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Piccolini M, Nosrati F, Adesso G, Morandotti R, Lo Franco R. Generating indistinguishability within identical particle systems: spatial deformations as quantum resource activators. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220104. [PMID: 37517442 DOI: 10.1098/rsta.2022.0104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2022] [Accepted: 08/05/2022] [Indexed: 08/01/2023]
Abstract
Identical quantum subsystems can possess a property which does not have any classical counterpart: indistinguishability. As a long-debated phenomenon, identical particles' indistinguishability has been shown to be at the heart of various fundamental physical results. When concerned with the spatial degree of freedom, identical constituents can be made indistinguishable by overlapping their spatial wave functions via appropriately defined spatial deformations. By the laws of quantum mechanics, any measurement designed to resolve a quantity which depends on the spatial degree of freedom only and performed on the regions of overlap is not able to assign the measured outcome to one specific particle within the system. The result is an entangled state where the measured property is shared between the identical constituents. In this work, we present a coherent formalization of the concept of deformation in a general [Formula: see text]-particle scenario, together with a suitable measure of the degree of indistinguishability. We highlight the basic differences with non-identical particles scenarios and discuss the inherent role of spatial deformations as entanglement activators within the spatially localized operations and classical communication operational framework. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.
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Affiliation(s)
- Matteo Piccolini
- Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, Palermo 90128, Italy
- INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, Québec, Canada J3X 1S2
| | - Farzam Nosrati
- Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, Palermo 90128, Italy
- INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, Québec, Canada J3X 1S2
| | - Gerardo Adesso
- School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham NG7 2RD, UK
| | - Roberto Morandotti
- INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, Québec, Canada J3X 1S2
| | - Rosario Lo Franco
- Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, Palermo 90128, Italy
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2
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Li X, Zhu J, Shi X, Wang Z, Chen X, Zhang X, Chen Y. Steric Hindrance On-Off Mass-Tagged Probe Set Enables Detection of Protein Homodimer in Living Cells. ACS APPLIED MATERIALS & INTERFACES 2022; 14:54517-54526. [PMID: 36449938 DOI: 10.1021/acsami.2c15010] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
The major challenge in the detection of protein homodimers is that the identical monomers in a homodimer are indistinguishable using most conventional methods and cannot be sequentially recognized. In this study, a steric hindrance on-off mass-tagged probe set strategy was developed for the quantification of HER2 homodimer in living cells. The probe set contained a hindrance probe and a detection probe. The hindrance probe had a DNA dendrimer as a hindrance group to achieve the steric hindrance on-off function and thus the assignment of monomer identity. The detection probe contained a mass tag released for mass spectrometric quantification. Using the steric hindrance on-off mass-tagged probe set, the level of HER2 homodimer in various breast cancer cell lines was quantified. This is the first report to determine the quantity of protein homodimers, and the steric hindrance on-off probe set developed herein can facilitate the illustration of protein function in cancer.
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Affiliation(s)
- Xiaoxu Li
- School of Pharmacy, Nanjing Medical University, Nanjing 211166, China
| | - Jianhua Zhu
- School of Pharmacy, Nanjing Medical University, Nanjing 211166, China
| | - Xiaoyu Shi
- School of Pharmacy, Nanjing Medical University, Nanjing 211166, China
| | - Zhongcheng Wang
- School of Pharmacy, Nanjing Medical University, Nanjing 211166, China
| | - Xi Chen
- School of Pharmacy, Nanjing Medical University, Nanjing 211166, China
| | - Xian Zhang
- School of Pharmacy, Nanjing Medical University, Nanjing 211166, China
| | - Yun Chen
- School of Pharmacy, Nanjing Medical University, Nanjing 211166, China
- State Key Laboratory of Reproductive Medicine, Nanjing Medical University, Nanjing 210029, China
- Key Laboratory of Cardiovascular & Cerebrovascular Medicine, Nanjing Medical University, Nanjing 211166, China
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3
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Lee D, Pramanik T, Hong S, Cho YW, Lim HT, Chin S, Kim YS. Entangling three identical particles via spatial overlap. OPTICS EXPRESS 2022; 30:30525-30535. [PMID: 36242154 DOI: 10.1364/oe.460866] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2022] [Accepted: 07/22/2022] [Indexed: 06/16/2023]
Abstract
Quantum correlations between identical particles are at the heart of quantum technologies. Several studies with two identical particles have shown that the spatial overlap and indistinguishability between the particles are necessary for generating bipartite entanglement. On the other hand, researches on the extension to more than two-particle systems are limited by the practical difficulty to control multiple identical particles in laboratories. In this work, we propose schemes to generate two fundamental classes of genuine tripartite entanglement, i.e., GHZ and W classes, which are experimentally demonstrated using linear optics with three identical photons. We also show that the tripartite entanglement class decays from the genuine entanglement to the full separability as the particles become more distinguishable from each other. Our results support the prediction that particle indistinguishability is a fundamental element for entangling identical particles.
