Source: http://aoot.osa.org/oe/abstract.cfm?uri=oe-27-6-8291
Timestamp: 2019-04-22 12:56:42+00:00

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We derive a closed-form analytical expression for the linear entropy of a multipartite qutrit state, providing a quantitative measure for quantum entanglement within the class of n-mode nonorthogonal qutrit states with any n. Conditions for enhanced and maximum quantum entanglement of multipartite qutrit states are identified. The usefulness of the introduced multipartite qutrit states as quantum communication channel resources is analyzed. The Hamiltonians allowing for the generation of multipartite qutrit states can be attained by combining optomechanical cavities with sequences of tunable beam splitters.
M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Phys. Rev. Lett. 99, 040403 (2007).
L. Sheridan and V. Scarani, “Security proof for quantum key distribution using qudit systems,” Phys. Rev. A 82, 030301 (2010).
T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
Y. Maleki and A. M. Zheltikov, “Generating maximally-path-entangled number states in two spin ensembles coupled to a superconducting flux qubit,” Phys. Rev. A 26, 012312 (2018).
Y. Maleki and A. M. Zheltikov, “Witnessing quantum entanglement in ensembles of nitrogen–vacancy centers coupled to a superconducting resonator,” Opt. Express 97, 17849–17858 (2018).
C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
S. L. Braunstein and P. V. Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally feasible quantum erasure-correcting code for continuous variables,” Phys. Rev. Lett. 101, 130503 (2008).
X. Wang, “Bipartite entangled non-orthogonal states,” J. Phys. A: Math. Gen. 35, 165 (2001).
A. Mann, B. C. Sanders, and W. J. Munro, “Bells inequality for an entanglement of nonorthogonal states,” Phys. Rev. A 51, 989 (1995).
M. Paternostro and H. Jeong, “Testing nonlocal realism with entangled coherent states,” Phys. Rev. A 81, 032115 (2010).
S. Bose and V. Vedral, “Mixedness and teleportation,” Phys. Rev. A 61, 040101 (2000).
W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245 (1998).
H. Fu, X. Wang, and A. I. Solomon, “Maximal entanglement of nonorthogonal states: classification,” Phys. Lett. A 291, 73–76 (2001).
P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, “Local environment can enhance fidelity of quantum teleportation,” Phys. Rev. A 62, 012311 (2000).
S. Sazim, S. Adhikari, S. Banerjee, and T. Pramanik, “Quantification of entanglement of teleportation in arbitrary dimensions,” Quantum Inf. Process. 13, 863–880 (2014).
S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
P. v. Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. v. Loock, and S. Massar, “Optimal cloning of coherent states with a linear amplifier and beam splitters,” Phys. Rev. Lett. 86, 4938 (2001).
S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 67, 022313 (2001).
Y. Maleki and A. Maleki, “Entangled multimode spin coherent states of trapped ions,” J. Opt. Soc. Am. B 35, 1211–1217 (2018).
Fig. 1 The density of the linear entropy of |Ψ〉− as a function of the overlap parameters p1 and p3 for p2 = 0.05 (a), 0.2 (b), 0.4 (c), and 0.5 (d).
Fig. 2 The density of the linear entropy of |Ψ〉+ as a function of overlap parameters p1 and p3 for p2 = 0.05 (a), 0.2 (b), 0.4 (c), and 0.5 (d).
Fig. 3 The linear entropy of two-qutrit coherent states |ψ〉 (a) and |ϕ〉 (b) as a function of α and θ.
Fig. 4 (a) The linear entropy of the multiqutrit state (37) as a function of p2 for p1 = p3 = 0 and k =1 (dotted line), k =3 (dash–dotted line), k =5 (dashed line), and k = 7 (solid line). (b) The linear entropy of the multiqutrit coherent state (39) as a function of α for k =1 (dotted line), k =2 (dash–dotted line), k =4 (dashed line), and k =8 (solid line). (c) The linear entropy of multiqutrit coherent (solid and dash–dotted lines) and squeezed (dashed and dotted lines) state as a function of α for k =1 (dotted and dash–dotted lines) and k =8 (solid and dashed lines).
Fig. 5 The concurrence of an amplitude-damped k-mode state as a function of the coherence parameter α with θ = 0 (a, b) and π (c), η = 1 (a) and 0.9 (b, c), k = 1 (black), 2 (red), 4 (green), and 8 (blue).
