Abstract

We demonstrate how to efficiently implement extremely high-dimensional compressive imaging of a bi-photon probability distribution. Our method uses fast-Hadamard-transform Kronecker-based compressive sensing to acquire the joint space distribution. We list, in detail, the operations necessary to enable fast-transform-based matrix-vector operations in the joint space to reconstruct a 16.8 million-dimensional image in less than 10 minutes. Within a subspace of that image exists a 3.2 million-dimensional bi-photon probability distribution. In addition, we demonstrate how the marginal distributions can aid in the accuracy of joint space distribution reconstructions.

© 2015 Optical Society of America

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2015 (2)

2014 (1)

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

2013 (4)

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref] [PubMed]

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref] [PubMed]

G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X 3, 011013 (2013).

J. Schneeloch, P. B. Dixon, G. A. Howland, C. J. Broadbent, and J. C. Howell, “Violation of continuous-variable Einstein-Podolsky-Rosen steering with discrete measurements,” Phys. Rev. Lett. 110, 130407 (2013).
[Crossref] [PubMed]

2012 (4)

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Trans. Image Process. 21, 494–504 (2012).
[Crossref]

S. L. Shishkin, “Fast and Robust Compressive Sensing Method Using Mixed Hadamard Sensing Matrix,” IEEE J. Emerging Sel. Top. Circuits Syst. 2, 353–361 (2012).
[Crossref]

2011 (2)

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

2009 (4)

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299 (2009).
[Crossref]

M. V. Fedorov, Y. M. Mikhailova, and P. A. Volkov, “Gaussian modelling and Schmidt modes of SPDC biphoton states,” J. Phys. B: At. Mol. Opt. Phys. 42, 175503 (2009).
[Crossref]

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imag. Sci. 2, 183–202 (2009).
[Crossref]

2008 (3)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process Mag. 25, 21–30 (2008).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process Mag. 25, 83 (2008).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

2006 (1)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

2005 (2)

I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process Mag. 22, 123–151 (2005).
[Crossref]

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

2004 (2)

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[Crossref] [PubMed]

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum-and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref]

1998 (1)

A. Steane, “Quantum computing,” Rep. Prog. Phys. 61, 117 (1998).
[Crossref]

1997 (1)

T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A 56, 1534 (1997).
[Crossref]

1996 (1)

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A 54, 5349 (1996).
[Crossref] [PubMed]

1995 (1)

D.L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[Crossref]

1994 (1)

D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 4255 (1994).
[Crossref]

1992 (1)

M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[Crossref] [PubMed]

1990 (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[Crossref]

Agnew, M.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

Almeida, M. P.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Antonini, M.

M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[Crossref] [PubMed]

Baraniuk, R. G.

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Trans. Image Process. 21, 494–504 (2012).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process Mag. 25, 83 (2008).
[Crossref]

I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process Mag. 22, 123–151 (2005).
[Crossref]

Barlaud, M.

M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[Crossref] [PubMed]

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imag. Sci. 2, 183–202 (2009).
[Crossref]

Bennink, R. S.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum-and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref]

Bentley, S. J.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum-and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref]

Boyd, R. W.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum-and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref]

Boyd, Robert W.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

Braunstein, S. L.

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

Broadbent, C. J.

J. Schneeloch, P. B. Dixon, G. A. Howland, C. J. Broadbent, and J. C. Howell, “Violation of continuous-variable Einstein-Podolsky-Rosen steering with discrete measurements,” Phys. Rev. Lett. 110, 130407 (2013).
[Crossref] [PubMed]

Broome, M. A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Buller, Gerald S.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

Candès, E. J.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process Mag. 25, 21–30 (2008).
[Crossref]

Chan, S.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

Daubechies, I.

M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[Crossref] [PubMed]

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[Crossref]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process Mag. 25, 83 (2008).
[Crossref]

Dixon, P. B.

J. Schneeloch, P. B. Dixon, G. A. Howland, C. J. Broadbent, and J. C. Howell, “Violation of continuous-variable Einstein-Podolsky-Rosen steering with discrete measurements,” Phys. Rev. Lett. 110, 130407 (2013).
[Crossref] [PubMed]

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 4255 (1994).
[Crossref]

Donoho, D.L.

D.L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[Crossref]

Dowling, J. P.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Duarte, M. F.

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Trans. Image Process. 21, 494–504 (2012).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process Mag. 25, 83 (2008).
[Crossref]

Dudley, A.

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref] [PubMed]

Eberly, J. H.

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[Crossref] [PubMed]

Edgar, M. P.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

Fedorov, M. V.

M. V. Fedorov, Y. M. Mikhailova, and P. A. Volkov, “Gaussian modelling and Schmidt modes of SPDC biphoton states,” J. Phys. B: At. Mol. Opt. Phys. 42, 175503 (2009).
[Crossref]

Fedrizzi, A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Fickler, R.

