DOCSLIB.ORG

Distance Bounds on Quantum Dynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Distance Bounds on Quantum Dynamics PHYSICAL REVIEW A 78, 012308 ͑2008͒ Distance bounds on quantum dynamics Daniel A. Lidar Departments of Chemistry, Electrical Engineering, and Physics, Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA Paolo Zanardi Department of Physics, Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA Kaveh Khodjasteh Department of Physics, Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA and Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA ͑Received 29 March 2008; published 7 July 2008͒ We derive rigorous upper bounds on the distance between quantum states in an open-system setting in terms of the operator norm between Hamiltonians describing their evolution. We illustrate our results with an example taken from protection against decoherence using dynamical decoupling. DOI: 10.1103/PhysRevA.78.012308 PACS number͑s͒: 03.67.Lx, 03.67.Pp I. INTRODUCTION extensively ͑Sec. II͒. We mention the trace norm and opera- When an open quantum system undergoes a dynamical tor norm, and a norm that mixes them via a duality between evolution generated by a Hamiltonian, how far does its state the Schrödinger and Heisenberg pictures. We then briefly evolve from itself compared to the magnitude of the Hamil- review the accepted distance measures between states so as tonian? This is a fundamental question that is central to to quantify the meaning of an expression such as ʈ␳ ͑ ͒ ␳0ʈ͑ ͒͑ ͒ quantum-information science ͓1͔ and quantum control ͓2,3͔. S t − S t Sec. III . Next, we discuss how to compute the In order to make it more precise, suppose that the unitary effective Hamiltonian ⍀͑t͒ using the Magnus expansion or propagator of the evolution, U͑t͒, which is related to the Thompson’s theorem, and introduce a generalized effective ˙ i superoperator generator ͑Sec. IV͒. Since in many applica- Hamiltonian H͑t͒ via the Schrödinger equation U=−ប HU,is written in terms of an effective Hamiltonian ⍀͑t͒ as U͑t͒ tions one is interested not in the distance between states gen- =exp͓−it⍀͑t͒/ប͔. The effective Hamiltonian can in turn by erated by unitary, closed-system evolution, but instead in the calculated from the Hamiltonian H͑t͒ using a Dyson or Mag- distance between states of systems undergoing nonunitary, nus expansion. If the system S is “open,” it is coupled to a open-system dynamics, we prove an upper bound on the dis- bath B and its time-evolved state is given via the partial trace tance between such system-only states in terms of the dis- ␳ ͑ ͒ ͑ ͒␳͑ ͒ ͑ ͒† ͓ ͔ ϩ ͑ ͒ operation by S t =TrB U t 0 U t 4,5 . In the context of tance between the full “system bath” states Sec. V .We quantum information, the question we have stated pertains to then come to our main result: an upper bound on the distance the problem of “quantum memory”: i.e., what is between system states in terms of the distance between ͑ef- ʈ␳ ͑ ͒ ␳ ͑ ͒ʈ ʈ⍀͑ ͒ʈ ʈ ͑ ͒ʈ ͒ ϩ S t − S 0 compared to t or H t ? When the issue fective Hamiltonians describing the system bath dynamics is quantum computation or general quantum control, one is ͑Sec. VI͒. We find the intuitive result that the distance is interested in comparing two time-evolved states: the “ideal” upper bounded by a function depending on the spectral di- ␳0͑ ͒ ameter of the difference between the effective Hamiltonians. state S t , which is error free and is described by an “ideal” effective Hamiltonian ⍀0͑t͒͑e.g., for a quantum algorithm͒, We present a discussion of our result in terms of an example ␳ ͑ ͒ and the “actual” state S t , which undergoes the full noisy borrowed from decoherence control using dynamical decou- dynamics described by the total effective Hamiltonian ⍀͑t͒. pling ͑Sec. VII͒. We conclude in Sec. VIII with some open Then the question becomes the following: what is questions. ʈ␳ ͑ ͒ ␳0͑ ͒ʈ ʈ⍀͑ ͒ ⍀0͑ ͒ʈ ʈ ͑ ͒ 0͑ ͒ʈ S t − S t compared to t − t or H t −H t ? The memory question can, of course, be seen as a special case of the computation question. II. UNITARILY INVARIANT NORMS Here we prove bounds that answer these fundamental questions. Our bounds have immediate applications to prob- Unitarily invariant norms are norms satisfying, for all uni- ͓ ͔ lems in decoherence control ͓6͔ and fault-tolerant quantum tary U,V 8 , computation ͓7͔, as they quantify the sense in which a dis- ͑ ͒ tance between effective Hamiltonians describing the evolu- ʈUAVʈ = ʈAʈ . ͑1͒ tion should be made small, in order to guarantee a small ui ui distance between a desired and actual state. To begin, we first recall the definition and key properties We list some important examples. of so-called unitarily invariant norms, as we use such norms ͑i͒ The trace norm: 1050-2947/2008/78͑1͒/012308͑7͒ 012308-1 ©2008 The American Physical Society LIDAR, ZANARDI, AND KHODJASTEH PHYSICAL REVIEW A 78, 012308 ͑2008͒ ͑ ͒ ʈ⌫ ⌳⌫†ʈ ʈ ͑⌳␳͒ † †ʈ ʈAʈ ϵ Tr͉A͉ = ͚ s ͑A͒, ͑2͒ i.e., V is unitary ; then, 1 2 ϱ,1=sup␳ V2V1 V1V2 1 1 i ʈ⌳␳ʈ ʈ⌳ʈ i =sup␳ 1 = ϱ,1. Therefore the OT norm is also submul- tiplicative. However, note that unlike the case of the standard ͉ ͉ϵͱ † ͑ ͒ ͑ where A A A and si A are the singular values eigenval- operator norm, there is no simple expression for ʈ⌳ʈϱ in ͉ ͉͒ ,1 ues of A . terms of the singular values of ⌳. ͑ ͒ V denotes the Hilbert-Schmidt dual ء⌳ ii The operator norm: Let be an inner product space On the other hand, if ʈ ʈϵͱ͚ ͉ ͉2͗ ͘ equipped with the Euclidean norm x i xi ei ,ei , of ⌳, i.e., Vۋ͚ ෈V V ͕ ͖ ⌳ V where x= ixiei and =Sp ei . Let : . The opera- X͔͒ ∀ ␳,X, ͑12͒͑ء⌳tor norm is Tr͓⌳͑␳͒X͔ =Tr͓␳ ʈ⌳xʈ then one can prove the following identity ͓see, e.g., Sec. 2.4 ʈ⌳ʈ ϵ ͑⌳͒ ͑ ͒ ϱ sup = max si . 3 of Ref. ͓10͔ or Eq. ͑9͒ of Ref. ͓11͔͔: x෈V ʈxʈ i 13͒͑ .͒ء⌳͑ X͒ʈϱ = maxs͑ء⌳ϱ: = sup ʈءʈ⌳ʈϱ = ʈ⌳ʈ Therefore ʈ⌳xʈՅʈ⌳ʈϱʈxʈ. In our applications V=L͑H͒ is the ,1 i ʈXʈϱ=1 i space of all linear operators on the Hilbert space H, ⌳ is a superoperator, and x=␳ is a normalized quantum state: For example, for completely positive maps ͓1,4,5͔ ⌳␳ ʈ␳ʈ ␳ ͚ ␳ † ͑ ͕ ͖ ͒ 1 =Tr =1. = iAi Ai where Ai are the Kraus operators , it is easily ⌳ ␳ ͚ †␳ ͑ ͒ ⌳ iAi Ai. The duality 12 between and = ء iii͒ The Frobenius, or Hilbert-Schmidt, norm: verified that͑ is the duality between the Heisenberg and Schrödinger ء⌳ ʈ ʈ ϵ ͱ † ͑ ͒2 ͑ ͒ A 2 Tr A A = ͱ͚ si A . 