(1) | Ψ 〉 = μ | α 1 〉 | β 1 〉 + λ | α 2 〉 | β 2 〉 + ν | α 3 〉 | β 3 〉 .
(8) M i = 1 − | q i | 2 , M = [ 1 − | q 1 | 2 − | q 2 | 2 − | q 3 | 2 + q 1 q ¯ 2 q 3 + q ¯ 1 q 2 q ¯ 3 ] 1 2 .
(9) | Ψ 〉 = ( μ + λ q 1 p 1 + ν p 2 q 2 ) | 0 0 ˜ 〉 + ( ν p 2 q 3 − q ¯ 1 q 2 M 1 + λ M 1 p 1 ) | 0 1 ˜ 〉 + ( ν q 2 p 3 − p ¯ 1 p 2 N 1 + λ N 1 q 1 ) | 1 0 ˜ 〉 + ( ν ( p 3 − p ¯ 1 p 2 ) N 1 ( q 3 − q ¯ 1 q 2 ) M 1 + λ N 1 M 1 ) | 1 1 ˜ 〉 + ν p 2 M M 1 | 0 2 ˜ 〉 + ν q 2 N N 1 | 2 0 ˜ 〉 + ( ν p 3 − p ¯ 1 p 2 N 1 M M 1 ) | 1 2 ˜ 〉 + ν q 3 − q ¯ 1 q 2 M 1 N N 1 | 2 1 ˜ 〉 + ν N N 1 M M 1 | 2 2 ˜ 〉 .
(10) I lin = d d − 1 ( 1 − Tr ρ 1 2 ) .
(12) Δ 1 = | μ λ | 2 M 1 2 N 1 2 + | μ ν | 2 M 2 2 N 2 2 + | λ ν | 2 M 3 3 N 3 2 , Δ 2 = | μ | 2 Re [ λ ¯ ν ( q ¯ 1 q 2 − q 3 ) ( p ¯ 1 p 2 − p 3 ) ] , Δ 3 = | λ | 2 Re [ μ ¯ ν ( q ¯ 1 q 3 − q 2 ) ( p 1 p 3 − p 2 ) ] , Δ 4 = | ν | 2 Re [ μ ¯ λ ( q ¯ 3 q 2 − q 1 ) ( p ¯ 3 p 2 − p 1 ) ] , 풩 = | μ | 2 + | λ | 2 + | ν | 2 + 2 Re ( μ ¯ λ q 1 p 1 + μ ¯ ν q 2 p 2 + λ ¯ ν q 3 p 3 ) .
(13) I lin = 4 | μ ν | 2 ( 1 − | p 2 | 2 ) ( 1 − | q 2 | 2 ) ( | μ | 2 + | ν | 2 + 2 Re ( μ ¯ ν q 2 p 2 ) ) 2 .
(14) I lin ≤ 4 | μ ν | 2 sin 2 θ 1 sin 2 θ 2 ( | μ | 2 + | ν | 2 − 2 | μ ν | cos θ 1 cos θ 2 ) 2 .
(15) I lin ≤ sin 2 θ 1 sin 2 θ 2 ( 1 − cos θ 1 cos θ 2 ) 2 .
(16) I 0 = 2 d d − 1 | μ λ | 2 + | μ ν | 2 + | λ ν | 2 ( | μ | 2 + | λ | 2 + | ν | 2 ) 2 .
(18) | Ψ 〉 = μ | α 1 〉 | 0 〉 + λ | α 2 〉 | 1 〉 + ν | α 3 〉 | 2 〉 .
(20) | Ψ 〉 ± = | α 1 〉 | β 1 〉 ± | α 2 〉 | β 2 〉 + | α 3 〉 | β 3 〉 .
(24) I lin = 3 2 ( p 2 − 1 ) 2 + ( p 8 − 1 ) 2 + cos θ ( p − p 5 ) 2 + 6 [ a ( p 4 − p 2 ) 2 + b ( p − p 5 ) 2 ] ( 3 + 2 ( b + cos θ ) p 2 + 2 a p 8 ) 2 .
