Many experiments in quantum information aim at creating graph states. Quantifying the purity of an experimentally achieved graph state could in principle be accomplished using full-state tomography. This method requires a number of measurement settings growing exponentially with the number of constituents involved. Thus, full-state tomography becomes experimentally infeasible even for a moderate number of qubits. In this paper, we present a method to estimate the purity of experimentally achieved graph states with simple measurements. The observables we consider are the stabilizers of the underlying graph. Then, we formulate the problem as: "What is the state with the least purity that is compatible with the measurement data?" We solve this problem analytically and compare the obtained bounds with results from full-state tomography for simulated data.Many experiments in quantum information aim at creating graph states. Quantifying the purity of an experimentally achieved graph state could in principle be accomplished using full-state tomography. This method requires a number of measurement settings growing exponentially with the number of constituents involved. Thus, full-state tomography becomes experimentally infeasible even for a moderate number of qubits. In this paper, we present a method to estimate the purity of experimentally achieved graph states with simple measurements. The observables we consider are the stabilizers of the underlying graph. Then, we formulate the problem as: "What is the state with the least purity that is compatible with the measurement data?" We solve this problem analytically and compare the obtained bounds with results from full-state tomography for simulated data.

International Journal of Quantum InformationFeb 1, 20104 citations

Quantitative verification of entanglement and fidelities from incomplete measurement data

Harald Wunderlich, Martin Bodo Plenio

Many experiments in quantum information aim at creating multi-partite entangled states. Quantifying the amount of entanglement that was actually generated can, in principle, be accomplished using full-state tomography. This method requires the determination of a parameter set that is growing exponentially with the number of qubits and becomes infeasible even for moderate numbers of particles. Non-trivial bounds on experimentally prepared entanglement can however be obtained from partial information on the density matrix. As introduced in the paper of K.M.R. Audenaert and M.B. Plenio [New J. Phys. 2006, 8, 266], the fundamental question is then formulated as: What is the entanglement content of the least entangled quantum state that is compatible with the available measurement data? We formulate the problem mathematically [Audenaert, K.M.R.; Plenio, M.B. New J. Phys. 2006, 8, 266] employing methods from the theory of semi-definite programming and then address this problem for the case, where the goal of the experiment is the creation of graph states. The observables that we consider are the generators of the stabilizer group, thus the number of measurement settings grows only linearly in the number of qubits. We provide analytical solutions as well as numerical methods that may be applied directly to experiments, and compare the obtained bounds with results from full-state tomography for simulated data.Many experiments in quantum information aim at creating multi-partite entangled states. Quantifying the amount of entanglement that was actually generated can, in principle, be accomplished using full-state tomography. This method requires the determination of a parameter set that is growing exponentially with the number of qubits and becomes infeasible even for moderate numbers of particles. Non-trivial bounds on experimentally prepared entanglement can however be obtained from partial information on the density matrix. As introduced in the paper of K.M.R. Audenaert and M.B. Plenio [New J. Phys. 2006, 8, 266], the fundamental question is then formulated as: What is the entanglement content of the least entangled quantum state that is compatible with the available measurement data? We formulate the problem mathematically [Audenaert, K.M.R.; Plenio, M.B. New J. Phys. 2006, 8, 266] employing methods from the theory of semi-definite programming and then address this problem for the case, where the goal of the experiment is the creation of graph states. The observables that we consider are the generators of the stabilizer group, thus the number of measurement settings grows only linearly in the number of qubits. We provide analytical solutions as well as numerical methods that may be applied directly to experiments, and compare the obtained bounds with results from full-state tomography for simulated data.

arXiv:0902.2093Journal of Modern OpticsAug 19, 200930 citations

Quantum state tomography and purity measurement of a multiqubit system with random single-pulse rotations

Yariv Yanay, Charles Tahan

Quantum state tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many-body physics through quantum simulators. Unfortunately, the number of experimental measurements of the system required to estimate these global quantities scales exponentially with system size. Here, we consider the use of random-axis measurements for quantum state tomography and state purity estimation. We perform a general analysis of the statistical deviation in such methods for any given algorithm. We then propose a simple protocol which relies on single-pulse rotations only. We find that it reduces the basis of the exponential growth, calculating the statistical variance to scale as ${\ensuremath{\sum}}_{\mathbit{a},\mathbit{b}}{|\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathbit{a}\mathbit{b}}|}^{2}\ensuremath{\sim}{5}^{N}/{N}_{\mathrm{tot}}$ for full tomography, and ${\mathrm{\ensuremath{\Delta}}\ensuremath{\mu}}^{2}\ensuremath{\sim}{7}^{N}/{N}_{\mathrm{tot}}^{2}$ for purity estimation, for $N$ qubits and ${N}_{\mathrm{tot}}$ measurements performed.Quantum state tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many-body physics through quantum simulators. Unfortunately, the number of experimental measurements of the system required to estimate these global quantities scales exponentially with system size. Here, we consider the use of random-axis measurements for quantum state tomography and state purity estimation. We perform a general analysis of the statistical deviation in such methods for any given algorithm. We then propose a simple protocol which relies on single-pulse rotations only. We find that it reduces the basis of the exponential growth, calculating the statistical variance to scale as ${\ensuremath{\sum}}_{\mathbit{a},\mathbit{b}}{|\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathbit{a}\mathbit{b}}|}^{2}\ensuremath{\sim}{5}^{N}/{N}_{\mathrm{tot}}$ for full tomography, and ${\mathrm{\ensuremath{\Delta}}\ensuremath{\mu}}^{2}\ensuremath{\sim}{7}^{N}/{N}_{\mathrm{tot}}^{2}$ for purity estimation, for $N$ qubits and ${N}_{\mathrm{tot}}$ measurements performed.