4 pictures. i All unitarily invariant norms satisfy the important property III. DISTANCE AND FIDELITY MEASURES of submultiplicativity ͓9͔: Various measures are known that quantify the notion of ʈABʈ Յ ʈAʈ ʈBʈ . ͑5͒ ui ui ui distance and fidelity between states. For example, the dis- ␳ ␳ The norms of interest to us are also multiplicative over ten- tance measure between quantum states 1 and 2 is the trace sor products and obey an ordering ͓9͔ distance ʈA Bʈ = ʈAʈ ʈBʈ , i =1,2,ϱ, 1 i i i D͑␳ ,␳ ͒ϵ ʈ␳ − ␳ ʈ . ͑14͒ 1 2 2 1 2 1 ʈAʈϱ Յ ʈAʈ Յ ʈAʈ , 2 1 The trace distance is the maximum probability of distinguishing ␳ from ␳ : namely, D͑␳ ,␳ ͒ ʈ ʈ Յ ʈ ʈ ʈ ʈ ʈ ʈ ʈ ʈ ͑ ͒ 1 2 1 2 AB ui A ϱ B ui, B ϱ A ui. 6 ͑͗ ͘ ͗ ͘ ͒ ͗ ͘ ␳ =max0ϽPϽI P 1 − P 2 , where P i =Tr iP and P is a pro- There is an interesting duality between the trace and operator jector, or more generally an element of a positive operator- norm ͓9͔: valued measure ͑POVM͓͒1͔. The fidelity between quantum states ␳ and ␳ is ʈ ʈ ͕͉ ͑ † ͉͒ ʈ ʈ Յ ͖ ͑ ͒ 1 2 A 1 = max Tr B A : B ϱ 1 , 7 ͑␳ ␳ ͒ϵʈͱ␳ ͱ␳ ʈ ͱͱ␳ ␳ ͱ␳ ͑ ͒ F 1, 2 1 2 1 = 1 2 1, 15 ʈ ʈ ͕͉ ͑ † ͉͒ ʈ ʈ Յ ͖ ͑ ͒ A ϱ = max Tr B A : B 1 1 , 8 ␳ ͉␺͗͘␺͉ ␳ ͉␾͗͘␾͉ which reduces for pure states 1 = and 2 = to F͉͑␺͘,͉␾͒͘=͉͗␺͉␾͉͘. The fidelity and distance bound each ͉ ͑ ͉͒ Յ ʈ ʈ ʈ †ʈ ʈ †ʈ ʈ ʈ ͑ ͒ Tr BA A ϱ B 1, B ϱ A 1. 9 other from above and below ͓12͔: In the last three inequalities A and B can map between spaces 1−D͑␳ ,␳ ͒ Յ F͑␳ ,␳ ͒ Յ ͱ1−D͑␳ ,␳ ͒2, ͑16͒ of different dimensions. If they map between spaces of the 1 2 1 2 1 2 same dimension, then so that knowing one bounds the other. Many other measures ͓ ͔ ʈ ʈ ͉ ͑ † ͉͒ ͑ ͒ exist and are useful in a variety of circumstances 1 . A 1 = max Tr B A . 10 B†B=I We now define another norm, which we call the “operator- IV. GENERATORS OF THE DYNAMICS trace” norm ͑O-T norm͒: A.
Load more

Recommended publications
  • Front Matter of the Book Contains a Detailed Table of Contents, Which We Encourage You to Browse
    Cambridge University Press 978-1-107-00217-3 - Quantum Computation and Quantum Information: 10th Anniversary Edition Michael A. Nielsen & Isaac L. Chuang Frontmatter More information Quantum Computation and Quantum Information 10th Anniversary Edition One of the most cited books in physics of all time, Quantum Computation and Quantum Information remains the best textbook in this exciting field of science. This 10th Anniversary Edition includes a new Introduction and Afterword from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography, and quantum error-correction. Quantum mechanics and computer science are introduced, before moving on to describe what a quantum computer is, how it can be used to solve problems faster than “classical” computers, and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering. MICHAEL NIELSEN was educated at the University of Queensland, and as a Fulbright Scholar at the University of New Mexico. He worked at Los Alamos National Laboratory, as the Richard Chace Tolman Fellow at Caltech, was Foundation Professor of Quantum Information Science and a Federation Fellow at the University of Queensland, and a Senior Faculty Member at the Perimeter Institute for Theoretical Physics. He left Perimeter Institute to write a book about open science and now lives in Toronto. ISAAC CHUANG is a Professor at the Massachusetts Institute of Technology, jointly appointed in Electrical Engineering & Computer Science, and in Physics.