(26) | Ψ 〉 = μ | α 1 〉 … | α 1 〉 ︸ k | β 1 〉 … | β 1 〉 ︸ k + λ | α 2 〉 … | α 2 〉 | β 2 〉 … | β 2 〉 + ν | α 3 〉 … | α 3 〉 | β 3 〉 … | β 3 〉 .
(27) | Ψ 〉 = μ | α 1 〉 … | α k 〉 ︸ k | β 1 〉 … | β n − k 〉 ︸ n − k + λ | α k + 1 〉 … | α 2 k 〉 | β n − k + 1 〉 … | β 2 ( n − k ) 〉 + ν | α 2 k + 1 〉 … | α 3 k 〉 | β 2 ( n − k ) + 1 〉 … | β 3 ( n − k ) 〉 .
(31) N = [ 1 − | p 1 | 2 k − | p 2 | 2 k − | p 3 | 2 k + p 1 k p ¯ 2 k p 3 k + p ¯ 1 k p 2 k p ¯ 3 k ] 1 2 .
(35) M = [ 1 − | q 1 | 2 k − | q 2 | 2 k − | q 3 | 2 k + q 1 k q ¯ 2 k q 3 k + q ¯ 1 k q 2 k q p ¯ 3 k ] 1 2 .
(37) I lin = 3 p 2 4 k + 3 ( 2 p 2 2 k + 3 ) 2 .
(39) I lin = 3 2 ( p 2 k − 1 ) 2 + ( p 8 k − 1 ) 2 + 2 ( p 4 k − p 2 k ) 2 − 4 ( p k − p 5 k ) 2 ( 3 − 4 p 2 k + 2 p 8 k ) 2 .
(40) | Ψ 〉 = | 0 〉 … | 0 〉 ︸ k | 0 〉 … | 0 〉 ︸ k − | 1 〉 … | 1 〉 | 1 〉 … | 1 〉 + | 2 〉 … | 2 〉 | 2 〉 … | 2 〉 .
(43) | Ψ ( t ) 〉 = e − | α 1 | 2 2 ∑ n = 0 ∞ [ α 1 e − i ζ t e i κ Im [ β η e − i t ] ] n n ! e i κ 2 n 2 ( t − sin t ) | n 〉 c ⊗ | β e − i t + κ n η 〉 m .
(45) | ψ 〉 c = 1 2 [ e i π 4 | α 〉 c + e − i π 4 | − α 〉 c ] .
(51) | Ψ 〉 = ( | η α 〉 A ⊗ k | 1 − η α 〉 E ⊗ k | ) ( | η α 〉 B ⊗ k | 1 − η α 〉 E ⊗ k ) + e i θ ( | η 3 α 〉 A ⊗ k | 1 − η 3 α 〉 E ⊗ k ) ( | η 3 α 〉 B ⊗ k | 1 − η 3 α 〉 E ⊗ k ) .
(53) C = 1 − e − 4 k η | α | 2 1 + cos ( θ ) e − 4 k | α | 2 e − 4 k ( 1 − η ) | α | 2 .
(54) | Ψ d n 〉 = μ 0 | α 1 〉 … | α n 〉 + μ 1 | β 1 〉 … | β n 〉 + … + μ d − 1 | γ 1 〉 … | γ n 〉 .
(55) | Ψ d n 〉 = ∑ i = 0 d − 1 μ i | α i A 〉 | β i B 〉 .
(58) ρ 1 = ( I ⊗ 〈 0 | ) | Ψ 〉 〈 Ψ | ( I ⊗ 〈 0 | ) + ( I ⊗ 〈 1 | ) | Ψ 〉 〈 Ψ | ( I ⊗ 〈 1 | ) + ( I ⊗ 〈 2 | ) | Ψ 〉 〈 Ψ | ( I ⊗ 〈 2 | ) .
