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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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2
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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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3
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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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Regula B, Takagi R. One-Shot Manipulation of Dynamical Quantum Resources. PHYSICAL REVIEW LETTERS 2021; 127:060402. [PMID: 34420334 DOI: 10.1103/physrevlett.127.060402] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [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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Theurer T, Satyajit S, Plenio MB. Quantifying Dynamical Coherence with Dynamical Entanglement. PHYSICAL REVIEW LETTERS 2020; 125:130401. [PMID: 33034475 DOI: 10.1103/physrevlett.125.130401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2020] [Revised: 07/23/2020] [Accepted: 09/03/2020] [Indexed: 06/11/2023]
Abstract
Coherent superposition and entanglement are two fundamental aspects of nonclassicality. Here we provide a quantitative connection between the two on the level of operations by showing that the dynamical coherence of an operation upper bounds the dynamical entanglement that can be generated from it with the help of additional incoherent operations. In case a particular choice of monotones based on the relative entropy is used for the quantification of these dynamical resources, this bound can be achieved. In addition, we show that an analog to the entanglement potential exists on the level of operations and serves as a valid quantifier for dynamical coherence.
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Affiliation(s)
- Thomas Theurer
- Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
| | | | - Martin B Plenio
- Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
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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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Xu H, Xu F, Theurer T, Egloff D, Liu ZW, Yu N, Plenio MB, Zhang L. Experimental Quantification of Coherence of a Tunable Quantum Detector. PHYSICAL REVIEW LETTERS 2020; 125:060404. [PMID: 32845691 DOI: 10.1103/physrevlett.125.060404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2019] [Revised: 07/17/2020] [Accepted: 07/17/2020] [Indexed: 06/11/2023]
Abstract
Quantum coherence is a fundamental resource that quantum technologies exploit to achieve performance beyond that of classical devices. A necessary prerequisite to achieve this advantage is the ability of measurement devices to detect coherence from the measurement statistics. Based on a recently developed resource theory of quantum operations, here we quantify experimentally the ability of a typical quantum-optical detector, the weak-field homodyne detector, to detect coherence. We derive an improved algorithm for quantum detector tomography and apply it to reconstruct the positive-operator-valued measures of the detector in different configurations. The reconstructed positive-operator-valued measures are then employed to evaluate how well the detector can detect coherence using two computable measures. As the first experimental investigation of quantum measurements from a resource theoretical perspective, our work sheds new light on the rigorous evaluation of the performance of a quantum measurement apparatus.
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Affiliation(s)
- Huichao Xu
- National Laboratory of Solid State Microstructures, Key Laboratory of Intelligent Optical Sensing and Manipulation, College of Engineering and Applied Sciences and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Feixiang Xu
- National Laboratory of Solid State Microstructures, Key Laboratory of Intelligent Optical Sensing and Manipulation, College of Engineering and Applied Sciences and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Thomas Theurer
- Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
| | - Dario Egloff
- Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
- Technical University Dresden, Institute of Theoretical Physics, D-01062 Dresden, Germany
| | - Zi-Wen Liu
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Nengkun Yu
- Centre for Quantum Computation and Intelligent Systems, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, New South Wales 2007, Australia
| | - Martin B Plenio
- Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
| | - Lijian Zhang
- National Laboratory of Solid State Microstructures, Key Laboratory of Intelligent Optical Sensing and Manipulation, College of Engineering and Applied Sciences and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
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Fang K, Liu ZW. No-Go Theorems for Quantum Resource Purification. PHYSICAL REVIEW LETTERS 2020; 125:060405. [PMID: 32845667 DOI: 10.1103/physrevlett.125.060405] [Citation(s) in RCA: 16] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2019] [Revised: 02/26/2020] [Accepted: 06/30/2020] [Indexed: 06/11/2023]
Abstract
The manipulation of quantum "resources" such as entanglement, coherence, and magic states lies at the heart of quantum science and technology, empowering potential advantages over classical methods. In practice, a particularly important kind of manipulation is to "purify" the quantum resources since they are inevitably contaminated by noise and thus often lose their power or become unreliable for direct usage. Here we prove fundamental limitations on how effectively generic noisy resources can be purified enforced by the laws of quantum mechanics, which universally apply to any reasonable kind of quantum resource. More explicitly, we derive nontrivial lower bounds on the error of converting any full-rank noisy state to any target pure resource state by any free protocol (including probabilistic ones)-it is impossible to achieve perfect resource purification, even probabilistically. Our theorems indicate strong limits on the efficiency of distillation, a widely used type of resource purification routine that underpins many key applications of quantum information science. In particular, this general result induces the first explicit lower bounds on the resource cost of magic state distillation, a leading scheme for realizing scalable fault-tolerant quantum computation. Implications for the standard error-correction-based methods are specifically discussed.
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Affiliation(s)
- Kun Fang
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, United Kingdom
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Zi-Wen Liu
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
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Burniston J, Grabowecky M, Scandolo CM, Chiribella G, Gour G. Necessary and sufficient conditions on measurements of quantum channels. Proc Math Phys Eng Sci 2020; 476:20190832. [PMID: 32398940 PMCID: PMC7209144 DOI: 10.1098/rspa.2019.0832] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2019] [Accepted: 03/24/2020] [Indexed: 11/12/2022] Open
Abstract
Quantum supermaps are a higher-order genera- lization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive and trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap. We link this result to the fact that the principle of causality fails in the theory of quantum supermaps.
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Affiliation(s)
- John Burniston
- Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada
- Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta, Canada
| | - Michael Grabowecky
- Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada
- Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta, Canada
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada
| | - Carlo Maria Scandolo
- Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada
- Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta, Canada
| | - Giulio Chiribella
- Department of Computer Science, The University of Hong Kong, Hong Kong, People’s Republic of China
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Gilad Gour
- Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada
- Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta, Canada
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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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Gour G, Winter A. How to Quantify a Dynamical Quantum Resource. PHYSICAL REVIEW LETTERS 2019; 123:150401. [PMID: 31702325 DOI: 10.1103/physrevlett.123.150401] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2019] [Indexed: 06/10/2023]
Abstract
We show that the generalization of the relative entropy of a resource from states to channels is not unique, and there are at least six such generalizations. Then, we show that two of these generalizations are asymptotically continuous, satisfy a version of the asymptotic equipartition property, and their regularizations appear in the power exponent of channel versions of the quantum Stein's lemma. To obtain our results, we use a new type of "smoothing" that can be applied to functions of channels (with no state analog). We call it "liberal smoothing" as it allows for more spread in the optimization. Along the way, we show that the diamond norm can be expressed as a max relative entropy distance to the set of quantum channels, and prove a variety of properties of all six generalizations of the relative entropy of a resource.
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Affiliation(s)
- Gilad Gour
- Department of Mathematics and Statistics, Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta T2N 1N4, Canada
| | - Andreas Winter
- ICREA and Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
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