    [Show full text]
  • Quantifying Bell Nonlocality with the Trace Distance
    PHYSICAL REVIEW A 97, 022111 (2018) Quantifying Bell nonlocality with the trace distance S. G. A. Brito,1 B. Amaral,1,2 and R. Chaves1 1International Institute of Physics, Federal University of Rio Grande do Norte, 59078-970, P. O. Box 1613, Natal, Brazil 2Departamento de Física e Matemática, CAP - Universidade Federal de São João del-Rei, 36.420-000, Ouro Branco, MG, Brazil (Received 15 September 2017; published 21 February 2018) Measurements performed on distant parts of an entangled quantum state can generate correlations incompatible with classical theories respecting the assumption of local causality. This is the phenomenon known as quantum nonlocality that, apart from its fundamental role, can also be put to practical use in applications such as cryptography and distributed computing. Clearly, developing ways of quantifying nonlocality is an important primitive in this scenario. Here, we propose to quantify the nonlocality of a given probability distribution via its trace distance to the set of classical correlations. We show that this measure is a monotone under the free operations of a resource theory and, furthermore, that it can be computed efficiently with a linear program. We put our framework to use in a variety of relevant Bell scenarios also comparing the trace distance to other standard measures in the literature. DOI: 10.1103/PhysRevA.97.022111 I. INTRODUCTION it can stand before becoming local. Interestingly, these two measures can be inversely related as demonstrated by the With the establishment of quantum information science, the fact that in the Clauser-Horne-Shimony-Holt (CHSH) scenario often-called counterintuitive features of quantum mechanics [28], the resistance against detection inefficiency increases as such as entanglement [1] and nonlocality [2] have been we decrease the entanglement of the state [13] (also reducing raised to the status of a physical resource that can be used the violation the of CHSH inequality).
    [Show full text]
  • Arxiv:2102.13630V1 [Quant-Ph] 26 Feb 2021
    Randomness Amplification under Simulated PT -symmetric Evolution Leela Ganesh Chandra Lakkaraju1, Shiladitya Mal1,2, Aditi Sen(De)1 1 Harish-Chandra Research Institute and HBNI, Chhatnag Road, Jhunsi, Allahabad - 211019, India 2 Department of Physics and Center for Quantum Frontiers of Research and Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan PT -symmetric quantum theory does not require the Hermiticity property of observables and hence allows a rich class of dynamics. Based on PT -symmetric quantum theory, various counter- intuitive phenomena like faster evolution than that allowed in standard quantum mechanics, single- shot discrimination of nonorthogonal states has been reported invoking Gedanken experiments. By exploiting open-system experimental set-up as well as by computing the probability of distinguish- ing two states, we prove here that if a source produces an entangled state shared between two parties, Alice and Bob, situated in a far-apart location, the information about the operations performed by Alice whose subsystem evolves according to PT -symmetric Hamiltonian can be gathered by Bob, if the density matrix is in complex Hilbert space. Employing quantum simulation of PT -symmetric evolution, feasible with currently available technologies, we also propose a scheme of sharing quan- tum random bit-string between two parties when one of them has access to a source generating pseudo-random numbers. We find evidences that the task becomes more efficient with the increase of dimension. I. INTRODUCTION shown to be violated in the Gedanken experimental set-up [27]. Violation of these fundamentally signif- Among the postulates of quantum mechanics, the re- icant no-go theorems have deep consequences in the quirement of the Hermiticity property for the observ- verification of the viability of quantum theory.
    [Show full text]
  • Application of Quantum Pinsker Inequality to Quantum Communications
    ISSN 2186-6570 Application of quantum Pinsker inequality to quantum communications Osamu Hirota Research Center for Quantum Communication, Quantum ICT Research Institute, Tamagawa University 6-1-1 Tamagawa-gakuen, Machida, Tokyo 194-8610, Japan Tamagawa University Quantum ICT Research Institute Bulletin, Vol.9, No.1, 1-7, 2019 ©Tamagawa University Quantum ICT Research Institute 2019 All rights reserved. No part of this publication may be reproduced in any form or by any means electrically, mechanically, by photocopying or otherwise, without prior permission of the copy right owner. Tamagawa University Quantum ICT Research Institute Bulletin 1 Vol.9 No.1 : 1―7（2019） Application of quantum Pinsker inequality to quantum communications Osamu Hirota Research Center for Quantum Communication, Quantum ICT Research Institute, Tamagawa University 6-1-1 Tamagawa-gakuen, Machida, Tokyo 194-8610, Japan E-mail: [email protected] Abstract—Back in the 1960s, based on Wiener’s thought, clearly defines very common signals processed in human Shikao Ikehara (first student of N.Wiener) encouraged the social activities and it is applied to the communication progress of Hisaharu Umegaki’s research from a pure system with the operational meanings for the relevant mathematical aspect in order to further develop the research on mathematical methods of quantum information at Tokyo information processing. Quantum information theory as Institute of Technology. Then, in the 1970s, based on the mathematics can be modeled on it. However, unlike Shan- results accomplished by Umegaki Group, Ikehara instructed non information, quantum entropy lacks versatility regard the author to develop and spread quantum information for operational meaning on information transmission.