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref] [PubMed]

Forbes, A.

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref] [PubMed]

Furusawa, A.

G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, “Continuous-variable entanglement on a chip,” Nat. Photonics 9, 316–319 (2015).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Giovannini, D.

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref] [PubMed]

Hashimoto, T.

G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, “Continuous-variable entanglement on a chip,” Nat. Photonics 9, 316–319 (2015).
[Crossref]

Horn, R. A.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge University Press, 1991.)
[Crossref]

Howell, J. C.

J. Schneeloch, S. H. Knarr, G. A. Howland, and J. C. Howell, “Demonstrating continuous variable Einstein-Podolsky-Rosen steering in spite of finite experimental capabilities using Fano steering bounds,” J. Opt. Soc. Am. B 32(4), A8–A14 (2015).
[Crossref]

J. Schneeloch, P. B. Dixon, G. A. Howland, C. J. Broadbent, and J. C. Howell, “Violation of continuous-variable Einstein-Podolsky-Rosen steering with discrete measurements,” Phys. Rev. Lett. 110, 130407 (2013).
[Crossref] [PubMed]

G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X 3, 011013 (2013).

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum-and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref]

Howland, G. A.

J. Schneeloch, S. H. Knarr, G. A. Howland, and J. C. Howell, “Demonstrating continuous variable Einstein-Podolsky-Rosen steering in spite of finite experimental capabilities using Fano steering bounds,” J. Opt. Soc. Am. B 32(4), A8–A14 (2015).
[Crossref]

J. Schneeloch, P. B. Dixon, G. A. Howland, C. J. Broadbent, and J. C. Howell, “Violation of continuous-variable Einstein-Podolsky-Rosen steering with discrete measurements,” Phys. Rev. Lett. 110, 130407 (2013).
[Crossref] [PubMed]

G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X 3, 011013 (2013).

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

Huver, S. D.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Izdebski, F.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

Johnson, C. R.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge University Press, 1991.)
[Crossref]

Johnstone, J. M.

D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 4255 (1994).
[Crossref]

Keller, T. E.

T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A 56, 1534 (1997).
[Crossref]

Kelly, K. E.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process Mag. 25, 83 (2008).
[Crossref]

Kingsbury, N. C.

I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process Mag. 22, 123–151 (2005).
[Crossref]

Knarr, S. H.

Kosut, R. L.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Krenn, M.

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref] [PubMed]

Lapkiewicz, R.

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref] [PubMed]

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process Mag. 25, 83 (2008).
[Crossref]

Law, C. K.

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[Crossref] [PubMed]

Leach, J.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref] [PubMed]

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

Li, C.

C. Li, “Compressive Sensing for 3D Data Processing Tasks: Applications, Models and Algorithms,” Ph.D. thesis, Rice University (2011).

C. Li, W. Yin, and Y. Zhang, “Users Guide for TVAL3: TV Minimization by Augmented Lagrangian and Alternating Direction Algorithms,” (2009).

Lindsay, A.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Lvovsky, A. I.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299 (2009).
[Crossref]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Masada, G.

G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, “Continuous-variable entanglement on a chip,” Nat. Photonics 9, 316–319 (2015).
[Crossref]

Mathieu, P.

M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[Crossref] [PubMed]

Mikhailova, Y. M.

M. V. Fedorov, Y. M. Mikhailova, and P. A. Volkov, “Gaussian modelling and Schmidt modes of SPDC biphoton states,” J. Phys. B: At. Mol. Opt. Phys. 42, 175503 (2009).
[Crossref]

Miyata, K.

G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, “Continuous-variable entanglement on a chip,” Nat. Photonics 9, 316–319 (2015).
[Crossref]

Mohseni, M.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

O’Brien, J. L.

G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, “Continuous-variable entanglement on a chip,” Nat. Photonics 9, 316–319 (2015).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Padgett, M. J.

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref] [PubMed]

Padgett, Miles J.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

Politi, A.

G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, “Continuous-variable entanglement on a chip,” Nat. Photonics 9, 316–319 (2015).
[Crossref]

Rabitz, H.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Ramelow, S.

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref] [PubMed]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299 (2009).
[Crossref]

Rivenson, Y.

Y. Rivenson and A. Stern, “Practical compressive sensing of large images,” in Proceedings of IEEE Conference on Digital Signal Processing (IEEE, 2009), pp. 1–8.

Romero, J.

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref] [PubMed]

Rubin, M. H.

T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A 56, 1534 (1997).
[Crossref]

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A 54, 5349 (1996).
[Crossref] [PubMed]

Schneeloch, J.

