Abstract

In this paper we propose feasibility demonstration of twin-field quantum key ditribution system based on multi-mode weak coherent phase-coded states. Their utilization provides indisputable advantages described in the paper. We also provide the detailed description of nontrivial interference scheme for those states and derive detection and quantum bit error rates. Since we propose the feasibility scheme we present in this paper only asymptotic secure key estimation and show that in principle it can beat well-known fundamental limit of repeaterless quantum communications , i.e., the secret key capacity of the lossy communication channel. Also we present here the experimental setup and provide the experimental values of detection rates dependence on the modulation signals phase difference.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (4)

M. Minder, M. Pittaluga, G. Roberts, M. Lucamarini, J. Dynes, Z. Yuan, and A. Shields, “Experimental quantum key distribution beyond the repeaterless secret key capacity,” Nat. Photonics 13(5), 334–338 (2019).
[Crossref]

S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

A. Gaidash, A. Kozubov, and G. Miroshnichenko, “Methods of decreasing the unambiguous state discrimination probability for subcarrier wave quantum key distribution systems,” J. Opt. Soc. Am. B 36(3), B16–B19 (2019).
[Crossref]

A. Gaidash, A. Kozubov, and G. Miroshnichenko, “Countermeasures for advanced unambiguous state discrimination attack on quantum key distribution protocol based on weak coherent states,” Phys. Scr. 94(12), 125102 (2019).
[Crossref]

2018 (4)

X. Ma, P. Zeng, and H. Zhou, “Phase-matching quantum key distribution,” Phys. Rev. X 8(3), 031043 (2018).
[Crossref]

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate–distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

S. Pirandola, S. L. Braunstein, R. Laurenza, C. Ottaviani, T. P. Cope, G. Spedalieri, and L. Banchi, “Theory of channel simulation and bounds for private communication,” Quantum Sci. Technol. 3(3), 035009 (2018).
[Crossref]

G. P. Miroshnichenko, A. V. Kozubov, A. A. Gaidash, A. V. Gleim, and D. B. Horoshko, “Security of subcarrier wave quantum key distribution against the collective beam-splitting attack,” Opt. Express 26(9), 11292–11308 (2018).
[Crossref]

2017 (2)

2016 (2)

2015 (1)

Z. Cao, Z. Zhang, H.-K. Lo, and X. Ma, “Discrete-phase-randomized coherent state source and its application in quantum key distribution,” New J. Phys. 17(5), 053014 (2015).
[Crossref]

2014 (1)

N. Jain, E. Anisimova, I. Khan, V. Makarov, C. Marquardt, and G. Leuchs, “Trojan-horse attacks threaten the security of practical quantum cryptography,” New J. Phys. 16(12), 123030 (2014).
[Crossref]

2013 (1)

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, C.-Z. Peng, Q. Zhang, and J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

2012 (4)

H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

S. L. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108(13), 130502 (2012).
[Crossref]

K. Tamaki, H.-K. Lo, C.-H. F. Fung, and B. Qi, “Phase encoding schemes for measurement-device-independent quantum key distribution with basis-dependent flaw,” Phys. Rev. A 85(4), 042307 (2012).
[Crossref]

X. Ma and M. Razavi, “Alternative schemes for measurement-device-independent quantum key distribution,” Phys. Rev. A 86(6), 062319 (2012).
[Crossref]

2011 (1)

L. Lydersen, M. K. Akhlaghi, A. H. Majedi, J. Skaar, and V. Makarov, “Controlling a superconducting nanowire single-photon detector using tailored bright illumination,” New J. Phys. 13(11), 113042 (2011).
[Crossref]

2010 (1)

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics 4(10), 686–689 (2010).
[Crossref]

2009 (2)

V. Makarov, “Controlling passively quenched single photon detectors by bright light,” New J. Phys. 11(6), 065003 (2009).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

2008 (1)

V. Scarani and R. Renner, “Quantum cryptography with finite resources: Unconditional security bound for discrete-variable protocols with one-way postprocessing,” Phys. Rev. Lett. 100(20), 200501 (2008).
[Crossref]

2005 (3)

R. Renner, N. Gisin, and B. Kraus, “Information-theoretic security proof for quantum-key-distribution protocols,” Phys. Rev. A 72(1), 012332 (2005).
[Crossref]

V. Makarov and D. R. Hjelme, “Faked states attack on quantum cryptosystems,” J. Mod. Opt. 52(5), 691–705 (2005).
[Crossref]

I. Devetak and A. Winter, “Distillation of secret key and entanglement from quantum states,” Proc. R. Soc. A 461(2053), 207–235 (2005).
[Crossref]

2003 (1)

K. Tamaki, M. Koashi, and N. Imoto, “Unconditionally secure key distribution based on two nonorthogonal states,” Phys. Rev. Lett. 90(16), 167904 (2003).
[Crossref]

Akhlaghi, M. K.

