C. Kadlec, H. Němec, P. Kužel, and F. Kadlec, “Transition between metamaterial and photonic-crystal behavior in arrays of dielectric rods,” Opt. Express 22(25), 30492–30503 (2014).
[Crossref] [PubMed]
J. Vucković, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(1 Pt 2), 016608 (2002).
[PubMed]
D. O. Güney and D. A. Meyer, “Creation of entanglement and implementation of quantum logic gate operations using a three-dimensional photonic crystal single-mode cavity,” J. Opt. Soc. Am. B 24(2), 283–294 (2007).
[Crossref]
D. O. Güney and D. A. Meyer, “Integrated conditional teleportation and readout circuit based on a photonic crystal single chip,” J. Opt. Soc. Am. B 24(2), 391–397 (2007).
[Crossref]
E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Plasmon-assisted transmission of entangled photons,” Nature 418(6895), 304–306 (2002).
[Crossref] [PubMed]
J. L. van Velsen, J. Tworzydlo, and C. W. J. Beenakker, “Scattering theory of plasmon-assisted entanglement transfer and distillation,” Phys. Rev. A 68(4), 043807 (2003).
[Crossref]
E. Moreno, F. J. García-Vidal, D. Erni, J. I. Cirac, and L. Martín-Moreno, “Theory of plasmon-assisted transmission of entangled photons,” Phys. Rev. Lett. 92(23), 236801 (2004).
[Crossref] [PubMed]
M. S. Tame, K. R. McEnery, S. K. Ozdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013).
[Crossref]
M. S. Tame, C. Lee, J. Lee, D. Ballester, M. Paternostro, A. V. Zayats, and M. S. Kim, “Single-photon excitation of surface plasmon polaritons,” Phys. Rev. Lett. 101(19), 190504 (2008).
[Crossref] [PubMed]
D. Ballester, M. S. Tame, C. Lee, J. Lee, and M. S. Kim, “Long-range surface plasmon-polariton excitation at the quantum level,” Phys. Rev. A 79(5), 053845 (2009).
[Crossref]
D. Ballester, M. S. Tame, and M. S. Kim, “Quantum theory of surface-plasmon polariton scattering,” Phys. Rev. A 82(1), 012325 (2010).
[Crossref]
B. J. Lawrie, P. G. Evans, and R. C. Pooser, “Extraordinary optical transmission of multimode quantum correlations via localized surface plasmons,” Phys. Rev. Lett. 110(15), 156802 (2013).
[Crossref] [PubMed]
D. O. Güney, Th. Koschny, and C. M. Soukoulis, “Surface plasmon driven electric and magnetic resonators for metamaterials,” Phys. Rev. B 83(4), 045107 (2011).
[Crossref]
D. A. Meyer and N. R. Wallach, “Global entanglement in multipartite systems,” J. Math. Phys. 43(9), 4273–4278 (2002).
[Crossref]
T. Yamamoto, M. Koashi, S. K. Özdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421(6921), 343–346 (2003).
[Crossref] [PubMed]
P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature 409(6823), 1014–1017 (2001).
[Crossref] [PubMed]
R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref] [PubMed]
T. Yamamoto, K. Hayashi, S. K. Özdemir, M. Koashi, and N. Imoto, “Robust photonic entanglement distribution by state-independent encoding onto decoherence-free subspace,” Nat. Photonics 2(8), 488–491 (2008).
[Crossref]
T. Tashima, S. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local expansion of photonic W state using a polarization-dependent beamsplitter,” New J. Phys. 11(2), 023024 (2009).
[Crossref]
T. Tashima, T. Wakatsuki, S. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local transformation of two Einstein-Podolsky-Rosen photon pairs into a three-photon w state,” Phys. Rev. Lett. 102(13), 130502 (2009).
[Crossref] [PubMed]
P. Walther, K. J. Resch, and A. Zeilinger, “Local conversion of Greenberger-Horne-Zeilinger states to approximate W states,” Phys. Rev. Lett. 94(24), 240501 (2005).
[Crossref]
Ş. K. Özdemir, E. Matsunaga, T. Tashima, T. Yamamoto, M. Koashi, and N. Imoto, “An optical fusion gate for W-states,” New J. Phys. 13(10), 103003 (2011).
[Crossref]
F. Ozaydin, S. Bugu, C. Yesilyurt, A. A. Altintas, M. Tame, and Ş. K. Özdemir, “Fusing multiple W states simultaneously with a Fredkin gate,” Phys. Rev. A 89(4), 042311 (2014).
[Crossref]
T. Tashima, T. Kitano, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Demonstration of local expansion toward large-scale entangled webs,” Phys. Rev. Lett. 105(21), 210503 (2010).
[Crossref] [PubMed]
B. Gu, D. Quan, and S. Xiao, “Multi-photon entanglement concentration protocol for partially entangled W states with projection measurement,” Int. J. Theor. Phys. 51(9), 2966–2973 (2012).
[Crossref]
B. Gu, “Single-photon-assisted entanglement concentration of partially entangled multiphoton W states with linear optics,” J. Opt. Soc. Am. B 29, 1685–1689 (2012).
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2014 (4)

