1
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Gluza M, Eisert J. Recovering Quantum Correlations in Optical Lattices from Interaction Quenches. PHYSICAL REVIEW LETTERS 2021; 127:090503. [PMID: 34506183 DOI: 10.1103/physrevlett.127.090503] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2020] [Revised: 03/29/2021] [Accepted: 07/13/2021] [Indexed: 06/13/2023]
Abstract
Quantum simulations with ultracold atoms in optical lattices open up an exciting path toward understanding strongly interacting quantum systems. Atom gas microscopes are crucial for this as they offer single-site density resolution, unparalleled in other quantum many-body systems. However, currently a direct measurement of local coherent currents is out of reach. In this Letter, we show how to achieve that by measuring densities that are altered in response to quenches to noninteracting dynamics, e.g., after tilting the optical lattice. For this, we establish a data analysis method solving the closed set of equations relating tunneling currents and atom number dynamics, allowing us to reliably recover the full covariance matrix, including off-diagonal terms representing coherent currents. The signal processing builds upon semidefinite optimization, providing bona fide covariance matrices optimally matching the observed data. We demonstrate how the obtained information about noncommuting observables allows one to quantify entanglement at finite temperature, which opens up the possibility to study quantum correlations in quantum simulations going beyond classical capabilities.
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Affiliation(s)
- Marek Gluza
- 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
- Helmholtz-Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany
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2
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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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3
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Fadel M, Usui A, Huber M, Friis N, Vitagliano G. Entanglement Quantification in Atomic Ensembles. PHYSICAL REVIEW LETTERS 2021; 127:010401. [PMID: 34270307 DOI: 10.1103/physrevlett.127.010401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2021] [Accepted: 06/02/2021] [Indexed: 06/13/2023]
Abstract
Entanglement measures quantify nonclassical correlations present in a quantum system, but can be extremely difficult to calculate, even more so, when information on its state is limited. Here, we consider broad families of entanglement criteria that are based on variances of arbitrary operators and analytically derive the lower bounds these criteria provide for two relevant entanglement measures: the best separable approximation and the generalized robustness. This yields a practical method for quantifying entanglement in realistic experimental situations, in particular, when only few measurements of simple observables are available. As a concrete application of this method, we quantify bipartite and multipartite entanglement in spin-squeezed Bose-Einstein condensates of ∼500 atoms, by lower bounding the best separable approximation and the generalized robustness only from measurements of first and second moments of the collective spin operator.
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Affiliation(s)
- Matteo Fadel
- Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Ayaka Usui
- Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, 904-0495 Okinawa, Japan
| | - Marcus Huber
- Atominstitut, Technische Universität Wien, 1020 Vienna, Austria
- Institute for Quantum Optics and Quantum Information-IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
| | - Nicolai Friis
- Atominstitut, Technische Universität Wien, 1020 Vienna, Austria
- Institute for Quantum Optics and Quantum Information-IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
| | - Giuseppe Vitagliano
- Institute for Quantum Optics and Quantum Information-IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
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4
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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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5
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Supremum of block entanglement for symmetric Gaussian states. Sci Rep 2018; 8:7394. [PMID: 29743518 PMCID: PMC5943598 DOI: 10.1038/s41598-018-25781-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2017] [Accepted: 04/25/2018] [Indexed: 11/12/2022] Open
Abstract
For a system composed of permutationally symmetric Gaussian modes, by identifying the boundary of valid states and making necessary change of variables, the existence and exact value of the supremum of logarithmic negativity (and negativity likewise) between any two blocks can be shown analytically. Involving only the total number of interchangeable modes and the sizes of respective blocks, this result is general and easy to be applied for such a class of states.
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6
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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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7
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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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8
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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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9
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Xin T, Lu D, Klassen J, Yu N, Ji Z, Chen J, Ma X, Long G, Zeng B, Laflamme R. Quantum State Tomography via Reduced Density Matrices. PHYSICAL REVIEW LETTERS 2017; 118:020401. [PMID: 28128623 DOI: 10.1103/physrevlett.118.020401] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2016] [Indexed: 06/06/2023]
Abstract
Quantum state tomography via local measurements is an efficient tool for characterizing quantum states. However, it requires that the original global state be uniquely determined (UD) by its local reduced density matrices (RDMs). In this work, we demonstrate for the first time a class of states that are UD by their RDMs under the assumption that the global state is pure, but fail to be UD in the absence of that assumption. This discovery allows us to classify quantum states according to their UD properties, with the requirement that each class be treated distinctly in the practice of simplifying quantum state tomography. Additionally, we experimentally test the feasibility and stability of performing quantum state tomography via the measurement of local RDMs for each class. These theoretical and experimental results demonstrate the advantages and possible pitfalls of quantum state tomography with local measurements.
