Abstract

Quantum state tomography is a key technology for fully determining a quantum state. Unfortunately, standard quantum state tomography is intractable for general many-body quantum states, because the number of measurements and the post-processing time increase exponentially with the size of the system. However, for the matrix product states (MPSs), there exists an efficient method using linearly scaled local measurements and polynomially scaled post-processing times. In this study, we demonstrate the validity of the method in practice by reconstructing a four-photon MPS from its local two- or three-photon reduced-density matrices with the presence of statistical errors and systematical errors in experiment.

© 2017 Optical Society of America

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References

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2016 (1)

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

2015 (1)

S. Dogra and K. Dorai, and Arvind, “Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor,” Phys. Rev. A 91, 022312 (2015).
[Crossref]

2014 (2)

B. Swingle and I. H. Kim, “Reconstructing quantum states from local data,” Phys. Rev. Lett. 113, 260501 (2014).
[Crossref]

Christian Schwemmer, Géza Tóth, Alexander Niggebaum, Tobias Moroder, and David Gross, Otfried Gühne, and Harald Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref]

2013 (2)

T. Baumgratz, A. Nüßeler, M. Cramer, and M. B. Plenio, “A scalable maximum likelihood method for quantum state tomography,” New J. Phys. 15, 125004 (2013).
[Crossref]

B. Qi, Z. B. Hou, L. Li, D. Dong, G. Y. Xiang, and G. C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

2011 (3)

D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory 57, 1548 (2011).
[Crossref]

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

S. T. Flammia and Y.-K. Liu, “Direct fidelity estimation from few pauli measurements,” Phys. Rev. Lett. 106, 230501 (2011).
[Crossref] [PubMed]

2010 (3)

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

J. F. Cai, E. J. Candes, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. on Optimization 201956 (2010).
[Crossref]

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).
[Crossref]

2008 (1)

S. N. Walck and D. W. Lyons, “Only n-qubit Greenberger-Horne-Zeilinger states are undetermined by their reduced density matrices,” Phys. Rev. Lett. 100, 050501 (2008).
[Crossref] [PubMed]

2007 (1)

Y. K. Liu, M. Christandl, and F. Verstraete, “Quantum computational complexity of the N-representability problem: QMA complete,” Phys. Rev. Lett. 98, 110503 (2007).
[Crossref] [PubMed]

2006 (3)

J. S. Xu, C. F. Li, and G. C. Guo, “Generation of a high-visibility four-photon entangled state and realization of a four-party quantum communication complexity scenario,” Phys. Rev. A 74, 052311 (2006).
[Crossref]

M. B. Hastings, “Solving gapped Hamiltonians locally,” Phys. Rev. B 73, 085115 (2006).
[Crossref]

A. A. Klyachko, “Quantum marginal problem and N-representability,” Journal of Physics: Conf. Series 36, 72 (2006).

2005 (1)

U. Schollwock, “The density-matrix renormalization group,” Rev. Mod. Phys. 77, 259 (2005).
[Crossref]

2004 (3)

Y. J. Han, Y. S. Zhang, and G. C. Guo, “Compatible conditions, entanglement, and invariants,” Phys. Rev. A 70, 042309 (2004).
[Crossref]

S. Bravyi, “Compatibility between local and multipartite states,” Quant. Inf. and Comput. 4, 12 (2004).

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a Controlled-NOT gate.,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

2003 (2)

G. Vidal, “Efficient classical simulation of slightly entangled quantum computations,” Phys. Rev. Lett. 91, 147902 (2003).
[Crossref] [PubMed]

M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental observation of four-photon entanglement from parametric down-conversion,” Phys. Rev. Lett. 90, 200403 (2003).
[Crossref] [PubMed]

2002 (2)

N. Linden, S. Popescu, and W. K. Wootters, “Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices,” Phys. Rev. Lett. 89, 207901 (2002).
[Crossref] [PubMed]

N. Linden and W. K. Wootters, “The parts determine the whole in a generic pure quantum state,” Phys. Rev. Lett. 89, 277906 (2002).
[Crossref]

2001 (2)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys.-Usp. 44, 131–136 (2001).
[Crossref]

1992 (1)

M. Fannes, B. Nachtergaele, and R. F. Werner, “Finitely correlated states on quantum spin chains,” Comm. Math. Phys. 144, 443–490 (1992).
[Crossref]

1988 (1)

I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Valence bond ground states in isotropic quantum antiferromagnets,” Commun. Math. Phys. 115, 477–528 (1988).
[Crossref]

1963 (1)

A. J. Coleman, “Structure of fermion density matrices,” Rev. Mod. Phys. 35, 668 (1963).
[Crossref]

Affleck, I.