L. Lydersen, M. K. Akhlaghi, A. H. Majedi, J. Skaar, and V. Makarov, “Controlling a superconducting nanowire single-photon detector using tailored bright illumination,” New J. Phys. 13(11), 113042 (2011).
[Crossref]

Andersen, U.

S. Pirandola, U. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” arXiv preprint arXiv:1906.01645 (2019).

Anisimov, A. A.

Anisimova, E.

N. Jain, E. Anisimova, I. Khan, V. Makarov, C. Marquardt, and G. Leuchs, “Trojan-horse attacks threaten the security of practical quantum cryptography,” New J. Phys. 16(12), 123030 (2014).
[Crossref]

Banchi, L.

S. Pirandola, S. L. Braunstein, R. Laurenza, C. Ottaviani, T. P. Cope, G. Spedalieri, and L. Banchi, “Theory of channel simulation and bounds for private communication,” Quantum Sci. Technol. 3(3), 035009 (2018).
[Crossref]

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
[Crossref]

S. Pirandola, U. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” arXiv preprint arXiv:1906.01645 (2019).

Bannik, O. I.

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Berta, M.

S. Pirandola, U. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” arXiv preprint arXiv:1906.01645 (2019).

Braunstein, S. L.

S. Pirandola, S. L. Braunstein, R. Laurenza, C. Ottaviani, T. P. Cope, G. Spedalieri, and L. Banchi, “Theory of channel simulation and bounds for private communication,” Quantum Sci. Technol. 3(3), 035009 (2018).
[Crossref]

S. L. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108(13), 130502 (2012).
[Crossref]

Buller, G. S.

Bunandar, D.

S. Pirandola, U. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” arXiv preprint arXiv:1906.01645 (2019).

Cao, Z.

Z. Cao, Z. Zhang, H.-K. Lo, and X. Ma, “Discrete-phase-randomized coherent state source and its application in quantum key distribution,” New J. Phys. 17(5), 053014 (2015).
[Crossref]

Cerf, N. J.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Chen, T.-Y.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, C.-Z. Peng, Q. Zhang, and J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Chen, W.

S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

Chistyakov, V. V.

Christandl, M.

M. Christandl, R. Renner, and A. Ekert, “A generic security proof for quantum key distribution,” arXiv preprint quant-ph/0402131 (2004).

Colbeck, R.

S. Pirandola, U. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” arXiv preprint arXiv:1906.01645 (2019).

Collins, R. J.

Cope, T. P.

S. Pirandola, S. L. Braunstein, R. Laurenza, C. Ottaviani, T. P. Cope, G. Spedalieri, and L. Banchi, “Theory of channel simulation and bounds for private communication,” Quantum Sci. Technol. 3(3), 035009 (2018).
[Crossref]

Cui, C.-H.

S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

Cui, K.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, C.-Z. Peng, Q. Zhang, and J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Curty, M.

H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

X. Zhong, J. Hu, M. Curty, L. Qian, and H.-K. Lo, “Proof-of-principle experimental demonstration of twin-field type quantum key distribution,” arXiv preprint arXiv:1902.10209 (2019).

Devetak, I.

I. Devetak and A. Winter, “Distillation of secret key and entanglement from quantum states,” Proc. R. Soc. A 461(2053), 207–235 (2005).
[Crossref]

Diamanti, E.

E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016).
[Crossref]

Dušek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Dynes, J.

M. Minder, M. Pittaluga, G. Roberts, M. Lucamarini, J. Dynes, Z. Yuan, and A. Shields, “Experimental quantum key distribution beyond the repeaterless secret key capacity,” Nat. Photonics 13(5), 334–338 (2019).
[Crossref]

Dynes, J. F.

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate–distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

Egorov, V. I.

Ekert, A.

M. Christandl, R. Renner, and A. Ekert, “A generic security proof for quantum key distribution,” arXiv preprint quant-ph/0402131 (2004).