arXiv:2101.05860Physical Review AJun 11, 20211 citation

Highly efficient estimation of entanglement measures for large experimentally created graph states via simple measurements

Harald Wunderlich, S. Virmani, Martin Bodo Plenio

Quantifying experimentally created entanglement could in principle be accomplished by measuring the entire density matrix and calculating an entanglement measure of choice thereafter. Due to the tensor structure of the Hilbert space, this approach becomes unfeasible even for medium-sized systems. Here we present methods for quantifying the entanglement of arbitrarily large two-colorable graph states from simple measurements. The presented methods provide non-trivial bounds on the entanglement for any state as long as there is sufficient fidelity with such a graph state. The measurement data considered here is merely given by stabilizer measurements, thus leading to an exponential reduction in the number of measurements required. We provide analytical results for the robustness of entanglement and the relative entropy of entanglement.Quantifying experimentally created entanglement could in principle be accomplished by measuring the entire density matrix and calculating an entanglement measure of choice thereafter. Due to the tensor structure of the Hilbert space, this approach becomes unfeasible even for medium-sized systems. Here we present methods for quantifying the entanglement of arbitrarily large two-colorable graph states from simple measurements. The presented methods provide non-trivial bounds on the entanglement for any state as long as there is sufficient fidelity with such a graph state. The measurement data considered here is merely given by stabilizer measurements, thus leading to an exponential reduction in the number of measurements required. We provide analytical results for the robustness of entanglement and the relative entropy of entanglement.

arXiv:1003.1681New Journal of PhysicsAug 10, 201010 citations

Quantifying Coherence and Entanglement via Simple Measurements

Graeme Smith, John A. Smolin, Xiao Yuan +3 more

Coherence and entanglement are fundamental properties of quantum systems, promising to power the near future quantum computers, sensors and simulators. Yet, their experimental detection is challenging, usually requiring full reconstruction of the system state. We show that one can extract quantitative bounds to the relative entropy of coherence and the coherent information, coherence and entanglement quantifiers respectively, by a limited number of purity measurements. The scheme is readily implementable with current technology to verify quantum computations in large scale registers, without carrying out expensive state tomography.Coherence and entanglement are fundamental properties of quantum systems, promising to power the near future quantum computers, sensors and simulators. Yet, their experimental detection is challenging, usually requiring full reconstruction of the system state. We show that one can extract quantitative bounds to the relative entropy of coherence and the coherent information, coherence and entanglement quantifiers respectively, by a limited number of purity measurements. The scheme is readily implementable with current technology to verify quantum computations in large scale registers, without carrying out expensive state tomography.

arXiv:1707.09928Jul 31, 20177 citations

Experimental Quantum State Tomography via Compressed Sampling

Wei-Tao Liu, Ting Zhang, Jiying Liu +2 more

A fundamental difficulty in demonstrating quantum state tomography is that the required resources grow exponentially with the system size. For pure states and nearly pure states, the task of tomography can be more efficient. We proposed two methods for state reconstruction, by (1) minimizing entropy and (2) maximizing likelihood. The algorithm of compressed sampling is employed to solve the optimization problem. Experiments are demonstrated considering 4-qubit photonic states. The results show that (1) much fewer measurements than the standard tomography are sufficient to obtain high fidelity, and (2) the method of maximizing likelihood is more accurate and noise robust than the original reconstruction method of compressed sampling. Furthermore, the physical meaning of the methods of minimizing entropy and maximizing likelihood is clear.A fundamental difficulty in demonstrating quantum state tomography is that the required resources grow exponentially with the system size. For pure states and nearly pure states, the task of tomography can be more efficient. We proposed two methods for state reconstruction, by (1) minimizing entropy and (2) maximizing likelihood. The algorithm of compressed sampling is employed to solve the optimization problem. Experiments are demonstrated considering 4-qubit photonic states. The results show that (1) much fewer measurements than the standard tomography are sufficient to obtain high fidelity, and (2) the method of maximizing likelihood is more accurate and noise robust than the original reconstruction method of compressed sampling. Furthermore, the physical meaning of the methods of minimizing entropy and maximizing likelihood is clear.

Physical Review LettersApr 24, 201283 citations

N

qubits and

{N}_{\mathrm{tot}}

measurements performed.