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Locality and entanglement of indistinguishable particles. Sci Rep 2021; 11:15478. [PMID: 34326438 PMCID: PMC8322085 DOI: 10.1038/s41598-021-94991-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2021] [Accepted: 07/13/2021] [Indexed: 02/07/2023] Open
Abstract
Entanglement is one of the strongest quantum correlation, and is a key ingredient in fundamental aspects of quantum mechanics and a resource for quantum technologies. While entanglement theory is well settled for distinguishable particles, there are five inequivalent approaches to entanglement of indistinguishable particles. We analyse the different definitions of indistinguishable particle entanglement in the light of the locality notion. This notion is specified by two steps: (i) the identification of subsystems by means of their local operators; (ii) the requirement that entanglement represent correlations between the above subsets of operators. We prove that three of the aforementioned five entanglement definitions are incompatible with any locality notion defined as above.
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5
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Piccolini M, Nosrati F, Compagno G, Livreri P, Morandotti R, Lo Franco R. Entanglement Robustness via Spatial Deformation of Identical Particle Wave Functions. ENTROPY (BASEL, SWITZERLAND) 2021; 23:708. [PMID: 34204915 PMCID: PMC8227133 DOI: 10.3390/e23060708] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/20/2021] [Revised: 05/26/2021] [Accepted: 06/01/2021] [Indexed: 11/30/2022]
Abstract
We address the problem of entanglement protection against surrounding noise by a procedure suitably exploiting spatial indistinguishability of identical subsystems. To this purpose, we take two initially separated and entangled identical qubits interacting with two independent noisy environments. Three typical models of environments are considered: amplitude damping channel, phase damping channel and depolarizing channel. After the interaction, we deform the wave functions of the two qubits to make them spatially overlap before performing spatially localized operations and classical communication (sLOCC) and eventually computing the entanglement of the resulting state. This way, we show that spatial indistinguishability of identical qubits can be utilized within the sLOCC operational framework to partially recover the quantum correlations spoiled by the environment. A general behavior emerges: the higher the spatial indistinguishability achieved via deformation, the larger the amount of recovered entanglement.
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Affiliation(s)
- Matteo Piccolini
- Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy; (M.P.); (F.N.); (P.L.)
- INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, QC J3X 1S2, Canada;
| | - Farzam Nosrati
- Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy; (M.P.); (F.N.); (P.L.)
- INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, QC J3X 1S2, Canada;
| | - Giuseppe Compagno
- Dipartimento di Fisica e Chimica—Emilio Segrè, Università di Palermo, via Archirafi 36, 90123 Palermo, Italy;
| | - Patrizia Livreri
- Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy; (M.P.); (F.N.); (P.L.)
| | | | - Rosario Lo Franco
- Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy; (M.P.); (F.N.); (P.L.)
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Mixing indistinguishable systems leads to a quantum Gibbs paradox. Nat Commun 2021; 12:1471. [PMID: 33674586 PMCID: PMC7935879 DOI: 10.1038/s41467-021-21620-7] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2020] [Accepted: 01/27/2021] [Indexed: 11/08/2022] Open
Abstract
The classical Gibbs paradox concerns the entropy change upon mixing two gases. Whether an observer assigns an entropy increase to the process depends on their ability to distinguish the gases. A resolution is that an "ignorant" observer, who cannot distinguish the gases, has no way of extracting work by mixing them. Moving the thought experiment into the quantum realm, we reveal new and surprising behaviour: the ignorant observer can extract work from mixing different gases, even if the gases cannot be directly distinguished. Moreover, in the macroscopic limit, the quantum case diverges from the classical ideal gas: as much work can be extracted as if the gases were fully distinguishable. We show that the ignorant observer assigns more microstates to the system than found by naive counting in semiclassical statistical mechanics. This demonstrates the importance of accounting for the level of knowledge of an observer, and its implications for genuinely quantum modifications to thermodynamics.
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Barros MR, Chin S, Pramanik T, Lim HT, Cho YW, Huh J, Kim YS. Entangling bosons through particle indistinguishability and spatial overlap. OPTICS EXPRESS 2020; 28:38083-38092. [PMID: 33379628 DOI: 10.1364/oe.410361] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2020] [Accepted: 11/05/2020] [Indexed: 06/12/2023]
Abstract
Particle identity and entanglement are two fundamental quantum properties that work as major resources for various quantum information tasks. However, it is still a challenging problem to understand the correlation of the two properties in the same system. While recent theoretical studies have shown that the spatial overlap between identical particles is necessary for nontrivial entanglement, the exact role of particle indistinguishability in the entanglement of identical particles has never been analyzed quantitatively before. Here, we theoretically and experimentally investigate the behavior of entanglement between two bosons as spatial overlap and indistinguishability simultaneously vary. The theoretical computation of entanglement for generic two bosons with pseudospins is verified experimentally in a photonic system. Our results show that the amount of entanglement is a monotonically increasing function of both quantities. We expect that our work provides an insight into deciphering the role of the entanglement in quantum networks that consist of identical particles.
