1
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Kuroiwa K, Takagi R, Adesso G, Yamasaki H. Every Quantum Helps: Operational Advantage of Quantum Resources beyond Convexity. PHYSICAL REVIEW LETTERS 2024; 132:150201. [PMID: 38682983 DOI: 10.1103/physrevlett.132.150201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/02/2023] [Accepted: 02/05/2024] [Indexed: 05/01/2024]
Abstract
Identifying what quantum-mechanical properties are useful to untap a superior performance in quantum technologies is a pivotal question. Quantum resource theories provide a unified framework to analyze and understand such properties, as successfully demonstrated for entanglement and coherence. While these are examples of convex resources, for which quantum advantages can always be identified, many physical resources are described by a nonconvex set of free states and their interpretation has so far remained elusive. Here we address the fundamental question of the usefulness of quantum resources without convexity assumption, by providing two operational interpretations of the generalized robustness measure in general resource theories. First, we characterize the generalized robustness in terms of a nonlinear resource witness and reveal that any state is more advantageous than a free one in some multicopy channel discrimination task. Next, we consider a scenario where a theory is characterized by multiple constraints and show that the generalized robustness coincides with the worst-case advantage in a single-copy channel discrimination setting. Based on these characterizations, we conclude that every quantum resource state shows a qualitative and quantitative advantage in discrimination problems in a general resource theory even without any specification on the structure of the free states.
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Affiliation(s)
- Kohdai Kuroiwa
- Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo, Ontario N2L 3G1, Canada
- Perimeter Institute for Theoretical Physics, Ontario N2L 2Y5, Canada
| | - Ryuji Takagi
- Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
| | - Gerardo Adesso
- School of Mathematical Sciences and Centre for the Mathematical and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Hayata Yamasaki
- Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
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2
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Stratton B, Hsieh CY, Skrzypczyk P. Dynamical Resource Theory of Informational Nonequilibrium Preservability. PHYSICAL REVIEW LETTERS 2024; 132:110202. [PMID: 38563949 DOI: 10.1103/physrevlett.132.110202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2023] [Accepted: 01/30/2024] [Indexed: 04/04/2024]
Abstract
Information is instrumental in our understanding of thermodynamics. Their interplay has been studied through completely degenerate Hamiltonians whereby the informational contributions to thermodynamic transformations can be isolated. In this setting, all states other than the maximally mixed state are considered to be in informational nonequilibrium. An important yet still open question is how to characterize the ability of quantum dynamics to preserve informational nonequilibrium. Here, the dynamical resource theory of informational nonequilibrium preservability is introduced to begin providing an answer to this question. A characterization of the allowed operations is given for qubit channels and the n-dimensional Weyl-covariant channels-a physically relevant subset of the general channels. An operational interpretation of a state discrimination game with Bell state measurements is given. Finally, an explicit link between a channel's classical capacity and its ability to preserve informational nonequilibrium is made.
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Affiliation(s)
- Benjamin Stratton
- Quantum Engineering Centre for Doctoral Training, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol BS8 1FD, United Kingdom
- H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
| | - Chung-Yun Hsieh
- H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
| | - Paul Skrzypczyk
- H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
- CIFAR Azrieli Global Scholars Program, CIFAR, Toronto, Canada
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3
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Tendick L, Kampermann H, Bruß D. Distributed Quantum Incompatibility. PHYSICAL REVIEW LETTERS 2023; 131:120202. [PMID: 37802938 DOI: 10.1103/physrevlett.131.120202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2023] [Revised: 07/03/2023] [Accepted: 08/23/2023] [Indexed: 10/08/2023]
Abstract
Incompatible, i.e., nonjointly measurable quantum measurements are a necessary resource for many information processing tasks. It is known that increasing the number of distinct measurements usually enhances the incompatibility of a measurement scheme. However, it is generally unclear how large this enhancement is and on what it depends. Here, we show that the incompatibility which is gained via additional measurements is upper and lower bounded by certain functions of the incompatibility of subsets of the available measurements. We prove the tightness of some of our bounds by providing explicit examples based on mutually unbiased bases. Finally, we discuss the consequences of our results for the nonlocality that can be gained by enlarging the number of measurements in a Bell experiment.