    [Show full text]
  • Non-Markovian Dynamics in Open Quantum Systems
    Non-Markovian dynamics in open quantum systems Heinz-Peter Breuer Physikalisches Institut, Universit¨atFreiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany Elsi-Mari Laine QCD Labs, COMP Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 13500, FI-00076 AALTO, Finland Jyrki Piilo Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun yliopisto, Finland Bassano Vacchini Dipartimento di Fisica, Universit`adegli Studi di Milano, Via Celoria 16, I-20133 Milan, Italy INFN, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy (Dated: May 7, 2015) The dynamical behavior of open quantum systems plays a key role in many applica- tions of quantum mechanics, examples ranging from fundamental problems, such as the environment-induced decay of quantum coherence and relaxation in many-body sys- tems, to applications in condensed matter theory, quantum transport, quantum chem- istry and quantum information. In close analogy to a classical Markovian stochastic process, the interaction of an open quantum system with a noisy environment is often modeled phenomenologically by means of a dynamical semigroup with a corresponding time-independent generator in Lindblad form, which describes a memoryless dynamics of the open system typically leading to an irreversible loss of characteristic quantum fea- tures. However, in many applications open systems exhibit pronounced memory effects and a revival of genuine quantum properties such as quantum coherence, correlations and entanglement. Here, recent theoretical results on the rich non-Markovian quantum arXiv:1505.01385v1 [quant-ph] 6 May 2015 dynamics of open systems are discussed, paying particular attention to the rigorous mathematical definition, to the physical interpretation and classification, as well as to the quantification of quantum memory effects.
    [Show full text]
  • The Monge Metric on the Sphere and Geometry of Quantum States
    INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 34 (2001) 6689–6722 PII: S0305-4470(01)18080-7 The Monge metric on the sphere and geometry of quantum states Karol Zyczkowski˙ 1,2 and Wojciech Słomczynski´ 3 1 Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotnikow´ 32, 02-668 Warszawa, Poland 2 Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagiellonski,´ ul. Reymonta 4, 30-059 Krakow,´ Poland 3 Instytut Matematyki, Uniwersytet Jagiellonski,´ ul. Reymonta 4, 30-059 Krakow,´ Poland E-mail: [email protected] and [email protected] Received 18 October 2000, in final form 14 June 2001 Published 17 August 2001 Online at stacks.iop.org/JPhysA/34/6689 Abstract Topological and geometrical properties of the set of mixed quantum states in the N-dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere we use the vector SU(2) coherent states and the generalized Husimi distributions to define the Monge distance between two arbitrary density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state ρ∗, proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized ‘chaotic’ states are close to ρ∗. This contrasts the geometry induced by the standard (trace, Hilbert–Schmidt or Bures) metrics, for which the distance from ρ∗ is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence.
