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Regula B, Lami L. Reversibility of quantum resources through probabilistic protocols. Nat Commun 2024; 15:3096. [PMID: 38632233 PMCID: PMC11024169 DOI: 10.1038/s41467-024-47243-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/12/2023] [Accepted: 03/26/2024] [Indexed: 04/19/2024] Open
Abstract
Among the most fundamental questions in the manipulation of quantum resources such as entanglement is the possibility of reversibly transforming all resource states. The key consequence of this would be the identification of a unique entropic resource measure that exactly quantifies the limits of achievable transformation rates. Remarkably, previous results claimed that such asymptotic reversibility holds true in very general settings; however, recently those findings have been found to be incomplete, casting doubt on the conjecture. Here we show that it is indeed possible to reversibly interconvert all states in general quantum resource theories, as long as one allows protocols that may only succeed probabilistically. Although such transformations have some chance of failure, we show that their success probability can be ensured to be bounded away from zero, even in the asymptotic limit of infinitely many manipulated copies. As in previously conjectured approaches, the achievability here is realised through operations that are asymptotically resource non-generating, and we show that this choice is optimal: smaller sets of transformations cannot lead to reversibility. Our methods are based on connecting the transformation rates under probabilistic protocols with strong converse rates for deterministic transformations, which we strengthen into an exact equivalence in the case of entanglement distillation.
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Affiliation(s)
- Bartosz Regula
- Mathematical Quantum Information RIKEN Hakubi Research Team, RIKEN Cluster for Pioneering Research (CPR) and RIKEN Center for Quantum Computing (RQC), Wako, Saitama, 351-0198, Japan.
| | - Ludovico Lami
- QuSoft, Science Park 123, Amsterdam, 1098 XG, The Netherlands.
- Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 105-107, Amsterdam, 1098 XG, The Netherlands.
- Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Amsterdam, 1098 XH, The Netherlands.
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Yuan X, Regula B, Takagi R, Gu M. Virtual Quantum Resource Distillation. Phys Rev Lett 2024; 132:050203. [PMID: 38364147 DOI: 10.1103/physrevlett.132.050203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2023] [Revised: 09/26/2023] [Accepted: 11/27/2023] [Indexed: 02/18/2024]
Abstract
Distillation, or purification, is central to the practical use of quantum resources in noisy settings often encountered in quantum communication and computation. Conventionally, distillation requires using some restricted "free" operations to convert a noisy state into one that approximates a desired pure state. Here, we propose to relax this setting by only requiring the approximation of the measurement statistics of a target pure state, which allows for additional classical postprocessing of the measurement outcomes. We show that this extended scenario, which we call "virtual resource distillation," provides considerable advantages over standard notions of distillation, allowing for the purification of noisy states from which no resources can be distilled conventionally. We show that general states can be virtually distilled with a cost (measurement overhead) that is inversely proportional to the amount of existing resource, and we develop methods to efficiently estimate such cost via convex and semidefinite programming, giving several computable bounds. We consider applications to coherence, entanglement, and magic distillation, and an explicit example in quantum teleportation (distributed quantum computing). This work opens a new avenue for investigating generalized ways to manipulate quantum resources.
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Affiliation(s)
- Xiao Yuan
- Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China
- School of Computer Science, Peking University, Beijing 100871, China
| | - Bartosz Regula
- Mathematical Quantum Information RIKEN Hakubi Research Team, RIKEN Cluster for Pioneering Research (CPR) and RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan
- Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Ryuji Takagi
- Department of Basic Science, The University of Tokyo, Tokyo 153-8902, Japan
- Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371, Singapore
| | - Mile Gu
- Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371, Singapore
- Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore
- CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore 117543, Singapore
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Regula B. Probabilistic Transformations of Quantum Resources. Phys Rev Lett 2022; 128:110505. [PMID: 35363021 DOI: 10.1103/physrevlett.128.110505] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2021] [Accepted: 02/23/2022] [Indexed: 06/14/2023]
Abstract
The difficulty in manipulating quantum resources deterministically often necessitates the use of probabilistic protocols, but the characterization of their capabilities and limitations has been lacking. We develop a general approach to this problem by introducing a new resource monotone that obeys a very strong type of monotonicity: it can rule out all transformations, probabilistic or deterministic, between states in any quantum resource theory. This allows us to place fundamental limitations on state transformations and restrict the advantages that probabilistic protocols can provide over deterministic ones, significantly strengthening previous findings and extending recent no-go theorems. We apply our results to obtain a substantial improvement in bounds for the errors and overheads of probabilistic distillation protocols, directly applicable to tasks such as entanglement or magic state distillation, and computable through convex optimization. In broad classes of resources, we strengthen our results to show that the monotone completely governs probabilistic transformations-it serves as a necessary and sufficient condition for state convertibility. This endows the monotone with a direct operational interpretation, as it can exactly quantify the highest fidelity achievable in resource distillation tasks by means of any probabilistic manipulation protocol.
