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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Permutationally invariant tomography of a four-qubit symmetric Dicke state

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

Published May 1, 2011• 0 citations
Publisher

Abstract

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%.

DOI
10.1109/cleoe.2011.5943393

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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%.
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.
Quantum state tomography provides complete information about a quantum state, but its measurement cost generally grows exponentially with system size. In many-particle quantum simulators, this challenge is further compounded by the limited accessibility of local measurements and controls. Here we develop a tomography protocol for permutation-invariant fermionic many-body states with U(1) particle-number symmetry. We show that any such state is completely determined by the distribution of the total particle number and the occupation of a single collective mode within each particle-number sector, both of which are accessible in current ultracold-atom experiments. The number of required observables scales only linearly with the system size. More generally, the protocol reconstructs the permutation-symmetrized component of arbitrary U(1)-symmetric fermionic states, which can still encode nontrivial many-body and state-level structure beyond conventional few-body observables. We demonstrate this protocol in interacting non-Gaussian states of the complex Sachdev-Ye-Kitaev model and in free-fermion chains across a Lifshitz transition. This framework opens a route toward information-theoretic characterization of strongly correlated itinerant quantum matter in experimentally realistic fermionic quantum simulators.
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.