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arXiv:1005.3313 · Physical Review Letters

Permutationally Invariant Quantum Tomography

Géza Tóth, Witlef Wieczorek, D. Gross, Roland Krischek, Christian Schwemmer, Harald Weinfurter

Published Dec 16, 2010• 232 citations
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Abstract

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.

DOI
10.1103/physrevlett.105.250403

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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.
Here, the authors present experimental results of the tomographic analysis of a photonic four-qubit symmetric Dicke state. Instead of 34=81 basis settings for full tomography only 15 basis settings have to be measured for permutationally invariant tomography. A further advantage of our method is that the single qubits do not have to be individually manipulable since for each measurement the same local setting is applied to all qubits. The best performance of our measurement algorithm is achieved when the basis settings are evenly distributed on the Bloch sphere. In our experiments we showed that the overlap between the density matrix obtained from full tomography and the one obtained from permutationally invariant tomography, and thus the permutationally invariant component of the observed quantum state was 94.7%.
Multi-partite entangled quantum states are promising candidates for potential applications like quantum metrology or quantum communication. Yet, efficient tools are needed to characterize these states and to evaluate their applicability. Standard quantum state tomography suffers from an exponential increase in the measurement effort with the number of qubits. Here, we show that by restricting to permutational invariant states like GHZ, W or symmetric Dicke states the problem can be recast such that the measurement effort scales only quadratically. We apply this method to experimentally analyze a six photon symmetric Dicke state generated by parametric down conversion where instead of 729 only 28 basis settings have to be measured.
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 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.
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.
Multimode Gaussian states are a versatile resource for quantum information technologies and have been realized across a wide range of physical platforms. Recent progress in the large-scale generation of such states provides a key ingredient for scalable quantum technologies. Despite the importance of accurately characterizing these states, conventional tomography methods are often impractical because they require large sample sizes and can yield unphysical states. Here we present a reliable and efficient tomography method for multimode Gaussian states based on maximum-likelihood estimation. By directly operating on covariance matrices, the method avoids the exponential overhead associated with density-matrix reconstruction. We consider two commonly used detection schemes--single and joint homodyne detection--and systematically analyze the reconstruction performance. Our method outperforms conventional approaches by ensuring physical covariance matrices and achieving better agreement with the true states. To demonstrate the experimental applicability of the method, we experimentally generate various multipartite entangled states--six-mode graph states with different connectivity, a six-mode GHZ state, and a fully connected ten-mode graph state--and reconstruct their covariance matrices. Using the reconstructed covariance matrices, we quantify fidelities, detect entanglement, and reveal the multimode structure of squeezing and noise. Our technique offers a practical diagnostic tool for developing scalable quantum technologies.