I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Valence bond ground states in isotropic quantum antiferromagnets,” Commun. Math. Phys. 115, 477–528 (1988).
[Crossref]

Barreiro, J. T.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Bartlett, S. D.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Baumgratz, T.

T. Baumgratz, A. Nüßeler, M. Cramer, and M. B. Plenio, “A scalable maximum likelihood method for quantum state tomography,” New J. Phys. 15, 125004 (2013).
[Crossref]

Becker, S.

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).
[Crossref]

Blatt, R.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Bourennane, M.

M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental observation of four-photon entanglement from parametric down-conversion,” Phys. Rev. Lett. 90, 200403 (2003).
[Crossref] [PubMed]

Bravyi, S.

S. Bravyi, “Compatibility between local and multipartite states,” Quant. Inf. and Comput. 4, 12 (2004).

Cai, J. F.

J. F. Cai, E. J. Candes, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. on Optimization 201956 (2010).
[Crossref]

Candes, E. J.

J. F. Cai, E. J. Candes, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. on Optimization 201956 (2010).
[Crossref]

Chen, C.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Chen, L. K.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Chen, Y. A.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Christandl, M.

Y. K. Liu, M. Christandl, and F. Verstraete, “Quantum computational complexity of the N-representability problem: QMA complete,” Phys. Rev. Lett. 98, 110503 (2007).
[Crossref] [PubMed]

Chwalla, M.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Coish, W. A.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Coleman, A. J.

A. J. Coleman, “Structure of fermion density matrices,” Rev. Mod. Phys. 35, 668 (1963).
[Crossref]

A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: Coulson’s Challenge (Springer-Verlag, 2000).
[Crossref]

Cramer, M.

T. Baumgratz, A. Nüßeler, M. Cramer, and M. B. Plenio, “A scalable maximum likelihood method for quantum state tomography,” New J. Phys. 15, 125004 (2013).
[Crossref]

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Dogra, S.

S. Dogra and K. Dorai, and Arvind, “Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor,” Phys. Rev. A 91, 022312 (2015).
[Crossref]

Dong, D.

B. Qi, Z. B. Hou, L. Li, D. Dong, G. Y. Xiang, and G. C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Dorai, K.

S. Dogra and K. Dorai, and Arvind, “Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor,” Phys. Rev. A 91, 022312 (2015).
[Crossref]

Eibl, M.

M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental observation of four-photon entanglement from parametric down-conversion,” Phys. Rev. Lett. 90, 200403 (2003).
[Crossref] [PubMed]

Eisert, J.

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).
[Crossref]

Fannes, M.

M. Fannes, B. Nachtergaele, and R. F. Werner, “Finitely correlated states on quantum spin chains,” Comm. Math. Phys. 144, 443–490 (1992).
[Crossref]

Flammia, S. T.

S. T. Flammia and Y.-K. Liu, “Direct fidelity estimation from few pauli measurements,” Phys. Rev. Lett. 106, 230501 (2011).
[Crossref] [PubMed]

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

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).
[Crossref]

Gaertner, S.

M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental observation of four-photon entanglement from parametric down-conversion,” Phys. Rev. Lett. 90, 200403 (2003).
[Crossref] [PubMed]

Gilchrist, A.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a Controlled-NOT gate.,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Gross, D.

D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory 57, 1548 (2011).
[Crossref]

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

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).
[Crossref]

Gross, David

Christian Schwemmer, Géza Tóth, Alexander Niggebaum, Tobias Moroder, and David Gross, Otfried Gühne, and Harald Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref]

Guo, G. C.

B. Qi, Z. B. Hou, L. Li, D. Dong, G. Y. Xiang, and G. C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

J. S. Xu, C. F. Li, and G. C. Guo, “Generation of a high-visibility four-photon entangled state and realization of a four-party quantum communication complexity scenario,” Phys. Rev. A 74, 052311 (2006).
[Crossref]

Y. J. Han, Y. S. Zhang, and G. C. Guo, “Compatible conditions, entanglement, and invariants,” Phys. Rev. A 70, 042309 (2004).
[Crossref]

Han, Y. J.