    [Show full text]
  • Generalized Trace-Distance Measure Connecting Quantum and Classical Non-Markovianity
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by AIR Universita degli studi di Milano PHYSICAL REVIEW A 92, 042108 (2015) Generalized trace-distance measure connecting quantum and classical non-Markovianity Steffen Wißmann and Heinz-Peter Breuer Physikalisches Institut, Universitat¨ Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany Bassano Vacchini Dipartimento di Fisica, Universita` degli Studi di Milano, Via Celoria 16, I-20133 Milan, Italy and INFN, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy (Received 31 July 2015; published 12 October 2015) We establish a direct connection of quantum Markovianity of an open system to its classical counterpart by generalizing the criterion based on the information flow. Here the flow is characterized by the time evolution of Helstrom matrices, given by the weighted difference of statistical operators, under the action of the quantum dynamical map. It turns out that the introduced criterion is equivalent to P divisibility of a quantum process, namely, divisibility in terms of positive maps, which provides a direct connection to classical Markovian stochastic processes. Moreover, it is shown that mathematical representations similar to those found for the original trace-distance-based measure hold true for the associated generalized measure for quantum non-Markovianity. That is, we prove orthogonality of optimal states showing a maximal information backflow and establish a local and universal representation of the measure. We illustrate some properties of the generalized criterion by means of examples. DOI: 10.1103/PhysRevA.92.042108 PACS number(s): 03.65.Yz, 03.65.Ta, 03.67.−a, 02.50.−r I.
    [Show full text]
  • Arxiv:1004.4110V1 [Quant-Ph]
    Computing the distance between quantum channels: Usefulness of the Fano representation 1,2, 3, Giuliano Benenti ∗ and Giuliano Strini † 1CNISM, CNR-INFM, and Center for Nonlinear and Complex Systems, Universit`adegli Studi dell’Insubria, via Valleggio 11, 22100 Como, Italy 2Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy 3Dipartimento di Fisica, Universit`adegli Studi di Milano, via Celoria 16, 20133 Milano, Italy (Dated: April 26, 2010) The diamond norm measures the distance between two quantum channels. From an operational vewpoint, this norm measures how well we can distinguish between two channels by applying them to input states of arbitrarily large dimensions. In this paper, we show that the diamond norm can be conveniently and in a physically transparent way computed by means of a Monte-Carlo algorithm based on the Fano representation of quantum states and quantum operations. The effectiveness of this algorithm is illustrated for several single-qubit quantum channels. PACS numbers: 03.65.Yz, 03.67.-a I. INTRODUCTION cated semidefinite programming or convex optimization. On the other hand, analytical solutions are limited to special classes of channels [7, 8]. In this paper, we pro- Quantum information processes in a noisy environment pose a simple and easily parallelizable Monte-Carlo al- quantum chan- are conveniently described in terms of gorithm based on the Fano representation of quantum nels , that is, linear, trace preserving, completely positive states and quantum operations. We show that our al- maps on the set of quantum states [1, 2]. The problem gorithm provides reliable results for the case, most sig- of discriminating quantum channels is of great interest.
    [Show full text]
  • Command of Quantum Systems Via Rigidity of CHSH Games
    A classical leash for a quantum system: Command of quantum systems via rigidity of CHSH games Ben W. Reichardt Falk Unger Umesh Vazirani University of Southern California Knight Capital Group UC Berkeley Abstract Can a classical system command a general adversarial quantum system to realize arbitrary quantum dynamics? If so, then we could realize the dream of device-independent quantum cryptography: using untrusted quantum devices to establish a shared random key, with security based on the correctness of quantum mechanics. It would also allow for testing whether a claimed quantum computer is truly quantum. Here we report a technique by which a classical system can certify the joint, entangled state of a bipartite quantum system, as well as command the application of specific operators on each subsystem. This is accomplished by showing a strong converse to Tsirelson's optimality result for the Clauser-Horne-Shimony-Holt (CHSH) game: the only way to win many games is if the bipartite state is close to the tensor product of EPR states, and the measurements are the optimal CHSH measurements on successive qubits. This leads directly to a scheme for device-independent quantum key distribution. Control over the state and operators can also be leveraged to create more elaborate protocols for realizing general quantum circuits, and to establish that QMIP = MIP∗. arXiv:1209.0448v1 [quant-ph] 3 Sep 2012 1 Contents 1 Introduction 3 2 Proof sketches 8 2.1 Rigidity of the CHSH game . .8 2.2 Tensor-product structure for repeated CHSH games . .8 2.2.1 Construction of the ideal strategy S^ .......................9 2.2.2 Ideal strategy S^ is close to S ...........................9 2.3 Verified quantum dynamics .