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Sun K, Wang Y, Liu ZH, Xu XY, Xu JS, Li CF, Guo GC, Castellini A, Nosrati F, Compagno G, Lo Franco R. Experimental quantum entanglement and teleportation by tuning remote spatial indistinguishability of independent photons. OPTICS LETTERS 2020; 45:6410-6413. [PMID: 33258824 DOI: 10.1364/ol.401735] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2020] [Accepted: 10/13/2020] [Indexed: 06/12/2023]
Abstract
Quantitative control of spatial indistinguishability of identical subsystems as a direct quantum resource at distant sites has not yet been experimentally proven. We design a setup capable of tuning remote spatial indistinguishability of two independent photons by individually adjusting their spatial distribution in two distant regions, leading to polarization entanglement from uncorrelated photons. This is achieved by spatially localized operations and classical communication on photons that meet only at the detectors. The amount of entanglement depends uniquely on the degree of spatial indistinguishability, quantified by an entropic measure I, which enables teleportation with fidelities above the classical threshold. The results open the way to viable indistinguishability-enhanced quantum information processing.
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9
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Becher JH, Sindici E, Klemt R, Jochim S, Daley AJ, Preiss PM. Measurement of Identical Particle Entanglement and the Influence of Antisymmetrization. PHYSICAL REVIEW LETTERS 2020; 125:180402. [PMID: 33196275 DOI: 10.1103/physrevlett.125.180402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2020] [Accepted: 09/25/2020] [Indexed: 06/11/2023]
Abstract
We explore the relationship between symmetrization and entanglement through measurements on few-particle systems in a multiwell potential. In particular, considering two or three trapped atoms, we measure and distinguish correlations arising from two different physical origins: antisymmetrization of the fermionic wave function and interaction between particles. We quantify this through the entanglement negativity of states, and the introduction of an antisymmetric negativity, which allows us to understand the role that symmetrization plays in the measured entanglement properties. We apply this concept both to pure theoretical states and to experimentally reconstructed density matrices of two or three mobile particles in an array of optical tweezers.
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Affiliation(s)
- J H Becher
- Physics Institute, Heidelberg University, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
| | - E Sindici
- Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
| | - R Klemt
- Physics Institute, Heidelberg University, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
| | - S Jochim
- Physics Institute, Heidelberg University, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
| | - A J Daley
- Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
| | - P M Preiss
- Physics Institute, Heidelberg University, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
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10
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11
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Blasiak P, Markiewicz M. Entangling three qubits without ever touching. Sci Rep 2019; 9:20131. [PMID: 31882584 PMCID: PMC6934615 DOI: 10.1038/s41598-019-55137-3] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2019] [Accepted: 10/30/2019] [Indexed: 11/09/2022] Open
Abstract
All identical particles are inherently correlated from the outset, regardless of how far apart their creation took place. In this paper, this fact is used for extraction of entanglement from independent particles unaffected by any interactions. Specifically, we are concerned with operational schemes for generation of all tripartite entangled states, essentially the GHZ state and the W state, which prevent the particles from touching one another over the entire evolution. The protocols discussed in the paper require only three particles in linear optical setups with equal efficiency for boson, fermion or anyon statistics. Within this framework indistinguishability of particles presents itself as a useful resource of entanglement accessible for practical applications.
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Affiliation(s)
- Pawel Blasiak
- Institute of Nuclear Physics Polish Academy of Sciences, PL-31342, Kraków, Poland.
- City, University of London, London, EC1V OHB, UK.
| | - Marcin Markiewicz
- Institute of Physics, Jagiellonian University, PL-30348, Kraków, Poland
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12
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Entropic nonclassicality and quantum non-Gaussianity tests via beam splitting. Sci Rep 2019; 9:17835. [PMID: 31780692 PMCID: PMC6882878 DOI: 10.1038/s41598-019-54110-4] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2019] [Accepted: 11/08/2019] [Indexed: 11/21/2022] Open
Abstract
We propose entropic nonclassicality criteria for quantum states of light that can be readily tested using homodyne detection with beam splitting operation. Our method draws on the fact that the entropy of quadrature distributions for a classical state is non-increasing under an arbitrary loss channel. We show that our test is strictly stronger than the variance-based squeezing condition and that it can also be extended to detect quantum non-Gaussianity in conjunction with phase randomization. Furthermore, we address how our criteria can be used to identify single-mode resource states to generate two-mode states demonstrating EPR paradox, i.e., quantum steering, via beam-splitter setting.
