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Verhandlungen der Deutschen Physikalischen Gesellschaft

Permutationally invariant tomography of a six qubit symmetric Dicke state

Christian Schwemmer, Alexander Niggebaum, Harald Weinfurter, Ludwig-Maximilians-Universitaet Fakultaet fuer Physik, Géza Tóth, Tobias Moroder, Philipp Hyllus, Otfried Gühne

Published Jul 1, 2012• 0 citations
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Abstract

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.

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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%.

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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.

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Entanglement is the crucial resource in the field of quantum information processing and consequently a better understanding of its properties is very desirable. While for bipartite entangled qubit states many tools for characterization exist and also the experimental implementation of the tools is feasible, the situation for multipartite entangled states is different: The available tools have to be suited for a much greater diversity of entanglement properties, e.g. to analyze all the different entanglement classes [1]. A full experimental characterization via state tomography is increasingly difficult as the number of necessary measurement settings grows exponentially with the number of particles. In this contribution we focus on efficient, non-tomographic methods to characterize experimentally observed, multipartite entangled states. First steps towards this goal were demonstrated recently, for instance, for the proof of genuine multipartite entanglement [2] or for the analysis of the persistency of entanglement under loss [3]. Such methods can be further optimized by tailoring them to particular properties of states. We show how three important characteristic quantities, the fidelity, the non-separability and the permutation symmetry, can be evaluated with a significantly lower number of measurement settings than required for a complete tomography. Although the determination of these particular properties provides less information than the reconstruction of the complete density matrix, it allows to judge how well state characteristics are reproduced experimentally. This might be important with respect to an experimental application of states in quantum information. Exemplarily we apply the analysis tools to two genuinely four-qubit entangled states: the Dicke state with two excitations |D 4 〉 [4] and the cluster state |C4 〉 [5, 6]. These states are relevant for particular tasks: The state |D 4 〉 is a resource for telecloning [7], cluster states find an application in one-way quantum computation [6, 8] and are a subgroup of graph states, which play an important role for quantum error correction.

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We present an experimental and theoretical characterization of the sym- metric four-qubit entangled Dicke state with two excitations D (2) . We investigate the state's violation of local realism and study its characteristic properties with re- spect to quantum information applications. For the experimental observation of the state we used photons obtained from parametric down conversion. This allowed, in a simple experimental set-up, quantum state tomography yielding a fidelity as high as 0.844 ± 0.008.

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Practical methods for witnessing genuine multi-qubit entanglement in the vicinity of symmetric states

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We present general numerical methods to construct witness operators for entanglement detection and estimation of the fidelity. Our methods are applied to detecting entanglement in the vicinity of a six-qubit Dicke state with three excitations and also to further entangled symmetric states. All our witnesses are designed to keep the measurement effort small. We also present general results on the efficient local decomposition of permutationally invariant operators, which makes it possible to measure projectors to symmetric states efficiently.

•Aug 4, 2009•19 citations
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.
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%.
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.
Entanglement is the crucial resource in the field of quantum information processing and consequently a better understanding of its properties is very desirable. While for bipartite entangled qubit states many tools for characterization exist and also the experimental implementation of the tools is feasible, the situation for multipartite entangled states is different: The available tools have to be suited for a much greater diversity of entanglement properties, e.g. to analyze all the different entanglement classes [1]. A full experimental characterization via state tomography is increasingly difficult as the number of necessary measurement settings grows exponentially with the number of particles. In this contribution we focus on efficient, non-tomographic methods to characterize experimentally observed, multipartite entangled states. First steps towards this goal were demonstrated recently, for instance, for the proof of genuine multipartite entanglement [2] or for the analysis of the persistency of entanglement under loss [3]. Such methods can be further optimized by tailoring them to particular properties of states. We show how three important characteristic quantities, the fidelity, the non-separability and the permutation symmetry, can be evaluated with a significantly lower number of measurement settings than required for a complete tomography. Although the determination of these particular properties provides less information than the reconstruction of the complete density matrix, it allows to judge how well state characteristics are reproduced experimentally. This might be important with respect to an experimental application of states in quantum information. Exemplarily we apply the analysis tools to two genuinely four-qubit entangled states: the Dicke state with two excitations |D 4 〉 [4] and the cluster state |C4 〉 [5, 6]. These states are relevant for particular tasks: The state |D 4 〉 is a resource for telecloning [7], cluster states find an application in one-way quantum computation [6, 8] and are a subgroup of graph states, which play an important role for quantum error correction.
We present an experimental and theoretical characterization of the sym- metric four-qubit entangled Dicke state with two excitations D (2) . We investigate the state's violation of local realism and study its characteristic properties with re- spect to quantum information applications. For the experimental observation of the state we used photons obtained from parametric down conversion. This allowed, in a simple experimental set-up, quantum state tomography yielding a fidelity as high as 0.844 ± 0.008.
We present general numerical methods to construct witness operators for entanglement detection and estimation of the fidelity. Our methods are applied to detecting entanglement in the vicinity of a six-qubit Dicke state with three excitations and also to further entangled symmetric states. All our witnesses are designed to keep the measurement effort small. We also present general results on the efficient local decomposition of permutationally invariant operators, which makes it possible to measure projectors to symmetric states efficiently.