DOCSLIB.ORG

The Critical Points of Coherent Information on the Manifold of Positive Definite Matrices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Critical Points of Coherent Information on the Manifold of Positive Definite Matrices The Critical Points of Coherent Information on the Manifold of Positive Definite Matrices by Alireza Tehrani A Thesis presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Mathematics Guelph, Ontario, Canada c Alireza Tehrani, January, 2020 ABSTRACT THE CRITICAL POINTS OF COHERENT INFORMATION ON THE MANIFOLD OF POSITIVE DEFINITE MATRICES Alireza Tehrani Advisors: University of Guelph, 2020 Dr. Bei Zeng Dr. Rajesh Pereira The coherent information of quantum channels plays a important role in quantum infor- mation theory as it can be used to calculate the quantum capacity of a channel. However, it is a non-linear, non-differentiable optimization problem. This thesis discusses that by restricting to the space of positive definite density matrices and restricting the class of quan- tum channels to be strictly positive, the coherent information becomes differentiable. This allows the computation of the Riemannian gradient and Hessian of the coherent information. It will be shown that the maximally mixed state is a critical point for the n-shot coherent information of the Pauli, dephrasure and Pauli-erasure channels. In addition, the classifica- tion of the maximally mixed state as a local maxima/minima and saddle-point will be solved for the one shot coherent information. The hope of this work is to provide a new avenue to explore the quantum capacity problem. Dedication To those who have inspired or encouraged me: Brother, Father, Randy, Farnaz and Paul. iii Acknowledgements I want to express my gratitude and thanks to Dr. Zeng for the guidance, advice and direction that was provided and for giving me the opportunity to work in quantum information theory and to be part of IQC and the Peng Cheng Laborotory. It has been a inspiring experience to witness the level of enthusiasm with respect to science that a person could have. In addition to Dr.Pereira, whom made sure I was on the right path, answering any questions and providing new suggestions. I'm thankful for the time spent doing research and words can never be enough to express the kindness that was shown. Couldn't ask for better office mates, David, Eric, Katrina, Momina and Ningping thanks for the time spent together. iv Contents Abstract ii Dedication iii Acknowledgements iv List of Figures viii 1 Introduction 1 1.1 Introduction to Problem . .1 1.1.1 Contributions . .2 1.2 Overview of Chapters . .3 2 Coherent Information 5 2.1 von Neumann Entropy . .5 2.1.1 Classical Entropy . .6 2.1.2 Density Matrices . .7 2.1.3 Quantum/von Neumann Entropy . .8 2.2 Coherent Information . 11 2.2.1 Quantum Channels . 11 2.2.2 Motivation . 15 2.2.3 Coherent Information of a channel . 17 2.3 Properties . 19 2.3.1 Data Processing and Error-Correction . 19 2.3.2 Quantum Capacity . 20 2.3.3 Lipschitz . 21 2.3.4 Conjecture: Optima on Positive Definite Density Matrices . 22 2.4 Known Results . 22 2.4.1 Amplitude-Damping Channel . 23 2.4.2 Pauli Channel . 23 2.4.3 Dephrasure Channel . 24 2.4.4 Degradable and Antidegradable Quantum Channels . 25 v 3 Introduction to Manifold Theory and The Gradient/Hessian 27 3.1 Preliminaries to Manifold Theory . 28 3.1.1 Manifolds . 28 3.1.2 Smooth Maps . 33 3.1.3 Tangent Space . 36 3.1.4 Differential . 37 3.1.5 Tangent Bundle . 43 3.1.6 Submanifolds . 44 3.1.7 Riemannian Manifolds . 46 3.2 Riemannian Gradient and Hessian . 48 3.2.1 Euclidean Gradient . 49 3.2.2 Riemannian Gradient . 50 3.2.3 Hessian . 53 4 Gradient/Hessian of Coherent Information on Positive Definite Matrices 59 4.1 Strictly Positive Quantum Channels . 60 4.1.1 Definition . 60 4.1.2 Dense . 63 4.1.3 Examples . 64 4.2 Differentiability of Coherent Information . 64 4.2.1 Differentiability of Entropy Exchange and Coherent Information . 65 4.3 Gradient and Hessian . 66 4.3.1 Gradient . 66 4.3.2 Hessian . 71 5 Critical Points and Local Maximas/Minimas 76 5.1 n-shot Coherent Information . 77 5.1.1 Critical Points of Product States . 77 5.2 Unital Channels . 80 5.3 Pauli Channel . 83 5.3.1 Dephasing Channel . 87 5.3.2 Depolarizing Channel . 88 5.4 Dephrasure Channel . 95 5.5 Pauli Erasure Channel . 103 6 Conclusion And Further Work 107 6.1 Further Work . 109 Bibliography 112 vi A Matrix Calculus 115 A.1 Matrix Functions . 115 A.2 Frechet and Gateux Derivatives . 117 A.3 Matrix Exponential and Logarithm . 118 A.4 Power Series . 120 vii List of Figures 2.1 The initial model and assumptions of the coherent information. 16 5.1 Coherent Information of Depolarizing Channel Evaluated at the Maximally Mixed State. 