Y. J. Han, Y. S. Zhang, and G. C. Guo, “Compatible conditions, entanglement, and invariants,” Phys. Rev. A 70, 042309 (2004).
[Crossref]

Hansel, W.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Harlander, M.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Hastings, M. B.

M. B. Hastings, “Solving gapped Hamiltonians locally,” Phys. Rev. B 73, 085115 (2006).
[Crossref]

Hennrich, M.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Hou, Z. B.

B. Qi, Z. B. Hou, L. Li, D. Dong, G. Y. Xiang, and G. C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Hu, Y.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Huang, H. L.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

James, D. F. V.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a Controlled-NOT gate.,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Jiang, X.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Kennedy, T.

I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Valence bond ground states in isotropic quantum antiferromagnets,” Commun. Math. Phys. 115, 477–528 (1988).
[Crossref]

Kim, I. H.

B. Swingle and I. H. Kim, “Reconstructing quantum states from local data,” Phys. Rev. Lett. 113, 260501 (2014).
[Crossref]

Kitaev, A. Y.

A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys.-Usp. 44, 131–136 (2001).
[Crossref]

Klyachko, A. A.

A. A. Klyachko, “Quantum marginal problem and N-representability,” Journal of Physics: Conf. Series 36, 72 (2006).

Kosut, R. L.

R. L. Kosut, “Quantum process tomography via l1-norm minimization,” https://arxiv.org/abs/0812.4323.

Kurtsiefer, C.

M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental observation of four-photon entanglement from parametric down-conversion,” Phys. Rev. Lett. 90, 200403 (2003).
[Crossref] [PubMed]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Landon-Cardinal, O.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Langford, N. K.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a Controlled-NOT gate.,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Li, C. F.

J. S. Xu, C. F. Li, and G. C. Guo, “Generation of a high-visibility four-photon entangled state and realization of a four-party quantum communication complexity scenario,” Phys. Rev. A 74, 052311 (2006).
[Crossref]

Li, L.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

B. Qi, Z. B. Hou, L. Li, D. Dong, G. Y. Xiang, and G. C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Li, W.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Li, Z. D.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Lieb, E. H.

I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Valence bond ground states in isotropic quantum antiferromagnets,” Commun. Math. Phys. 115, 477–528 (1988).
[Crossref]

Linden, N.

N. Linden, S. Popescu, and W. K. Wootters, “Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices,” Phys. Rev. Lett. 89, 207901 (2002).
[Crossref] [PubMed]

N. Linden and W. K. Wootters, “The parts determine the whole in a generic pure quantum state,” Phys. Rev. Lett. 89, 277906 (2002).
[Crossref]

Liu, C.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Liu, N. L.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Liu, Y. K.

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).
[Crossref]

Y. K. Liu, M. Christandl, and F. Verstraete, “Quantum computational complexity of the N-representability problem: QMA complete,” Phys. Rev. Lett. 98, 110503 (2007).
[Crossref] [PubMed]

Liu, Y.-K.

S. T. Flammia and Y.-K. Liu, “Direct fidelity estimation from few pauli measurements,” Phys. Rev. Lett. 106, 230501 (2011).
[Crossref] [PubMed]

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Lu, C. Y.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Lu, H.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Luo, Y. H.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Lyons, D. W.

S. N. Walck and D. W. Lyons, “Only n-qubit Greenberger-Horne-Zeilinger states are undetermined by their reduced density matrices,” Phys. Rev. Lett. 100, 050501 (2008).
[Crossref] [PubMed]

Monz, T.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Moroder, Tobias

Christian Schwemmer, Géza Tóth, Alexander Niggebaum, Tobias Moroder, and David Gross, Otfried Gühne, and Harald Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref]

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Nachtergaele, B.

M. Fannes, B. Nachtergaele, and R. F. Werner, “Finitely correlated states on quantum spin chains,” Comm. Math. Phys. 144, 443–490 (1992).
[Crossref]

Nigg, D.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Niggebaum, Alexander

Christian Schwemmer, Géza Tóth, Alexander Niggebaum, Tobias Moroder, and David Gross, Otfried Gühne, and Harald Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref]

Nüßeler, A.