(59) ρ ˜ 1 = [ | μ | 2 + | λ | 2 | p 1 | 2 + | ν | 2 | p 2 | 2 + μ λ ¯ q ¯ 1 p ¯ 1 + μ ν ¯ q ¯ 2 p ¯ 2 + λ ν ¯ q ¯ 3 p 1 p ¯ 2 + μ λ ¯ q ¯ 1 p ¯ 1 + μ ¯ ν q 2 p 2 + λ ¯ ν q 3 p ¯ 1 p 2 ] | 0 〉 〈 0 | + [ | λ | 2 p 1 N 1 + μ λ ¯ q ¯ 1 N 1 | ν | 2 p 2 p ¯ 3 − p ¯ 2 p 1 N 1 + μ ν ¯ q ¯ 2 ( p ¯ 3 − p ¯ 2 p 1 ) N 1 + λ ν ¯ q ¯ 3 p 1 ( p ¯ 3 − p ¯ 2 p 1 ) N 1 + λ ¯ ν q 3 N 1 p 2 ] | 0 〉 〈 1 | + [ | λ 2 | p ¯ 1 N 1 + | ν | 2 p ¯ 2 p 3 − p 2 p ¯ 1 N 1 μ ¯ λ q 1 N 1 + μ ¯ ν q 2 ( p 3 − p 2 p ¯ 1 ) N 1 + λ ¯ ν q 3 p ¯ 1 ( p 3 − p 2 p ¯ 1 ) + λ ν ¯ q ¯ 3 N 1 p ¯ 2 ] | 1 〉 〈 0 | + [ | λ | 2 N 1 2 + | ν | 2 | p 3 − p 2 p ¯ 1 | 2 N 1 2 + λ ν ¯ q ¯ 3 ( p ¯ 3 − p ¯ 2 p 1 ) + λ ¯ ν q 3 ( p 3 − p 2 p ¯ 1 ) ] | 1 〉 〈 1 | + [ ( | ν | 2 p 2 N N 1 + ( μ ν ¯ q ¯ 2 + λ ν ¯ q ¯ 3 p 1 ) N N 1 ] | 0 〉 〈 2 | + [ μ ¯ ν q 2 + λ ¯ ν q 3 p ¯ 1 ) N N 1 + | ν | 2 p ¯ 2 N N 1 | 2 〉 〈 0 | + [ | ν | 2 ( p 3 − p 2 p ¯ 1 ) N 1 N N 1 ] | 1 〉 〈 2 | + [ | ν | 2 ( p ¯ 3 − p ¯ 2 p 1 ) N 1 N N 1 ] | 2 〉 〈 1 | + [ | ν | 2 N 2 N 1 2 ] | 2 〉 〈 2 | .
(62) B = ( μ λ ¯ q ¯ 1 p ¯ 1 + μ ν ¯ q ¯ 2 p ¯ 2 + λ ν ¯ q ¯ 3 p 1 p ¯ 2 ) | 0 〉 〈 0 | + ( λ ν ¯ q ¯ 3 N 1 p ¯ 2 ) | 1 〉 〈 0 | + ( μ λ ¯ q ¯ 1 N + μ ν ¯ q ¯ 2 ( p ¯ 3 − p ¯ 2 p 1 ) N 1 + λ ν ¯ q ¯ 3 p 1 ( p ¯ 3 − p ¯ 2 p 1 ) N 1 | 0 〉 〈 1 | + ( λ ν ¯ q ¯ 3 ( p ¯ 3 − p ¯ 2 p 1 ) ) | 1 〉 〈 1 | + ( μ ν ¯ q ¯ 2 + λ ν ¯ q ¯ 3 p 1 ) N N 1 ) | 0 〉 〈 2 | .
(63) ρ 1 = ρ ˜ 1 Tr ρ ˜ 1 .
(64) I lin = d d − 1 ( 1 − Tr ρ ˜ 1 2 Tr 2 ρ ˜ 1 ) = d d − 1 Tr 2 ρ ˜ 1 − Tr ρ ˜ 1 2 Tr 2 ρ ˜ 1 .
(66) Tr ρ ˜ 1 2 = Tr A 2 + Tr B 2 + Tr B † 2 + 2 Tr ( A B ) + 2 Tr ( A B † ) + 2 Tr ( B B † ) .
(67) I lin = d ( d − 1 ) Tr 2 ρ ˜ 1 [ ( Tr 2 A − Tr A 2 ) + ( Tr 2 B − Tr B 2 ) + ( Tr 2 B † − Tr B † 2 ) + 2 ( Tr A Tr B − Tr ( A B ) ) + 2 ( Tr A Tr B † − Tr A B † ) + 2 ( Tr B Tr B † − Tr B B † ) ] .