91 5.2 Maximum of Coherent Information of Depolarizing Channel. 92 5.3 von Neumann entropy of the optimal solution of maximum of coherent infor- mation . 93 5.4 The eigenvalue of the Hessian at the Maximally Mixed state of coherent in- formation. 94 viii Chapter 1 Introduction 1.1 Introduction to Problem One of the most challenging problems of Quantum Information is the evaluation of quantum capacity Q(N ) of a quantum channel N , the maximal amount of quantum information that can be sent through N . It was shown in [22], [11], and [28], known as the LSD theorem, that it can be calculated as I (N ⊗n) Q(N ) = lim c ; n!1 n c where Ic(N ) = maxρ2D S(N (ρ)) − S(N (ρ)) is the coherent information of the quantum ⊗n channel N and D is the space of all density matrices. The term Ic(N ) is called the n-shot coherent information. It is the quantum analogue of mutual information found in classical information theory. Unlike the classical case, quantum capacity does not have a single letter expression [27]. This is due to the existence of superadditivity ⊗n Ic(N ) > nIc(N ); for some channel use n. Note that in general, for all n the following inequality holds ⊗n Ic(N ) ≥ nIc(N ). It was shown in 2005 by Devetak and Shor [10], that if the quantum channel is degradable then the quantum capacity has a single letter formula as Q(N ) = Ic(N ). This is due to ⊗n the fact that for all n, the coherent information is subadditive Ic(N ) = nIc(N ): If the quantum channel is antidegradable, then the quantum capacity is known to be zero. The notion of approximate degradable quantum channels were introduced in 2017 and bounds 1 on the quantum capacities were found [32] which were further studied in [18]. The quantum capacity is generally an extremely difficult problem. It is shown in 2015, that an unbounded number of channel uses of optimizing the coherent information may be required to find non-zero quantum capacity [7]. Due to this result, the problem shifted to understanding examples of when superadditivity occurs. This is further studied in the following papers [21] (2017) and [19] (2018), respectively. The overall goal of this thesis is to investigate the critical points (ie when its gradient is zero) of the n-shot coherent information in a coordinate invariant way. This is done by first investigating when the coherent information becomes a smooth map on the space of positive definite density matrices D++. The reasoning behind this restriction is that the global optima is conjectured to generically be positive definite. Additionally, the set of positive definite density matrices is dense inside the space of all density matrices and it has a smooth manifold structure. This allows the calculation of the gradient and Hessian of the coherent information. It will have the advantage of being coordinate invariant and will be shown that critical points can be completely studied on the space of positive definite matrices M++ (rather than D++). 1.1.1 Contributions The contributions of this thesis is as follows. • The restriction of the coherent information to the manifold of postive definite matrices and the conditions on the quantum channel N and the complementary channel N c to insure smoothness of coherent information is explored. This is heavily tied to the notion of a strictly positive map [4]. This is discussed in chapter four. • The Riemannian Gradient of the coherent information has been solved within the dense class of strictly positive quantum channels. It will be shown that a positive definite density matrix ρ is a critical point of the coherent information Ic of a strictly positive quantum channel N iff −N y log(N (ρ)) + (N c)y log(N c(ρ)) 2 spanfIg; where log is the matrix logarithm, N c is the complementary channel, N y is the adjoint of a channel and I is the identity matrix. This is the statement of Theorem 4.3.2. 2 • It will be shown that the coherent information Ic(N ; ρ) can be written using the gra- dient term above, ie Ic(N ; ρ) = hgrad(Ic(N ; ρ))jρi; y where hAjBi = T r(A B) is the Frobenius inner product and the gradient grad(Ic(N ; ρ)) is −N y(log(N (ρ))) + N cy(log(N c(ρ))) on the manifold of positive definite matrices. It will be shown that the coherent information is a positively homogeneous function of degree zero. • In addition, the Riemannian Hessian Hess(Ic(N ; ρ)) of the coherent information Ic at a positive definite density matrix ρ is solved. It is a linear map from trace zero Hermitian matrices H0 to itself such that y c y c Hess(Ic(N ; ρ))[V ] = P − N d logN (ρ) N (V ) + (N ) d logN c(ρ) N (V ) ; where V is a trace zero Hermitian matrix, P is the orthogonal projection from Her- 0 mitian matrices H to trace zero, Hermitian matrices H and d logX : H ! H is the differential of the matrix logarithm.
