Abstract

In quantum illumination (QI), a signal beam initially entangled with an idler beam held at the receiver interrogates a target region bathed in thermal background light. The returned beam is measured jointly with the idler in order to determine whether a weakly reflecting target is present. Using tools from quantum information theory, we derive lower bounds on the average error probability of detecting both specular and fading targets and on the mean squared error of estimating the reflectance of a detected target, which are obeyed by any QI transmitter satisfying a signal energy constraint. For bright thermal backgrounds, we show that the QI system using multiple copies of low-brightness two-mode squeezed vacuum states is nearly optimal. More generally, our results place limits on the best possible performance achievable using QI systems at all wavelengths, and at all signal and background noise levels.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

Corrections

6 July 2020: Typographical corrections were made to Eqs. (5) and (14).

References

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2020 (3)

S. Barzanjeh, S. Pirandola, D. Vitali, and J. M. Fink, Sci. Adv. 6, eabb0451 (2020).
[Crossref]

J. H. Shapiro, IEEE Aerosp. Electron. Syst. Mag. 35, 8 (2020).
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H. Shi, Z. Zhang, and Q. Zhuang, Phys. Rev. Appl. 13, 034029 (2020).
[Crossref]

2019 (3)

S. Pirandola, R. Laurenza, C. Lupo, and J. L. Pereira, npj Quantum Inf. 5, 1 (2019).
[Crossref]

C. W. S. Chang, A. M. Vadiraj, J. Bourassa, B. Balaji, and C. M. Wilson, Appl. Phys. Lett. 114, 112601 (2019).
[Crossref]

H. Liu, D. Giovannini, H. He, D. England, B. J. Sussman, B. Balaji, and A. S. Helmy, Optica 6, 1349 (2019).
[Crossref]

2018 (3)

G. De Palma and J. Borregaard, Phys. Rev. A 98, 012101 (2018).
[Crossref]

S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Nat. Photonics 12, 724 (2018).
[Crossref]

R. Nair, Phys. Rev. Lett. 121, 230801 (2018).
[Crossref]

2017 (5)

M. Sanz, U. Las Heras, J. J. García-Ripoll, E. Solano, and R. Di Candia, Phys. Rev. Lett. 118, 070803 (2017).
[Crossref]

Q. Zhuang, Z. Zhang, and J. H. Shapiro, Phys. Rev. Lett. 118, 040801 (2017).
[Crossref]

Q. Zhuang, Z. Zhang, and J. H. Shapiro, Phys. Rev. A 96, 020302 (2017).
[Crossref]

Q. Zhuang, Z. Zhang, and J. H. Shapiro, J. Opt. Soc. Am. B 34, 1567 (2017).
[Crossref]

M. M. Wilde, M. Tomamichel, S. Lloyd, and M. Berta, Phys. Rev. Lett. 119, 120501 (2017).
[Crossref]

2016 (2)

C. Weedbrook, S. Pirandola, J. Thompson, V. Vedral, and M. Gu, New J. Phys. 18, 043027 (2016).
[Crossref]

T. Cooney, M. Mosonyi, and M. M. Wilde, Commun. Math. Phys. 344, 797 (2016).
[Crossref]

2015 (2)

Z. Zhang, S. Mouradian, F. N. C. Wong, and J. H. Shapiro, Phys. Rev. Lett. 114, 110506 (2015).
[Crossref]

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, Phys. Rev. Lett. 114, 080503 (2015).
[Crossref]

2014 (1)

K. M. R. Audenaert, Quantum Inf. Comput. 14, 31 (2014).

2013 (3)

E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, Phys. Rev. Lett. 110, 153603 (2013).
[Crossref]

Z. Zhang, M. Tengner, T. Zhong, F. N. C. Wong, and J. H. Shapiro, Phys. Rev. Lett. 111, 010501 (2013).
[Crossref]

M. Tsang, New J. Phys. 15, 073005 (2013).
[Crossref]

2012 (3)

