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Generating arbitrary photon-number entangled states for continuous-variable quantum informatics

Su-Yong Lee, Jiyong Park, Hai-Woong Lee, and Hyunchul Nha

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Abstract

We propose two experimental schemes that can produce an arbitrary photon-number entangled state (PNES) in a finite dimension. This class of entangled states naturally includes non-Gaussian continuous-variable (CV) states that may provide some practical advantages over the Gaussian counterparts (two-mode squeezed states). We particularly compare the entanglement characteristics of the Gaussian and the non-Gaussian states in view of the degree of entanglement and the Einstein-Podolsky-Rosen correlation, and further discuss their applications to the CV teleportation and the nonlocality test. The experimental imperfection due to the on-off photodetectors with nonideal efficiency is also considered in our analysis to show the feasibility of our schemes within existing technologies.

© 2012 Optical Society of America

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Figures (7)
Fig. 1
Fig. 1 (a) Degree of entanglement and (b) EPR correlation for the states: |TMSS〉 (blue solid) as a function of the squeezing parameter s, and ∑ n = 0 N C n | n 〉 a | n 〉 b at N = 1 (red dotted), N = 2 (red dashed), N = 10 (red dot-dashed).

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Fig. 2
Fig. 2 (a) Average fidelity in teleporting a coherent state and (b) Bell parameter BBW as a function of the squeezing parameter s for the |TMSS〉 (blue solid) and the PNES ∑ n = 0 N C n | n 〉 a | n 〉 b at N = 1 (red dotted), N = 2 (red dashed) and N = 3 (red dot-dashed). The coefficients of the PNESs are optimized for each N.

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Fig. 3
Fig. 3 (a) Experimental scheme to implement the operation S ^ a b † ( ξ ) ( t a ^ a ^ † + r a ^ † a ^ ) S ^ a b ( ξ ) on an arbitrary state. BS1, BS2, and BS3 are beam splitters with transmissivities T1, T2 and tn, respectively. PD0, PD1 and PD2: photo detectors. The operation is successfully achieved under the detection of a single photon at only one of two detectors PD1 and PD2, with PD0 clicked. (b) For a vacuum input state, the sequence of operations Ôn can yield a finite dimensional PNES, ∑ n = 0 N C n | n 〉 a | n 〉 b.

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Fig. 4
Fig. 4 Experimental scheme to implement the operation (t2nâ + r2nb̂†)(t2n−1b̂ + r2n−1â†) on an input state |ψ〉ab. BS1, BS2, BS3 and BS4 are beam splitters with transmissivities T1, T2, t2n−1, and t2n, respectively. PD1, PD2, PD3 and PD4: photo detectors. The operation is successfully achieved under the detection of a single photon at only one of two detectors PD1 and PD2 and the detection of a single-photon at only one of two detectors PD3 and PD4.

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Fig. 5
Fig. 5 Fidelity between the ideal state C0|0〉a|0〉b + C1|1〉a|1〉b and the output state ρout obtained by applying S ^ a b † ( ξ ) ( t a ^ a ^ † + r a ^ † a ^ ) S ^ a b ( ξ ) (blue circle) or (t2â + r2b̂†)(t1b̂ + r1â†) (red square), using on-off detectors with efficiency η to the input state ρin = |0〉a|0〉b as a function of |C0|2 for η = 0.66. Black triangle represents the output fidelity using the scissor scheme of [57], with the input two-mode squeezed state (s = 0.1) and the on-off detectors (η = 0.66).

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Fig. 6
Fig. 6 Fidelity between the ideal state C0|0〉a|0〉b +C1|1〉a|1〉b +C2|2〉a|2〉b and the output state ρout obtained by applying twice (a) S ^ a b † ( ξ 2 ) ( t 2 a ^ a ^ † + r 2 a ^ † a ^ ) S ^ a b ( ξ 2 ) S ^ a b † ( ξ 1 ) ( t 1 a ^ a ^ † + r 1 a ^ † a ^ ) S ^ a b ( ξ 1 ) or (b) (t4â+r4b̂†)(t3b̂+r3â†)(t2â+r2b̂†)(t1b̂+r1â†), using on-off detectors with efficiency η to the input state ρin = |0〉a|0〉b as a function of |C1|2 and |C2|2 for η = 0.66.

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Fig. 7
Fig. 7 Fidelity between the ideal state C0|0〉a|0〉b + C1|1〉a|1〉b and the output state with the error Δti = ±0.01 of the beam-splitter transmissivity (i = 1,2). Other parameters are the same as those in Fig. 6.