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Affiliation(s)
- Lucas Tendick
- Institute for Theoretical Physics III, Heinrich Heine University Düsseldorf, D-40225 Düsseldorf, Germany
| | - Hermann Kampermann
- Institute for Theoretical Physics III, Heinrich Heine University Düsseldorf, D-40225 Düsseldorf, Germany
| | - Dagmar Bruß
- Institute for Theoretical Physics III, Heinrich Heine University Düsseldorf, D-40225 Düsseldorf, Germany
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4
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Datta C, Ganardi R, Kondra TV, Streltsov A. Is There a Finite Complete Set of Monotones in Any Quantum Resource Theory? PHYSICAL REVIEW LETTERS 2023; 130:240204. [PMID: 37390426 DOI: 10.1103/physrevlett.130.240204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2022] [Accepted: 05/22/2023] [Indexed: 07/02/2023]
Abstract
Entanglement quantification aims to assess the value of quantum states for quantum information processing tasks. A closely related problem is state convertibility, asking whether two remote parties can convert a shared quantum state into another one without exchanging quantum particles. Here, we explore this connection for quantum entanglement and for general quantum resource theories. For any quantum resource theory which contains resource-free pure states, we show that there does not exist a finite set of resource monotones which completely determines all state transformations. We discuss how these limitations can be surpassed, if discontinuous or infinite sets of monotones are considered, or by using quantum catalysis. We also discuss the structure of theories which are described by a single resource monotone and show equivalence with totally ordered resource theories. These are theories where a free transformation exists for any pair of quantum states. We show that totally ordered theories allow for free transformations between all pure states. For single-qubit systems, we provide a full characterization of state transformations for any totally ordered resource theory.
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Affiliation(s)
- Chandan Datta
- Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
| | - Ray Ganardi
- Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
| | - Tulja Varun Kondra
- Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
| | - Alexander Streltsov
- Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
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5
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Kurdziałek S, Demkowicz-Dobrzański R. Measurement Noise Susceptibility in Quantum Estimation. PHYSICAL REVIEW LETTERS 2023; 130:160802. [PMID: 37154663 DOI: 10.1103/physrevlett.130.160802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2022] [Revised: 12/02/2022] [Accepted: 03/30/2023] [Indexed: 05/10/2023]
Abstract
Fisher information is a key notion in the whole field of quantum metrology. It allows for a direct quantification of the maximal achievable precision of the estimation of the parameters encoded in quantum states using the most general quantum measurement. It fails, however, to quantify the robustness of quantum estimation schemes against measurement imperfections, which are always present in any practical implementations. Here, we introduce a new concept of Fisher information measurement noise susceptibility that quantifies the potential loss of Fisher information due to small measurement disturbance. We derive an explicit formula for the quantity, and demonstrate its usefulness in the analysis of paradigmatic quantum estimation schemes, including interferometry and superresolution optical imaging.
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6
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Ahnefeld F, Theurer T, Egloff D, Matera JM, Plenio MB. Coherence as a Resource for Shor's Algorithm. PHYSICAL REVIEW LETTERS 2022; 129:120501. [PMID: 36179183 DOI: 10.1103/physrevlett.129.120501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2022] [Revised: 07/10/2022] [Accepted: 08/01/2022] [Indexed: 06/16/2023]
Abstract
Shor's factoring algorithm provides a superpolynomial speedup over all known classical factoring algorithms. Here, we address the question of which quantum properties fuel this advantage. We investigate a sequential variant of Shor's algorithm with a fixed overall structure and identify the role of coherence for this algorithm quantitatively. We analyze this protocol in the framework of dynamical resource theories, which capture the resource character of operations that can create and detect coherence. This allows us to derive a lower and an upper bound on the success probability of the protocol, which depend on rigorously defined measures of coherence as a dynamical resource. We compare these bounds with the classical limit of the protocol and conclude that within the fixed structure that we consider, coherence is the quantum resource that determines its performance by bounding the success probability from below and above. Therefore, we shine new light on the fundamental role of coherence in quantum computation.