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13
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Makhalov V, Satoor T, Evrard A, Chalopin T, Lopes R, Nascimbene S. Probing Quantum Criticality and Symmetry Breaking at the Microscopic Level. PHYSICAL REVIEW LETTERS 2019; 123:120601. [PMID: 31633983 DOI: 10.1103/physrevlett.123.120601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2019] [Indexed: 06/10/2023]
Abstract
We report on an experimental study of the Lipkin-Meshkov-Glick model of quantum spins interacting at infinite range in a transverse magnetic field, which exhibits a ferromagnetic phase transition in the thermodynamic limit. We use dysprosium atoms of electronic spin J=8, subjected to a quadratic Zeeman light shift, to simulate 2J=16 interacting spins 1/2. We probe the system microscopically using single magnetic sublevel resolution, giving access to the spin projection parity, which is the collective observable characterizing the underlying Z_{2} symmetry. We measure the thermodynamic properties and dynamical response of the system, and we study the quantum critical behavior around the transition point. In the ferromagnetic phase, we achieve coherent tunneling between symmetry-broken states, and we test the link between symmetry breaking and the appearance of a finite order parameter.
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Affiliation(s)
- Vasiliy Makhalov
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Tanish Satoor
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Alexandre Evrard
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Thomas Chalopin
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Raphael Lopes
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - Sylvain Nascimbene
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
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14
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Reid MD, He QY. Quantifying the Mesoscopic Nature of Einstein-Podolsky-Rosen Nonlocality. PHYSICAL REVIEW LETTERS 2019; 123:120402. [PMID: 31633988 DOI: 10.1103/physrevlett.123.120402] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/05/2018] [Revised: 05/21/2019] [Indexed: 06/10/2023]
Abstract
Evidence for Bell's nonlocality is so far mainly restricted to microscopic systems, where the elements of reality that are negated predetermine results of measurements to within one spin unit. Any observed nonlocal effect (or lack of classical predetermination) is then limited to no more than the difference of a single photon or electron being detected or not (at a given detector). In this paper, we analyze experiments that report the Einstein-Podolsky-Rosen (EPR) steering form of nonlocality for mesoscopic photonic or Bose-Einstein condensate systems. Using an EPR steering parameter, we show how the EPR nonlocalities involved can be quantified for four-mode states, to give evidence of EPR-nonlocal effects corresponding to a two-mode number difference of 10^{5} photons, or of several tens of atoms (at a given site). Applying to experiments, we also show how the variance criterion of Duan, Giedke, Cirac and Zoller for EPR entanglement can be used to determine a lower bound on the number of particles in a pure two-mode EPR-entangled or steerable state.
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Affiliation(s)
- M D Reid
- Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne, Victoria 3122, Australia
- Institute of Theoretical Atomic, Molecular and Optical Physics (ITAMP), Harvard University, Cambridge, Massachusetts 02138, USA
| | - Q Y He
- State Key Laboratory of Mesoscopic Physics, School of Physics, Peking University, Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
- Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
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Compagno G, Castellini A, Lo Franco R. Dealing with indistinguishable particles and their entanglement. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:rsta.2017.0317. [PMID: 29807896 DOI: 10.1098/rsta.2017.0317] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 04/16/2018] [Indexed: 06/08/2023]
Abstract
Here, we discuss a particle-based approach to deal with systems of many identical quantum objects (particles) that never employs labels to mark them. We show that it avoids both methodological problems and drawbacks in the study of quantum correlations associated with the standard quantum mechanical treatment of identical particles. The core of this approach is represented by the multiparticle probability amplitude, whose structure in terms of single-particle amplitudes we derive here by first principles. To characterize entanglement among the identical particles, this new method uses the same notions, such as partial trace, adopted for non-identical ones. We highlight the connection between our approach and second quantization. We also define spin-exchanged multipartite states which contain a generalization of W states to identical particles. We prove that particle spatial overlap plays a role in the distributed entanglement within multipartite systems and is responsible for the appearance of non-local quantum correlations.This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.