    [Show full text]
  • Computing the Distance Between Quantum Channels: Usefulness of the Fano Representation Giuliano Benenti, Giuliano Strini
    Computing the distance between quantum channels: usefulness of the Fano representation Giuliano Benenti, Giuliano Strini To cite this version: Giuliano Benenti, Giuliano Strini. Computing the distance between quantum channels: usefulness of the Fano representation. Journal of Physics B: Atomic, Molecular and Optical Physics, IOP Publish- ing, 2010, 43 (21), pp.215508. 10.1088/0953-4075/43/21/215508. hal-00569863 HAL Id: hal-00569863 https://hal.archives-ouvertes.fr/hal-00569863 Submitted on 25 Feb 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Confidential: not for distribution. Submitted to IOP Publishing for peer review 24 June 2010 Computing the distance between quantum channels: Usefulness of the Fano representation Giuliano Benenti1,2,∗ and Giuliano Strini3, † 1CNISM, CNR-INFM, and Center for Nonlinear and Complex Systems, Universit`adegliStudidell’Insubria,viaValleggio11,22100Como,Italy 2Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy 3Dipartimento di Fisica, Universit`adegliStudidiMilano,viaCeloria16,20133Milano,Italy (Dated: June 24, 2010) The diamond norm measures the distance between two quantum channels. From an operational vewpoint, this norm measures how well we can distinguish between two channels by applying them to input states of arbitrarily large dimensions.
    [Show full text]
  • Lecture Notes Live Here
    Physics 239/139: Quantum information is physical Spring 2018 Lecturer: McGreevy These lecture notes live here. Please email corrections to mcgreevy at physics dot ucsd dot edu. THESE NOTES ARE SUPERSEDED BY THE NOTES HERE Schr¨odinger's cat and Maxwell's demon, together at last. Last updated: 2021/05/09, 15:30:29 1 Contents 0.1 Introductory remarks............................4 0.2 Conventions.................................7 0.3 Lightning quantum mechanics reminder..................8 1 Hilbert space is a myth 11 1.1 Mean field theory is product states.................... 14 1.2 The local density matrix is our friend................... 16 1.3 Complexity and the convenient illusion of Hilbert space......... 19 2 Quantifying information 26 2.1 Relative entropy............................... 33 2.2 Data compression.............................. 37 2.3 Noisy channels............................... 43 2.4 Error-correcting codes........................... 49 3 Information is physical 53 3.1 Cost of erasure............................... 53 3.2 Second Laws of Thermodynamics..................... 59 4 Quantifying quantum information and quantum ignorance 64 4.1 von Neumann entropy........................... 64 4.2 Quantum relative entropy......................... 67 4.3 Purification, part 1............................. 69 4.4 Schumacher compression.......................... 71 4.5 Quantum channels............................. 73 4.6 Channel duality............................... 79 4.7 Purification, part 2............................. 84 4.8 Deep
    [Show full text]
  • Quantum Hypothesis Testing in Many-Body Systems Abstract Contents
    SciPost Phys. Core 4, 019 (2021) Quantum hypothesis testing in many-body systems 1? 1† 2,4‡ 2,3‘ Jan de Boer , Victor Godet , Jani Kastikainen and Esko Keski-Vakkuri ◦ 1 Institute for Theoretical Physics, University of Amsterdam, PO Box 94485, 1090 GL Amsterdam, The Netherlands 2 Department of Physics, P.O.Box 64, FIN-00014 University of Helsinki, Finland 3 Helsinki Institute of Physics, P.O.Box 64, FIN-00014 University of Helsinki, Finland 4 APC, AstroParticule et Cosmologie, Université de Paris, CNRS/IN2P3, CEA/IRFU, Observatoire de Paris, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France ? [email protected],† [email protected], ‡ jani.kastikainen@helsinki.fi, esko.keski-vakkuri@helsinki.fi ◦ Abstract One of the key tasks in physics is to perform measurements in order to determine the state of a system. Often, measurements are aimed at determining the values of physical parameters, but one can also ask simpler questions, such as “is the system in state A or state B?”. In quantum mechanics, the latter type of measurements can be studied and optimized using the framework of quantum hypothesis testing. In many cases one can explicitly find the optimal measurement in the limit where one has simultaneous access to a large number n of identical copies of the system, and estimate the expected error as n becomes large. Interestingly, error estimates turn out to involve various quantum information theoretic quantities such as relative entropy, thereby giving these quantities operational meaning. In this paper we consider the application of quantum hypothesis testing to quantum many-body systems and quantum field theory.
    [Show full text]