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, Nat. Commun. 3, 1063 (2012).
[Crossref]

M. Tsang and R. Nair, Phys. Rev. A 86, 042115 (2012).
[Crossref]

R. García-Patrón, C. Navarrete-Benlloch, S. Lloyd, J. H. Shapiro, and N. J. Cerf, Phys. Rev. Lett. 108, 110505 (2012).
[Crossref]

2011 (3)

R. Nair, Phys. Rev. A 84, 032312 (2011).
[Crossref]

S. Pirandola, Phys. Rev. Lett. 106, 090504 (2011).
[Crossref]

B. Escher, R. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011).
[Crossref]

2009 (2)

J. H. Shapiro and S. Lloyd, New J. Phys. 11, 063045 (2009).
[Crossref]

S. Guha and B. I. Erkmen, Phys. Rev. A 80, 052310 (2009).
[Crossref]

2008 (3)

S. Lloyd, Science 321, 1463 (2008).
[Crossref]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, Phys. Rev. Lett. 101, 253601 (2008).
[Crossref]

S. Pirandola and S. Lloyd, Phys. Rev. A 78, 012331 (2008).
[Crossref]

2007 (1)

K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, L. Masanes, A. Acín, and F. Verstraete, Phys. Rev. Lett. 98, 160501 (2007).
[Crossref]

2006 (1)

F. Caruso, V. Giovannetti, and A. S. Holevo, New J. Phys. 8, 310 (2006).
[Crossref]

1999 (1)

C. Fuchs and J. van de Graaf, IEEE Trans. Inf. Theory 45, 1216 (1999).
[Crossref]

1994 (1)

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
[Crossref]

1982 (1)

Acín, A.

K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, L. Masanes, A. Acín, and F. Verstraete, Phys. Rev. Lett. 98, 160501 (2007).
[Crossref]

Audenaert, K. M. R.

K. M. R. Audenaert, Quantum Inf. Comput. 14, 31 (2014).

K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, L. Masanes, A. Acín, and F. Verstraete, Phys. Rev. Lett. 98, 160501 (2007).
[Crossref]

Bagan, E.

K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, L. Masanes, A. Acín, and F. Verstraete, Phys. Rev. Lett. 98, 160501 (2007).
[Crossref]

Balaji, B.

C. W. S. Chang, A. M. Vadiraj, J. Bourassa, B. Balaji, and C. M. Wilson, Appl. Phys. Lett. 114, 112601 (2019).
[Crossref]

H. Liu, D. Giovannini, H. He, D. England, B. J. Sussman, B. Balaji, and A. S. Helmy, Optica 6, 1349 (2019).
[Crossref]

Bardhan, B. R.

S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Nat. Photonics 12, 724 (2018).
[Crossref]

Barzanjeh, S.

S. Barzanjeh, S. Pirandola, D. Vitali, and J. M. Fink, Sci. Adv. 6, eabb0451 (2020).
[Crossref]

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, Phys. Rev. Lett. 114, 080503 (2015).
[Crossref]

Berta, M.

M. M. Wilde, M. Tomamichel, S. Lloyd, and M. Berta, Phys. Rev. Lett. 119, 120501 (2017).
[Crossref]

M. Berta, C. Hirche, E. Kaur, and M. M. Wilde, “Amortized channel divergence for asymptotic quantum channel discrimination,”arXiv:1808.01498 (2018).

Borregaard, J.

G. De Palma and J. Borregaard, Phys. Rev. A 98, 012101 (2018).
[Crossref]

Bourassa, J.

C. W. S. Chang, A. M. Vadiraj, J. Bourassa, B. Balaji, and C. M. Wilson, Appl. Phys. Lett. 114, 112601 (2019).
[Crossref]

Braunstein, S. L.

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
[Crossref]

Brida, G.

E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, Phys. Rev. Lett. 110, 153603 (2013).
[Crossref]

Calsamiglia, J.

K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, L. Masanes, A. Acín, and F. Verstraete, Phys. Rev. Lett. 98, 160501 (2007).
[Crossref]

Caruso, F.

