1
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Englbrecht M, Kraft T, Dittel C, Buchleitner A, Giedke G, Kraus B. Indistinguishability of Identical Bosons from a Quantum Information Theory Perspective. Phys Rev Lett 2024; 132:050201. [PMID: 38364122 DOI: 10.1103/physrevlett.132.050201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2023] [Revised: 12/14/2023] [Accepted: 01/03/2024] [Indexed: 02/18/2024]
Abstract
Using tools from quantum information theory, we present a general theory of indistinguishability of identical bosons in experiments consisting of passive linear optics followed by particle number detection. Our results do neither rely on additional assumptions on the input state of the interferometer, such as, for instance, a fixed mode occupation, nor on any assumption on the degrees of freedom that potentially make the particles distinguishable. We identify the expectation value of the projector onto the N-particle symmetric subspace as an operationally meaningful measure of indistinguishability, and derive tight lower bounds on it that can be efficiently measured in experiments. Moreover, we present a consistent definition of perfect distinguishability and characterize the corresponding set of states. In particular, we show that these states are diagonal in the computational basis up to a permutationally invariant unitary. Moreover, we find that convex combinations of states that describe partially distinguishable and perfectly indistinguishable particles can lead to perfect distinguishability, which itself is not preserved under convex combinations.
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Affiliation(s)
- Matthias Englbrecht
- Department of Physics, QAA, Technical University of Munich, James-Franck-Straße 1, D-85748 Garching, Germany
- Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria
| | - Tristan Kraft
- Department of Physics, QAA, Technical University of Munich, James-Franck-Straße 1, D-85748 Garching, Germany
- Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria
| | - Christoph Dittel
- Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
- EUCOR Centre for Quantum Science and Quantum Computing, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany
- Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universität Freiburg, Albertstraße 19, D-79104 Freiburg, Germany
| | - Andreas Buchleitner
- Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
- EUCOR Centre for Quantum Science and Quantum Computing, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany
| | - Geza Giedke
- Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
- IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, E-48009 Bilbao, Spain
| | - Barbara Kraus
- Department of Physics, QAA, Technical University of Munich, James-Franck-Straße 1, D-85748 Garching, Germany
- Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria
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2
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Koutný D, Ginés L, Moczała-Dusanowska M, Höfling S, Schneider C, Predojević A, Ježek M. Deep learning of quantum entanglement from incomplete measurements. Sci Adv 2023; 9:eadd7131. [PMID: 37467336 DOI: 10.1126/sciadv.add7131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2022] [Accepted: 06/16/2023] [Indexed: 07/21/2023]
Abstract
The quantification of the entanglement present in a physical system is of paramount importance for fundamental research and many cutting-edge applications. Now, achieving this goal requires either a priori knowledge on the system or very demanding experimental procedures such as full state tomography or collective measurements. Here, we demonstrate that, by using neural networks, we can quantify the degree of entanglement without the need to know the full description of the quantum state. Our method allows for direct quantification of the quantum correlations using an incomplete set of local measurements. Despite using undersampled measurements, we achieve a quantification error of up to an order of magnitude lower than the state-of-the-art quantum tomography. Furthermore, we achieve this result using networks trained using exclusively simulated data. Last, we derive a method based on a convolutional network input that can accept data from various measurement scenarios and perform, to some extent, independently of the measurement device.
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Affiliation(s)
- Dominik Koutný
- Department of Optics, Faculty of Science, Palacký University, 17. listopadu 12, 77146 Olomouc, Czechia
| | - Laia Ginés
- Department of Physics, Stockholm University, 10691 Stockholm, Sweden
| | | | - Sven Höfling
- Technische Physik, Physikalisches Institut and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
| | | | - Ana Predojević
- Department of Physics, Stockholm University, 10691 Stockholm, Sweden
| | - Miroslav Ježek
- Department of Optics, Faculty of Science, Palacký University, 17. listopadu 12, 77146 Olomouc, Czechia
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3
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Hai VT, Ho LB. Universal compilation for quantum state tomography. Sci Rep 2023; 13:3750. [PMID: 36879023 PMCID: PMC9988891 DOI: 10.1038/s41598-023-30983-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Accepted: 03/03/2023] [Indexed: 03/08/2023] Open
Abstract
Universal compilation is a training process that compiles a trainable unitary into a target unitary. It has vast potential applications from depth-circuit compressing to device benchmarking and quantum error mitigation. Here we propose a universal compilation algorithm for quantum state tomography in low-depth quantum circuits. We apply the Fubini-Study distance as a trainable cost function and employ various gradient-based optimizations. We evaluate the performance of various trainable unitary topologies and the trainability of different optimizers for getting high efficiency and reveal the crucial role of the circuit depth in robust fidelity. The results are comparable with the shadow tomography method, a similar fashion in the field. Our work expresses the adequate capability of the universal compilation algorithm to maximize the efficiency in the quantum state tomography. Further, it promises applications in quantum metrology and sensing and is applicable in the near-term quantum computers for various quantum computing tasks.
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Affiliation(s)
- Vu Tuan Hai
- University of Information Technology, Ho Chi Minh City, 700000, Vietnam.,Vietnam National University, Ho Chi Minh City, 700000, Vietnam
| | - Le Bin Ho
- Ho Chi Minh City Institute of Physics, National Institute of Applied Mechanics and Informatics, Vietnam Academy of Science and Technology, Ho Chi Minh City, 700000, Vietnam. .,Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai, 980-8578, Japan. .,Department of Applied Physics, Graduate School of Engineering, Tohoku University, Sendai, 980-8579, Japan.
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4
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Zhu Y, Wu YD, Bai G, Wang DS, Wang Y, Chiribella G. Flexible learning of quantum states with generative query neural networks. Nat Commun 2022; 13:6222. [PMID: 36266334 PMCID: PMC9584912 DOI: 10.1038/s41467-022-33928-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2022] [Accepted: 10/07/2022] [Indexed: 12/24/2022] Open
Abstract
Deep neural networks are a powerful tool for characterizing quantum states. Existing networks are typically trained with experimental data gathered from the quantum state that needs to be characterized. But is it possible to train a neural network offline, on a different set of states? Here we introduce a network that can be trained with classically simulated data from a fiducial set of states and measurements, and can later be used to characterize quantum states that share structural similarities with the fiducial states. With little guidance of quantum physics, the network builds its own data-driven representation of a quantum state, and then uses it to predict the outcome statistics of quantum measurements that have not been performed yet. The state representations produced by the network can also be used for tasks beyond the prediction of outcome statistics, including clustering of quantum states and identification of different phases of matter.