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Affiliation(s)
- Bartosz Regula
- Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
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Tan KC, Narasimhachar V, Regula B. Fisher Information Universally Identifies Quantum Resources. Phys Rev Lett 2021; 127:200402. [PMID: 34860070 DOI: 10.1103/physrevlett.127.200402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2021] [Revised: 07/31/2021] [Accepted: 09/28/2021] [Indexed: 06/13/2023]
Abstract
We show that both the classical as well as the quantum definitions of the Fisher information faithfully identify resourceful quantum states in general quantum resource theories, in the sense that they can always distinguish between states with and without a given resource. This shows that all quantum resources confer an advantage in metrology, and establishes the Fisher information as a universal tool to probe the resourcefulness of quantum states. We provide bounds on the extent of this advantage, as well as a simple criterion to test whether different resources are useful for the estimation of unitarily encoded parameters. Finally, we extend the results to show that the Fisher information is also able to identify the dynamical resourcefulness of quantum operations.
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Affiliation(s)
- Kok Chuan Tan
- School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Republic of Singapore
| | - Varun Narasimhachar
- School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Republic of Singapore
| | - Bartosz Regula
- School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Republic of Singapore
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Regula B, Takagi R. One-Shot Manipulation of Dynamical Quantum Resources. Phys Rev Lett 2021; 127:060402. [PMID: 34420334 DOI: 10.1103/physrevlett.127.060402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2021] [Accepted: 06/21/2021] [Indexed: 06/13/2023]
Abstract
We develop a unified framework to characterize one-shot transformations of dynamical quantum resources in terms of resource quantifiers, establishing universal conditions for exact and approximate transformations in general resource theories. Our framework encompasses all dynamical resources represented as quantum channels, including those with a specific structure-such as boxes, assemblages, and measurements-thus immediately applying in a vast range of physical settings. For the particularly important manipulation tasks of distillation and dilution, we show that our conditions become necessary and sufficient for broad classes of important theories, enabling an exact characterization of these tasks and establishing a precise connection between operational problems and resource monotones based on entropic divergences. We exemplify our results by considering explicit applications to quantum communication, where we obtain exact expressions for one-shot quantum capacity and simulation cost assisted by no-signaling, separability-preserving, and positive partial transpose-preserving codes; as well as to nonlocality, contextuality, and measurement incompatibility, where we present operational applications of a number of relevant resource measures.
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Affiliation(s)
- Bartosz Regula
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
| | - Ryuji Takagi
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
- Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Regula B, Lami L, Ferrari G, Takagi R. Operational Quantification of Continuous-Variable Quantum Resources. Phys Rev Lett 2021; 126:110403. [PMID: 33798371 DOI: 10.1103/physrevlett.126.110403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2020] [Accepted: 02/08/2021] [Indexed: 06/12/2023]
Abstract
The diverse range of resources which underlie the utility of quantum states in practical tasks motivates the development of universally applicable methods to measure and compare resources of different types. However, many of such approaches were hitherto limited to the finite-dimensional setting or were not connected with operational tasks. We overcome this by introducing a general method of quantifying resources for continuous-variable quantum systems based on the robustness measure, applicable to a plethora of physically relevant resources such as optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence. We demonstrate in particular that the measure has a direct operational interpretation as the advantage enabled by a given state in a class of channel discrimination tasks. We show that the robustness constitutes a well-behaved, bona fide resource quantifier in any convex resource theory, contrary to a related negativity-based measure known as the standard robustness. Furthermore, we show the robustness to be directly observable-it can be computed as the expectation value of a single witness operator-and establish general methods for evaluating the measure. Explicitly applying our results to the relevant resources, we demonstrate the exact computability of the robustness for several classes of states.