Load more

Recommended publications
  • Lecture 6: Quantum Error Correction and Quantum Capacity
    Lecture 6: Quantum error correction and quantum capacity Mark M. Wilde∗ The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally \quantum" theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum com- munication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem provides the intuition behind quantum error correction. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information|otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H(B) − H(E)|a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain.
    [Show full text]
  • Lecture 18 — October 26, 2015 1 Overview 2 Quantum Entropy
    PHYS 7895: Quantum Information Theory Fall 2015 Lecture 18 | October 26, 2015 Prof. Mark M. Wilde Scribe: Mark M. Wilde This document is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 1 Overview In the previous lecture, we discussed classical entropy and entropy inequalities. In this lecture, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy. It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state. The von Neumann entropy gives meaning to a notion of the information qubit. This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information.
    [Show full text]
  • Quantum Information Science
    Quantum Information Science Seth Lloyd Professor of Quantum-Mechanical Engineering Director, WM Keck Center for Extreme Quantum Information Theory (xQIT) Massachusetts Institute of Technology Article Outline: Glossary I. Definition of the Subject and Its Importance II. Introduction III. Quantum Mechanics IV. Quantum Computation V. Noise and Errors VI. Quantum Communication VII. Implications and Conclusions 1 Glossary Algorithm: A systematic procedure for solving a problem, frequently implemented as a computer program. Bit: The fundamental unit of information, representing the distinction between two possi- ble states, conventionally called 0 and 1. The word ‘bit’ is also used to refer to a physical system that registers a bit of information. Boolean Algebra: The mathematics of manipulating bits using simple operations such as AND, OR, NOT, and COPY. Communication Channel: A physical system that allows information to be transmitted from one place to another. Computer: A device for processing information. A digital computer uses Boolean algebra (q.v.) to processes information in the form of bits. Cryptography: The science and technique of encoding information in a secret form. The process of encoding is called encryption, and a system for encoding and decoding is called a cipher. A key is a piece of information used for encoding or decoding. Public-key cryptography operates using a public key by which information is encrypted, and a separate private key by which the encrypted message is decoded. Decoherence: A peculiarly quantum form of noise that has no classical analog. Decoherence destroys quantum superpositions and is the most important and ubiquitous form of noise in quantum computers and quantum communication channels.
    [Show full text]
  • On the Complementary Quantum Capacity of the Depolarizing Channel
    On the complementary quantum capacity of the depolarizing channel Debbie Leung1 and John Watrous2 1Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo 2Institute for Quantum Computing and School of Computer Science, University of Waterloo October 2, 2018 The qubit depolarizing channel with noise parameter η transmits an input qubit perfectly with probability 1 − η, and outputs the completely mixed state with probability η. We show that its complementary channel has positive quantum capacity for all η > 0. Thus, we find that there exists a single parameter family of channels having the peculiar property of having positive quantum capacity even when the outputs of these channels approach a fixed state independent of the input. Comparisons with other related channels, and implications on the difficulty of studying the quantum capacity of the depolarizing channel are discussed. 1 Introduction It is a fundamental problem in quantum information theory to determine the capacity of quantum channels to transmit quantum information. The quantum capacity of a channel is the optimal rate at which one can transmit quantum data with high fidelity through that channel when an asymptotically large number of channel uses is made available. In the classical setting, the capacity of a classical channel to transmit classical data is given by Shannon’s noisy coding theorem [12]. Although the error correcting codes that allow one to approach the capacity of a channel may involve increasingly large block lengths, the capacity expression itself is a simple, single letter formula involving an optimization over input distributions maximizing the input/output mutual information over one use of the channel.