F. Caruso, V. Giovannetti, and A. S. Holevo, New J. Phys. 8, 310 (2006).
[Crossref]

Caves, C. M.

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
[Crossref]

Cerf, N. J.

R. García-Patrón, C. Navarrete-Benlloch, S. Lloyd, J. H. Shapiro, and N. J. Cerf, Phys. Rev. Lett. 108, 110505 (2012).
[Crossref]

Chang, C. W. S.

C. W. S. Chang, A. M. Vadiraj, J. Bourassa, B. Balaji, and C. M. Wilson, Appl. Phys. Lett. 114, 112601 (2019).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Cooney, T.

T. Cooney, M. Mosonyi, and M. M. Wilde, Commun. Math. Phys. 344, 797 (2016).
[Crossref]

Davidovich, L.

B. Escher, R. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011).
[Crossref]

de Matos Filho, R.

B. Escher, R. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011).
[Crossref]

De Palma, G.

G. De Palma and J. Borregaard, Phys. Rev. A 98, 012101 (2018).
[Crossref]

Degiovanni, I. P.

E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, Phys. Rev. Lett. 110, 153603 (2013).
[Crossref]

Demkowicz-Dobrzanski, R.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, Nat. Commun. 3, 1063 (2012).
[Crossref]

Di Candia, R.

M. Sanz, U. Las Heras, J. J. García-Ripoll, E. Solano, and R. Di Candia, Phys. Rev. Lett. 118, 070803 (2017).
[Crossref]

England, D.

Erkmen, B. I.

S. Guha and B. I. Erkmen, Phys. Rev. A 80, 052310 (2009).
[Crossref]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, Phys. Rev. Lett. 101, 253601 (2008).
[Crossref]

Escher, B.

B. Escher, R. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011).
[Crossref]

Fink, J. M.

S. Barzanjeh, S. Pirandola, D. Vitali, and J. M. Fink, Sci. Adv. 6, eabb0451 (2020).
[Crossref]

Fuchs, C.

C. Fuchs and J. van de Graaf, IEEE Trans. Inf. Theory 45, 1216 (1999).
[Crossref]

García-Patrón, R.

R. García-Patrón, C. Navarrete-Benlloch, S. Lloyd, J. H. Shapiro, and N. J. Cerf, Phys. Rev. Lett. 108, 110505 (2012).
[Crossref]

García-Ripoll, J. J.

M. Sanz, U. Las Heras, J. J. García-Ripoll, E. Solano, and R. Di Candia, Phys. Rev. Lett. 118, 070803 (2017).
[Crossref]

Gehring, T.

S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Nat. Photonics 12, 724 (2018).
[Crossref]

Genovese, M.

E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, Phys. Rev. Lett. 110, 153603 (2013).
[Crossref]

Giovannetti, V.

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, Phys. Rev. Lett. 101, 253601 (2008).
[Crossref]

F. Caruso, V. Giovannetti, and A. S. Holevo, New J. Phys. 8, 310 (2006).
[Crossref]

Giovannini, D.

Gu, M.

C. Weedbrook, S. Pirandola, J. Thompson, V. Vedral, and M. Gu, New J. Phys. 18, 043027 (2016).
[Crossref]

Guha, S.

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, Phys. Rev. Lett. 114, 080503 (2015).
[Crossref]

S. Guha and B. I. Erkmen, Phys. Rev. A 80, 052310 (2009).
[Crossref]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, Phys. Rev. Lett. 101, 253601 (2008).
[Crossref]

Guta, M.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, Nat. Commun. 3, 1063 (2012).
[Crossref]

He, H.

Helmy, A. S.

Helstrom, C. W.

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

Hirche, C.

M. Berta, C. Hirche, E. Kaur, and M. M. Wilde, “Amortized channel divergence for asymptotic quantum channel discrimination,”arXiv:1808.01498 (2018).

Holevo, A. S.