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Equations (14)

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F = 1 π ∫ d 2 λ C out ( λ ) C in ( − λ ) ,
C E ( λ 2 , λ 3 ) = e − ( | λ 2 | 2 + | λ 3 | 2 ) / 2 [ | C 0 | 2 + | C 1 | 2 ( 1 − | λ 2 | 2 ) ( 1 − | λ 3 | 2 ) + | C 2 | 2 4 ( 2 − 4 | λ 2 | 2 + | λ 2 | 4 ) ( 2 − 4 | λ 3 | 2 + | λ 3 | 4 ) + C 0 * C 1 λ 2 * λ 3 * + C 0 C 1 * λ 2 λ 3 + C 0 * C 2 2 λ 2 * 2 λ 3 * 2 + C 0 C 2 * 2 λ 2 2 λ 3 2 + 1 2 ( C 1 * C 2 λ 2 * λ 3 * + C 1 C 2 * λ 2 λ 3 ) ( | λ 2 | 2 − 2 ) ( | λ 3 | 2 − 2 ) ] ,
B BW = π 2 4 | W ( α , β ) + W ( α , β ′ ) + W ( α ′ , β ) − W ( α ′ , β ′ ) | ≤ 2 ,
O ^ n ≡ S ^ a b † ( ξ n ) ( t n a ^ a ^ † + r n a ^ † a ^ ) S ^ a b ( ξ n )             = A n + ( t n + r n ) ( a ^ † a ^ cosh 2 s n + b ^ † b ^ sinh 2 s n )                       − ( t n + r n ) cosh s n sinh s n [ exp ( − i φ n ) a ^ b ^ + exp ( i φ n ) a ^ † b ^ † ] ,
A n = t n cosh 2 s n + r n sinh 2 s n ,
B ^ a c S ^ a b ( ξ n ) | ψ 〉 a b | 0 〉 c ≈ ( 1 − R 1 * T 1 a ^ c ^ † ) S ^ a b ( ξ n ) | ψ 〉 a b | 0 〉 c .
〈 1 | e S ^ a e ( 1 − R 1 * T 1 a ^ c ^ † ) S ^ a b ( ξ n ) | ψ 〉 a b | 0 〉 c | 0 〉 e ≈ − s a ^ † ( 1 − R 1 * T 1 a ^ c ^ † ) S ^ a b ( ξ n ) | ψ 〉 a b | 0 〉 c ,
( − s ) B ^ a d a ^ † ( 1 − R 1 * T 1 a ^ c ^ † ) S ^ a b ( ξ n ) | ψ 〉 a b | 0 〉 c d ≈ ( − s ) ( 1 − R 2 * T 2 a ^ d ^ † ) a ^ † ( 1 − R 1 * T 1 a ^ c ^ † ) S ^ a b ( ξ n ) | ψ 〉 a b | 0 〉 c d ,
| S | ψ 〉 〉 ≡ ( − s ) [ 1 − R 2 * T 2 a ^ ( t n d ^ † − r n c ^ † ) ] a ^ † [ 1 − R 1 * T 1 a ^ ( t n c ^ † + r n d ^ † ) ] S ^ a b ( ξ n ) | ψ 〉 a b | 0 〉 c d .
O ^ n ′ ≡ ( t 2 n a ^ + r 2 n b ^ † ) ( t 2 n − 1 b ^ + r 2 n − 1 a ^ † )             = t 2 n − 1 t 2 n a ^ b ^ + r 2 n − 1 r 2 n a ^ † b ^ † + r 2 n − 1 t 2 n a ^ a ^ † + t 2 n − 1 r 2 n b ^ † b ^ ,
[ 1 − R 1 * T 1 b ^ ( t 2 n − 1 d ^ † − r 2 n − 1 c ^ † ) ] [ 1 − s 1 a ^ † ( t 2 n − 1 c ^ † + r 2 n − 1 d ^ † ) ] | ψ 〉 a b | 0 〉 c d .
| S | ψ 〉 ′ 〉 ≡ [ 1 − R 2 * T 2 a ^ ( t 2 n e ^ † + r 2 n f ^ † ) ] [ 1 − s 2 b ^ † ( t 2 n f ^ † + r 2 n e ^ † ) ] | Φ 〉 a b | 0 〉 e f .
ρ out = Tr c d e [ Π ^ 0 c Π ^ 1 d Π ^ 1 e U ^ 1 ρ in U ^ 1 † ] Tr a b c d e [ Π ^ 0 c Π ^ 1 d Π ^ 1 e U ^ 1 ρ in U ^ 1 † ] ,
ρ out ′ = Tr c d e f [ Π ^ 0 e Π ^ 1 f Π ^ 0 c Π ^ 1 d U ^ 2 ρ in ′ U ^ 2 † ] Tr a b c d e f [ Π ^ 0 e Π ^ 1 f Π ^ 0 c Π ^ 1 d U ^ 2 ρ in ′ U ^ 2 † ] ,
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