A. Vora, J. Gwamuri, N. Pala, A. Kulkarni, J. M. Pearce, and D. O. Güney, “Exchanging Ohmic losses in metamaterial absorbers with useful optical absorption for photovoltaics,” Sci Rep 4, 4901 (2014).
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F. Dominec, C. Kadlec, H. Němec, P. Kužel, and F. Kadlec, “Transition between metamaterial and photonic-crystal behavior in arrays of dielectric rods,” Opt. Express 22(25), 30492–30503 (2014).
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F. Ozaydin, S. Bugu, C. Yesilyurt, A. A. Altintas, M. Tame, and Ş. K. Özdemir, “Fusing multiple W states simultaneously with a Fredkin gate,” Phys. Rev. A 89(4), 042311 (2014).
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K. R. McEnery, M. S. Tame, S. A. Maier, and M. S. Kim, “Tunable negative permeability in a quantum plasmonic metamaterial,” Phys. Rev. A 89(1), 013822 (2014).
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2013 (8)

L. Zhou, Y.-B. Sheng, W.-W. Cheng, L.-Y. Gong, and S.-M. Zhao, “Efficient entanglement concentration for arbitrary single-photon multimode W state,” J. Opt. Soc. Am. B 30(1), 71–78 (2013).
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M. S. Tame, K. R. McEnery, S. K. Ozdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013).
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M. I. Aslam and D. O. Güney, “On negative index metamaterial spacers and their unusual optical properties,” Prog. Electromagn. Res. B 47, 203–217 (2013).
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H. Odabasi, F. Teixeira, and D. O. Güney, “Electrically small, complementary electric-field-coupled resonator antennas,” J. Appl. Phys. 113(8), 084903 (2013).
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2012 (4)

M. I. Aslam and D. O. Güney, “Dual band double-negative polarization independent metamaterial for the visible spectrum,” J. Opt. Soc. Am. B 29(10), 2839–2847 (2012).
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B. Gu, D. Quan, and S. Xiao, “Multi-photon entanglement concentration protocol for partially entangled W states with projection measurement,” Int. J. Theor. Phys. 51(9), 2966–2973 (2012).
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2011 (5)

Ş. K. Özdemir, E. Matsunaga, T. Tashima, T. Yamamoto, M. Koashi, and N. Imoto, “An optical fusion gate for W-states,” New J. Phys. 13(10), 103003 (2011).
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M. Ligare, “Propagation of quantized fields through negative-index media,” J. Mod. Opt. 58(17), 1551–1559 (2011).
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2010 (4)

D. O. Güney, T. Koschny, and C. M. Soukoulis, “Intra-connected three-dimensionally isotropic bulk negative index photonic metamaterial,” Opt. Express 18(12), 12348–12353 (2010).
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2009 (9)

T. Tashima, S. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local expansion of photonic W state using a polarization-dependent beamsplitter,” New J. Phys. 11(2), 023024 (2009).
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D. O. Güney, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Connected bulk negative index photonic metamaterials,” Opt. Lett. 34(4), 506–508 (2009).
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D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
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D. Ballester, M. S. Tame, C. Lee, J. Lee, and M. S. Kim, “Long-range surface plasmon-polariton excitation at the quantum level,” Phys. Rev. A 79(5), 053845 (2009).
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2008 (8)

M. S. Tame, C. Lee, J. Lee, D. Ballester, M. Paternostro, A. V. Zayats, and M. S. Kim, “Single-photon excitation of surface plasmon polaritons,” Phys. Rev. Lett. 101(19), 190504 (2008).
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T. Yamamoto, K. Hayashi, S. K. Özdemir, M. Koashi, and N. Imoto, “Robust photonic entanglement distribution by state-independent encoding onto decoherence-free subspace,” Nat. Photonics 2(8), 488–491 (2008).
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2007 (7)

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
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D. O. Güney and D. A. Meyer, “Integrated conditional teleportation and readout circuit based on a photonic crystal single chip,” J. Opt. Soc. Am. B 24(2), 391–397 (2007).
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2006 (2)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
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2005 (3)

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2004 (2)

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).
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2003 (3)

J. L. van Velsen, J. Tworzydlo, and C. W. J. Beenakker, “Scattering theory of plasmon-assisted entanglement transfer and distillation,” Phys. Rev. A 68(4), 043807 (2003).
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2001 (1)

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T. H. Hand and S. A. Cummer, “Frequency tunable electromagnetic metamaterial using ferroelectric loaded split rings,” J. Appl. Phys. 103(6), 066105 (2008).
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Grzegorczyk, T. M.