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Affiliation(s)
- Giuseppe Compagno
- Dipartimento di Fisica e Chimica, Università di Palermo, Via Archirafi 36, 90123 Palermo, Italy
| | - Alessia Castellini
- Dipartimento di Fisica e Chimica, Università di Palermo, Via Archirafi 36, 90123 Palermo, Italy
- INFN Sezione di Catania, Catania, Italy
| | - Rosario Lo Franco
- Dipartimento di Fisica e Chimica, Università di Palermo, Via Archirafi 36, 90123 Palermo, Italy
- Dipartimento di Energia, Ingegneria dell'Informazione e Modelli Matematici, Università di Palermo, Viale delle Scienze, Edificio 9, 90128 Palermo, Italy
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16
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Lo Franco R, Compagno G. Indistinguishability of Elementary Systems as a Resource for Quantum Information Processing. PHYSICAL REVIEW LETTERS 2018; 120:240403. [PMID: 29957003 DOI: 10.1103/physrevlett.120.240403] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2017] [Revised: 03/14/2018] [Indexed: 06/08/2023]
Abstract
Typical elements of quantum networks are made by identical systems, which are the basic particles constituting a resource for quantum information processing. Whether the indistinguishability due to particle identity is an exploitable quantum resource remains an open issue. Here we study independently prepared identical particles showing that, when they spatially overlap, an operational entanglement exists that can be made manifest by means of separated localized measurements. We prove this entanglement is physical in that it can be directly exploited to activate quantum information protocols, such as teleportation. These results establish that particle indistinguishability is a utilizable quantum feature and open the way to new quantum-enhanced applications.
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Affiliation(s)
- Rosario Lo Franco
- Dipartimento di Energia, Ingegneria dell'Informazione e Modelli Matematici, Università di Palermo, Viale delle Scienze, Edificio 9, 90128 Palermo, Italy
- Dipartimento di Fisica e Chimica, Università di Palermo, via Archirafi 36, 90123 Palermo, Italy
| | - Giuseppe Compagno
- Dipartimento di Fisica e Chimica, Università di Palermo, via Archirafi 36, 90123 Palermo, Italy
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17
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Lange K, Peise J, Lücke B, Kruse I, Vitagliano G, Apellaniz I, Kleinmann M, Tóth G, Klempt C. Entanglement between two spatially separated atomic modes. Science 2018; 360:416-418. [PMID: 29700263 DOI: 10.1126/science.aao2035] [Citation(s) in RCA: 35] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2017] [Accepted: 03/22/2018] [Indexed: 02/15/2024]
Abstract
Modern quantum technologies in the fields of quantum computing, quantum simulation, and quantum metrology require the creation and control of large ensembles of entangled particles. In ultracold ensembles of neutral atoms, nonclassical states have been generated with mutual entanglement among thousands of particles. The entanglement generation relies on the fundamental particle-exchange symmetry in ensembles of identical particles, which lacks the standard notion of entanglement between clearly definable subsystems. Here, we present the generation of entanglement between two spatially separated clouds by splitting an ensemble of ultracold identical particles prepared in a twin Fock state. Because the clouds can be addressed individually, our experiments open a path to exploit the available entangled states of indistinguishable particles for quantum information applications.
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Affiliation(s)
- Karsten Lange
- Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
| | - Jan Peise
- Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
| | - Bernd Lücke
- Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
| | - Ilka Kruse
- Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
| | - Giuseppe Vitagliano
- Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
- Department of Theoretical Physics, University of the Basque Country UPV/EHU, Post Office Box 644, E-48080 Bilbao, Spain
| | - Iagoba Apellaniz
- Department of Theoretical Physics, University of the Basque Country UPV/EHU, Post Office Box 644, E-48080 Bilbao, Spain
| | - Matthias Kleinmann
- Department of Theoretical Physics, University of the Basque Country UPV/EHU, Post Office Box 644, E-48080 Bilbao, Spain
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany
| | - Géza Tóth
- Department of Theoretical Physics, University of the Basque Country UPV/EHU, Post Office Box 644, E-48080 Bilbao, Spain
- Ikerbasque, Basque Foundation for Science, E-48013 Bilbao, Spain
- Wigner Research Centre for Physics, Hungarian Academy of Sciences, Post Office Box 49, H-1525 Budapest, Hungary
| | - Carsten Klempt
- Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany.
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Fadel M, Zibold T, Décamps B, Treutlein P. Spatial entanglement patterns and Einstein-Podolsky-Rosen steering in Bose-Einstein condensates. Science 2018; 360:409-413. [DOI: 10.1126/science.aao1850] [Citation(s) in RCA: 139] [Impact Index Per Article: 23.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2017] [Accepted: 03/13/2018] [Indexed: 11/03/2022]
Abstract
Many-particle entanglement is a fundamental concept of quantum physics that still presents conceptual challenges. Although nonclassical states of atomic ensembles were used to enhance measurement precision in quantum metrology, the notion of entanglement in these systems was debated because the correlations among the indistinguishable atoms were witnessed by collective measurements only. Here, we use high-resolution imaging to directly measure the spin correlations between spatially separated parts of a spin-squeezed Bose-Einstein condensate. We observe entanglement that is strong enough for Einstein-Podolsky-Rosen steering: We can predict measurement outcomes for noncommuting observables in one spatial region on the basis of corresponding measurements in another region with an inferred uncertainty product below the Heisenberg uncertainty bound. This method could be exploited for entanglement-enhanced imaging of electromagnetic field distributions and quantum information tasks.