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Affiliation(s)
- Yan Zhu
- grid.194645.b0000000121742757QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong
| | - Ya-Dong Wu
- grid.194645.b0000000121742757QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong
| | - Ge Bai
- grid.194645.b0000000121742757QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong
| | - Dong-Sheng Wang
- grid.9227.e0000000119573309CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190 P.R. China
| | - Yuexuan Wang
- grid.194645.b0000000121742757QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong ,grid.13402.340000 0004 1759 700XCollege of Computer Science and Technology, Zhejiang University, Hangzhou, China
| | - Giulio Chiribella
- grid.194645.b0000000121742757QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong ,grid.4991.50000 0004 1936 8948Department of Computer Science, Oxford, OX1 3QD UK ,grid.420198.60000 0000 8658 0851Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5 Canada
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5
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Pereira L, García-Ripoll JJ, Ramos T. Complete Physical Characterization of Quantum Nondemolition Measurements via Tomography. Phys Rev Lett 2022; 129:010402. [PMID: 35841584 DOI: 10.1103/physrevlett.129.010402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2021] [Accepted: 06/03/2022] [Indexed: 06/15/2023]
Abstract
We introduce a self-consistent tomography for arbitrary quantum nondemolition (QND) detectors. Based on this, we build a complete physical characterization of the detector, including the measurement processes and a quantification of the fidelity, ideality, and backaction of the measurement. This framework is a diagnostic tool for the dynamics of QND detectors, allowing us to identify errors, and to improve their calibration and design. We illustrate this on a realistic Jaynes-Cummings simulation of a superconducting qubit readout. We characterize nondispersive errors, quantify the backaction introduced by the readout cavity, and calibrate the optimal measurement point.
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Affiliation(s)
- L Pereira
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
| | - J J García-Ripoll
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
| | - T Ramos
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
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6
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Hsieh H, Ning J, Chen Y, Wu H, Chen HL, Wu C, Lee R. Direct Parameter Estimations from Machine Learning-Enhanced Quantum State Tomography. Symmetry (Basel) 2022; 14:874. [DOI: 10.3390/sym14050874] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
With the power to find the best fit to arbitrarily complicated symmetry, machine-learning (ML)-enhanced quantum state tomography (QST) has demonstrated its advantages in extracting complete information about the quantum states. Instead of using the reconstruction model in training a truncated density matrix, we develop a high-performance, lightweight, and easy-to-install supervised characteristic model by generating the target parameters directly. Such a characteristic model-based ML-QST can avoid the problem of dealing with a large Hilbert space, but cab keep feature extractions with high precision, capturing the underlying symmetry in data. With the experimentally measured data generated from the balanced homodyne detectors, we compare the degradation information about quantum noise squeezed states predicted by the reconstruction and characteristic models; both are in agreement with the empirically fitting curves obtained from the covariance method. Such a ML-QST with direct parameter estimations illustrates a crucial diagnostic toolbox for applications with squeezed states, from quantum information process, quantum metrology, advanced gravitational wave detectors, to macroscopic quantum state generation.
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7
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Jansen ND, Loucks M, Gilbert S, Fleming-Dittenber C, Egbert J, Hunt KLC. Shannon and von Neumann entropies of multi-qubit Schrödinger's cat states. Phys Chem Chem Phys 2022; 24:7666-7681. [PMID: 35297927 DOI: 10.1039/d1cp05255a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Using IBM's publicly accessible quantum computers, we have analyzed the entropies of Schrödinger's cat states, which have the form Ψ = (1/2)1/2 [|0 0 0⋯0〉 + |1 1 1⋯1〉]. We have obtained the average Shannon entropy SSo of the distribution over measurement outcomes from 75 runs of 8192 shots, for each of the numbers of entangled qubits, on each of the quantum computers tested. For the distribution over N fault-free measurements on pure cat states, SSo would approach one as N → ∞, independent of the number of qubits; but we have found that SSo varies nearly linearly with the number of qubits n. The slope of SSoversus the number of qubits differs among computers with the same quantum volumes. We have developed a two-parameter model that reproduces the near-linear dependence of the entropy on the number of qubits, based on the probabilities of observing the output 0 when a qubit is set to |0〉 and 1 when it is set to |1〉. The slope increases as the error rate increases. The slope provides a sensitive measure of the accuracy of a quantum computer, so it serves as a quickly determinable index of performance. We have used tomographic methods with error mitigation as described in the qiskit documentation to find the density matrix ρ and evaluate the von Neumann entropies of the cat states. From the reduced density matrices for individual qubits, we have calculated the entanglement entropies. The reduced density matrices represent mixed states with approximately 50/50 probabilities for states |0〉 and |1〉. The entanglement entropies are very close to one.
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Affiliation(s)
- Nathan D Jansen
- Department of Chemistry, Michigan State University, East Lansing, Michigan, 48824, USA.
| | - Matthew Loucks
- Department of Chemistry, Michigan State University, East Lansing, Michigan, 48824, USA.
| | - Scott Gilbert
- Department of Chemistry, Michigan State University, East Lansing, Michigan, 48824, USA.
| | | | - Julia Egbert
- Department of Chemistry, Michigan State University, East Lansing, Michigan, 48824, USA.
| | - Katharine L C Hunt
- Department of Chemistry, Michigan State University, East Lansing, Michigan, 48824, USA.