Gu, B.

B. Gu, “Single-photon-assisted entanglement concentration of partially entangled multiphoton W states with linear optics,” J. Opt. Soc. Am. B 29, 1685–1689 (2012).

Güney, D. O.

M. I. Aslam and D. O. Güney, “On negative index metamaterial spacers and their unusual optical properties,” Prog. Electromagn. Res. B 47, 203–217 (2013).
[Crossref]

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[Crossref]

M. I. Aslam and D. O. Güney, “Dual band double-negative polarization independent metamaterial for the visible spectrum,” J. Opt. Soc. Am. B 29(10), 2839–2847 (2012).
[Crossref]

M. I. Aslam and D. O. Güney, “Surface plasmon driven scalable low-loss negative-index metamaterial in the visible spectrum,” Phys. Rev. B 84(19), 195465 (2011).
[Crossref]

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[Crossref] [PubMed]

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Appl. Opt. (1)

Appl. Phys. Lett. (2)

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Electron. Lett. (1)

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IEEE Antenn. Propag. M. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

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Int. J. Theor. Phys. (1)

J. Appl. Phys. (3)

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J. Math. Phys. (1)

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Fig. 1 (a) Distillation of partially entangled photons using (b) appropriately designed plasmonic metamaterials.

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a / \sqrt{2}

in their planes with respect to each other where

a

is the unit cell size for the square lattice. The metamaterial is designed to be functional under normally incident light indicated by wave vector k and polarizations

| 0 〉

and

| 1 〉 .

The metamaterial can be designed as polarization-independent (or polarization dependent) by choosing the strip widths

w_{0}

and

w_{1}

equal (or slightly different).

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ε_{r} = {ε^{'}}_{r} + i {ε^{″}}_{r}

) and relative magnetic permeability (

μ_{r} = {μ^{'}}_{r} + i {μ^{″}}_{r}

). The strip widths

w_{0} = w_{1} = 40 nm .

The lattice constant

a = 80 nm .

The thicknesses of the thin film and the strips are 5nm and 11nm, respectively. The strips are separated from the thin film in the middle by 8nm. The thickness of the unit cell along the direction of propagation is 100nm. The dashed green line in (b) indicates the first Brillouin zone edge.

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Fig. 4 Transmittance for horizontally and vertically polarized light.

w_{0} = 39 nm,

w_{1} = 45 nm .

Other parameters are the same as in Fig. 3.

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Fig. 5 Distillation of partially entangled Bell states

| Φ_{2} 〉 .

The first photon (i.e., Photon 1) in the partially entangled state travels through the metamaterial of Design I and the second photon (i.e., Photon 2) travels through the metamaterial of Design I*. The transmittance of the metamaterial with Design I is

| α |

for the horizontal polarization

| 0 〉

and

| β |

for the vertical polarization

| 1 〉 .

The metamaterial of Design I* is obtained by rotating the Design I around k by

π / 2 .

For example, for

| α | = 0.6

and

| β | = 0.8,

Design I refers to the metamaterial structure operating at 396THz, considered in Fig. 4.

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Fig. 6 Distillation of partially entangled

3

-photon W states

| Φ_{3} 〉 .

The first photon (i.e., Photon 1) in the partially entangled state travels through the metamaterial of Design I², the second (i.e., Photon 2) and third photons (i.e., Photon 3) travel through polarization-independent metamaterials of Design II. The transmittance of the metamaterial with Design I² is

{| α |}^{2}

for the horizontal polarization

| 0 〉

and

{| β |}^{2}

for the vertical polarization

| 1 〉

(i.e., compare the required transmittances for Design I and Design I² for the naming). The metamaterial with Design II has polarization independent transmittance

{| γ |}^{2} .

For example, for

{| α |}^{2} = 0.8

and

{| β |}^{2} = 0.2,

Design I² refers to the metamaterial structure operating at 418THz, considered in Fig. 4, while Design II refers to the polarization independent metamaterial structure considered in Fig. 3, operating at the same frequency.

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Equations (31)

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Q (| ψ 〉) = \frac{4}{n} \sum_{j = 1}^{n} D (μ_{j} (0) | ψ 〉, μ_{j} (1) | ψ 〉) .

