Abstract

The field of quantum information relies on the crucial issue of characterizing quantum states from measurements. This is performed through a process called quantum state tomography (QST). However, QST requires a large number of measurements, each derived from a different physical observable corresponding to a different experimental setup. Changing the setup results in unwanted changes to the data, prolongs the measurement, and impairs assumptions made about noise. Here, we propose to overcome these drawbacks by performing QST with a single observable. A single observable can often be realized by a single setup, thereby considerably reducing the experimental effort. However, the information contained in a single observable is insufficient for full QST. To overcome the lack of sufficient measurements in a single observable, we increase the system dimension by adding an ancilla that couples to the information in the system and exploit the fact that the sought state is often close to a pure state. We demonstrate our approach on multiphoton states by recovering structured quantum states from a single observable in a single experimental setup.

© 2017 Optical Society of America

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2017 (2)

G. Carleo and M. Troyer, “Solving the quantum many-body problem with artificial neural networks,” Science 355, 602–606 (2017).
[Crossref]

H. Wang, Y. He, Y.-H. Li, Z.-E. Su, B. Li, H.-L. Huang, X. Ding, M.-C. Chen, C. Liu, J. Qin, J.-P. Li, Y.-M. He, C. Schneider, M. Kamp, C.-Z. Peng, S. Höfling, C.-Y. Lu, and J.-W. Pan, “High-efficiency multiphoton boson sampling,” Nat. Photonics 11, 361–365 (2017).
[Crossref]

2016 (10)

P. Sidorenko and O. Cohen, “Single-shot ptychography,” Optica 3, 9–14 (2016).
[Crossref]

D. Oren, Y. Shechtman, M. Mutzafi, Y. C. Eldar, and M. Segev, “Sparsity-based recovery of three-photon quantum states from two-fold correlations,” Optica 3, 226–232 (2016).
[Crossref]

R. Heilmann, J. Sperling, A. Perez-Leija, M. Gräfe, M. Heinrich, S. Nolte, W. Vogel, and A. Szameit, “Harnessing click detectors for the genuine characterization of light states,” Sci. Rep. 6, 19489 (2016).
[Crossref]

J. G. Titchener, A. S. Solntsev, and A. A. Sukhorukov, “Two-photon tomography using on-chip quantum walks,” Opt. Lett. 41, 4079–4082 (2016).
[Crossref]

J. Sperling, T. J. Bartley, G. Donati, M. Barbieri, X.-M. Jin, A. Datta, W. Vogel, and I. A. Walmsley, “Quantum correlations from the conditional statistics of incomplete data,” Phys. Rev. Lett. 117, 083601 (2016).
[Crossref]

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walsmley, “Optimal design for universal multiport interferometers,” Optica 3, 1460–1465 (2016).
[Crossref]

G. I. Struchalin, I. A. Pogorelov, S. S. Straupe, K. S. Kravtsov, I. V. Radchenko, and S. P. Kulik, “Experimental adaptive quantum tomography of two-qubit states,” Phys. Rev. A 93, 012103 (2016).
[Crossref]

P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis, “Scalable quantum simulation of molecular energies,” Phys. Rev. X 6, 031007 (2016).
[Crossref]

S. Debnath, N. M. Linke, C. Figgatt, K. A. Landsman, K. Wright, and C. Monroe, “Demonstration of a small programmable quantum computer with atomic qubits,” Nature 536, 63–66 (2016).
[Crossref]

I. Schwartz, D. Cogan, E. R. Schmidgall, Y. Don, L. Gantz, O. Kenneth, N. H. Lindner, and D. Gershoni, “Deterministic generation of a cluster state of entangled photons,” Science 49, 1804–1807 (2016).

2015 (5)

J. Carolan, C. Harrold, C. Sparrow, E. Martin-Lopez, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

P. Sidorenko, O. Kfir, Y. Shechtman, A. Fleischer, Y. C. Eldar, M. Segev, and O. Cohen, “Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects,” Nat. Commun. 6, 8209 (2015).
[Crossref]

M. Mutzafi, Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based ankylography for recovering 3D molecular structures from single-shot 2D scattered light intensity,” Nat. Commun. 6, 7950 (2015).
[Crossref]

2014 (3)

G. A. Howland, J. Schneeloch, D. J. Lum, and J. C. Howell, “Simultaneous measurement of complementary observables with compressive sensing,” Phys. Rev. Lett. 112, 253602 (2014).
[Crossref]

M. Mirhosseini, O. S. Magaña-Loaiza, S. M. Hashemi Rafsanjani, and R. W. Boyd, “Compressive direct measurement of the quantum wave function,” Phys. Rev. Lett. 113, 090402 (2014).
[Crossref]

G. A. Howland, D. J. Lum, and J. C. Howell, “Compressive wavefront sensing with weak values,” Opt. Express 22, 18870–18880 (2014).
[Crossref]