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8
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Hsieh HY, Chen YR, Wu HC, Chen HL, Ning J, Huang YC, Wu CM, Lee RK. Extract the Degradation Information in Squeezed States with Machine Learning. Phys Rev Lett 2022; 128:073604. [PMID: 35244420 DOI: 10.1103/physrevlett.128.073604] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2021] [Revised: 12/18/2021] [Accepted: 01/25/2022] [Indexed: 06/14/2023]
Abstract
In order to leverage the full power of quantum noise squeezing with unavoidable decoherence, a complete understanding of the degradation in the purity of squeezed light is demanded. By implementing machine-learning architecture with a convolutional neural network, we illustrate a fast, robust, and precise quantum state tomography for continuous variables, through the experimentally measured data generated from the balanced homodyne detectors. Compared with the maximum likelihood estimation method, which suffers from time-consuming and overfitting problems, a well-trained machine fed with squeezed vacuum and squeezed thermal states can complete the task of reconstruction of the density matrix in less than one second. Moreover, the resulting fidelity remains as high as 0.99 even when the antisqueezing level is higher than 20 dB. Compared with the phase noise and loss mechanisms coupled from the environment and surrounding vacuum, experimentally, the degradation information is unveiled with machine learning for low and high noisy scenarios, i.e., with the antisqueezing levels at 12 dB and 18 dB, respectively. Our neural network enhanced quantum state tomography provides the metrics to give physical descriptions of every feature observed in the quantum state with a single scan measurement just by varying the local oscillator phase from 0 to 2π and paves a way of exploring large-scale quantum systems in real time.
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Affiliation(s)
- Hsien-Yi Hsieh
- Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Yi-Ru Chen
- Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Hsun-Chung Wu
- Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Hua Li Chen
- Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Jingyu Ning
- Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Yao-Chin Huang
- Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Chien-Ming Wu
- Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Ray-Kuang Lee
- Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan
- Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
- Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
- Center for Quantum Technology, Hsinchu 30013, Taiwan
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9
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Ahmed S, Sánchez Muñoz C, Nori F, Kockum AF. Quantum State Tomography with Conditional Generative Adversarial Networks. Phys Rev Lett 2021; 127:140502. [PMID: 34652197 DOI: 10.1103/physrevlett.127.140502] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2020] [Revised: 05/21/2021] [Accepted: 06/10/2021] [Indexed: 06/13/2023]
Abstract
Quantum state tomography (QST) is a challenging task in intermediate-scale quantum devices. Here, we apply conditional generative adversarial networks (CGANs) to QST. In the CGAN framework, two dueling neural networks, a generator and a discriminator, learn multimodal models from data. We augment a CGAN with custom neural-network layers that enable conversion of output from any standard neural network into a physical density matrix. To reconstruct the density matrix, the generator and discriminator networks train each other on data using standard gradient-based methods. We demonstrate that our QST-CGAN reconstructs optical quantum states with high fidelity, using orders of magnitude fewer iterative steps, and less data, than both accelerated projected-gradient-based and iterative maximum-likelihood estimation. We also show that the QST-CGAN can reconstruct a quantum state in a single evaluation of the generator network if it has been pretrained on similar quantum states.
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Affiliation(s)
- Shahnawaz Ahmed
- Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
| | - Carlos Sánchez Muñoz
- Departamento de Fisica Teorica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autonoma de Madrid, Madrid 28049, Spain
| | - Franco Nori
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
| | - Anton Frisk Kockum
- Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
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10
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Zhang X, Luo M, Wen Z, Feng Q, Pang S, Luo W, Zhou X. Direct Fidelity Estimation of Quantum States Using Machine Learning. Phys Rev Lett 2021; 127:130503. [PMID: 34623828 DOI: 10.1103/physrevlett.127.130503] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2021] [Revised: 08/04/2021] [Accepted: 08/17/2021] [Indexed: 06/13/2023]
Abstract
In almost all quantum applications, one of the key steps is to verify that the fidelity of the prepared quantum state meets expectations. In this Letter, we propose a new approach solving this problem using machine-learning techniques. Compared to other fidelity estimation methods, our method is applicable to arbitrary quantum states, the number of required measurement settings is small, and this number does not increase with the size of the system. For example, for a general five-qubit quantum state, only four measurement settings are required to predict its fidelity with ±1% precision in a nonadversarial scenario. This machine-learning-based approach for estimating quantum state fidelity has the potential to be widely used in the field of quantum information.
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Affiliation(s)
- Xiaoqian Zhang
- School of Physics and State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510000, China
| | - Maolin Luo
- School of Physics and State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510000, China
| | - Zhaodi Wen
- College of Information Science and Technology, College of Cyber Security, Jinan University, Guangzhou 510632, China
| | - Qin Feng
- School of Physics and State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510000, China
| | - Shengshi Pang
- School of Physics and State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510000, China
| | - Weiqi Luo
- College of Information Science and Technology, College of Cyber Security, Jinan University, Guangzhou 510632, China
| | - Xiaoqi Zhou
- School of Physics and State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510000, China
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11
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Abstract
Entanglement plays a central role in quantum information processing and quantum physics. However, few effective ways are known to detect the amount of entanglement of an unknown quantum state. Here, we propose a scheme to estimate the entanglement negativity for any bipartition of a composite system. The proposed scheme is based on the random unitary evolution and local measurements on a single-copy quantum state, which is more practical compared to former methods based on collective measurements on many copies of the identical state. Meanwhile, we generalize the scheme to quantify the total correlation. We demonstrate the efficiency of the scheme with statistical analyses and numerical simulations. Our scheme is quite suitable for state-of-the-art quantum platforms, which can serve as a useful benchmarking tool to advance quantum technologies and a probe to study fundamental quantum physics like entanglement dynamics.