T. Baumgratz, A. Nüßeler, M. Cramer, and M. B. Plenio, “A scalable maximum likelihood method for quantum state tomography,” New J. Phys. 15, 125004 (2013).
[Crossref]

O’Brien, J. L.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a Controlled-NOT gate.,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Pan, J. W.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Paris, M.

M. Paris and J. Řeháček, Quantum State Estimation, Lecture Notes in Physics (Springer, 2004).
[Crossref]

Peng, C. Z.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Plenio, M. B.

T. Baumgratz, A. Nüßeler, M. Cramer, and M. B. Plenio, “A scalable maximum likelihood method for quantum state tomography,” New J. Phys. 15, 125004 (2013).
[Crossref]

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Popescu, S.

N. Linden, S. Popescu, and W. K. Wootters, “Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices,” Phys. Rev. Lett. 89, 207901 (2002).
[Crossref] [PubMed]

Poulin, D.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Pryde, G. J.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a Controlled-NOT gate.,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Qi, B.

B. Qi, Z. B. Hou, L. Li, D. Dong, G. Y. Xiang, and G. C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Ralph, T. C.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a Controlled-NOT gate.,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Rehácek, J.

M. Paris and J. Řeháček, Quantum State Estimation, Lecture Notes in Physics (Springer, 2004).
[Crossref]

Schindler, P.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

Schollwock, U.

U. Schollwock, “The density-matrix renormalization group,” Rev. Mod. Phys. 77, 259 (2005).
[Crossref]

Schwemmer, Christian

Christian Schwemmer, Géza Tóth, Alexander Niggebaum, Tobias Moroder, and David Gross, Otfried Gühne, and Harald Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref]

Shen, Z.

J. F. Cai, E. J. Candes, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. on Optimization 201956 (2010).
[Crossref]

Somma, R.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Su, Z. E.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Swingle, B.

B. Swingle and I. H. Kim, “Reconstructing quantum states from local data,” Phys. Rev. Lett. 113, 260501 (2014).
[Crossref]

Tasaki, H.

I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Valence bond ground states in isotropic quantum antiferromagnets,” Commun. Math. Phys. 115, 477–528 (1988).
[Crossref]

Tóth, Géza

Christian Schwemmer, Géza Tóth, Alexander Niggebaum, Tobias Moroder, and David Gross, Otfried Gühne, and Harald Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref]

Verstraete, F.

Y. K. Liu, M. Christandl, and F. Verstraete, “Quantum computational complexity of the N-representability problem: QMA complete,” Phys. Rev. Lett. 98, 110503 (2007).
[Crossref] [PubMed]

Vidal, G.

G. Vidal, “Efficient classical simulation of slightly entangled quantum computations,” Phys. Rev. Lett. 91, 147902 (2003).
[Crossref] [PubMed]

Walck, S. N.

S. N. Walck and D. W. Lyons, “Only n-qubit Greenberger-Horne-Zeilinger states are undetermined by their reduced density matrices,” Phys. Rev. Lett. 100, 050501 (2008).
[Crossref] [PubMed]

Wang, X. L.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Weinfurter, H.

M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental observation of four-photon entanglement from parametric down-conversion,” Phys. Rev. Lett. 90, 200403 (2003).
[Crossref] [PubMed]

Werner, R. F.

M. Fannes, B. Nachtergaele, and R. F. Werner, “Finitely correlated states on quantum spin chains,” Comm. Math. Phys. 144, 443–490 (1992).
[Crossref]

White, A. G.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a Controlled-NOT gate.,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Wootters, W. K.

N. Linden, S. Popescu, and W. K. Wootters, “Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices,” Phys. Rev. Lett. 89, 207901 (2002).
[Crossref] [PubMed]

N. Linden and W. K. Wootters, “The parts determine the whole in a generic pure quantum state,” Phys. Rev. Lett. 89, 277906 (2002).
[Crossref]

Wu, D.

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Xiang, G. Y.

B. Qi, Z. B. Hou, L. Li, D. Dong, G. Y. Xiang, and G. C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Xu, J. S.

J. S. Xu, C. F. Li, and G. C. Guo, “Generation of a high-visibility four-photon entangled state and realization of a four-party quantum communication complexity scenario,” Phys. Rev. A 74, 052311 (2006).
[Crossref]

Yukalov, V. I.