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Affiliation(s)
- Matteo Fadel
- Department of Physics and Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Tilman Zibold
- Department of Physics and Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Boris Décamps
- Department of Physics and Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Philipp Treutlein
- Department of Physics and Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
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19
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Kunkel P, Prüfer M, Strobel H, Linnemann D, Frölian A, Gasenzer T, Gärttner M, Oberthaler MK. Spatially distributed multipartite entanglement enables EPR steering of atomic clouds. Science 2018; 360:413-416. [DOI: 10.1126/science.aao2254] [Citation(s) in RCA: 125] [Impact Index Per Article: 20.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2017] [Accepted: 02/16/2018] [Indexed: 11/02/2022]
Abstract
A key resource for distributed quantum-enhanced protocols is entanglement between spatially separated modes. However, the robust generation and detection of entanglement between spatially separated regions of an ultracold atomic system remain a challenge. We used spin mixing in a tightly confined Bose-Einstein condensate to generate an entangled state of indistinguishable particles in a single spatial mode. We show experimentally that this entanglement can be spatially distributed by self-similar expansion of the atomic cloud. We used spatially resolved spin read-out to reveal a particularly strong form of quantum correlations known as Einstein-Podolsky-Rosen (EPR) steering between distinct parts of the expanded cloud. Based on the strength of EPR steering, we constructed a witness, which confirmed genuine 5-partite entanglement.
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Affiliation(s)
- Philipp Kunkel
- Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Maximilian Prüfer
- Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Helmut Strobel
- Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Daniel Linnemann
- Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Anika Frölian
- Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Thomas Gasenzer
- Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Martin Gärttner
- Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Markus K. Oberthaler
- Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
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20
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Wasak T, Smerzi A, Chwedeńczuk J. Role of Particle Entanglement in the Violation of Bell Inequalities. Sci Rep 2018; 8:1777. [PMID: 29379056 PMCID: PMC5788934 DOI: 10.1038/s41598-018-20034-8] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2017] [Accepted: 01/08/2018] [Indexed: 11/25/2022] Open
Abstract
Entanglement between two separate systems is a necessary resource to violate a Bell inequality in a test of local realism. We demonstrate that to overcome the Bell bound, this correlation must be accompanied by the entanglement between the constituent particles. This happens whenever a super-selection rule prohibits coherences between states with different total number of particles and thus imposes a constraint on feasible local operations in each sub-system. We show that the necessary entanglement between the particles might solely result from their indistinguishability. We also give an example of both mode and particle-entangled pure state, which does not violate any Bell inequality. Our result reveals a fundamental relation between the non-locality and the particle entanglement.
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Affiliation(s)
- Tomasz Wasak
- Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093, Warszawa, Poland
| | - Augusto Smerzi
- QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125, Firenze, Italy
| | - Jan Chwedeńczuk
- Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093, Warszawa, Poland.
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21
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Universality of Schmidt decomposition and particle identity. Sci Rep 2017; 7:44675. [PMID: 28333163 PMCID: PMC5363071 DOI: 10.1038/srep44675] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2016] [Accepted: 02/03/2017] [Indexed: 11/25/2022] Open
Abstract
Schmidt decomposition is a widely employed tool of quantum theory which plays a key role for distinguishable particles in scenarios such as entanglement characterization, theory of measurement and state purification. Yet, its formulation for identical particles remains controversial, jeopardizing its application to analyze general many-body quantum systems. Here we prove, using a newly developed approach, a universal Schmidt decomposition which allows faithful quantification of the physical entanglement due to the identity of particles. We find that it is affected by single-particle measurement localization and state overlap. We study paradigmatic two-particle systems where identical qubits and qutrits are located in the same place or in separated places. For the case of two qutrits in the same place, we show that their entanglement behavior, whose physical interpretation is given, differs from that obtained before by different methods. Our results are generalizable to multiparticle systems and open the way for further developments in quantum information processing exploiting particle identity as a resource.
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22
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Krenn M, Hochrainer A, Lahiri M, Zeilinger A. Entanglement by Path Identity. PHYSICAL REVIEW LETTERS 2017; 118:080401. [PMID: 28282180 DOI: 10.1103/physrevlett.118.080401] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2016] [Indexed: 05/11/2023]
Abstract
Quantum entanglement is one of the most prominent features of quantum mechanics and forms the basis of quantum information technologies. Here we present a novel method for the creation of quantum entanglement in multipartite and high-dimensional systems. The two ingredients are (i) superposition of photon pairs with different origins and (ii) aligning photons such that their paths are identical. We explain the experimentally feasible creation of various classes of multiphoton entanglement encoded in polarization as well as in high-dimensional Hilbert spaces-starting only from nonentangled photon pairs. For two photons, arbitrary high-dimensional entanglement can be created. The idea of generating entanglement by path identity could also apply to quantum entities other than photons. We discovered the technique by analyzing the output of a computer algorithm. This shows that computer designed quantum experiments can be inspirations for new techniques.