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Affiliation(s)
- You Zhou
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Pei Zeng
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - Zhenhuan Liu
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
- School of Physics, Peking University, Beijing 100871, China
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12
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Zhang WH, Zhang C, Chen Z, Peng XX, Xu XY, Yin P, Yu S, Ye XJ, Han YJ, Xu JS, Chen G, Li CF, Guo GC. Experimental Optimal Verification of Entangled States Using Local Measurements. Phys Rev Lett 2020; 125:030506. [PMID: 32745389 DOI: 10.1103/physrevlett.125.030506] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2020] [Accepted: 07/01/2020] [Indexed: 06/11/2023]
Abstract
The initialization of a quantum system into a certain state is a crucial aspect of quantum information science. While a variety of measurement strategies have been developed to characterize how well the system is initialized, for a given one, there is in general a trade-off between its efficiency and the accessible information of the quantum state. Conventional quantum state tomography can characterize unknown states while requiring exponentially expensive time-consuming postprocessing. Alternatively, recent theoretical breakthroughs show that quantum state verification provides a technique to quantify the prepared state with significantly fewer samples, especially for multipartite entangled states. In this Letter, we modify the original proposal to be robust to practical imperfections, and experimentally implement a scalable quantum state verification on two-qubit and four-qubit entangled states with nonadaptive local measurements. For all the tested states, the estimated infidelity is inversely proportional to the number of samples, which illustrates the power to characterize a quantum state with a small number of samples. Compared to the globally optimal strategy which requires nonlocal measurements, the efficiency in our experiment is only worse by a small constant factor (<2.5). We compare the performance difference between quantum state verification and quantum state tomography in an experiment to characterize a four-photon Greenberger-Horne-Zeilinger state, and the results indicate the advantage of quantum state verification in both the achieved efficiency and precision.
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Affiliation(s)
- Wen-Hao Zhang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Chao Zhang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zhe Chen
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xing-Xiang Peng
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xiao-Ye Xu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Peng Yin
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Shang Yu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xiang-Jun Ye
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yong-Jian Han
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jin-Shi Xu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Geng Chen
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Chuan-Feng Li
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Guang-Can Guo
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
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13
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Torlai G, Timar B, van Nieuwenburg EPL, Levine H, Omran A, Keesling A, Bernien H, Greiner M, Vuletić V, Lukin MD, Melko RG, Endres M. Integrating Neural Networks with a Quantum Simulator for State Reconstruction. Phys Rev Lett 2019; 123:230504. [PMID: 31868463 DOI: 10.1103/physrevlett.123.230504] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2019] [Revised: 09/15/2019] [Indexed: 06/10/2023]
Abstract
We demonstrate quantum many-body state reconstruction from experimental data generated by a programmable quantum simulator by means of a neural-network model incorporating known experimental errors. Specifically, we extract restricted Boltzmann machine wave functions from data produced by a Rydberg quantum simulator with eight and nine atoms in a single measurement basis and apply a novel regularization technique to mitigate the effects of measurement errors in the training data. Reconstructions of modest complexity are able to capture one- and two-body observables not accessible to experimentalists, as well as more sophisticated observables such as the Rényi mutual information. Our results open the door to integration of machine learning architectures with intermediate-scale quantum hardware.
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Affiliation(s)
- Giacomo Torlai
- Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA
- Department of Physics and Astronomy, University of Waterloo, Ontario N2L 3G1, Canada
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - Brian Timar
- Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
| | - Evert P L van Nieuwenburg
- Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
| | - Harry Levine
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Ahmed Omran
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Alexander Keesling
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Hannes Bernien
- Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA
| | - Markus Greiner
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Vladan Vuletić
- Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Mikhail D Lukin
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Roger G Melko
- Department of Physics and Astronomy, University of Waterloo, Ontario N2L 3G1, Canada
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - Manuel Endres
- Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
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14
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Abstract
Ptychography is an imaging technique in which a localized illumination scans overlapping regions of an object and generates a set of diffraction intensities used to computationally reconstruct its complex-valued transmission function. We propose a quantum analogue of this technique designed to reconstruct d-dimensional pure states. A set of n rank-r projectors “scans” overlapping parts of an input state and the moduli of the d Fourier amplitudes of each part are measured. These nd outcomes are fed into an iterative phase retrieval algorithm that estimates the state. Using d up to 100 and r around d / 2, we performed numerical simulations for single systems in an economic (n = 4) and a costly (n = d) scenario, as well as for multiqubit systems (n = 6logd). This numeric study included realistic amounts of depolarization and poissonian noise, and all scenarios yielded, in general, reconstructions with infidelities below 10−2. The method is shown, therefore, to be resilient to noise and, for any d, requires a simple and fast postprocessing algorithm. We show that the algorithm is equivalent to an alternating gradient search, which ensures that it does not suffer from local-minima stagnation. Unlike traditional approaches to state reconstruction, the ptychographic scheme uses a single measurement basis; the diversity and redundancy in the measured data—key for its success—are provided by the overlapping projections. We illustrate the simplicity of this scheme with the paradigmatic multiport interferometer.
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Affiliation(s)
- Mário Foganholi Fernandes
- Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 31270-901, Brazil.
| | - Leonardo Neves
- Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 31270-901, Brazil.
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15
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Wang J, Scholz VB, Renner R. Confidence Polytopes in Quantum State Tomography. Phys Rev Lett 2019; 122:190401. [PMID: 31144942 DOI: 10.1103/physrevlett.122.190401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2018] [Indexed: 06/09/2023]
Abstract
Quantum state tomography is the task of inferring the state of a quantum system from measurement data. A reliable tomography scheme should not only report an estimate for that state, but also well-justified error bars. These may be specified in terms of confidence regions, i.e., subsets of the state space which contain the system's state with high probability. Here, building upon a quantum generalization of Clopper-Pearson confidence intervals-a notion known from classical statistics-we present a simple and reliable scheme for generating confidence regions. These have the shape of a polytope and can be computed efficiently. We provide several examples to demonstrate the practical usability of the scheme in experiments.
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Affiliation(s)
- Jinzhao Wang
- Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
| | - Volkher B Scholz
- Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
| | - Renato Renner
- Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
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16
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Stefano QP, Rebón L, Ledesma S, Iemmi C. Set of 4d-3 observables to determine any pure qudit state. Opt Lett 2019; 44:2558-2561. [PMID: 31090731 DOI: 10.1364/ol.44.002558] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2019] [Accepted: 04/22/2019] [Indexed: 06/09/2023]
Abstract
We present a tomographic method which requires only 4d-3 measurement outcomes to reconstruct any pure quantum state of arbitrary dimension d. Using the proposed scheme, we have experimentally reconstructed a large number of pure states of dimension d=7, obtaining a mean fidelity of 0.94. Moreover, we performed numerical simulations of the reconstruction process, verifying the feasibility of the method for higher dimensions. In addition, the a priori assumption of purity can be certified within the same set of measurements, which represents an improvement with respect to other similar methods and contributes to answering the question of how many observables are needed to uniquely determine any pure state.