F. Caruso, V. Giovannetti, and A. S. Holevo, New J. Phys. 8, 310 (2006).
[Crossref]

Kaur, E.

M. Berta, C. Hirche, E. Kaur, and M. M. Wilde, “Amortized channel divergence for asymptotic quantum channel discrimination,”arXiv:1808.01498 (2018).

Kolodynski, J.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, Nat. Commun. 3, 1063 (2012).
[Crossref]

Las Heras, U.

M. Sanz, U. Las Heras, J. J. García-Ripoll, E. Solano, and R. Di Candia, Phys. Rev. Lett. 118, 070803 (2017).
[Crossref]

Laurenza, R.

S. Pirandola, R. Laurenza, C. Lupo, and J. L. Pereira, npj Quantum Inf. 5, 1 (2019).
[Crossref]

Liu, H.

Lloyd, S.

S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Nat. Photonics 12, 724 (2018).
[Crossref]

M. M. Wilde, M. Tomamichel, S. Lloyd, and M. Berta, Phys. Rev. Lett. 119, 120501 (2017).
[Crossref]

R. García-Patrón, C. Navarrete-Benlloch, S. Lloyd, J. H. Shapiro, and N. J. Cerf, Phys. Rev. Lett. 108, 110505 (2012).
[Crossref]

J. H. Shapiro and S. Lloyd, New J. Phys. 11, 063045 (2009).
[Crossref]

S. Pirandola and S. Lloyd, Phys. Rev. A 78, 012331 (2008).
[Crossref]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, Phys. Rev. Lett. 101, 253601 (2008).
[Crossref]

S. Lloyd, Science 321, 1463 (2008).
[Crossref]

Lopaeva, E. D.

E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, Phys. Rev. Lett. 110, 153603 (2013).
[Crossref]

Lupo, C.

S. Pirandola, R. Laurenza, C. Lupo, and J. L. Pereira, npj Quantum Inf. 5, 1 (2019).
[Crossref]

Maccone, L.

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, Phys. Rev. Lett. 101, 253601 (2008).
[Crossref]

Masanes, L.

K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, L. Masanes, A. Acín, and F. Verstraete, Phys. Rev. Lett. 98, 160501 (2007).
[Crossref]

Meng, F.

M.-H. Yung, F. Meng, and M.-J. Zhao, “One-shot detection limits of quantum illumination with discrete signals,” arXiv:1801.07591 (2018).

Mosonyi, M.

T. Cooney, M. Mosonyi, and M. M. Wilde, Commun. Math. Phys. 344, 797 (2016).
[Crossref]

Mouradian, S.

Z. Zhang, S. Mouradian, F. N. C. Wong, and J. H. Shapiro, Phys. Rev. Lett. 114, 110506 (2015).
[Crossref]

Muñoz Tapia, R.

K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, L. Masanes, A. Acín, and F. Verstraete, Phys. Rev. Lett. 98, 160501 (2007).
[Crossref]

Nair, R.

R. Nair, Phys. Rev. Lett. 121, 230801 (2018).
[Crossref]

M. Tsang and R. Nair, Phys. Rev. A 86, 042115 (2012).
[Crossref]

R. Nair, Phys. Rev. A 84, 032312 (2011).
[Crossref]

Navarrete-Benlloch, C.

R. García-Patrón, C. Navarrete-Benlloch, S. Lloyd, J. H. Shapiro, and N. J. Cerf, Phys. Rev. Lett. 108, 110505 (2012).
[Crossref]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Olivares, S.

E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, Phys. Rev. Lett. 110, 153603 (2013).
[Crossref]

Pereira, J. L.

S. Pirandola, R. Laurenza, C. Lupo, and J. L. Pereira, npj Quantum Inf. 5, 1 (2019).
[Crossref]

Pirandola, S.

