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Amosov G. On Constructing Informationally Complete Covariant Positive Operator-Valued Measures. ENTROPY (BASEL, SWITZERLAND) 2023; 25:e25050783. [PMID: 37238538 DOI: 10.3390/e25050783] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2023] [Revised: 04/28/2023] [Accepted: 05/10/2023] [Indexed: 05/28/2023]
Abstract
We study a projective unitary representation of the product G=G˜×G, where G is a locally compact Abelian group and G^ is its dual consisting of characters on G. It is proven that the representation is irreducible, which allows us to define a covariant positive operator-valued measure (covariant POVM) generated by orbits of projective unitary representations of G. The quantum tomography associated with the representation is discussed. It is shown that the integration over such a covariant POVM defines a family of contractions which are multiples of unitary operators from the representation. Using this fact, it is proven that the measure is informationally complete. The obtained results are illustrated by optical tomography on groups and by a measure with a density that has a value in the set of coherent states.
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Affiliation(s)
- Grigori Amosov
- Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow 119991, Russia
- Center of Pure Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny 141701, Russia
- Saint Petersburg State University, Saint Petersburg 199034, Russia
- Institute of Mathematics with Computer Center of the Ufa Science Center of Russian Academy of Sciences, Ufa 450008, Russia
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Ahmed S, Sánchez Muñoz C, Nori F, Kockum AF. Quantum State Tomography with Conditional Generative Adversarial Networks. PHYSICAL REVIEW LETTERS 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] [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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Macchiavello C, Sacchi MF. Efficient Accessible Bounds to the Classical Capacity of Quantum Channels. PHYSICAL REVIEW LETTERS 2019; 123:090503. [PMID: 31524487 DOI: 10.1103/physrevlett.123.090503] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2019] [Indexed: 06/10/2023]
Abstract
We present a method to detect lower bounds to the classical capacity of quantum communication channels by means of few local measurements (i.e., without complete process tomography), reconstruction of sets of conditional probabilities, and classical optimization. The method does not require any a priori information about the channel. We illustrate its performance for significant forms of noisy channels.
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Affiliation(s)
- Chiara Macchiavello
- Quit group, Dipartimento di Fisica, Università di Pavia, via A. Bassi 6, I-27100 Pavia, Italy
- Istituto Nazionale di Fisica Nucleare, Gruppo IV, via A. Bassi 6, I-27100 Pavia, Italy
| | - Massimiliano F Sacchi
- Quit group, Dipartimento di Fisica, Università di Pavia, via A. Bassi 6, I-27100 Pavia, Italy
- Istituto di Fotonica e Nanotecnologie-CNR, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy
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Macchiavello C, Sacchi MF. Detecting Lower Bounds to Quantum Channel Capacities. PHYSICAL REVIEW LETTERS 2016; 116:140501. [PMID: 27104688 DOI: 10.1103/physrevlett.116.140501] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2015] [Indexed: 06/05/2023]
Abstract
We propose a method to detect lower bounds to quantum capacities of a noisy quantum communication channel by means of a few measurements. The method is easily implementable and does not require any knowledge about the channel. We test its efficiency by studying its performance for most well-known single-qubit noisy channels and for the generalized Pauli channel in an arbitrary finite dimension.
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Affiliation(s)
- Chiara Macchiavello
- Quit group, Dipartimento di Fisica, Università di Pavia, via A. Bassi 6, I-27100 Pavia, Italy
- Istituto Nazionale di Fisica Nucleare, Gruppo IV, via A. Bassi 6, I-27100 Pavia, Italy
| | - Massimiliano F Sacchi
- Quit group, Dipartimento di Fisica, Università di Pavia, via A. Bassi 6, I-27100 Pavia, Italy
- Istituto di Fotonica e Nanotecnologie-CNR, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
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Rastegin AE. On the Brukner–Zeilinger approach to information in quantum measurements. Proc Math Phys Eng Sci 2015. [DOI: 10.1098/rspa.2015.0435] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We address the problem of properly quantifying information in quantum theory. Brukner and Zeilinger proposed the concept of an operationally invariant measure based on measurement statistics. Their measure of information is calculated with probabilities generated in a complete set of mutually complementary observations. This approach was later criticized for several reasons. We show that some critical points can be overcome by means of natural extension or reformulation of the Brukner–Zeilinger approach. In particular, this approach is connected with symmetric informationally complete measurements. The ‘total information’ of Brukner and Zeilinger can further be treated in the context of mutually unbiased measurements as well as general symmetric informationally complete measurements. The Brukner–Zeilinger measure of information is also examined in the case of detection inefficiencies. It is shown to be decreasing under the action of bistochastic maps. The Brukner–Zeilinger total information can be used for estimating the map norm of quantum operations.
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Bisio A, Chiribella G, D'Ariano GM, Facchini S, Perinotti P. Optimal quantum tomography of States, measurements, and transformations. PHYSICAL REVIEW LETTERS 2009; 102:010404. [PMID: 19257173 DOI: 10.1103/physrevlett.102.010404] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2008] [Indexed: 05/27/2023]
Abstract
We present the first complete optimization of quantum tomography, for states, positive operator value measures, and various classes of transformations, for arbitrary prior ensemble and arbitrary representation, giving corresponding feasible experimental schemes in terms of random Bell measurements.
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Affiliation(s)
- A Bisio
- Quit group, Dipartimento di Fisica A. Volta, and INFN Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy
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Buscemi F. Channel correction via quantum erasure. PHYSICAL REVIEW LETTERS 2007; 99:180501. [PMID: 17995387 DOI: 10.1103/physrevlett.99.180501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2006] [Indexed: 05/25/2023]
Abstract
By exploiting a generalization of recent results on environment-assisted channel correction, we show that, whenever a quantum system undergoes a channel realized as an interaction with a probe, the more efficiently the information about the input state can be erased from the probe, the higher the corresponding entanglement fidelity of the corrected channel, and vice-versa. The present analysis also applies to channels for which perfect quantum erasure is impossible, thus extending the original quantum eraser arrangement, and naturally embodies a general information-disturbance trade-off.
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Affiliation(s)
- Francesco Buscemi
- ERATO-SORST Quantum Computation and Information Project, Japan Science and Technology Agency, Daini Hongo White Building 201, 5-28-3 Hongo, Tokyo 113-0033, Japan.
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D'Ariano GM, Perinotti P. Optimal data processing for quantum measurements. PHYSICAL REVIEW LETTERS 2007; 98:020403. [PMID: 17358587 DOI: 10.1103/physrevlett.98.020403] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2006] [Indexed: 05/14/2023]
Abstract
We consider the general measurement scenario in which the ensemble average of an operator is determined via suitable data processing of the outcomes of a quantum measurement described by a positive operator-valued measure. We determine the optimal processing that minimizes the statistical error of the estimation.
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Affiliation(s)
- G M D'Ariano
- QUIT Group, Dipartimento di Fisica, Pavia, Italy
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