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Affiliation(s)
- Bartosz Regula
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
| | - Ludovico Lami
- Institut für Theoretische Physik und IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
| | - Giovanni Ferrari
- Institut für Theoretische Physik und IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
- Dipartimento di Fisica e Astronomia Galileo Galilei, Università degli studi di Padova, via Marzolo 8, 35131 Padova, Italy
| | - Ryuji Takagi
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
- Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Lami L, Regula B, Adesso G. Generic Bound Coherence under Strictly Incoherent Operations. Phys Rev Lett 2019; 122:150402. [PMID: 31050502 DOI: 10.1103/physrevlett.122.150402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2018] [Revised: 01/25/2019] [Indexed: 06/09/2023]
Abstract
We compute analytically the maximal rates of distillation of quantum coherence under strictly incoherent operations (SIO) and physically incoherent operations (PIO), showing that they coincide for all states, and providing a complete description of the phenomenon of bound coherence. In particular, we establish a simple, analytically computable necessary and sufficient criterion for the asymptotic distillability under SIO and PIO. We use this result to show that almost every quantum state is undistillable-only pure states as well as states whose density matrix contains a rank-one submatrix allow for coherence distillation under SIO or PIO, while every other quantum state exhibits bound coherence. This demonstrates the fundamental operational limitations of SIO and PIO in the resource theory of quantum coherence. We show that the fidelity of distillation of a single bit of coherence under SIO can be efficiently computed as a semidefinite program, and investigate the generalization of this result to provide an understanding of asymptotically achievable distillation fidelity.
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Affiliation(s)
- Ludovico Lami
- 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, United Kingdom
| | - Bartosz Regula
- 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, United Kingdom
| | - 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, United Kingdom
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Takagi R, Regula B, Bu K, Liu ZW, Adesso G. Operational Advantage of Quantum Resources in Subchannel Discrimination. Phys Rev Lett 2019; 122:140402. [PMID: 31050492 DOI: 10.1103/physrevlett.122.140402] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2018] [Revised: 01/15/2019] [Indexed: 06/09/2023]
Abstract
One of the central problems in the study of quantum resource theories is to provide a given resource with an operational meaning, characterizing physical tasks in which the resource can give an explicit advantage over all resourceless states. We show that this can always be accomplished for all convex resource theories. We establish in particular that any resource state enables an advantage in a channel discrimination task, allowing for a strictly greater success probability than any state without the given resource. Furthermore, we find that the generalized robustness measure serves as an exact quantifier for the maximal advantage enabled by the given resource state in a class of subchannel discrimination problems, providing a universal operational interpretation to this fundamental resource quantifier. We also consider a wider range of subchannel discrimination tasks and show that the generalized robustness still serves as the operational advantage quantifier for several well-known theories such as entanglement, coherence, and magic.
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Affiliation(s)
- Ryuji Takagi
- Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Bartosz Regula
- 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, United Kingdom
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
- Complexity Institute, Nanyang Technological University, 637335, Singapore
| | - Kaifeng Bu
- School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Zi-Wen Liu
- Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - 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, United Kingdom
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Fang K, Wang X, Lami L, Regula B, Adesso G. Probabilistic Distillation of Quantum Coherence. Phys Rev Lett 2018; 121:070404. [PMID: 30169098 DOI: 10.1103/physrevlett.121.070404] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2018] [Revised: 07/17/2018] [Indexed: 06/08/2023]
Abstract
The ability to distill quantum coherence is pivotal for optimizing the performance of quantum technologies; however, such a task cannot always be accomplished with certainty. Here we develop a general framework of probabilistic distillation of quantum coherence in a one-shot setting, establishing fundamental limitations for different classes of free operations. We first provide a geometric interpretation for the maximal success probability, showing that under maximally incoherent operations (MIO) and dephasing-covariant incoherent operations (DIO) the problem can be simplified into efficiently computable semidefinite programs. Exploiting these results, we find that DIO and its subset of strictly incoherent operations have equal power in the probabilistic distillation of coherence from pure input states, while MIO are strictly stronger. We then prove a fundamental no-go result: Distilling coherence from any full-rank state is impossible even probabilistically. We further find that in some conditions the maximal success probability can vanish suddenly beyond a certain threshold in the distillation fidelity. Finally, we consider probabilistic coherence distillation assisted by a catalyst and demonstrate, with specific examples, its superiority to the unassisted and deterministic cases.