(74) | Ψ 〉 = μ | α 1 〉 … | α k 〉 ︸ k | β 1 〉 … | β n − k 〉 ︸ n − k + λ | α k + 1 〉 … | α 2 k 〉 | β n − k + 1 〉 … | β 2 ( n − k ) 〉 + ν | α 2 k + 1 〉 … | α 3 k 〉 | β 2 ( n − k ) + 1 〉 … | β 3 ( n − k ) 〉 .
(78) N = [ 1 − | ∏ i = 1 k p i , k + i | 2 − | ∏ i = 1 k p i , 2 k + i | 2 − | ∏ i = 1 k p k + i , 2 k + i | 2 + ∏ i = 1 k p i , k + i ∏ i = 1 k p ¯ i , 2 k + i ∏ i = 1 k p k + i , 2 k + i + ∏ i = 1 k p ¯ i , k + i ∏ i = 1 k p i , 2 k + i ∏ i = 1 k p ¯ k + i , 2 k + i ] ] 1 2 .
(82) M = [ 1 − | ∏ i = 1 ( n − k ) q i , n − k + i | 2 − | ∏ i = 1 ( n − k ) q i , 2 ( n − k ) + i | 2 − | ∏ i = 1 ( n − k ) q k + i , 2 ( n − k ) + i | 2 + ∏ i = 1 ( n − k ) q i , n − k + i ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ∏ i = 1 ( n − k ) q ¯ i , 2 ( n − k ) + i ∏ i = 1 k q n − k + i , 2 ( n − k ) + i ∏ i = 1 ( n − k ) q ¯ i , ( n − k ) + i ∏ i = 1 ( n − k ) q i , 2 ( n − k ) + i ∏ i = 1 ( n − k ) q ¯ n − k + i , 2 ( n − k ) + i ] 1 2 .
(83) | Ψ 〉 = ( μ + λ ∏ i = 1 ( n − k ) q i , n − k + i ∏ i = 1 k p i , k + i + ν ∏ i = 1 ( n − k ) q i , 2 ( n − k ) + i ∏ i = 1 k p i , 2 k + i ) | 0 0 ˜ 〉 + ( ν M 1 ∏ i = 1 k p i , 2 k + i ( ∏ i = 1 ( n − k ) q n − k + i , 2 ( n − k ) + i − ∏ i = 1 ( n − k ) q ¯ i , n − k + i ∏ i = 1 ( n − k ) q i , 2 ( n − k ) − i ) + λ M 1 ∏ i = 1 k p i , k + i ) | 0 1 ˜ 〉 + ( ν N 1 ∏ i = 1 ( n − k ) q i , 2 ( n − k ) + i ( ∏ i = 1 k p k + i , 2 k + i − ∏ i = 1 k p ¯ i , k + i ∏ i = 1 k p i , 2 k + i ) + λ N 1 ∏ i = 1 ( n − k ) q i , n − k + i ) | 1 0 ˜ 〉 + ( ν ( ∏ i = 1 k p k + i , 2 k + i − ∏ i = 1 k p ¯ i , k + i ∏ i = 1 k p i , 2 k + i ) N 1 ( ∏ i = 1 ( n − k ) q n − k + i , 2 ( n − k ) + i − ∏ i = 1 ( n − k ) q ¯ i , n − k + i ∏ i = 1 ( n − k ) q i , 2 ( n − k ) + i ) M 1 + λ N 1 M 1 ) | 1 1 ˜ 〉 + ν ∏ i = 1 k p i , 2 k + i M M 1 | 0 2 ˜ 〉 + ν ∏ i = 1 ( n − k ) q i , 2 ( n − k ) + i N N 1 | 2 0 ˜ 〉 ( ν ∏ i = 1 k p k + i , 2 k + i − ∏ i = 1 k p ¯ i , k + i ∏ i = 1 k p i , 2 k + i N 1 M M 1 ) | 1 2 ˜ 〉 + ν ∏ i = 1 ( n − k ) q n − k + i , 2 ( n − k ) + i − ∏ i = 1 ( n − k ) q ¯ i , n − k + i ∏ i = 1 ( n − k ) q i , 2 ( n − k ) + i M 1 N N 1 | 2 1 ˜ 〉 + ν N N 1 M M 1 | 2 2 ˜ 〉 .

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