A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: Coulson’s Challenge (Springer-Verlag, 2000).
[Crossref]

Zhang, Y. S.

Y. J. Han, Y. S. Zhang, and G. C. Guo, “Compatible conditions, entanglement, and invariants,” Phys. Rev. A 70, 042309 (2004).
[Crossref]

Zukowski, M.

M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental observation of four-photon entanglement from parametric down-conversion,” Phys. Rev. Lett. 90, 200403 (2003).
[Crossref] [PubMed]

Comm. Math. Phys. (1)

M. Fannes, B. Nachtergaele, and R. F. Werner, “Finitely correlated states on quantum spin chains,” Comm. Math. Phys. 144, 443–490 (1992).
[Crossref]

Commun. Math. Phys. (1)

I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Valence bond ground states in isotropic quantum antiferromagnets,” Commun. Math. Phys. 115, 477–528 (1988).
[Crossref]

IEEE Trans. Inf. Theory (1)

D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory 57, 1548 (2011).
[Crossref]

Journal of Physics: Conf. Series (1)

A. A. Klyachko, “Quantum marginal problem and N-representability,” Journal of Physics: Conf. Series 36, 72 (2006).

Nat. Commun. (1)

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

New J. Phys. (1)

T. Baumgratz, A. Nüßeler, M. Cramer, and M. B. Plenio, “A scalable maximum likelihood method for quantum state tomography,” New J. Phys. 15, 125004 (2013).
[Crossref]

Phys. Rev. A (4)

J. S. Xu, C. F. Li, and G. C. Guo, “Generation of a high-visibility four-photon entangled state and realization of a four-party quantum communication complexity scenario,” Phys. Rev. A 74, 052311 (2006).
[Crossref]

Y. J. Han, Y. S. Zhang, and G. C. Guo, “Compatible conditions, entanglement, and invariants,” Phys. Rev. A 70, 042309 (2004).
[Crossref]

S. Dogra and K. Dorai, and Arvind, “Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor,” Phys. Rev. A 91, 022312 (2015).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Phys. Rev. B (1)

M. B. Hastings, “Solving gapped Hamiltonians locally,” Phys. Rev. B 73, 085115 (2006).
[Crossref]

Phys. Rev. Lett. (13)

S. T. Flammia and Y.-K. Liu, “Direct fidelity estimation from few pauli measurements,” Phys. Rev. Lett. 106, 230501 (2011).
[Crossref] [PubMed]

Christian Schwemmer, Géza Tóth, Alexander Niggebaum, Tobias Moroder, and David Gross, Otfried Gühne, and Harald Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref]

M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental observation of four-photon entanglement from parametric down-conversion,” Phys. Rev. Lett. 90, 200403 (2003).
[Crossref] [PubMed]

Y. K. Liu, M. Christandl, and F. Verstraete, “Quantum computational complexity of the N-representability problem: QMA complete,” Phys. Rev. Lett. 98, 110503 (2007).
[Crossref] [PubMed]

G. Vidal, “Efficient classical simulation of slightly entangled quantum computations,” Phys. Rev. Lett. 91, 147902 (2003).
[Crossref] [PubMed]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a Controlled-NOT gate.,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

B. Swingle and I. H. Kim, “Reconstructing quantum states from local data,” Phys. Rev. Lett. 113, 260501 (2014).
[Crossref]

N. Linden, S. Popescu, and W. K. Wootters, “Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices,” Phys. Rev. Lett. 89, 207901 (2002).
[Crossref] [PubMed]

X. L. Wang, L. K. Chen, W. Li, H. L. Huang, C. Liu, C. Chen, Y. H. Luo, Z. E. Su, D. Wu, Z. D. Li, H. Lu, Y. Hu, X. Jiang, C. Z. Peng, L. Li, N. L. Liu, Y. A. Chen, C. Y. Lu, and J. W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hansel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref] [PubMed]

N. Linden and W. K. Wootters, “The parts determine the whole in a generic pure quantum state,” Phys. Rev. Lett. 89, 277906 (2002).
[Crossref]

S. N. Walck and D. W. Lyons, “Only n-qubit Greenberger-Horne-Zeilinger states are undetermined by their reduced density matrices,” Phys. Rev. Lett. 100, 050501 (2008).
[Crossref] [PubMed]