    [Show full text]
  • Quantum Information Chapter 10. Quantum Shannon Theory
    Quantum Information Chapter 10. Quantum Shannon Theory John Preskill Institute for Quantum Information and Matter California Institute of Technology Updated June 2016 For further updates and additional chapters, see: http://www.theory.caltech.edu/people/preskill/ph219/ Please send corrections to [email protected] Contents 10 Quantum Shannon Theory 1 10.1 Shannon for Dummies 2 10.1.1 Shannon entropy and data compression 2 10.1.2 Joint typicality, conditional entropy, and mutual infor- mation 6 10.1.3 Distributed source coding 8 10.1.4 The noisy channel coding theorem 9 10.2 Von Neumann Entropy 16 10.2.1 Mathematical properties of H(ρ) 18 10.2.2 Mixing, measurement, and entropy 20 10.2.3 Strong subadditivity 21 10.2.4 Monotonicity of mutual information 23 10.2.5 Entropy and thermodynamics 24 10.2.6 Bekenstein’s entropy bound. 26 10.2.7 Entropic uncertainty relations 27 10.3 Quantum Source Coding 30 10.3.1 Quantum compression: an example 31 10.3.2 Schumacher compression in general 34 10.4 Entanglement Concentration and Dilution 38 10.5 Quantifying Mixed-State Entanglement 45 10.5.1 Asymptotic irreversibility under LOCC 45 10.5.2 Squashed entanglement 47 10.5.3 Entanglement monogamy 48 10.6 Accessible Information 50 10.6.1 How much can we learn from a measurement? 50 10.6.2 Holevo bound 51 10.6.3 Monotonicity of Holevo χ 53 10.6.4 Improved distinguishability through coding: an example 54 10.6.5 Classical capacity of a quantum channel 58 ii Contents iii 10.6.6 Entanglement-breaking channels 62 10.7 Quantum Channel Capacities and Decoupling
    [Show full text]
  • The Squashed Entanglement of a Quantum Channel
    The squashed entanglement of a quantum channel Masahiro Takeoka∗y Saikat Guhay Mark M. Wildez January 22, 2014 Abstract This paper defines the squashed entanglement of a quantum channel as the maximum squashed entanglement that can be registered by a sender and receiver at the input and output of a quantum channel, respectively. A new subadditivity inequality for the original squashed entanglement measure of Christandl and Winter leads to the conclusion that the squashed en- tanglement of a quantum channel is an additive function of a tensor product of any two quantum channels. More importantly, this new subadditivity inequality, along with prior results of Chri- standl, Winter, et al., establishes the squashed entanglement of a quantum channel as an upper bound on the quantum communication capacity of any channel assisted by unlimited forward and backward classical communication. A similar proof establishes this quantity as an upper bound on the private capacity of a quantum channel assisted by unlimited forward and backward public classical communication. This latter result is relevant as a limitation on rates achievable in quantum key distribution. As an important application, we determine that these capacities can never exceed log((1 + η)=(1 − η)) for a pure-loss bosonic channel for which a fraction η of the input photons make it to the output on average. The best known lower bound on these capacities is equal to log(1=(1 − η)). Thus, in the high-loss regime for which η 1, this new upper bound demonstrates that the protocols corresponding to the above lower bound are nearly optimal.
    [Show full text]
  • Quantum Channel Capacities
    Quantum Channel Capacities Graeme Smith IBM TJ Watson Research Center 1101 Kitchawan Road Yorktown NY 10598 [email protected] Abstract—A quantum communication channel can be put to must consider the quantum capacity of our channel, and if we many uses: it can transmit classical information, private classical have access to arbitrary quantum correlations between sender information, or quantum information. It can be used alone, with and receiver the relevant capacity is the entanglement assisted shared entanglement, or together with other channels. For each of these settings there is a capacity that quantifies a channel’s capacity. In general, these capacities are all different, which potential for communication. In this short review, I summarize gives a variety of inequivalent ways to quantify the value of what is known about the various capacities of a quantum channel, a quantum channel for communication. including a discussion of the relevant additivity questions. I The communication capacities of a quantum channel are also give some indication of potentially interesting directions for not nearly as well understood as their classical counterparts, future research. and many basic questions about quantum capacities remain open. The purpose of this paper is to give an introduction to a I. INTRODUCTION quantum channel’s capacities, summarize what we know about The capacity of a noisy communication channel for noise- them, and point towards some important unsolved problems. less information transmission is a central quantity in the study of information theory [1]. This capacity establishes the II. QUANTUM STATES AND CHANNELS ultimate boundary between communication rates which are The states of least uncertainty in quantum mechanics are achievable in principle and those which are not.