Elser, D.

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics 4(10), 686–689 (2010).
[Crossref]

Englund, D.

S. Pirandola, U. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” arXiv preprint arXiv:1906.01645 (2019).

Fejer, M. M.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, C.-Z. Peng, Q. Zhang, and J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Fung, C.-H. F.

K. Tamaki, H.-K. Lo, C.-H. F. Fung, and B. Qi, “Phase encoding schemes for measurement-device-independent quantum key distribution with basis-dependent flaw,” Phys. Rev. A 85(4), 042307 (2012).
[Crossref]

Gaidash, A.

A. Gaidash, A. Kozubov, and G. Miroshnichenko, “Countermeasures for advanced unambiguous state discrimination attack on quantum key distribution protocol based on weak coherent states,” Phys. Scr. 94(12), 125102 (2019).
[Crossref]

A. Gaidash, A. Kozubov, and G. Miroshnichenko, “Methods of decreasing the unambiguous state discrimination probability for subcarrier wave quantum key distribution systems,” J. Opt. Soc. Am. B 36(3), B16–B19 (2019).
[Crossref]

A. Kozubov, A. Gaidash, and G. Miroshnichenko, “Finite-key security for quantum key distribution systems utilizing weak coherent states,” arXiv preprint arXiv:1903.04371 (2019).

Gaidash, A. A.

Gehring, T.

S. Pirandola, U. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” arXiv preprint arXiv:1906.01645 (2019).

Gisin, N.

R. Renner, N. Gisin, and B. Kraus, “Information-theoretic security proof for quantum-key-distribution protocols,” Phys. Rev. A 72(1), 012332 (2005).
[Crossref]

Gleim, A. V.

Guo, G.-C.

S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

Han, Z.-F.

S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

He, D.-Y.

S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

Hjelme, D. R.

V. Makarov and D. R. Hjelme, “Faked states attack on quantum cryptosystems,” J. Mod. Opt. 52(5), 691–705 (2005).
[Crossref]

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Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, C.-Z. Peng, Q. Zhang, and J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
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Figures (3)

Fig. 1.
Fig. 1. Simulation of TF QKD protocol based on multi-mode phase-coded weak coherent states. For the considered simulation parameters, the key rate of proposed TF QKD exceeds the fundamental PLOB bound [20] when $\eta _c \gtrsim$ 40 dB (200 km). In addition, our protocol is also able to achieve a long transmission distance of $\eta _c\approx$ 83 dB (460 km).
Fig. 2.
Fig. 2. Experimental setup for testing the Twin-Field SCW QKD. For the sake of simplicity we use only one laser and beam splitter in order to imitate separated Alice and Bob. The upper an the lower optical paths of the picture denotes the encoding parts of Alice and Bob. Here $L1$ is the continuous-wave laser, $I$ is the isolator, $VOA$ is the variable optical attenuator, which allows to control the intensity of the beam, $PC$ is the polarization controller, $BS$ is the beam splitter (both beam splitters here assumed to be 50:50). $PM$ is the phase modulator, which is used for state encoding, $C$ is the circulator, $OF$ is the optical filter and $D$ is the single photon detector. Electrical (denoted as double line) inputs of $PM1$ and $PM2$ were connected with digital-to-analog converters ($DAC1$ and $DAC2$) outputs with modulating RF-signals. Relative optical phase adjustment is implemented by the DC voltage applied to modulators of two independent electrical outputs from waveform generator denoted as $GEN1$ and $GEN2$. The latter are connected (wide black solid line) to D2 and D3 respectively in order to provide feedback that adjusts optical phase.
Fig. 3.
Fig. 3. Experimentally measured values (four rounds) of detection rates ($F\mathcal {P}_{1,2}^{+}$) dependent on phase difference ($\Delta \varphi$) compared to numerical simulations are shown in the figure. Results for the first and the second channel are denoted as symbols $\blacktriangledown$ and $\blacktriangle$ respectively. Numerical simulations for the first and the second channel are denoted as solid and dashed lines respectively and were performed in according to Eq. (12).