2013 (7)

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref]

Y. Rivenson, A. Stern, and B. Javidi, “Overview of compressive sensing techniques applied in holography [Invited],” Appl. Opt. 52, A423–A432 (2013).
[Crossref]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[Crossref]

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[Crossref]

M. Tillmann, B. Dakic, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7, 540–544 (2013).
[Crossref]

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Integrated multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
[Crossref]

Y. Shechtman, E. Small, Y. Lahini, M. Verbin, Y. C. Eldar, Y. Silberberg, and M. Segev, “Sparsity-based super-resolution and phase-retrieval in waveguide arrays,” Opt. Express 21, 24015–24024 (2013).
[Crossref]

2012 (6)

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85, 023820 (2012).
[Crossref]

S. T. Flammia, D. Gross, Y.-K. Liu, and J. Eisert, “Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators,” New J. Phys. 14, 095022 (2012).
[Crossref]

E. Candes and B. Recht, “Exact matrix completion via convex optimization,” Commun. ACM 55, 111–119 (2012).
[Crossref]

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, 455–459 (2012).
[Crossref]

L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express 20, 8296–8308 (2012).
[Crossref]

F. Huszár and N. M. T. Houlsby, “Adaptive Bayesian quantum tomography,” Phys. Rev. A 85, 052120 (2012).
[Crossref]

2011 (3)

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]

Y. Shechtman, Y. C. Eldar, A. Szameit, and M. Segev, “Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing,” Opt. Express 19, 14807–14822 (2011).
[Crossref]

M. Mishali and Y. C. Eldar, “Sub-Nyquist sampling,” IEEE Signal Process. Mag. 28(6), 98–124 (2011).
[Crossref]

2010 (3)

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. (a) A system of N photons in m ports, with N=3 and m=4. The input state is assumed to be sparse, that is close to a pure state. (b) In our scheme, the dimension is increased by the addition of an ancilla, taking the form of vacuum ports in a photonic system. (c) The mixing between the degrees of freedom is realized by a random, linear coupler in the larger system.
Fig. 2.
Fig. 2. (a) Recovery example of a rank-2 density matrix describing N=3 photons in m=3 input ports and M=7 output ports. The state is recovered with fidelity 0.96. (b) Mean fidelity of the state recovered from measurements of a single observable versus rank of input state. Solid curves: recovery with practically no noise (SNR of 100 dB) using 7, 9, and 11 output ports. Dotted curves: same as the solid curves but with depolarization noise added to the state and measurement noise of 25 dB added to the measurements. The plots show the average over 200 realizations of the density matrix and 10 realizations of the random coupler for each point. The measurements used here are only a portion (21%–71%) of the measurements required for full QST. (c) Mean fidelity versus rank of input state, describing N=3 photons in m=7 input ports and M=16 output ports, averaged over 15 realizations of the density matrix. Here, the dimension of the system is large, d=84. The number of measurements in a single setup in this scenario is 11% of the measurements required for full QST. (d) Comparison between a fully mixing coupler (randomly sampled from the Haar measure) and a simpler coupler, consisting of identical evanescently coupled waveguides. The simpler coupler does not facilitate recovery from a single observable, whereas the Haar coupler performs well up to rank 6.
Fig. 3.
Fig. 3. Fidelity of the recovery of the quantum states from Fig. 2(b) in settings with (a) seven and (b) nine output ports. Solid lines: recovery fidelity obtained by comparing to the original states, known in our simulation but not available in experiments. Shaded areas: the ranks for which our scheme to gain confidence in the solution yields the answer “trustworthy” without knowing the input state. Without knowledge of the original state, the shaded areas match the ranks for which perfect recovery is achieved from only a fraction of the measurements, using the single observable, in a single setup.
Fig. 4.
Fig. 4. (a) Recovery example of a rank-2 density matrix describing N=3 photons in m=3 input ports, M=8 output ports, and with 25 dB of measurement noise. The state is recovered with 0.93 fidelity from measurements using click detectors. (b) Recovery fidelity with click detectors of density matrices of N=3 photons in m=4 output ports and M=11 output ports, corresponding to 41% of the measurements required for QST, in a single setup.

Equations (5)

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ρ0|00|U(ρ0|00|)U,
A=iiU|{n}i{n}i|UCD×D,
yi=Tr(ρU|{n}i{n}i|U)={n}i|UρU|{n}i,i{1,,D}.
minρ0rank(ρ0)subject to  ρ0=ρ0,ρ00,Tr(ρ0)=1|Tr(ρ0|00|Ai)yi|ϵ,i=1,,D.
minXkTr(Xk1+δI)1Xk,subject to  Xk0,Tr(Xk)=1,Xk=Xk,|Tr(Xk|00|Ai)yi|ϵ,i=1,,D.

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