S. Barzanjeh, S. Pirandola, D. Vitali, and J. M. Fink, Sci. Adv. 6, eabb0451 (2020).
[Crossref]

S. Pirandola, R. Laurenza, C. Lupo, and J. L. Pereira, npj Quantum Inf. 5, 1 (2019).
[Crossref]

S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Nat. Photonics 12, 724 (2018).
[Crossref]

C. Weedbrook, S. Pirandola, J. Thompson, V. Vedral, and M. Gu, New J. Phys. 18, 043027 (2016).
[Crossref]

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, Phys. Rev. Lett. 114, 080503 (2015).
[Crossref]

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Supplementary Material (1)

NameDescription
» Supplement 1       Supplemental document with supporting calculations.

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Figures (2)

Fig. 1.
Fig. 1. General setup for QI: An entangled state $\Psi$ of $M$ signal and idler modes is prepared. Each signal mode $\hat a_S^{(m)}$ interrogates a distant target region bathed in thermal radiation $\hat a_B^{(m)}$, which may contain a target of effective reflectance $\eta \ll 1$. The modes $\{\hat a_R^{(m)}\} _{m = 1}^M$ returned from the target region are measured jointly with the unperturbed idler modes $\{\hat a_I^{(m)}\} _{m = 1}^M$ in order to decide whether or not the target was present.
Fig. 2.
Fig. 2. Left: A noisy attenuator channel ${{\cal L}_{\eta ,N}}$ mixes the signal mode $S$ with a background mode $B$ in a thermal state of energy $N$ at a beam splitter of transmittance $\eta$. Right: ${{\cal L}_{\eta ,N}}$ can be realized as a cascade of a quantum-limited (mode ${E_1}$ in a vacuum) loss channel ${{\cal L}_{\tilde \eta}}$ with $\tilde \eta = \eta /G$ and a quantum-limited (mode ${E_2}$ in a vacuum) amplifier (two-mode squeezer) ${{\cal A}_G}$ of gain $G = (1 - \eta)N + 1$.

Equations (21)

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a ^ R ( m ) = η h e i ϕ a ^ S ( m ) + 1 η h a ^ B ( m ) .
a ^ B ( m ) a ^ B ( m ) h N B ( h ) = N B / ( 1 η h )
| ψ IS = n p n | χ n I | n S
ρ h = [ id I ( U ϕ L η h , N B ( h ) ) M ] ( Ψ ) ,
P e [ ρ 0 , ρ 1 ] = 1 / 2 π 0 ρ 0 π 1 ρ 1 1 / 2 ,
P e cl π 0 π 1 exp [ η N S ( N B + 1 N B ) 2 ] .
P e TMSV π 0 π 1 exp ( η N S / N B )
L η , N = A G L η ~ ,
L η h , N B ( h ) = A N B + 1 L η h / ( N B + 1 ) A N B + 1 L η ~ h .
F ( ρ 0 , ρ 1 ) F ( [ i d I L η ~ 0 M ] ( Ψ ) , [ i d I L η ~ 1 M ] ( Ψ ) ) ,
n p n μ n ,
P e Ψ π 0 π 1 { n = 0 p n [ 1 η / ( N B + 1 ) ] n / 2 } 2
P e QI π 0 π 1 exp ( β N S ) ,
P e TMSV π 0 π 1 [ 1 + N S ( 1 1 η / ( N B + 1 ) ) / M ] 2 M .
ρ 1 = 1 2 π 0 1 d η P ( η ) 0 2 π d ϕ [ id I ( U ϕ L η , N B ( η ) ) M ] ( Ψ ) ,
P e QI ; fading π 0 π 1 / [ 1 + η ¯ N S ln ( 1 + 1 / N B ) ]
ρ η = [ id I ( U ϕ L η , N B ( η ) ) M ] ( Ψ ) ,
MS E η K η 1 = [ 4 2 F ( ρ η , ρ η ) / η 2 | η = η ] 1 ,
K η QI N S / [ η ( N B + 1 η ) ]
K η cl N S / [ η ( 2 N B + 1 ) ]
K η TMSV = [ N S ( 1 + N S ) ] / [ η ( 2 N S N B + N S + N B + 1 ) ] .

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