Abstract

We present a scheme for recovering the complex input field launched into a waveguide array, from partial measurements of its output intensity, given advance knowledge that the input is sparse. In spite of the fact that in general the inversion problem is ill-conditioned, we demonstrate experimentally and in simulations that the prior knowledge of sparsity helps overcome the loss of information. Our method is based on GESPAR, a recently proposed efficient phase retrieval algorithm. Possible applications include optical interconnects and quantum state tomography, and the ideas are extendable to other multiple input and multiple output (MIMO) communication schemes.

© 2013 OSA

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2013 (2)

Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Efficient coherent diffractive imaging for sparsely varying objects,” Opt. Express 21(5), 6327–6338 (2013).
[Crossref] [PubMed]

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process. 22(4), 1382–1394 (2013).
[Crossref] [PubMed]

2012 (2)

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A 86(1), 011603 (2012).
[Crossref]

2011 (1)

2010 (2)

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35(8), 1148–1150 (2010).
[Crossref] [PubMed]

2009 (3)

D. A. B. Miller, “Device Requirements for Optical Interconnects to Silicon Chips,” Proc. IEEE 97(7), 1166–1185 (2009).
[Crossref]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[Crossref] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102(25), 253904 (2009).
[Crossref] [PubMed]

2006 (2)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

2005 (1)

2004 (1)

M. A. Nielsen, “Optical Quantum Computation Using Cluster States,” Phys. Rev. Lett. 93(4), 040503 (2004).
[Crossref] [PubMed]

2001 (1)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[Crossref] [PubMed]

2000 (1)

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000).
[Crossref]

1965 (1)

Blömer, D.

Bromberg, Y.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A 86(1), 011603 (2012).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102(25), 253904 (2009).
[Crossref] [PubMed]

Bullkich, E.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Burghoff, J.

Candes, E. J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Efficient coherent diffractive imaging for sparsely varying objects,” Opt. Express 21(5), 6327–6338 (2013).
[Crossref] [PubMed]

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Cohen-Hyams, T.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Dana, H.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Dong, W.

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process. 22(4), 1382–1394 (2013).
[Crossref] [PubMed]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

Eldar, Y. C.

Gazit, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35(8), 1148–1150 (2010).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[Crossref] [PubMed]

Huber, S. D.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A 86(1), 011603 (2012).
[Crossref]

Ismail, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Jones, A. L.

Kley, E. B.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Knill, E.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[Crossref] [PubMed]

Laflamme, R.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[Crossref] [PubMed]

Lahini, Y.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A 86(1), 011603 (2012).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102(25), 253904 (2009).
[Crossref] [PubMed]

Lederer, F.

Lobino, M.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Lukac, R.

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process. 22(4), 1382–1394 (2013).
[Crossref] [PubMed]

Matsuda, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Matthews, J. C. F.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Milburn, G. J.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[Crossref] [PubMed]

Miller, D. A. B.

D. A. B. Miller, “Device Requirements for Optical Interconnects to Silicon Chips,” Proc. IEEE 97(7), 1166–1185 (2009).
[Crossref]

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000).
[Crossref]

Morandotti, R.

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102(25), 253904 (2009).
[Crossref] [PubMed]

Nielsen, M. A.

M. A. Nielsen, “Optical Quantum Computation Using Cluster States,” Phys. Rev. Lett. 93(4), 040503 (2004).
[Crossref] [PubMed]

Nolte, S.

OBrien, J. L.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Osherovich, E.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Pertsch, T.

Peruzzo, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Politi, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Poulios, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Pugatch, R.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A 86(1), 011603 (2012).
[Crossref]

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Schreiber, T.

Segev, M.

Shechtman, Y.

Shi, G.

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process. 22(4), 1382–1394 (2013).
[Crossref] [PubMed]

Shoham, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Sidorenko, P.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Silberberg, Y.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A 86(1), 011603 (2012).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and Classical Correlations in Waveguide Lattices,” Phys. Rev. Lett. 102(25), 253904 (2009).
[Crossref] [PubMed]

Steiner, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Szameit, A.

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Thompson, M. G.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Tünnermann, A.

Verbin, M.

Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, “Quantum walk of two interacting bosons,” Phys. Rev. A 86(1), 011603 (2012).
[Crossref]

Wörhoff, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Yavneh, I.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Zhang, L.

W. Dong, L. Zhang, R. Lukac, and G. Shi, “Sparse Representation Based Image Interpolation With Nonlocal Autoregressive Modeling,” IEEE Trans. Image Process. 22(4), 1382–1394 (2013).
[Crossref] [PubMed]

Zhou, X.-Q.

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A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
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A. Beck and Y. C. Eldar, “Sparsity constrained nonlinear optimization: Optimality conditions and algorithms,” arXiv preprint arXiv:1203.4580 (2012).

Y. C. Eldar and S. Mendelson, “Phase retrieval: Stability and recovery guarantees,” arXiv preprint arXiv:1211.0872 (2012).

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Figures (5)

Fig. 1
Fig. 1

(a) Impulse response for light propagating in a periodic waveguide array (intensity). The light is launched into the central waveguide (#0), and evolves such that the envelope corresponds to the absolute value squared of the expression in Eq. (1), from left to right. Each row represents an individual waveguide. (b) A different input signal would yield a correspondingly different propagation pattern.

Fig. 2
Fig. 2

Experimental scheme. The EM field emerging from the 1D SLM (working in amplitude mode between two crossed polarizers) is imaged onto the input facet of the waveguide array. The light undergoes discrete diffraction in the array. The output intensity is imaged onto a CCD camera.

Fig. 3
Fig. 3

Recovery from the complete set of measurements: (a)measured output intensity, (b) discretized measurements vector, and recovery results of (c) amplitude and (d) phase.

Fig. 4
Fig. 4

Recovery from partial measurements. (a) Point spread function of the system. (b) Input and recovered amplitudes. (c) Measured output intensities. (d) Input and recovered phases. (e) The 11 measurements used and the consistency of the recovered signal at the system's output. (f) Super-resolution: accurate recovery of the non-measured output intensities (blue) using half of the output intensities (dashed blue).

Fig. 5
Fig. 5

(a) Reconstruction error vs. sparsity level (k) for various SNR values. (b) Support location vs. sparsity levels for different values of m - number of evenly spaced intensity measurements

Equations (6)

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i z a k +c( a k1 + a k+1 )=0
h[ k ]= i k J k ( 2c z 0 )
y i = | h i x | 2 = x * H ˜ i x,iM
x ^ = argmin x i=1 m ( y i x * H ˜ i x ) 2 subjectto x 0 s
n c = q a ' k q p ' k ( m k )= q a ' k q p ' k m! k!( mk )!
α= n c n uc = q a ' k q p ' k m! q m k!( mk )! >1

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