J. Schneeloch, S. H. Knarr, G. A. Howland, and J. C. Howell, “Demonstrating continuous variable Einstein-Podolsky-Rosen steering in spite of finite experimental capabilities using Fano steering bounds,” J. Opt. Soc. Am. B 32(4), A8–A14 (2015).
[Crossref]

J. Schneeloch, P. B. Dixon, G. A. Howland, C. J. Broadbent, and J. C. Howell, “Violation of continuous-variable Einstein-Podolsky-Rosen steering with discrete measurements,” Phys. Rev. Lett. 110, 130407 (2013).
[Crossref] [PubMed]

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

Selesnick, I. W.

I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process Mag. 22, 123–151 (2005).
[Crossref]

Shabani, A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Shishkin, S. L.

S. L. Shishkin, “Fast and Robust Compressive Sensing Method Using Mixed Hadamard Sensing Matrix,” IEEE J. Emerging Sel. Top. Circuits Syst. 2, 353–361 (2012).
[Crossref]

Steane, A.

A. Steane, “Quantum computing,” Rep. Prog. Phys. 61, 117 (1998).
[Crossref]

Stern, A.

Y. Rivenson and A. Stern, “Practical compressive sensing of large images,” in Proceedings of IEEE Conference on Digital Signal Processing (IEEE, 2009), pp. 1–8.

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process Mag. 25, 83 (2008).
[Crossref]

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process Mag. 25, 83 (2008).
[Crossref]

Tasca, D. S.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

Teboulle, M.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imag. Sci. 2, 183–202 (2009).
[Crossref]

Tonolini, F.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

Van Loock, P.

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

Volkov, P. A.

M. V. Fedorov, Y. M. Mikhailova, and P. A. Volkov, “Gaussian modelling and Schmidt modes of SPDC biphoton states,” J. Phys. B: At. Mol. Opt. Phys. 42, 175503 (2009).
[Crossref]

Vuckovic, J.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Wakin, M. B.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process Mag. 25, 21–30 (2008).
[Crossref]

Warburton, R. E.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

White, A. G.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Wildfeuer, C. F.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Yarlagadda, R. K.

R. K. Yarlagadda and R. R. Yarlagadda, Hadamard Matrix Analysis and Synthesis (Kluwer Academic Publishers, 1997).
[Crossref]

Yarlagadda, R. R.

R. K. Yarlagadda and R. R. Yarlagadda, Hadamard Matrix Analysis and Synthesis (Kluwer Academic Publishers, 1997).
[Crossref]

Yin, W.

C. Li, W. Yin, and Y. Zhang, “Users Guide for TVAL3: TV Minimization by Augmented Lagrangian and Alternating Direction Algorithms,” (2009).

Zeilinger, A.

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref] [PubMed]

Zhang, Y.

C. Li, W. Yin, and Y. Zhang, “Users Guide for TVAL3: TV Minimization by Augmented Lagrangian and Alternating Direction Algorithms,” (2009).

Biometrika (1)

D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 4255 (1994).
[Crossref]

IEEE J. Emerging Sel. Top. Circuits Syst. (1)

S. L. Shishkin, “Fast and Robust Compressive Sensing Method Using Mixed Hadamard Sensing Matrix,” IEEE J. Emerging Sel. Top. Circuits Syst. 2, 353–361 (2012).
[Crossref]

IEEE Signal Process Mag. (3)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process Mag. 25, 21–30 (2008).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process Mag. 25, 83 (2008).
[Crossref]

I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process Mag. 22, 123–151 (2005).
[Crossref]

IEEE Trans. Image Process. (2)

M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Process. 1, 205–220 (1992).
[Crossref] [PubMed]

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Trans. Image Process. 21, 494–504 (2012).
[Crossref]

IEEE Trans. Inf. Theory (3)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[Crossref]

D.L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. B: At. Mol. Opt. Phys. (1)

M. V. Fedorov, Y. M. Mikhailova, and P. A. Volkov, “Gaussian modelling and Schmidt modes of SPDC biphoton states,” J. Phys. B: At. Mol. Opt. Phys. 42, 175503 (2009).
[Crossref]

Nat. Commun. (1)

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, Gerald S. Buller, Robert W. Boyd, and Miles J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref] [PubMed]

Nat. Photonics (3)

G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, “Continuous-variable entanglement on a chip,” Nat. Photonics 9, 316–319 (2015).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Phys. Rev. A (3)

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A 54, 5349 (1996).
[Crossref] [PubMed]

T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A 56, 1534 (1997).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Phys. Rev. Lett. (6)

J. Schneeloch, P. B. Dixon, G. A. Howland, C. J. Broadbent, and J. C. Howell, “Violation of continuous-variable Einstein-Podolsky-Rosen steering with discrete measurements,” Phys. Rev. Lett. 110, 130407 (2013).
[Crossref] [PubMed]

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum-and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref]

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[Crossref] [PubMed]

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref] [PubMed]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Phys. Rev. X (1)

G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X 3, 011013 (2013).