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17
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18
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Chalopin T, Bouazza C, Evrard A, Makhalov V, Dreon D, Dalibard J, Sidorenkov LA, Nascimbene S. Quantum-enhanced sensing using non-classical spin states of a highly magnetic atom. Nat Commun 2018; 9:4955. [PMID: 30470745 DOI: 10.1038/s41467-018-07433-1] [Citation(s) in RCA: 31] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2018] [Accepted: 10/26/2018] [Indexed: 11/08/2022] Open
Abstract
Coherent superposition states of a mesoscopic quantum object play a major role in our understanding of the quantum to classical boundary, as well as in quantum-enhanced metrology and computing. However, their practical realization and manipulation remains challenging, requiring a high degree of control of the system and its coupling to the environment. Here, we use dysprosium atoms-the most magnetic element in its ground state-to realize coherent superpositions between electronic spin states of opposite orientation, with a mesoscopic spin size J = 8. We drive coherent spin states to quantum superpositions using non-linear light-spin interactions, observing a series of collapses and revivals of quantum coherence. These states feature highly non-classical behavior, with a sensitivity to magnetic fields enhanced by a factor 13.9(1.1) compared to coherent spin states-close to the Heisenberg limit 2J = 16-and an intrinsic fragility to environmental noise.
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19
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Barthel T, Lu J. Fundamental Limitations for Measurements in Quantum Many-Body Systems. Phys Rev Lett 2018; 121:080406. [PMID: 30192598 DOI: 10.1103/physrevlett.121.080406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2018] [Revised: 06/20/2018] [Indexed: 06/08/2023]
Abstract
Dynamical measurement schemes are an important tool for the investigation of quantum many-body systems, especially in the age of quantum simulation. Here, we address the question whether generic measurements can be implemented efficiently if we have access to a certain set of experimentally realizable measurements and can extend it through time evolution. For the latter, two scenarios are considered: (a) evolution according to unitary circuits and (b) evolution due to Hamiltonians that we can control in a time-dependent fashion. We find that the time needed to realize a certain measurement to a predefined accuracy scales in general exponentially with the system size-posing a fundamental limitation. The argument is based on the construction of ϵ-packings for manifolds of observables with identical spectra and a comparison of their cardinalities to those of ϵ-coverings for quantum circuits and unitary time-evolution operators. The former is related to the study of Grassmann manifolds.
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Affiliation(s)
- Thomas Barthel
- Department of Physics, Duke University, Durham, North Carolina 27708, USA
| | - Jianfeng Lu
- Department of Physics, Duke University, Durham, North Carolina 27708, USA
- Department of Mathematics, Duke University, Durham, North Carolina 27708, USA
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20
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Abstract
The experimental interest and developments in quantum spin-1/2 chains has increased uninterruptedly over the past decade. In many instances, the target quantum simulation belongs to the broader class of noninteracting fermionic models, constituting an important benchmark. In spite of this class being analytically efficiently tractable, no direct certification tool has yet been reported for it. In fact, in experiments, certification has almost exclusively relied on notions of quantum state tomography scaling very unfavorably with the system size. Here, we develop experimentally friendly fidelity witnesses for all pure fermionic Gaussian target states. Their expectation value yields a tight lower bound to the fidelity and can be measured efficiently. We derive witnesses in full generality in the Majorana-fermion representation and apply them to experimentally relevant spin-1/2 chains. Among others, we show how to efficiently certify strongly out-of-equilibrium dynamics in critical Ising chains. At the heart of the measurement scheme is a variant of importance sampling specially tailored to overlaps between covariance matrices. The method is shown to be robust against finite experimental-state infidelities.
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Affiliation(s)
- M Gluza
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
| | - M Kliesch
- Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Poland
- Institute for Theoretical Physics, Heinrich Heine University Düsseldorf, 40225 Düsseldorf, Germany
| | - J Eisert
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
| | - L Aolita
- Instituto de Física, Universidade Federal do Rio de Janeiro, P.O. Box 68528, Rio de Janeiro, Rio de Janeiro 21941-972, Brazil
- International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil
- ICTP South American Institute for Fundamental Research, Instituto de Física Teórica, UNESP-Universidade Estadual Paulista R. Dr. Bento T. Ferraz 271, Bl. II, São Paulo 01140-070, São Paulo, Brazil
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21
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Daoud M, Kibler MR. Generalized Weyl-Heisenberg Algebra, Qudit Systems and Entanglement Measure of Symmetric States via Spin Coherent States. Entropy (Basel) 2018; 20:E292. [PMID: 33265383 DOI: 10.3390/e20040292] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 03/26/2018] [Revised: 04/13/2018] [Accepted: 04/13/2018] [Indexed: 11/17/2022]
Abstract
A relation is established in the present paper between Dicke states in a d-dimensional space and vectors in the representation space of a generalized Weyl–Heisenberg algebra of finite dimension d. This provides a natural way to deal with the separable and entangled states of a system of N=d−1 symmetric qubit states. Using the decomposition property of Dicke states, it is shown that the separable states coincide with the Perelomov coherent states associated with the generalized Weyl–Heisenberg algebra considered in this paper. In the so-called Majorana scheme, the qudit (d-level) states are represented by N points on the Bloch sphere; roughly speaking, it can be said that a qudit (in a d-dimensional space) is describable by a N-qubit vector (in a N-dimensional space). In such a scheme, the permanent of the matrix describing the overlap between the N qubits makes it possible to measure the entanglement between the N qubits forming the qudit. This is confirmed by a Fubini–Study metric analysis. A new parameter, proportional to the permanent and called perma-concurrence, is introduced for characterizing the entanglement of a symmetric qudit arising from N qubits. For d=3 (⇔N=2), this parameter constitutes an alternative to the concurrence for two qubits. Other examples are given for d=4 and 5. A connection between Majorana stars and zeros of a Bargmmann function for qudits closes this article.