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Affiliation(s)
- Felix Ahnefeld
- Institute of Theoretical Physics, Universität Ulm, Albert-Einstein-Allee 11, D-89081 Ulm, Germany
| | - Thomas Theurer
- Department of Mathematics and Statistics, Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada
| | - Dario Egloff
- Institute of Theoretical Physics, Technical University Dresden, D-01062 Dresden, Germany
| | - Juan Mauricio Matera
- IFLP-CONICET, Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 67, La Plata 1900, Argentina
| | - Martin B Plenio
- Institute of Theoretical Physics, Universität Ulm, Albert-Einstein-Allee 11, D-89081 Ulm, Germany
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Siudzińska K. Indecomposability of entanglement witnesses constructed from symmetric measurements. Sci Rep 2022; 12:10785. [PMID: 35750801 PMCID: PMC9232524 DOI: 10.1038/s41598-022-14920-5] [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: 02/10/2022] [Accepted: 06/15/2022] [Indexed: 12/04/2022] Open
Abstract
We propose a family of positive maps constructed from a recently introduced class of symmetric measurements. These maps are used to define entanglement witnesses, which include other popular approaches with mutually unbiased bases and mutually unbiased measurements. A particular interest is given to indecomposable witnesses that can be used to detect entanglement of quantum states with positive partial transposition. We present several examples for different number of measurements.
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Affiliation(s)
- Katarzyna Siudzińska
- Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Toruń, ul. Grudziądzka 5, 87-100, Toruń, Poland.
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Haapasalo E, Kraft T, Pellonpää JP, Uola R. Operational Characterization of Infinite-Dimensional Quantum Resources. PHYSICAL REVIEW LETTERS 2021; 127:250401. [PMID: 35029426 DOI: 10.1103/physrevlett.127.250401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2020] [Revised: 10/11/2021] [Accepted: 10/22/2021] [Indexed: 06/14/2023]
Abstract
Recently, various nonclassical properties of quantum states and channels have been characterized through an advantage they provide in quantum information tasks over their classical counterparts. Such advantage can be typically proven to be quantitative, in that larger amounts of quantum resources lead to better performance in the corresponding tasks. So far, these characterizations have been established only in the finite-dimensional setting, hence, leaving out central resources in continuous variable systems such as entanglement and nonclassicality of states as well as entanglement breaking and broadcasting channels. In this Letter, we present a fully general framework for resource quantification in infinite-dimensional systems. The framework is applicable to a wide range of resources with the only premises being that classical randomness cannot create a resource and that the resourceless objects form a closed set in an appropriate sense. As the latter may be hard to establish for the abstract topologies of continuous variable systems, we provide a relaxation of the condition with no reference to topology. This envelopes the aforementioned resources and various others, hence, giving them an interpretation as performance enhancement in so-called input-output games.
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Affiliation(s)
- Erkka Haapasalo
- Centre for Quantum Technologies, National University of Singapore, Science Drive 2 Block S15-03-18, Singapore 117543
| | - Tristan Kraft
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany
| | - Juha-Pekka Pellonpää
- Department of Physics and Astronomy, University of Turku, FI-20014 Turun yliopisto, Finland
| | - Roope Uola
- Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland
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Designolle S, Uola R, Luoma K, Brunner N. Set Coherence: Basis-Independent Quantification of Quantum Coherence. PHYSICAL REVIEW LETTERS 2021; 126:220404. [PMID: 34152163 DOI: 10.1103/physrevlett.126.220404] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2020] [Accepted: 05/04/2021] [Indexed: 06/13/2023]
Abstract
The coherence of an individual quantum state can be meaningfully discussed only when referring to a preferred basis. This arbitrariness can, however, be lifted when considering sets of quantum states. Here we introduce the concept of set coherence for characterizing the coherence of a set of quantum systems in a basis-independent way. We construct measures for quantifying set coherence of sets of quantum states as well as quantum measurements. These measures feature an operational meaning in terms of discrimination games and capture precisely the advantage offered by a given set over incoherent ones. Along the way, we also connect the notion of set coherence to various resource-theoretic approaches recently developed for quantum systems.