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).
[Crossref]

Phys.-Usp. (1)

A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys.-Usp. 44, 131–136 (2001).
[Crossref]

Quant. Inf. and Comput. (1)

S. Bravyi, “Compatibility between local and multipartite states,” Quant. Inf. and Comput. 4, 12 (2004).

Rev. Mod. Phys. (2)

U. Schollwock, “The density-matrix renormalization group,” Rev. Mod. Phys. 77, 259 (2005).
[Crossref]

A. J. Coleman, “Structure of fermion density matrices,” Rev. Mod. Phys. 35, 668 (1963).
[Crossref]

Sci. Rep. (1)

B. Qi, Z. B. Hou, L. Li, D. Dong, G. Y. Xiang, and G. C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

SIAM J. on Optimization (1)

J. F. Cai, E. J. Candes, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. on Optimization 201956 (2010).
[Crossref]

Other (3)

R. L. Kosut, “Quantum process tomography via l1-norm minimization,” https://arxiv.org/abs/0812.4323.

A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: Coulson’s Challenge (Springer-Verlag, 2000).
[Crossref]

M. Paris and J. Řeháček, Quantum State Estimation, Lecture Notes in Physics (Springer, 2004).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Experimental configuration: an ultraviolet (UV) pulse pumps a 2 mm beta-barium-borate (BBO) crystal to produce two pairs of entangled photons; beam splitter 1 (BS1) and BS2 each split one pair of photons into two arms; four sets of quarter-wave plates (λ/4), half-wave plates (λ/2) and polarizing beam splitters (PBS) are used to perform quantum state tomography; the birefringence of o and e in the BBO is compensated by BBO1 and BBO2, and the phase difference between the H and V polarizations in the reflected path after BS1 (BS2) is compensated by Q1 (Q2) generated by tilting the quarter-wave plate to the left or the right; quartz plates with different lengths are used as dephasing channels

Fig. 2
Fig. 2

Comparison of an ideal state, a prepared state (with a fidelity of 0.9422 ± 0.0041) and pure states that are constructed from two-photon (MPS2) and three-photon (MPS3) reduced-density matrices; the pure states here and in Fig. 3 (Fig. 4) are obtained using 10,000 MPS-SVT iterations for the two-photon reduced matrices and 2000 MPS-SVT iterations for the three-photon reduced matrices. The real part of these states shows that the ideal state, MPS2 and MPS3 are almost the same; a fidelity of up to 0.9862 ± 0.0012 is found between MPS2 and the ideal state as well as MPS3 and the ideal state (see also Fig. 4)

Fig. 3
Fig. 3

Results from experimental reconstruction: the Y axis corresponds to Rfidelity, which is the fidelity between the reconstructed pure states and the four-photon prepared states obtained using standard quantum tomography; the X axis corresponds to the fidelity of the prepared states; the data points shown in red and blue correspond to the pure states reconstructed from the measured three-photon and two-photon reduced-density matrices, respectively; the solid and dashed lines indicate the pure states reconstructed from the theoretical two-photon and three-photon reduced matrices from dephasing noise on the ideal state, respectively; our experimental results agree with the theoretical results; it shows that the Rfidelity will asymptotically achieve 1 when the prepared state is approaching the ideal state; the error bars in this figure and Fig. 4 are obtained by performing 200 Monte Carlo simulation runs.

Fig. 4
Fig. 4

Fidelity between the reconstructed pure states and the ideal state: the results show that the efficient quantum state tomography is robust to noise; the pure states reconstructed from the reduced matrices of different prepared states are almost the same as the ideal state, and the lowest fidelity of the reconstructed state is 0.9674 ± 0.0022; the saturation of the size of the reduced matrices is also clearly shown.

Equations (4)

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| Ψ MPS = s 1 s N = 1 d Tr ( A 1 s 1 A 2 s 2 A N s N ) | s 1 s N ,
X ^ n = y n m . i y n | P m i | y n 2 N P m i Y ^ n + 1 = Y ^ n + δ n ( R ^ X ^ n ) .
| Ψ = ( a H a H + a V a V ) 2 | 0 .
| Ψ 4 = 3 3 ( | H H H H + | V V V V ) + 3 6 ( | H H V V + | H V V H + | V H H V + | V V H H ) .

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