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Theory of Quantum Information with Memory

Name: Theory of Quantum Information with Memory
Author: Mou-Hsiung Chang
ISBN: 9783110788198

by Mou-Hsiung Chang

August 2022

Intermediate to advanced

502 pages

13h 42m

English

De Gruyter

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1 Basic notation and preliminaries
1.1 Complex Hilbert and Banach spaces1.2 Linear operators and their adjoints1.2.1 Adjoint and self-adjoint operators1.2.2 Bounded linear operators1.3 Positive operators1.4 Resolvent set and spectrum1.5 Unitary operators1.6 Projection operators1.7 Finite rank and compact linear operators1.7.1 Finite rank linear operators1.7.2 Compact linear operators1.8 Trace-class and Hilbert–Schmidt operators1.8.1 Trace-class operators1.8.2 Hilbert–Schmidt operator
2 Formulation of quantum systems
2.1 Operator topologies2.2 Operator algebras2.2.1 C∗-algebras2.2.2 von Neumann algebras2.3 Bounded linear functionals2.3.1 Bounded linear functionals on ℍ2.3.2 𝒜∗ and 𝒜∗2.4 Quantum states2.5 GNS representation2.6 Quantum observables and measurements2.6.1 Positive operator valued measures2.6.2 Quantum observables2.6.3 Quantum measurements2.7 Tensor products and direct sums2.7.1 Tensor products of Hilbert spaces and operators2.7.2 Direct sum of Hilbert spaces and operators2.8 Composite quantum systems2.8.1 Composite system as tensor products2.8.2 Partial traces
3 Probability measures and convex functions on 𝒮(ℍ)
3.1 Probability measures on 𝒮(ℍ)3.1.1 Support of Borel measures3.2 Some compactness criteria3.2.1 Compactness criteria on 𝒮(ℍ)3.2.2 Compactness criteria on 𝒫(𝒮(ℍ))3.3 Barycenters3.3.1 A brief review of convex analysis3.3.2 Properties of barycenter3.4 Convex functions on 𝒮(ℍ)3.4.1 σ-convexity and μ-convexity3.4.2 Convex hull and convex closure
4 Completely positive maps
4.1 Definitions and properties4.2 Some technical results4.3 Stinespring representation4.4 Kraus representation
5 Quantum channels and operations
5.1 Quantum operations5.2 Quantum channels5.2.1 Quantum channels in the Schrodinger picture5.2.2 Quantum channels in the Heisenberg picture5.3 Representations and dilations of channels5.3.1 Stinespring representation5.3.2 Unitary dilation5.3.3 Kraus representation5.4 Structural properties of quantum channels5.4.1 Convexity of channels5.4.2 Concatenation of channels5.4.3 Tensor product of channels5.5 Purification5.6 Reversible quantum channels5.7 Complementary channels
6 Approximation and convergence of quantum channels
6.1 Distinguishability of quantum states6.1.1 Fidelity of quantum states6.1.2 Bures distance6.2 Channel fidelity6.2.1 Monotonicity of channel fidelity6.2.2 Ensemble average and entanglement fidelities6.3 Norms on unconstrained channels6.3.1 Diamond norm6.3.2 Complete boundedness norm6.3.3 Bures distance6.4 Norms on constrained channels6.4.1 Energy-constrained operator norm6.4.2 Constrained diamond norms and Bures distances6.5 Approximations of quantum channels6.6 Convergences of channels6.6.1 Strong and uniform convergence6.6.2 Strong∗ convergence of channels
7 von Neumann entropy
7.1 von Neumann entropy on 𝒮(ℍ)7.2 Properties of H(·) on 𝒮(ℍ)7.3 H(·) on subsets of 𝒮(ℍ)7.3.1 Closed convex subsets 𝒜⊂𝒮(ℍ)7.3.2 𝒦H(h)7.3.3 H(·) on an arbitrary closed subset 𝒦7.4 H(·) extended to 𝔗+(ℍ)7.4.1 Properties of H(·) on 𝔗+(ℍ)7.4.2 Discontinuity of extended quantum entropies
8 Relative and conditional entropies
8.1 Quantum relative entropy8.2 H(·‖·) extended to 𝔗+(ℍ)×𝔗+(ℍ)8.3 Quantum conditional entropy8.3.1 Definitions8.3.2 Properties of conditional entropy
9 Channel output entropies
9.1 Monotonicity of output relative entropy and Petz’s theorem9.1.1 Monotonicity of output relative entropy9.1.2 Petz’s theorem on output relative entropy9.2 Continuity of output entropies9.2.1 Positive linear map Φ9.2.2 Quantum operations9.3 Convex closure of output entropy
10 Quantum entanglement
10.1 Schmidt decomposability of statesA. Finite-dimensional caseB. Infinite-dimensional case10.2 Separability of states10.2.1 Matrix of amplitudes10.2.2 Separability10.3 Measurements of entanglement10.3.1 Entropy of entanglement10.3.2 Relative entropy of entanglement10.3.3 Entanglement of formation