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Affiliation(s)
- Mario Krenn
- Vienna Center for Quantum Science & Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria and Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
| | - Armin Hochrainer
- Vienna Center for Quantum Science & Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria and Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
| | - Mayukh Lahiri
- Vienna Center for Quantum Science & Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria and Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
| | - Anton Zeilinger
- Vienna Center for Quantum Science & Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria and Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
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23
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Luo XY, Zou YQ, Wu LN, Liu Q, Han MF, Tey MK, You L. Deterministic entanglement generation from driving through quantum phase transitions. Science 2017; 355:620-623. [DOI: 10.1126/science.aag1106] [Citation(s) in RCA: 143] [Impact Index Per Article: 20.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2016] [Accepted: 01/12/2017] [Indexed: 11/02/2022]
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24
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Chang Y, González-Tudela A, Sánchez Muñoz C, Navarrete-Benlloch C, Shi T. Deterministic Down-Converter and Continuous Photon-Pair Source within the Bad-Cavity Limit. PHYSICAL REVIEW LETTERS 2016; 117:203602. [PMID: 27886465 DOI: 10.1103/physrevlett.117.203602] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/02/2015] [Indexed: 06/06/2023]
Abstract
The development, characterization, and control of N-photon sources are instrumental for quantum technological applications. This work constitutes a step forward in this direction, where we propose a cavity quantum electrodynamics setup designed for the generation of photon pairs. We identify both the regime where our system works as a deterministic down-converter of a single input photon and as an optimal two-photon source under weak continuous driving. We use both the scattering and master equation formalisms to characterize the system, and from their connection naturally arises a physical criterion characterizing when weakly driven systems behave as continuous antibunched two-photon sources. We also show that the outgoing photons share nontrivial quantum correlations in general. We provide a specific implementation based on state-of-the-art superconducting circuits, showing how our proposal is within the reach of current technologies. As an outlook, we show the proposal can be extended to achieve deterministic conversion of a single photon into N photons.
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Affiliation(s)
- Yue Chang
- Max-Planck Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
| | | | - Carlos Sánchez Muñoz
- Max-Planck Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
- Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, 28049 Madrid, Spain
| | - Carlos Navarrete-Benlloch
- Max-Planck Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
- Institute for Theoretical Physics II, Universität Erlangen-Nürnberg, Staudtstrasse 7, 91058 Erlangen, Germany
| | - Tao Shi
- Max-Planck Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
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25
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Nonclassicality Invariant of General Two-Mode Gaussian States. Sci Rep 2016; 6:26523. [PMID: 27210547 PMCID: PMC4876421 DOI: 10.1038/srep26523] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2016] [Accepted: 05/05/2016] [Indexed: 11/18/2022] Open
Abstract
We introduce a new quantity for describing nonclassicality of an arbitrary optical two-mode Gaussian state which remains invariant under any global photon-number preserving unitary transformation of the covariance matrix of the state. The invariant naturally splits into an entanglement monotone and local-nonclassicality quantifiers applied to the reduced states. This shows how entanglement can be converted into local squeezing and vice versa. Twin beams and their transformations at a beam splitter are analyzed as an example providing squeezed light. An extension of this approach to pure three-mode Gaussian states is given.
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26
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Killoran N, Steinhoff FES, Plenio MB. Converting Nonclassicality into Entanglement. PHYSICAL REVIEW LETTERS 2016; 116:080402. [PMID: 26967398 DOI: 10.1103/physrevlett.116.080402] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2015] [Indexed: 06/05/2023]
Abstract
Quantum mechanics exhibits a wide range of nonclassical features, of which entanglement in multipartite systems takes a central place. In several specific settings, it is well known that nonclassicality (e.g., squeezing, spin squeezing, coherence) can be converted into entanglement. In this work, we present a general framework, based on superposition, for structurally connecting and converting nonclassicality to entanglement. In addition to capturing the previously known results, this framework also allows us to uncover new entanglement convertibility theorems in two broad scenarios, one which is discrete and one which is continuous. In the discrete setting, the classical states can be any finite linearly independent set. For the continuous setting, the pertinent classical states are "symmetric coherent states," connected with symmetric representations of the group SU(K). These results generalize and link convertibility properties from the resource theory of coherence, spin coherent states, and optical coherent states, while also revealing important connections between local and nonlocal pictures of nonclassicality.