D (| u 〉, | v 〉) = \sum_{x < y} {| u_{x} v_{y} - u_{y} v_{x} |}^{2}

μ_{j} (b) | b_{1} \dots b_{n} 〉 = δ_{b b_{j}} | b_{1} \dots \hat{b_{j}} \dots b_{n} 〉 .

\begin{array}{l} | W_{n} 〉 = \frac{1}{\sqrt{n}} ({| 1 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} + {| 0 〉}_{1} {| 1 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} \\ + {| 0 〉}_{1} {| 0 〉}_{2} {| 1 〉}_{3} ... {| 0 〉}_{n} + ... + {| 0 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 1 〉}_{n}) . \end{array}

\begin{array}{l} | {W^{'}}_{n} 〉 = \frac{1}{\sqrt{n}} (t^{(1)} {| 1 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} + t^{(2)} {| 0 〉}_{1} {| 1 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} \\ ​ ​ ​ ​ ​ ​ ​ ​ ​ + t^{(3)} {| 0 〉}_{1} {| 0 〉}_{2} {| 1 〉}_{3} ... {| 0 〉}_{n} + ... ​ + t^{(n)} {| 0 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 1 〉}_{n}), \end{array}

t^{(i)} = \frac{t_{01} t_{02} \dots t_{1 i} \dots t_{0 n}}{Z} .

Q (| {W^{'}}_{n} 〉) = \frac{8 \sum_{i < j} {| t^{[i]} |}^{2} {| t^{[j]} |}^{2}}{n {(\sum_{i = 1}^{n} {| t^{[i]} |}^{2})}^{2}},

t^{[i]} = t^{(i)} Z = t_{01} t_{02} \dots t_{1 i} \dots t_{0 n}

Q (| {W^{'}}_{3} 〉) = \frac{8}{3} \frac{({| t^{[1]} |}^{2} {| t^{[2]} |}^{2} + {| t^{[1]} |}^{2} {| t^{[3]} |}^{2} + {| t^{[2]} |}^{2} {| t^{[3]} |}^{2})}{{({| t^{[1]} |}^{2} + {| t^{[2]} |}^{2} + {| t^{[3]} |}^{2})}^{2}}

Q (| {W^{'}}_{n} 〉) = Q (| W_{n} 〉) = \frac{4 (n - 1)}{n^{2}} .

\begin{array}{l} | Φ_{n} 〉 = α {| 1 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} \\ + β ({| 0 〉}_{1} {| 1 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} + {| 0 〉}_{1} {| 0 〉}_{2} {| 1 〉}_{3} ... {| 0 〉}_{n} + ... + {| 0 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 1 〉}_{n}), \end{array}

{| α |}^{2} + (n - 1) {| β |}^{2} = 1.

\begin{array}{l} | {Φ^{'}}_{n} 〉 = α t^{(1)} {| 1 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} \\ + β (t^{(2)} {| 0 〉}_{1} {| 1 〉}_{2} {| 0 〉}_{3} ... {| 0 〉}_{n} + t^{(3)} {| 0 〉}_{1} {| 0 〉}_{2} {| 1 〉}_{3} ... {| 0 〉}_{n} \\ + ... + t^{(n)} {| 0 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} ... {| 1 〉}_{n}) . \end{array}

Q (| {Φ^{'}}_{n} 〉) = \frac{8}{n} ({| α β t^{(1)} |}^{2} \sum_{i = 2}^{n} | t^{(i)} |^{2} + {| β |}^{4} \sum_{i = 2}^{n} \sum_{i < j} {| t^{(i)} t^{(j)} |}^{2}) .

Q (| {Φ^{'}}_{2} 〉) = \frac{4 {| α β |}^{2} τ_{1}}{{({| α |}^{2} + {| β |}^{2} τ_{1})}^{2}},

| t_{01} | = | t_{12} | = \sqrt{| α |},

| t_{11} | = | t_{02} | = \sqrt{| β |} .

Q (| {Φ^{'}}_{3} 〉) = \frac{8}{3} \frac{{| α β |}^{2} τ_{2} + {| β |}^{4} τ_{3}^{2}}{{({| α |}^{2} + {| β |}^{2} τ_{2})}^{2}},

τ_{2} = \frac{({| t^{[2]} |}^{2} + {| t^{[3]} |}^{2})}{{| t^{[1]} |}^{2}},

τ_{3} = \frac{| t^{[2]} | | t^{[3]} |}{{| t^{[1]} |}^{2}} .

τ_{2} = u^{2} + v^{2},

τ_{3} = u v,

u = \frac{| t^{[2]} |}{| t^{[1]} |},

v = \frac{| t^{[3]} |}{| t^{[1]} |} .

u = \sqrt{τ_{2}} \cos θ,

v = \sqrt{τ_{2}} \sin θ,

Q (| {Φ^{'}}_{3} 〉) = \frac{8}{3} \frac{{| α β |}^{2} τ_{2} + {| β |}^{4} {(\frac{τ_{2}}{2} \sin (2 θ))}^{2}}{{({| α |}^{2} + {| β |}^{2} τ_{2})}^{2}} .

u = v = \frac{| α |}{| β |} .

| t_{02} | = | t_{03} | = | t_{12} | = | t_{13} |,

| t_{01} | = | α |,

| t_{11} | = | β |,