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Affiliation(s)
| | - Roope Uola
- Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland
| | - Kimmo Luoma
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
| | - Nicolas Brunner
- Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland
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10
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Regula B, Lami L, Ferrari G, Takagi R. Operational Quantification of Continuous-Variable Quantum Resources. PHYSICAL REVIEW LETTERS 2021; 126:110403. [PMID: 33798371 DOI: 10.1103/physrevlett.126.110403] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [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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Ducuara AF, Skrzypczyk P. Operational Interpretation of Weight-Based Resource Quantifiers in Convex Quantum Resource Theories. PHYSICAL REVIEW LETTERS 2020; 125:110401. [PMID: 32976009 DOI: 10.1103/physrevlett.125.110401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2019] [Revised: 04/26/2020] [Accepted: 07/21/2020] [Indexed: 06/11/2023]
Abstract
We introduce the resource quantifier of weight of resource for convex quantum resource theories of states and measurements with arbitrary resources. We show that it captures the advantage that a resourceful state (measurement) offers over all possible free states (measurements) in the operational task of exclusion of subchannels (states). Furthermore, we introduce information-theoretic quantities related to exclusion for quantum channels and find a connection between the weight of resource of a measurement and the exclusion-type information of quantum-to-classical channels. Our results apply to the resource theory of entanglement in which the weight of resource is known as the best-separable approximation or Lewenstein-Sanpera decomposition introduced in 1998. Consequently, the results found here provide an operational interpretation to this 21-year-old entanglement quantifier.
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Affiliation(s)
- Andrés F Ducuara
- Quantum Engineering Centre for Doctoral Training, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, Bristol BS8 1FD, United Kingdom
- Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, Bristol BS8 1FD, United Kingdom
| | - Paul Skrzypczyk
- H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
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Takagi R, Wang K, Hayashi M. Application of the Resource Theory of Channels to Communication Scenarios. PHYSICAL REVIEW LETTERS 2020; 124:120502. [PMID: 32281862 DOI: 10.1103/physrevlett.124.120502] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/07/2019] [Accepted: 02/19/2020] [Indexed: 06/11/2023]
Abstract
We introduce a resource theory of channels relevant to communication via quantum channels, in which the set of constant channels-useless channels for communication tasks-are considered as free resources. We find that our theory with such a simple structure is useful to address central problems in quantum Shannon theory-in particular, we provide a converse bound for the one-shot nonsignaling assisted classical capacity that naturally leads to its strong converse property, as well as obtain the one-shot channel simulation cost with nonsignaling assistance. We clarify an intimate connection between the nonsignaling assistance and our formalism by identifying the nonsignaling assisted channel coding with the channel transformation under the maximal set of resource nongenerating superchannels, providing a physical characterization of the latter. Our results provide new perspectives and concise arguments to those problems, connecting the recently developed fields of resource theories to "classic" settings in quantum information theory and shedding light on the validity of resource theories of channels as effective tools to address practical problems.
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Affiliation(s)
- Ryuji Takagi
- Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Kun Wang
- Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
- Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518000, China
| | - Masahito Hayashi
- Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
- Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518000, China
- Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
- Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117542, Singapore
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13
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Bae J, Chruściński D, Piani M. More Entanglement Implies Higher Performance in Channel Discrimination Tasks. PHYSICAL REVIEW LETTERS 2019; 122:140404. [PMID: 31050453 DOI: 10.1103/physrevlett.122.140404] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2018] [Indexed: 06/09/2023]
Abstract
We show that every entangled state provides an advantage in ancilla-assisted bi- and multichannel discrimination that singles out its degree of entanglement, quantified in terms of the Schmidt number. The Schmidt-number robustness provides a compelling quantification of such an advantage, and, remarkably, the well-known robustness of entanglement exactly provides the largest multiplicative advantage an entangled state can provide compared to the case where no ancilla is used in a channel discrimination task.
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Affiliation(s)
- Joonwoo Bae
- School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
| | - Dariusz Chruściński
- Faculty of Physics, Astronomy, and Informatics, Institute of Physics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland
| | - Marco Piani
- SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom
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Takagi R, Regula B, Bu K, Liu ZW, Adesso G. Operational Advantage of Quantum Resources in Subchannel Discrimination. PHYSICAL REVIEW LETTERS 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] [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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