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22
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Abstract
In a recent publication we showed that permutation symmetry reduces the numerical complexity of Lindblad quantum master equations for identical multi-level systems from exponential to polynomial scaling. This is important for open system dynamics including realistic system bath interactions and dephasing in, for instance, the Dicke model, multi-Λ system setups etc. Here we present an object-oriented C++ library that allows to setup and solve arbitrary quantum optical Lindblad master equations, especially those that are permutationally symmetric in the multi-level systems. PsiQuaSP (Permutation symmetry for identical Quantum Systems Package) uses the PETSc package for sparse linear algebra methods and differential equations as basis. The aim of PsiQuaSP is to provide flexible, storage efficient and scalable code while being as user friendly as possible. It is easily applied to many quantum optical or quantum information systems with more than one multi-level system. We first review the basics of the permutation symmetry for multi-level systems in quantum master equations. The application of PsiQuaSP to quantum dynamical problems is illustrated with several typical, simple examples of open quantum optical systems.
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Affiliation(s)
- Michael Gegg
- Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universität Berlin, Hardenbergstr, 36 EW 7-1, 10623, Berlin, Germany.
| | - Marten Richter
- Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universität Berlin, Hardenbergstr, 36 EW 7-1, 10623, Berlin, Germany
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23
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Knips L, Schwemmer C, Klein N, Wieśniak M, Weinfurter H. Multipartite Entanglement Detection with Minimal Effort. Phys Rev Lett 2016; 117:210504. [PMID: 27911541 DOI: 10.1103/physrevlett.117.210504] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2015] [Indexed: 06/06/2023]
Abstract
Certifying entanglement of a multipartite state is generally considered a demanding task. Since an N qubit state is parametrized by 4^{N}-1 real numbers, one might naively expect that the measurement effort of generic entanglement detection also scales exponentially with N. Here, we introduce a general scheme to construct efficient witnesses requiring a constant number of measurements independent of the number of qubits for states like, e.g., Greenberger-Horne-Zeilinger states, cluster states, and Dicke states. For four qubits, we apply this novel method to experimental realizations of the aforementioned states and prove genuine four-partite entanglement with two measurement settings only.
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Affiliation(s)
- Lukas Knips
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
- Fakultät für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| | - Christian Schwemmer
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
- Fakultät für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| | - Nico Klein
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
- Fakultät für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| | - Marcin Wieśniak
- Institute of Theoretical Physics and Astrophysics, University of Gdańsk, PL-80-952 Gdańsk, Poland
| | - Harald Weinfurter
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
- Fakultät für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
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24
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Abstract
Precise characterization of quantum devices is usually achieved with quantum tomography. However, most methods which are currently widely used in experiments, such as maximum likelihood estimation, lack a well-justified error analysis. Promising recent methods based on confidence regions are difficult to apply in practice or yield error bars which are unnecessarily large. Here, we propose a practical yet robust method for obtaining error bars. We do so by introducing a novel representation of the output of the tomography procedure, the quantum error bars. This representation is (i) concise, being given in terms of few parameters, (ii) intuitive, providing a fair idea of the "spread" of the error, and (iii) useful, containing the necessary information for constructing confidence regions. The statements resulting from our method are formulated in terms of a figure of merit, such as the fidelity to a reference state. We present an algorithm for computing this representation and provide ready-to-use software. Our procedure is applied to actual experimental data obtained from two superconducting qubits in an entangled state, demonstrating the applicability of our method.
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Affiliation(s)
- Philippe Faist
- Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
| | - Renato Renner
- Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
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25
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Bolduc E, Gariepy G, Leach J. Direct measurement of large-scale quantum states via expectation values of non-Hermitian matrices. Nat Commun 2016; 7:10439. [PMID: 26780858 DOI: 10.1038/ncomms10439] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2015] [Accepted: 12/10/2015] [Indexed: 11/26/2022] Open
Abstract
In quantum mechanics, predictions are made by way of calculating expectation values of observables, which take the form of Hermitian operators. Non-Hermitian operators, however, are not necessarily devoid of physical significance, and they can play a crucial role in the characterization of quantum states. Here we show that the expectation values of a particular set of non-Hermitian matrices, which we call column operators, directly yield the complex coefficients of a quantum state vector. We provide a definition of the state vector in terms of measurable quantities by decomposing these column operators into observables. The technique we propose renders very-large-scale quantum states significantly more accessible in the laboratory, as we demonstrate by experimentally characterizing a 100,000-dimensional entangled state. This represents an improvement of two orders of magnitude with respect to previous phase-and-amplitude characterizations of discrete entangled states. Quantum state tomography is the process of retrieving the values that define a quantum system, but realizing it experimentally can be burdensome. Here, the authors provide an alternative approach via the expectation values of a set of non-Hermitian matrices, and characterize a 100,000-dimensional state.
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26
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Aolita L, Gogolin C, Kliesch M, Eisert J. Reliable quantum certification of photonic state preparations. Nat Commun 2015; 6:8498. [PMID: 26577800 PMCID: PMC4673657 DOI: 10.1038/ncomms9498] [Citation(s) in RCA: 64] [Impact Index Per Article: 7.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/25/2014] [Accepted: 08/27/2015] [Indexed: 11/09/2022] Open
Abstract
Quantum technologies promise a variety of exciting applications. Even though impressive progress has been achieved recently, a major bottleneck currently is the lack of practical certification techniques. The challenge consists of ensuring that classically intractable quantum devices perform as expected. Here we present an experimentally friendly and reliable certification tool for photonic quantum technologies: an efficient certification test for experimental preparations of multimode pure Gaussian states, pure non-Gaussian states generated by linear-optical circuits with Fock-basis states of constant boson number as inputs, and pure states generated from the latter class by post-selecting with Fock-basis measurements on ancillary modes. Only classical computing capabilities and homodyne or hetorodyne detection are required. Minimal assumptions are made on the noise or experimental capabilities of the preparation. The method constitutes a step forward in many-body quantum certification, which is ultimately about testing quantum mechanics at large scales.