11 Quantum mutual and coherent information
11.1 Quantum mutual information11.1.1 Finite-dimensional case11.1.2 Infinite-dimensional case11.2 Coherent information
12 Holevo χ-capacity
12.1 The χ-functions12.1.1 Input χ-function12.1.2 Output χ-function12.2 Holevo χ-capacities12.2.1 Optimal ensembles12.2.2 Compact constraint 𝒜12.2.3 Convex constraint 𝒜12.3 Continuity of Holevo χ-capacity12.4 Holevo χ-quantity and coherent information12.5 Additivity of Holevo χ-capacities12.5.1 Hastings’ counterexamples to additivity12.5.2 Additivity of unconstrained χ-quantity12.5.3 Tensor product of constrained channels12.5.4 Additivity of constrained Holevo χ-capacity
13 Classical capacities of memoryless channels
13.1 Transmissions of classical information13.1.1 Preparation of classical information13.1.2 Encoding and decoding of classical information13.2 Unconstrained classical capacity13.2.1 Holevo–Schumacher–Westmoreland theorem13.2.2 Quantum version of Feinstein’s lemma13.3 Classical capacity of constrained channels13.4 Entanglement-assisted classical capacity13.4.1 Finite-dimensional channels13.4.2 Constrained infinite-dimensional channels13.4.3 Continuity of Cea(·):QC(A,B)→[0,+∞]13.5 Comparison of classical capacities13.5.1 Finite-dimensional unconstrained capacities13.5.2 Infinite-dimensional constrained capacities
14 Structure of quantum memory channels
14.1 Representations of memoryless channels14.1.1 Kraus representation in the Schrodinger picture14.1.2 Kraus representation in the Heisenberg picture14.1.3 Unitary representation in the Schrodinger picture14.2 Constructive approach to memory channels14.2.1 Unitary representation in the Schrodinger picture14.2.2 Unitary representation in the Heisenberg picture14.2.3 Kraus representation of the memory channel14.3 Quasilocal algebras approach14.3.1 Construction of quasilocal algebras14.3.2 Structure of causal channels14.4 Forgetful channels14.4.1 Definition of forgetful channels14.4.2 Topological properties
15 Channels with Markovian memory
15.1 A brief review on Markov chainsA. Probability transition matrixB. Stationary distributionC. Classification of statesD. Ergodic Markov chains15.2 Constructions of Markov memory models15.3 Channels with ergodic Markovian memory15.3.1 Classical capacity15.4 Channels with general Markov memory15.4.1 Preamble for Markovian memory15.4.2 Classical capacity
16 Channels with long-term memory
16.1 The model16.2 Classical product state capacity16.2.1 Infinite-dimensional classical capacity16.2.2 Constrained classical capacity in infinite dimensions16.3 Entanglement-assisted classical capacity16.3.1 Unconstrained case16.3.2 Energy constrained caseBibliography
Subject Index

Content preview from Theory of Quantum Information with Memory

14 Structure of quantum memory channels

In Chapter 13, we investigated n uses of memoryless quantum channels and studied their various properties and classical capacities associated with them. This chapter is devoted to the study of structures of memory quantum channels. As a special case, a class of memory channels, namely forgetful channels will also be explored. This chapter, however, does not address topics on the coding theorem and classical capacity. Special cases of quantum memory channels such as channels with Markovian and long-term memory and their classical capacities are the topics of interest in the next two chapters.

For each $n∈ℕ$ , we use the block coding of length n and size $N_{n}$ as considered in the previous chapter. Let the ...

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Theory of Quantum Information with Memory

by Mou-Hsiung Chang

14 Structure of quantum memory channels

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