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Affiliation(s)
- Tao Xin
- State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Dawei Lu
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Joel Klassen
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Department of Physics, University of Guelph, Guelph, N1G 2W1 Ontario, Canada
| | - Nengkun Yu
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Centre for Quantum Computation and Intelligent Systems, Faculty of Engineering and Information Technology, University of Technology Sydney, New South Wales 2007, Australia
- Department of Mathematics and Statistics, University of Guelph, Guelph, N1G 2W1 Ontario, Canada
| | - Zhengfeng Ji
- Centre for Quantum Computation and Intelligent Systems, Faculty of Engineering and Information Technology, University of Technology Sydney, New South Wales 2007, Australia
- State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, 100190 Beijing, China
| | - Jianxin Chen
- Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland, USA
| | - Xian Ma
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Guilu Long
- State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
| | - Bei Zeng
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Department of Mathematics and Statistics, University of Guelph, Guelph, N1G 2W1 Ontario, Canada
- Canadian Institute for Advanced Research, Toronto, M5G 1Z8 Ontario, Canada
| | - Raymond Laflamme
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Canadian Institute for Advanced Research, Toronto, M5G 1Z8 Ontario, Canada
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
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10
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Marty O, Cramer M, Plenio MB. Practical Entanglement Estimation for Spin-System Quantum Simulators. PHYSICAL REVIEW LETTERS 2016; 116:105301. [PMID: 27015489 DOI: 10.1103/physrevlett.116.105301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2015] [Indexed: 06/05/2023]
Abstract
We present practical methods to measure entanglement for quantum simulators that can be realized with trapped ions, cold atoms, and superconducting qubits. Focusing on long- and short-range Ising-type Hamiltonians, we introduce schemes that are applicable under realistic experimental conditions including mixedness due to, e.g., noise or temperature. In particular, we identify a single observable whose expectation value serves as a lower bound to entanglement and that may be obtained by a simple quantum circuit. As such circuits are not (yet) available for every platform, we investigate the performance of routinely measured observables as quantitative entanglement witnesses. Possible applications include experimental studies of entanglement scaling in critical systems and the reliable benchmarking of quantum simulators.
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Affiliation(s)
- O Marty
- Institut für Theoretische Physik & IQST, Albert-Einstein-Allee 11, Universität Ulm, Germany
| | - M Cramer
- Institut für Theoretische Physik & IQST, Albert-Einstein-Allee 11, Universität Ulm, Germany
| | - M B Plenio
- Institut für Theoretische Physik & IQST, Albert-Einstein-Allee 11, Universität Ulm, Germany
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11
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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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12
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Islam R, Ma R, Preiss PM, Tai ME, Lukin A, Rispoli M, Greiner M. Measuring entanglement entropy in a quantum many-body system. Nature 2016; 528:77-83. [PMID: 26632587 DOI: 10.1038/nature15750] [Citation(s) in RCA: 194] [Impact Index Per Article: 24.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2015] [Accepted: 09/16/2015] [Indexed: 12/28/2022]
Abstract
Entanglement is one of the most intriguing features of quantum mechanics. It describes non-local correlations between quantum objects, and is at the heart of quantum information sciences. Entanglement is now being studied in diverse fields ranging from condensed matter to quantum gravity. However, measuring entanglement remains a challenge. This is especially so in systems of interacting delocalized particles, for which a direct experimental measurement of spatial entanglement has been elusive. Here, we measure entanglement in such a system of itinerant particles using quantum interference of many-body twins. Making use of our single-site-resolved control of ultracold bosonic atoms in optical lattices, we prepare two identical copies of a many-body state and interfere them. This enables us to directly measure quantum purity, Rényi entanglement entropy, and mutual information. These experiments pave the way for using entanglement to characterize quantum phases and dynamics of strongly correlated many-body systems.
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Affiliation(s)
- Rajibul Islam
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Ruichao Ma
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Philipp M Preiss
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - M Eric Tai
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Alexander Lukin
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Matthew Rispoli
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Markus Greiner
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
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13
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Fukuhara T, Hild S, Zeiher J, Schauß P, Bloch I, Endres M, Gross C. Spatially Resolved Detection of a Spin-Entanglement Wave in a Bose-Hubbard Chain. PHYSICAL REVIEW LETTERS 2015; 115:035302. [PMID: 26230800 DOI: 10.1103/physrevlett.115.035302] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/10/2015] [Indexed: 06/04/2023]
Abstract
Entanglement is an essential property of quantum many-body systems. However, its local detection is challenging and was so far limited to spin degrees of freedom in ion chains. Here we measure entanglement between the spins of atoms located on two lattice sites in a one-dimensional Bose-Hubbard chain which features both local spin- and particle-number fluctuations. Starting with an initially localized spin impurity, we observe an outwards propagating entanglement wave and show quantitatively how entanglement in the spin sector rapidly decreases with increasing particle-number fluctuations in the chain.
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Affiliation(s)
- Takeshi Fukuhara
- Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
- RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
| | - Sebastian Hild
- Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
| | - Johannes Zeiher
- Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
| | - Peter Schauß
- Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
| | - Immanuel Bloch
- Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
- Ludwig-Maximilians-Universität, Fakultät für Physik, 80799 München, Germany
| | - Manuel Endres
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Christian Gross
- Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
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Tóth G, Moroder T, Gühne O. Evaluating convex roof entanglement measures. PHYSICAL REVIEW LETTERS 2015; 114:160501. [PMID: 25955038 DOI: 10.1103/physrevlett.114.160501] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2014] [Indexed: 06/04/2023]
Abstract
We show a powerful method to compute entanglement measures based on convex roof constructions. In particular, our method is applicable to measures that, for pure states, can be written as low order polynomials of operator expectation values. We show how to compute the linear entropy of entanglement, the linear entanglement of assistance, and a bound on the dimension of the entanglement for bipartite systems. We discuss how to obtain the convex roof of the three-tangle for three-qubit states. We also show how to calculate the linear entropy of entanglement and the quantum Fisher information based on partial information or device independent information. We demonstrate the usefulness of our method by concrete examples.
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Affiliation(s)
- Géza Tóth
- Department of Theoretical Physics, University of the Basque Country UPV/EHU, E-48080 Bilbao, Spain
- IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain
- Wigner Research Centre for Physics, Hungarian Academy of Sciences, H-1525 Budapest, Hungary
| | - Tobias Moroder
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany
| | - Otfried Gühne
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany
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