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Affiliation(s)
- Leandro Aolita
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
- Instituto de Física, Universidade Federal do Rio de Janeiro, P. O. Box 68528, Rio de Janeiro 21941-972, Brazil
| | - Christian Gogolin
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
| | - Martin Kliesch
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
| | - Jens Eisert
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
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27
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Schwemmer C, Knips L, Tran MC, de Rosier A, Laskowski W, Paterek T, Weinfurter H. Genuine Multipartite Entanglement without Multipartite Correlations. Phys Rev Lett 2015; 114:180501. [PMID: 26000990 DOI: 10.1103/physrevlett.114.180501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2014] [Indexed: 06/04/2023]
Abstract
Nonclassical correlations between measurement results make entanglement the essence of quantum physics and the main resource for quantum information applications. Surprisingly, there are n-particle states which do not exhibit n-partite correlations at all but still are genuinely n-partite entangled. We introduce a general construction principle for such states, implement them in a multiphoton experiment and analyze their properties in detail. Remarkably, even without multipartite correlations, these states do violate Bell inequalities showing that there is no classical, i.e., local realistic model describing their properties.
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Affiliation(s)
- Christian Schwemmer
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
- Department für Physik, Ludwig-Maximilians-Universität, 80797 München, Germany
| | - Lukas Knips
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
- Department für Physik, Ludwig-Maximilians-Universität, 80797 München, Germany
| | - Minh Cong Tran
- School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371 Singapore
| | - Anna de Rosier
- Institute of Theoretical Physics and Astrophysics, University of Gdańsk, PL-80-952 Gdańsk, Poland
| | - Wiesław Laskowski
- Institute of Theoretical Physics and Astrophysics, University of Gdańsk, PL-80-952 Gdańsk, Poland
| | - Tomasz Paterek
- School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371 Singapore
- Center for Quantum Technologies, National University of Singapore, Singapore, 117543 Singapore
- MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, Singapore, UMI 3654 Singapore
| | - Harald Weinfurter
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
- Department für Physik, Ludwig-Maximilians-Universität, 80797 München, Germany
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28
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Abstract
One of the greatest challenges in the fields of quantum information processing and quantum technologies is the detailed coherent control over each and every constituent of quantum systems with an ever increasing number of particles. Within this endeavor, harnessing of many-body entanglement against the detrimental effects of the environment is a major pressing issue. Besides being an important concept from a fundamental standpoint, entanglement has been recognized as a crucial resource for quantum speed-ups or performance enhancements over classical methods. Understanding and controlling many-body entanglement in open systems may have strong implications in quantum computing, quantum simulations of many-body systems, secure quantum communication or cryptography, quantum metrology, our understanding of the quantum-to-classical transition, and other important questions of quantum foundations.In this paper we present an overview of recent theoretical and experimental efforts to underpin the dynamics of entanglement under the influence of noise. Entanglement is thus taken as a dynamic quantity on its own, and we survey how it evolves due to the unavoidable interaction of the entangled system with its surroundings. We analyze several scenarios, corresponding to different families of states and environments, which render a very rich diversity of dynamical behaviors.In contrast to single-particle quantities, like populations and coherences, which typically vanish only asymptotically in time, entanglement may disappear at a finite time. In addition, important classes of entanglement display an exponential decay with the number of particles when subject to local noise, which poses yet another threat to the already-challenging scaling of quantum technologies. Other classes, however, turn out to be extremely robust against local noise. Theoretical results and recent experiments regarding the difference between local and global decoherence are summarized. Control and robustness-enhancement techniques, scaling laws, statistical and geometrical aspects of multipartite-entanglement decay are also reviewed; all in order to give a broad picture of entanglement dynamics in open quantum systems addressed to both theorists and experimentalists inside and outside the field of quantum information.
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Affiliation(s)
- Leandro Aolita
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
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29
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Schwemmer C, Knips L, Richart D, Weinfurter H, Moroder T, Kleinmann M, Gühne O. Systematic errors in current quantum state tomography tools. Phys Rev Lett 2015; 114:080403. [PMID: 25768740 DOI: 10.1103/physrevlett.114.080403] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2013] [Indexed: 06/04/2023]
Abstract
Common tools for obtaining physical density matrices in experimental quantum state tomography are shown here to cause systematic errors. For example, using maximum likelihood or least squares optimization to obtain physical estimates for the quantum state, we observe a systematic underestimation of the fidelity and an overestimation of entanglement. Such strongly biased estimates can be avoided using linear evaluation of the data or by linearizing measurement operators yielding reliable and computational simple error bounds.
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Affiliation(s)
- Christian Schwemmer
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany and Department für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| | - Lukas Knips
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany and Department für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| | - Daniel Richart
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany and Department für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| | - Harald Weinfurter
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany and Department für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| | - Tobias Moroder
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany
| | - Matthias Kleinmann
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany
- Departamento de Matemtica, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, Minas Gerais 31270-901, Brazil
- Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao, Spain
| | - Otfried Gühne
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany
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30
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Abstract
We introduce a self-learning tomographic technique in which the experiment guides itself to an estimate of its own state. Self-guided quantum tomography uses measurements to directly test hypotheses in an iterative algorithm which converges to the true state. We demonstrate through simulation on many qubits that Self-guided quantum tomography is a more efficient and robust alternative to the usual paradigm of taking a large amount of informationally complete data and solving the inverse problem of postprocessed state estimation.
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Affiliation(s)
- Christopher Ferrie
- Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87131-0001, USA
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31
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Abstract
Data compression is a ubiquitous aspect of modern information technology, and the advent of quantum information raises the question of what types of compression are feasible for quantum data, where it is especially relevant given the extreme difficulty involved in creating reliable quantum memories. We present a protocol in which an ensemble of quantum bits (qubits) can in principle be perfectly compressed into exponentially fewer qubits. We then experimentally implement our algorithm, compressing three photonic qubits into two. This protocol sheds light on the subtle differences between quantum and classical information. Furthermore, since data compression stores all of the available information about the quantum state in fewer physical qubits, it could allow for a vast reduction in the amount of quantum memory required to store a quantum ensemble, making even today's limited quantum memories far more powerful than previously recognized.