    [Show full text]
  • Reverse Coherent Information
    Reverse Coherent Information Ra´ul Garc´ıa-Patr´on,1 Stefano Pirandola,1 Seth Lloyd,1 and Jeffrey H. Shapiro1 1Research Laboratory of Electronics, MIT, Cambridge, MA 02139 In this letter we define a family of entanglement distribution protocols assisted by feedback clas- sical communication that gives an operational interpretation to reverse coherent information, i.e., the symmetric counterpart of the well known coherent information. This lead to the definition of a new entanglement distribution capacity that exceeds the unassisted capacity for some interesting channels. PACS numbers: 03.67.-a, 03.67.Hk Shannon’s great result was proving that sending in- The channel Λ is called degradable if there exists a formation through a noisy channel can be achieved map that transforms Bob’s output ρ into the en- N M B with vanishing error, in the limit of many uses of the vironment state ρE, i.e., (ρB) = ρE, where ρE = channel [1]. Shannon’s key idea was to add redundancy Tr [ φ φ ] and φ M is the purification of ρ . RB | ih |RBE | iRBE RB to the message in order to compensate for the channel’s Similarly if there is a map such that (ρE) = ρB the noise. He showed that the channel’s communication ca- channel is called antidegradableG and (Λ)G = 0. pacity ( ) between two partners, called Alice and Bob, Having free access to a classical communicationQ chan- is givenC byN the maximal mutual information between Al- nel Alice and Bob can improve the quantum communi- ice’s input a and Bob’s output b = (a), i.e., cation protocol, as opposed to Shannon’s theory where N using feedback gives no improvement [7].
    [Show full text]
  • Quantum Information Can Be Negative
    Quantum information can be negative Michal Horodecki1, Jonathan Oppenheim 2 & Andreas Winter3 1Institute of Theoretical Physics and Astrophysics, University of Gda´nsk, 80–952 Gda´nsk, Poland 2Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cam- bridge CB3 0WA, U.K. 3Department of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. Submitted Feb. 20th, 2005 Given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communica- tion measures the partial information one system needs conditioned on it’s prior information. It turns out to be given by an extremely simple formula, the conditional entropy. In the classi- cal case, partial information must always be positive, but we find that in the quantum world this physical quantity can be negative. If the partial information is positive, its sender needs to communicate this number of quantum bits to the receiver; if it is negative, the sender and receiver instead gain the corresponding potential for future quantum communication. We in- troduce a primitive quantum state merging which optimally transfers partial information. We show how it enables a systematic understanding of quantum network theory, and discuss sev- eral important applications including distributed compression, multiple access channels and multipartite assisted entanglement distillation (localizable entanglement). Negative channel capacities also receive a natural interpretation. ’Ignorance is strength’ is one of the three cyn- of information originating from a source is the ical mottos of Big Brother in George Orwell’s memory required to faithfully represent its out- 1984.
    [Show full text]
  • Quantum Entropy and Its Applications to Quantum Communication and Statistical Physics
    Entropy 2010, 12, 1194-1245; doi:10.3390/e12051194 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Review Quantum Entropy and Its Applications to Quantum Communication and Statistical Physics Masanori Ohya ? and Noboru Watanabe ? Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan ? Author to whom correspondence should be addressed; E-Mails: [email protected] (M.O.); [email protected] (N.W.). Received: 10 February 2010 / Accepted: 30 April 2010 / Published: 7 May 2010 Abstract: Quantum entropy is a fundamental concept for quantum information recently developed in various directions. We will review the mathematical aspects of quantum entropy (entropies) and discuss some applications to quantum communication, statistical physics. All topics taken here are somehow related to the quantum entropy that the present authors have been studied. Many other fields recently developed in quantum information theory, such as quantum algorithm, quantum teleportation, quantum cryptography, etc., are totally discussed in the book (reference number 60). Keywords: quantum entropy; quantum information 1. Introduction Theoretical foundation supporting today’s information-oriented society is Information Theory founded by Shannon [1] about 60 years ago. Generally, this theory can treat the efficiency of the information transmission by using measures of complexity, that is, the entropy, in the commutative system of signal space. The information theory is based on the entropy theory that is formulated mathematically. Before Shannon’s work, the entropy was first introduced in thermodynamics by Clausius and in statistical mechanics by Boltzmann. These entropies are the criteria to characterize a property of the physical systems.