Rep. Prog. Phys. (1)

A. Steane, “Quantum computing,” Rep. Prog. Phys. 61, 117 (1998).
[Crossref]

Rev. Mod. Phys. (2)

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299 (2009).
[Crossref]

Sci. Rep. (2)

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref] [PubMed]

SIAM J. Imag. Sci. (1)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imag. Sci. 2, 183–202 (2009).
[Crossref]

Other (7)

J. Schneeloch and J. C. Howell, “Introduction to the transverse spatial correlations in spontaneous parametric down-conversion through the biphoton birth zone,” http://arxiv.org/abs/1502.06996 .

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge University Press, 1991.)
[Crossref]

R. K. Yarlagadda and R. R. Yarlagadda, Hadamard Matrix Analysis and Synthesis (Kluwer Academic Publishers, 1997).
[Crossref]

C. Li, “Compressive Sensing for 3D Data Processing Tasks: Applications, Models and Algorithms,” Ph.D. thesis, Rice University (2011).

C. Li, W. Yin, and Y. Zhang, “Users Guide for TVAL3: TV Minimization by Augmented Lagrangian and Alternating Direction Algorithms,” (2009).

Y. Rivenson and A. Stern, “Practical compressive sensing of large images,” in Proceedings of IEEE Conference on Digital Signal Processing (IEEE, 2009), pp. 1–8.

E. Bolduc, G. Gariepy, and J. Leach, “Direct measurement of large-scale quantum states,” http://arxiv.org/abs/1506.00851 .

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Figures (3)

Fig. 1
Fig. 1 The above experimental diagram demonstrates how to image a joint two-particle system. In this paper, the joint system is composed of highly-correlated signal and idler photons from a SPDC source. The experiment samples the position distribution of the joint system by taking random projections of signal and idler intensities with a spatial light modulator within an image plane of the crystal. An avalanche photodiode (APD) detects photon arrivals while the photon counters measure photon coincidences.
Fig. 2
Fig. 2 Image (a) depicts a zoomed in view that captures the largest components of the reconstructed 16.8 × 106 joint space distribution. The largest components are contained within the resulting 4-megapixel image shown. The 1/e2 intensity profile, as seen by the signal’s SLM, is presented as the signal’s measured marginal distribution in (b). The reconstructed marginal distribution was obtained through the reconstructed joint space distribution and is displayed in (c) for comparison. This data was obtained compressively with M = 20, 000 (.00119 × 644) samples in approximately 44 hours and was reconstructed in under ten minutes.
Fig. 3
Fig. 3 The mutual information, obtained through reconstructions of the joint space distribution, as a function of the number of projections M is depicted above. The figure lists the theoretical maximum mutual information and the corresponding measurement results either utilizing or neglecting available information from the marginal distributions. Neglecting information found in the marginals clearly increases the inaccuracy, especially for significantly small M, while including this information leads to more realistic values.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

min x τ g ( x ) subject to y A ( Ψ 1 x ) 2 2 < ε ,
x p : = ( i = 1 n | x i | p ) 1 / p
A = [ P S [ 1 ] P I [ 1 ] P S [ 2 ] P I [ 2 ] P S [ M ] P I [ M ] ]
a b = [ a 11 b a 1 n b a m 1 b a m n b ] .
H 1 = [ 1 ] H 2 = [ 1 1 1 1 ] .
H 2 k = H 2 H 2 k 1 = [ H 2 k 1 H 2 k 1 H 2 k 1 H 2 k 1 ] .
P S = H N [ r S , p S ] P I = H N [ r I , p I ]
r SI [ i ] = N ( r S [ i ] 1 ) + r I [ i ]
p SI [ N ( i 1 ) + j ] = N ( p S [ i ] 1 ) + p I [ j ]
P S = H N [ r S , p S ] P I = H N [ r I , p I ] A = H N 2 [ r SI , p SI ]
H [ r [ i ] , : ] = [ α [ r [ i ] ] ] P [ i ] = H [ r [ i ] , p ]
H N 2 = ( H N + H N + ) + ( H N H N ) + ( H N + H N ) + ( H N H N + ) .
q [ p [ i ] ] = i
y = [ x [ q SI ] ] y = y [ r SI ] .
β [ r SI ] = y x [ q SI ] = [ β ] .
I ( X S , X I ) = x S X S x I X I p ( x S , x I ) log 2 ( p ( x S , x I ) p ( x S ) p ( x I ) ) ,
I ( x S , x I ) = log 2 ( 9 π σ p 2 + L z λ p 2 σ p 9 π L z λ p )
x 0 = c x t + 1 = η ^ 2 [ x t { η ^ 1 [ A T ( y A x t ) ] } + x t min ( x t ) ]

Metrics