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Affiliation(s)
- Lee A Rozema
- Centre for Quantum Information and Quantum Control and Department of Physics, University of Toronto, 60 Saint George Street, Toronto, Ontario M5S 1A7, Canada
| | - Dylan H Mahler
- Centre for Quantum Information and Quantum Control and Department of Physics, University of Toronto, 60 Saint George Street, Toronto, Ontario M5S 1A7, Canada
| | - Alex Hayat
- Centre for Quantum Information and Quantum Control and Department of Physics, University of Toronto, 60 Saint George Street, Toronto, Ontario M5S 1A7, Canada and Department of Electrical Engineering, Technion, Haifa 32000, Israel and Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
| | - Peter S Turner
- Department of Physics, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Aephraim M Steinberg
- Centre for Quantum Information and Quantum Control and Department of Physics, University of Toronto, 60 Saint George Street, Toronto, Ontario M5S 1A7, Canada and Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
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32
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Schwemmer C, Tóth G, Niggebaum A, Moroder T, Gross D, Gühne O, Weinfurter H. Experimental comparison of efficient tomography schemes for a six-qubit state. Phys Rev Lett 2014; 113:040503. [PMID: 25105604 DOI: 10.1103/physrevlett.113.040503] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2014] [Indexed: 06/03/2023]
Abstract
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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Affiliation(s)
- Christian Schwemmer
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany and Department für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
| | - Géza Tóth
- Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao, Spain and IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain and Wigner Research Centre for Physics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Alexander Niggebaum
- School of Physics and Astronomy, University of Birmingham, B15 2TT Birmingham, United Kingdom
| | - Tobias Moroder
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany
| | - David Gross
- Physikalisches Institut & FDM, Universität Freiburg, Rheinstraße 10, D-79104 Freiburg, Germany
| | - Otfried Gühne
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany
| | - Harald Weinfurter
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany and Department für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany
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33
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Gao T, Yan F, van Enk SJ. Permutationally invariant part of a density matrix and nonseparability of N-qubit states. Phys Rev Lett 2014; 112:180501. [PMID: 24856681 DOI: 10.1103/physrevlett.112.180501] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2013] [Indexed: 06/03/2023]
Abstract
We consider the concept of "the permutationally invariant (PI) part of a density matrix," which has proven very useful for both efficient quantum state estimation and entanglement characterization of N-qubit systems. We show here that the concept is, in fact, basis dependent but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering the PI part ρ(PI) of a general (mixed) N-qubit state ρ, we obtain (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multipartite separability criteria (one of which entails a sufficient criterion for genuine N-partite entanglement) that can be experimentally determined by just 2N+1 measurement settings, (iii) a definition of an efficiently measurable degree of separability, which can be used for quantifying a novel aspect of decoherence of N qubits, and (iv) an explicit example that shows there are, for increasing N, genuinely N-partite entangled states lying closer and closer to the maximally mixed state. Moreover, we show that if the PI part of a state is k nonseparable, then so is the actual state. We further argue to add as requirement on any multipartite entanglement measure E that it satisfy E(ρ)≥E(ρ(PI)), even though the operation that maps ρ→ρ(PI) is not local.
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Affiliation(s)
- Ting Gao
- College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
| | - Fengli Yan
- College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050024, China and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
| | - S J van Enk
- Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
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34
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Abstract
A simple yet efficient state reconstruction algorithm of linear regression estimation (LRE) is presented for quantum state tomography. In this method, quantum state reconstruction is converted into a parameter estimation problem of a linear regression model and the least-squares method is employed to estimate the unknown parameters. An asymptotic mean squared error (MSE) upper bound for all possible states to be estimated is given analytically, which depends explicitly upon the involved measurement bases. This analytical MSE upper bound can guide one to choose optimal measurement sets. The computational complexity of LRE is O(d4) where d is the dimension of the quantum state. Numerical examples show that LRE is much faster than maximum-likelihood estimation for quantum state tomography.
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Affiliation(s)
- Bo Qi
- Key Laboratory of Systems and Control, ISS, and National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, CAS, Beijing 100190, P. R. China
| | - Zhibo Hou
- Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei 230026, P. R. China
| | - Li Li
- Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei 230026, P. R. China
| | - Daoyi Dong
- School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia
| | - Guoyong Xiang
- Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei 230026, P. R. China
| | - Guangcan Guo
- Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei 230026, P. R. China
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35
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Abstract
Recent contributions in the field of quantum state tomography have shown that, despite the exponential growth of Hilbert space with the number of subsystems, tomography of one-dimensional quantum systems may still be performed efficiently by tailored reconstruction schemes. Here, we discuss a scalable method to reconstruct mixed states that are well approximated by matrix product operators. The reconstruction scheme only requires local information about the state, giving rise to a reconstruction technique that is scalable in the system size. It is based on a constructive proof that generic matrix product operators are fully determined by their local reductions. We discuss applications of this scheme for simulated data and experimental data obtained in an ion trap experiment.
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Affiliation(s)
- T Baumgratz
- Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany
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36
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Liu WT, Zhang T, Liu JY, Chen PX, Yuan JM. Experimental quantum state tomography via compressed sampling. Phys Rev Lett 2012; 108:170403. [PMID: 22680842 DOI: 10.1103/physrevlett.108.170403] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2011] [Indexed: 06/01/2023]
Abstract
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.
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Affiliation(s)
- Wei-Tao Liu
- Department of Physics, College of Science, National University of Defense Technology, Changsha, People's Republic of China.
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37
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Abstract
We describe a simple method for certifying that an experimental device prepares a desired quantum state ρ. Our method is applicable to any pure state ρ, and it provides an estimate of the fidelity between ρ and the actual (arbitrary) state in the lab, up to a constant additive error. The method requires measuring only a constant number of Pauli expectation values, selected at random according to an importance-weighting rule. Our method is faster than full tomography by a factor of d, the dimension of the state space, and extends easily and naturally to quantum channels.
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Affiliation(s)
- Steven T Flammia
- Institute for Quantum Information, California Institute of Technology, Pasadena, California, USA
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