Equations (34)

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| ψ 0 ( φ j ) = k = S S | α k ( φ j ) k ,
α k ( φ j ) = μ 0 d 0 k S ( β ) e i φ j k ,
cos ( β ) = 1 1 2 ( m S + 0.5 ) 2 ,
η c α k ( φ j ) = μ 0 η c d 0 k S ( β ) e i φ j k ,
| α A ± α B e i φ 0 2 1 , 2 = k = S S | μ 0 η c 2 d 0 k S ( β ) ( e i φ A k ± e i ( φ B k + φ 0 ) ) k .
n ( Δ φ ) 1 , 2 = μ 0 η c ( k 0 | d 0 k S ( β ) | 2 + ϑ | d 0 k S ( β ) | 2 ± ± cos ( φ 0 ) ( k 0 | d 0 k S ( β ) | 2 e i Δ φ k + ϑ | d 0 k S ( β ) | 2 ) ) ,
n ( Δ φ ) 1 , 2 = μ 0 η c ( 1 ( 1 ϑ ) ( 1 cos ( φ 0 ) ) | d 00 S ( β ) | 2 ± ± cos ( φ 0 ) d 00 S ( β ) ) ,
cos ( β ) = cos 2 ( β ) sin 2 ( β ) cos ( Δ φ ) .
P 1 , 2 + ( Δ φ ) = ( n ( Δ φ ) 1 , 2 η D F + γ d a r k ) Δ t ,
d n k S ( β ) S J n k ( m ) ,
β m ,
P 1 , 2 + ( Δ φ ) = μ η ( 1 ± cos ( Δ φ ) cos ( φ 0 ) ) + + ϑ ( μ c η ( 1 ± cos ( φ 0 ) ) ) + p d a r k ,
μ = μ 0 k 0 | d 0 k S ( β ) | 2 ,
μ c = μ 0 μ = μ 0 ( 1 k 0 | d 0 k S ( β ) | 2 ) .
P B | A ( b = 0 | a = 0 ) = P B | A ( b = 1 | a = 1 ) = M ,
P B | A ( b = 1 | a = 0 ) = P B | A ( b = 0 | a = 1 ) = N ,
M = P 1 + P 2 ( Δ φ = φ m ) + P 1 P 2 + ( Δ φ = π + φ m ) ,
N = P 1 P 2 + ( Δ φ = φ m ) + P 1 + P 2 ( Δ φ = π + φ m ) ,
I ( a ; b ) = ( M + N ) ( 1 h ( N M + N ) ) ,
R = M + N ,
Q = N M + N ,
C = F R ( 1 h ( Q ) ) .
P n = Tr ( ρ | n n | ) = Tr ( | α e i φ α e i φ | n n | ) = Tr ( k = 0 m = 0 e | α | 2 ( α e i φ ) k ( α e i φ ) m k ! m ! | k m | n n | ) = j = 0 k = 0 m = 0 e | α | 2 ( α e i φ ) k ( α e i φ ) m k ! m ! j | k m | n n | j = e | α | 2 ( | α | 2 n ) n ! ,
ρ ~ = | n n | | α e i φ α e i φ | | n n | P n .
| n n | | m = k = 0 ( 1 ) k ( 2 k ) ! ( 1 2 k ) ( k ! ) 2 4 k ( | n n | I ^ ) k | m ,
I ^ | m = | m ,
( | n n | ) k | m = ( | n n | ) k 1 | n n | m = 0 ,
| n n | | m = k = 0 ( 1 ) k ( 2 k ) ! ( 1 2 k ) ( k ! ) 2 4 k ( 1 ) k | m = 0 | m .
( | n n | ) k | n = ( | n n | ) k 1 = = | n n | n = ( | n n | ) k 1 | n = = | n ,
| n n | | n = k = 0 ( 1 ) k ( 2 k ) ! ( 1 2 k ) ( k ! ) 2 4 k ( | n n | I ^ ) k | n = | n + k = 1 ( 1 ) k ( 2 k ) ! ( 1 2 k ) ( k ! ) 2 4 k ( | n n | I ^ ) k | n = | n + 0 = | n .
ρ ~ = 1 P n k = 0 m = 0 e | α | 2 ( α e i φ ) k ( α e i φ ) m k ! m ! | n n | | k m | | n n | = 1 P n P n | n n | = | n n | .
χ ~ = 2 ( 1 χ ) χ + χ 2 .
χ = h ( 1 e μ 0 ( 1 d 00 S ( 2 β ) ) 2 ) h ( 1 e 2 μ 2 ) .
K = F R ( 1 ξ h ( Q ) χ ~ ) ,