Permutationally Invariant Quantum Tomography

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1. G. Tóth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, Permutationally invariant quantum tomography, Phys. Rev. Lett. 105, 250403 (2010); arxiv:1005.3313. [pdf,pdf2]

We present a scalable method for the tomography of large multiqubit quantum registers. It acquires information about the permutationally invariant part of the density operator, which is a good approximation to the true state in many relevant cases. Our method gives the best measurement strategy to minimize the experimental effort as well as the uncertainties of the reconstructed density matrix. We apply our method to the experimental tomography of a photonic four-qubit symmetric Dicke state.

2. T. Moroder, P. Hyllus, G. Tóth, C. Schwemmer, A. Niggebaum, S. Gaile, O. Gühne, and H. Weinfurter, Permutationally invariant state reconstruction, New J. Phys. 14, 105001 (2012), Focus issue on Quantum Tomography; arxiv:1205.4941. [pdf]

Feasible tomography schemes for large particle numbers must possess, besides an appropriate data acquisition protocol, an efﬁcient way to reconstruct the density operator from the observed ﬁnite data set. Since state reconstruction typically requires the solution of a nonlinear large-scale optimization problem, this is a major challenge in the design of scalable tomography schemes. Here we present an efﬁcient state reconstruction scheme for permutationally invariant quantum state tomography. It works for all common state-of-the-art reconstruction principles, including, in particular, maximum likelihood and least squares methods, which are the preferred choices in today’s experiments. This high efﬁciency is achieved by greatly reducing the dimensionality of the problem employing a particular representation of permutationally invariant states known from spin coupling combined with convex optimization, which has clear advantages regarding speed, control and accuracy in comparison to commonly employed numerical routines. First prototype implementations easily allow reconstruction of a state of 20 qubits in a few minutes on a standard computer.

3. C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, Efficient Tomographic Analysis of a Six Photon State, Phys. Rev. Lett. 113, 040503 (2014); arxiv:1401.7526. [pdf,pdf2]

Quantum state tomography suffers from the measurement effort increasing exponentially with the number of qubits. Here, we demonstrate permutationally invariant tomography for which, contrary to conventional tomography, all resources scale polynomially with the number of qubits both in terms of the measurement effort as well as the computational power needed to process and store the recorded data. We evaluate permutationally invariant tomography by comparing it to full tomography for six-photon states obtained from spontaneous parametric down-conversion. We show that their results are compatible within the statistical errors. For low rank states, we further optimize both schemes using compressed sensing. We demonstrate the benefits of combining permutationally invariant tomography with compressed sensing by studying the influence of the pump power on the noise present in a six-qubit symmetric Dicke state, a case where full tomography is possible only for very high pump powers.

LaTeX to make handling accents easier

G. T\'oth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, Phys. Rev. Lett. {\bf 105}, 250403 (2010);

T. Moroder, P. Hyllus, G. T\'oth, C. Schwemmer, A. Niggebaum, S. Gaile, O. G\"uhne, and H. Weinfurter, New J. Phys. {\bf 14}, 105001 (2012);

C. Schwemmer, G. T\'oth, A. Niggebaum, T. Moroder, D. Gross, O. G\"uhne, and H. Weinfurter, Phys. Rev. Lett., {\bf 113}, 040503 (2014); arxiv:1401.7526.

Related publications

G. M. D'Ariano, L. Maccone, M. Paini, Spin tomography, J. Opt. B: Quantum Semiclass. Opt. 5, 77 (2003); arxiv:quant-ph/0210105.

B. A. Chase and J. M. Geremia, Collective processes of an ensemble of spin-1∕2 particles, Phys. Rev. A 78, 052101 (2008).

R. B. A. Adamson, P. S. Turner, M. W. Mitchell, and A. M. Steinberg, Detecting Hidden Differences via Permutation Symmetries, Phys. Rev. A 78, 033832 (2008); arXiv:quant-ph/0612081.

R. B. A. Adamson , L. K. Shalm, M. W. Mitchell, and A. M. Steinberg, Multiparticle State Tomography: Hidden Differences, Phys. Rev. Lett. 98, 043601 (2007); arXiv:quant-ph/0601134.

L. K. Shalm, R. B. A. Adamson, and A. M. Steinberg, Squeezing and over-squeezing of triphoton, Nature 457, 67-70 (2009).

A. B. Klimov, G. Björk, and L. L. Sánchez-Soto, Optimal quantum tomography of permutationally invariant qubits, Phys. Rev. A 87, 012109 (2013); arXiv:1301.2123.

L. Novo, T. Moroder, and O. Gühne, Genuine multiparticle entanglement of permutationally invariant states, Phys. Rev. A 88, 012305 (2013); arXiv:1302.4100.