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Affiliation(s)
- Kun Fang
- Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, New South Wales 2007, Australia
| | - Xin Wang
- Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, New South Wales 2007, Australia
| | - Ludovico Lami
- 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, United Kingdom
| | - Bartosz Regula
- 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, United Kingdom
| | - 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, United Kingdom
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Abstract
We characterize the distillation of quantum coherence in the one-shot setting, that is, the conversion of general quantum states into maximally coherent states under different classes of quantum operations. We show that the maximally incoherent operations (MIO) and the dephasing-covariant incoherent operations (DIO) have the same power in the task of one-shot coherence distillation. We establish that the one-shot distillable coherence under MIO and DIO is efficiently computable with a semidefinite program, which we show to correspond to a quantum hypothesis testing problem. Further, we introduce a family of coherence monotones generalizing the robustness of coherence as well as the modified trace distance of coherence, and show that they admit an operational interpretation in characterizing the fidelity of distillation under different classes of operations. By providing an explicit formula for these quantities for pure states, we show that the one-shot distillable coherence under MIO, DIO, strictly incoherent operations, and incoherent operations is equal for all pure states.
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Affiliation(s)
- Bartosz Regula
- 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, United Kingdom
| | - Kun Fang
- Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology, University of Technology Sydney, New South Wales 2007, Australia
| | - Xin Wang
- Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology, University of Technology Sydney, New South Wales 2007, Australia
| | - 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, United Kingdom
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Regula B, Adesso G. Entanglement Quantification Made Easy: Polynomial Measures Invariant under Convex Decomposition. Phys Rev Lett 2016; 116:070504. [PMID: 26943522 DOI: 10.1103/physrevlett.116.070504] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2015] [Indexed: 06/05/2023]
Abstract
Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are available in only a few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-2 states obeying such a condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and we show that several representative classes of four-qubit pure states have marginals that enjoy this property.
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Affiliation(s)
- Bartosz Regula
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Gerardo Adesso
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
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Regula B, Di Martino S, Lee S, Adesso G. Erratum: Strong Monogamy Conjecture for Multiqubit Entanglement: The Four-Qubit Case [Phys. Rev. Lett. 113, 110501 (2014)]. Phys Rev Lett 2016; 116:049902. [PMID: 26871361 DOI: 10.1103/physrevlett.116.049902] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/15/2016] [Indexed: 06/05/2023]
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Regula B, Di Martino S, Lee S, Adesso G. Strong monogamy conjecture for multiqubit entanglement: the four-qubit case. Phys Rev Lett 2014; 113:110501. [PMID: 25259963 DOI: 10.1103/physrevlett.113.110501] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/05/2014] [Indexed: 06/03/2023]
Abstract
We investigate the distribution of bipartite and multipartite entanglement in multiqubit states. In particular, we define a set of monogamy inequalities sharpening the conventional Coffman-Kundu-Wootters constraints, and we provide analytical proofs of their validity for relevant classes of states. We present extensive numerical evidence validating the conjectured strong monogamy inequalities for arbitrary pure states of four qubits.
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Affiliation(s)
- Bartosz Regula
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Sara Di Martino
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom and Dipartimento di Matematica, Universitá di Bari, I-70125 Bari, Italy
| | - Soojoon Lee
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom and Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea
| | - Gerardo Adesso
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
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