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Affiliation(s)
- N Killoran
- Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm D-89069 Ulm, Germany
| | - F E S Steinhoff
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Strasse 3, D-57068 Siegen, Germany
- Instituto de Física, Universidade Federal de Goiás, 74001-970 Goiânia, Goiás, Brazil
| | - M B Plenio
- Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm D-89069 Ulm, Germany
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27
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Quantum entanglement of identical particles by standard information-theoretic notions. Sci Rep 2016; 6:20603. [PMID: 26857475 PMCID: PMC4746635 DOI: 10.1038/srep20603] [Citation(s) in RCA: 76] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2015] [Accepted: 01/07/2016] [Indexed: 11/23/2022] Open
Abstract
Quantum entanglement of identical particles is essential in quantum information theory. Yet, its correct determination remains an open issue hindering the general understanding and exploitation of many-particle systems. Operator-based methods have been developed that attempt to overcome the issue. Here we introduce a state-based method which, as second quantization, does not label identical particles and presents conceptual and technical advances compared to the previous ones. It establishes the quantitative role played by arbitrary wave function overlaps, local measurements and particle nature (bosons or fermions) in assessing entanglement by notions commonly used in quantum information theory for distinguishable particles, like partial trace. Our approach furthermore shows that bringing identical particles into the same spatial location functions as an entangling gate, providing fundamental theoretical support to recent experimental observations with ultracold atoms. These results pave the way to set and interpret experiments for utilizing quantum correlations in realistic scenarios where overlap of particles can count, as in Bose-Einstein condensates, quantum dots and biological molecular aggregates.
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28
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Shahandeh F, Sperling J, Vogel W. Structural quantification of entanglement. PHYSICAL REVIEW LETTERS 2014; 113:260502. [PMID: 25615292 DOI: 10.1103/physrevlett.113.260502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2014] [Indexed: 06/04/2023]
Abstract
We introduce an approach which allows a full structural and quantitative analysis of multipartite entanglement. The sets of states with different structures are convex and nested. Hence, they can be distinguished from each other using appropriate measurable witnesses. We derive equations for the construction of optimal witnesses and discuss general properties arising from our approach. As an example, we formulate witnesses for a 4-cluster state and perform a full quantitative analysis of the entanglement structure in the presence of noise and losses. The strength of the method in multimode continuous variable systems is also demonstrated for a dephased Greenberger-Horne-Zeilinger-type state.
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Affiliation(s)
- F Shahandeh
- Arbeitsgruppe Theoretische Quantenoptik, Institut für Physik, Universität Rostock, D-18055 Rostock, Germany
| | - J Sperling
- Arbeitsgruppe Theoretische Quantenoptik, Institut für Physik, Universität Rostock, D-18055 Rostock, Germany
| | - W Vogel
- Arbeitsgruppe Theoretische Quantenoptik, Institut für Physik, Universität Rostock, D-18055 Rostock, Germany
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29
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Herdman CM, Inglis S, Roy PN, Melko RG, Del Maestro A. Path-integral Monte Carlo method for Rényi entanglement entropies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:013308. [PMID: 25122411 DOI: 10.1103/physreve.90.013308] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2014] [Indexed: 06/03/2023]
Abstract
We introduce a quantum Monte Carlo algorithm to measure the Rényi entanglement entropies in systems of interacting bosons in the continuum. This approach is based on a path-integral ground state method that can be applied to interacting itinerant bosons in any spatial dimension with direct relevance to experimental systems of quantum fluids. We demonstrate how it may be used to compute spatial mode entanglement, particle partitioned entanglement, and the entanglement of particles, providing insights into quantum correlations generated by fluctuations, indistinguishability, and interactions. We present proof-of-principle calculations and benchmark against an exactly soluble model of interacting bosons in one spatial dimension. As this algorithm retains the fundamental polynomial scaling of quantum Monte Carlo when applied to sign-problem-free models, future applications should allow for the study of entanglement entropy in large-scale many-body systems of interacting bosons.
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Affiliation(s)
- C M Herdman
- Department of Physics, University of Vermont, Burlington, Vermont 05405, USA
| | - Stephen Inglis
- Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, and Center for Nanoscience, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 Munich, Germany and Department of Physics and Astronomy, University of Waterloo, Ontario, Canada N2L 3G1
| | - P-N Roy
- Department of Chemistry, University of Waterloo, Ontario, Canada N2L 3G1
| | - R G Melko
- Department of Physics and Astronomy, University of Waterloo, Ontario, Canada N2L 3G1 and Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5
| | - A Del Maestro
- Department of Physics, University of Vermont, Burlington, Vermont 05405, USA and Vermont Complex Systems Center, University of Vermont, Burlington, Vermont 05405, USA
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