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Concha D, Pereira L, Zambrano L, Delgado A. Training a quantum measurement device to discriminate unknown non-orthogonal quantum states. Sci Rep 2023; 13:7460. [PMID: 37156829 PMCID: PMC10167228 DOI: 10.1038/s41598-023-34327-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2022] [Accepted: 04/27/2023] [Indexed: 05/10/2023] Open
Abstract
Here, we study the problem of decoding information transmitted through unknown quantum states. We assume that Alice encodes an alphabet into a set of orthogonal quantum states, which are then transmitted to Bob. However, the quantum channel that mediates the transmission maps the orthogonal states into non-orthogonal states, possibly mixed. If an accurate model of the channel is unavailable, then the states received by Bob are unknown. In order to decode the transmitted information we propose to train a measurement device to achieve the smallest possible error in the discrimination process. This is achieved by supplementing the quantum channel with a classical one, which allows the transmission of information required for the training, and resorting to a noise-tolerant optimization algorithm. We demonstrate the training method in the case of minimum-error discrimination strategy and show that it achieves error probabilities very close to the optimal one. In particular, in the case of two unknown pure states, our proposal approaches the Helstrom bound. A similar result holds for a larger number of states in higher dimensions. We also show that a reduction of the search space, which is used in the training process, leads to a considerable reduction in the required resources. Finally, we apply our proposal to the case of the phase flip channel reaching an accurate value of the optimal error probability.
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Affiliation(s)
- D Concha
- Instituto Milenio de Investigación en Óptica y Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
| | - L Pereira
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid, 28006, Spain.
| | - L Zambrano
- Instituto Milenio de Investigación en Óptica y Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860, Castelldefels, Barcelona, Spain
| | - A Delgado
- Instituto Milenio de Investigación en Óptica y Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
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Rambach M, Qaryan M, Kewming M, Ferrie C, White AG, Romero J. Robust and Efficient High-Dimensional Quantum State Tomography. PHYSICAL REVIEW LETTERS 2021; 126:100402. [PMID: 33784128 DOI: 10.1103/physrevlett.126.100402] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2020] [Revised: 01/12/2021] [Accepted: 02/05/2021] [Indexed: 05/25/2023]
Abstract
The exponential growth in Hilbert space with increasing size of a quantum system means that accurately characterizing the system becomes significantly harder with system dimension d. We show that self-guided tomography is a practical, efficient, and robust technique of measuring higher-dimensional quantum states. The achieved fidelities are over 99.9% for qutrits (d=3) and ququints (d=5), and 99.1% for quvigints (d=20)-the highest values ever realized for qudit pure states. We also show excellent performance for mixed states, achieving average fidelities of 96.5% for qutrits. We demonstrate robustness against experimental sources of noise, both statistical and environmental. The technique is applicable to any higher-dimensional system, from a collection of qubits through to individual qudits, and any physical realization, be it photonic, superconducting, ionic, or spin.
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Affiliation(s)
- Markus Rambach
- Australian Research Council Centre of Excellence for Engineered Quantum Systems, Brisbane, Queensland 4072, Australia
- School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia
| | - Mahdi Qaryan
- Australian Research Council Centre of Excellence for Engineered Quantum Systems, Brisbane, Queensland 4072, Australia
- School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia
| | - Michael Kewming
- Australian Research Council Centre of Excellence for Engineered Quantum Systems, Brisbane, Queensland 4072, Australia
- School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia
| | - Christopher Ferrie
- Centre for Quantum Software and Information, University of Technology Sydney, Sydney, New South Wales 2007, Australia
| | - Andrew G White
- Australian Research Council Centre of Excellence for Engineered Quantum Systems, Brisbane, Queensland 4072, Australia
- School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia
| | - Jacquiline Romero
- Australian Research Council Centre of Excellence for Engineered Quantum Systems, Brisbane, Queensland 4072, Australia
- School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia
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Estimation of pure quantum states in high dimension at the limit of quantum accuracy through complex optimization and statistical inference. Sci Rep 2020; 10:12781. [PMID: 32728142 PMCID: PMC7391742 DOI: 10.1038/s41598-020-69646-z] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2020] [Accepted: 07/15/2020] [Indexed: 11/29/2022] Open
Abstract
Quantum tomography has become a key tool for the assessment of quantum states, processes, and devices. This drives the search for tomographic methods that achieve greater accuracy. In the case of mixed states of a single 2-dimensional quantum system adaptive methods have been recently introduced that achieve the theoretical accuracy limit deduced by Hayashi and Gill and Massar. However, accurate estimation of higher-dimensional quantum states remains poorly understood. This is mainly due to the existence of incompatible observables, which makes multiparameter estimation difficult. Here we present an adaptive tomographic method and show through numerical simulations that, after a few iterations, it is asymptotically approaching the fundamental Gill–Massar lower bound for the estimation accuracy of pure quantum states in high dimension. The method is based on a combination of stochastic optimization on the field of the complex numbers and statistical inference, exceeds the accuracy of any mixed-state tomographic method, and can be demonstrated with current experimental capabilities. The proposed method may lead to new developments in quantum metrology.
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