    [Show full text]
  • Quantum Information Chapter 10. Quantum Shannon Theory
    Quantum Information Chapter 10. Quantum Shannon Theory John Preskill Institute for Quantum Information and Matter California Institute of Technology Updated January 2018 For further updates and additional chapters, see: http://www.theory.caltech.edu/people/preskill/ph219/ Please send corrections to [email protected] Contents page v Preface vi 10 Quantum Shannon Theory 1 10.1 Shannon for Dummies 1 10.1.1Shannonentropyanddatacompression 2 10.1.2 Joint typicality, conditional entropy, and mutual information 4 10.1.3 Distributed source coding 6 10.1.4 The noisy channel coding theorem 7 10.2 Von Neumann Entropy 12 10.2.1 Mathematical properties of H(ρ) 14 10.2.2Mixing,measurement,andentropy 15 10.2.3 Strong subadditivity 16 10.2.4Monotonicityofmutualinformation 18 10.2.5Entropyandthermodynamics 19 10.2.6 Bekenstein’s entropy bound. 20 10.2.7Entropicuncertaintyrelations 21 10.3 Quantum Source Coding 23 10.3.1Quantumcompression:anexample 24 10.3.2 Schumacher compression in general 27 10.4 EntanglementConcentrationandDilution 30 10.5 QuantifyingMixed-StateEntanglement 35 10.5.1 AsymptoticirreversibilityunderLOCC 35 10.5.2 Squashed entanglement 37 10.5.3 Entanglement monogamy 38 10.6 Accessible Information 39 10.6.1 Howmuchcanwelearnfromameasurement? 39 10.6.2 Holevo bound 40 10.6.3 Monotonicity of Holevo χ 41 10.6.4 Improved distinguishability through coding: an example 42 10.6.5 Classicalcapacityofaquantumchannel 45 10.6.6 Entanglement-breaking channels 49 10.7 Quantum Channel Capacities and Decoupling 50 10.7.1 Coherent information and the
    [Show full text]
  • Narrow Bounds for the Quantum Capacity of Thermal Attenuators
    Narrow Bounds for the Quantum Capacity of Thermal Attenuators 1, 2, 2 2 Matteo Rosati, ∗ Andrea Mari, and Vittorio Giovannetti 1F´ısica Te`orica: Informaci´oi Fen`omensQu`antics,Departament de F´ısica, Universitat Aut`onomade Barcelona, 08193 Bellaterra, Spain 2NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy. ABSTRACT As usual in information theory, although a formula ex- ists for this quantity, it is quite difficult to compute for Thermal attenuator channels model the decoherence of general channels, due for example to the necessary reg- quantum systems interacting with a thermal bath, e.g., a ularization that takes into account the use of entangled two-level system subject to thermal noise and an electro- inputs over multiple channel uses [27]. The problem sim- magnetic signal travelling through a fiber or in free-space. plifies considerably for the so called degradable channels Hence determining the quantum capacity of these chan- [19, 28{31], but it also exhibits some striking features in nels is an outstanding open problem for quantum compu- other cases [32{36]. An important class of channels is tation and communication. Here we derive several upper that of Thermal Attenuators (TAs), which describe the bounds on the quantum capacity of qubit and bosonic effect of energy loss due to the interaction with a ther- thermal attenuators. We introduce an extended version mal environment. Common examples of this class are the of such channels which is degradable and hence has a qubit thermal attenuator or generalized amplitude damp- single-letter quantum capacity, bounding that of the orig- ing channel [7, 19] and its infinite-dimensional counter- inal thermal attenuators.
    [Show full text]