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arXiv:cond-mat/0508488

Electron-hole entanglement in the Fermi sea

C. W. J. Beenakker

Published Aug 20, 2005• 16 citations
arXivHTMLPDF

Abstract

I. Introduction (Preface, Exciton entanglers, Photon entanglers) II. Entanglement basics (Quantum versus classical correlations, Bell inequality, Entanglement measures for pure states, Entanglement measures for mixed states, Particle conservation, Phase reference) III. How to entangle free particles (Free bosons, Free fermions) IV. Spin versus orbital entanglement V. Entanglement detection by noise measurements (Tunneling regime, Beyond the tunneling regime, Full counting statistics) VI. Loss of entanglement by dephasing VII. Quantum entanglement pump VIII. Teleportation by electron-hole annihilation IX. Three-qubit entanglement X. The experimental challenge

DOI
10.48550/arxiv.cond-mat/0508488

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•Nov 29, 2004•13 citations
I. Introduction (Preface, Exciton entanglers, Photon entanglers) II. Entanglement basics (Quantum versus classical correlations, Bell inequality, Entanglement measures for pure states, Entanglement measures for mixed states, Particle conservation, Phase reference) III. How to entangle free particles (Free bosons, Free fermions) IV. Spin versus orbital entanglement V. Entanglement detection by noise measurements (Tunneling regime, Beyond the tunneling regime, Full counting statistics) VI. Loss of entanglement by dephasing VII. Quantum entanglement pump VIII. Teleportation by electron-hole annihilation IX. Three-qubit entanglement X. The experimental challenge
Entanglement is one of most peculiar characteristic of Quantum Mechanics (probably the other most mentioned is the particle tunneling through a barrier) because it entails correlations that supersedes classical correlations. It is conceived as the genuine quantum characteristic by many researchers and quite recently it has emerged as a physical resource to produce non-classical task, like quantum teleportation, quantum cryptography and quantum information. Entanglement could be distributed (efficient quantum communication is equivalent to efficient entanglement distribution (Plenio & Virmani, 2007)), concentrated (given an amount of entanglement, equal to the full content of entanglement in n pairs of identical entangled pure state systems, it is possible, using local operations on each system, to concentrate the total amount of entanglement into a smaller number m, m < n, of maximally pure entangled state system (Bennett, et al.; 1996)) and used to perform many quantum information tasks useful to overcome technical restrictions present on classical communication. In this chapter, we review the entanglement of quantum mechanical systems. First, in Section 2, we give a characterization of entanglement in terms of its special features as resource and its mathematical structure. Then, in section 3, we review some of the Bell inequalities. In Sections 4 and 5, we review some of the most important entanglement measures published for two and three entangled systems. After that, in Section 6, we give a characterization of quantum gates and operators by its entanglement power, i. e. the amount of entanglement that they can produce when acting on an arbitrary state. Finally, in Section 7, we briefly review the experimental detection of entanglement.
Preface.- I Introduction and Motivation.- II Diffusion Processes and their Transformations.- III Duality and Time Reversal of Diffusion Processes.- IV Equivalence of Diffusion and Schrodinger Equations.- V Variational Principle.- VI Diffusion Processes in q-Representation.- VII Segregation of a Population.- VIII The Schrodinger Equation can be a Boltzmann Equation.- IX Applications of the Statistical Model for Schrodinger Equations.- X Relative Entropy and Csiszar's Projection.- XI Large Deviations.- XII Non-Linearity Induced by the Branching Property.- Appendix.- References.- Index.
Quantum information technology largely relies on a precious and fragile resource, quantum entanglement, a highly nontrivial manifestation of the coherent superposition of states of composite quantum systems. However, our knowledge on the time evolution of this resource under realistic conditions, when corrupted by environment-induced decoherence, is hitherto rather limited, and general statements on entanglement dynamics in open systems are scarce. The lecture will start from a short recollection of the basic elements of entanglement theory for finite dimensional composite mixed quantum states, spell out the central difficulties for the formulation of a general theory, and then address various aspects of open system entanglement dynamics in variable dimensions. Also the issue of direct experimental probing of the entanglement inscribed into an unknown mixed quantum state will be briefly discussed.
A Survey by the Editors.- I.1 Violation of the Noncrossing Rule: The Hubbard Hamiltonian for Benzene.- I.2 Two Theorems on the Hubbard Model.- I.3 The Hubbard Model: Some Rigorous Results and Open Problems.- I.4 Flux Phase of the Half-Filled Band.- II.1 Proof of the Peierls Instability in One Dimension.- II.2 Stability of the Peierls Instability for Ring Shaped Molecules.- III.1 A Model for Crystallization: A Variation on the Hubbard Model.- III.2 Phase Separation due to Quantum Mechanical Correlations.- IV.1 Theory of Ferromagnetism and the Ordering of Electronic Energy Levels.- IV.2 Ordering Energy Levels of Interacting Spin Systems.- IV.3 Magnetic Properties of Some Itinerant-Electron Systems at T > 0.- V.1 Phase Transitions in Reservoir-Driven Open Systems with Applications to Lasers and Superconductors.- V.2 Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field.- V.3 Constructive Macroscopic Quantum Electrodynamics.- V.4 The Laser: A Reversible Quantum Dynamical System with Irreversible Classical Macroscopic Motion.- VI.1 A Proof of Part of Haldane's Conjecture on Spin Chains.- VI.2 Rigorous Results on Valence-Bond Ground States in Antiferromagnets.- VI.3 Valence Bond Ground States in Isotropic Quantum Antiferromagnets.- VI.4 Ground State Properties of a Fully Frustrated Quantum Spin System.- VII.1 Exact Ground State Energy of the Strong-Coupling Polaron.- VIII.1 Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas.- VIII.2 Bose-Einstein Quantum Phase Transition in an Optical Lattice Model.- VIII.3 The Quantum-Mechanical Many-Body Problem: The Bose Gas.- Some of the Early History of Exactly Soluble Models.- IX.1 Exact Solution of the Problem of the Entropy of Two-Dimensional Ice.- IX.2 Exact Solution of the F Model of an Antiferroelectric.- IX.3 Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric.- IX.4 Residual Entropy of Square Ice.- IX.5 Two-Dimensional Ferroelectric Models.- IX.6 Relations Between the 'Percolation' and 'Colouring' Problem and Other Graph-Theoretical Problems Associated with Regular Planar Lattices: Some Exact Results for the 'Percolation' Problem.- IX.7 Analytic Properties of the Free Energy for the Ice Models.- X.1 Two-Dimensional Ising Model as a Soluble Problem of Many Fermions.- X.2 Solution of the Dimer Problem by the Transfer Matrix Method.- XI.1 Two Soluble Models of an Antiferromagnetic Chain.- XII.1 Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State.- XII.2 Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum.- XIII.1 Delta-Function Fermi Gas with Two-Spin Deviates.- XIII.2 Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model on One Dimension.- XIV.1 Exact Solution of a Many-Fermion System and Its Associated Boson Field.- Selecta of Elliott H. Lieb.- Publications of Elliott H. Lieb.