The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography

Abstract

This thesis presents a study of the structure of bipartite quantum states. In the first part, the representation theory of the unitary and symmetric groups is used to analyse the spectra of quantum states. In particular, it is shown how to derive a one-to-one relation between the spectra of a bipartite quantum state and its reduced states, and the Kronecker coefficients of the symmetric group. In the second part, the focus lies on the entanglement of bipartite quantum states. Drawing on an analogy between entanglement distillation and secret-key agreement in classical cryptography, a new entanglement measure, `squashed entanglement', is introduced.

PhD thesis, February 2006, University of Cambridge. Part I contains results from quant-ph/0409016 and quant-ph/0511029, and analyses Horn's problem in this context. Part II reviews entanglement measures, presents results from quant-ph/0308088 and quant-ph/0501090, and provides new material on entanglement measures, information-gain versus disturbance tradeoffs and cheat sensitive quantum string commitment

References in corpus (14)

• On the quantum, classical and total amount of correlations in a quantum state
• Quantum information can be negative
• Randomizing quantum states: Constructions and applications
• Classical and quantum communication without a shared reference frame
• Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
• The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
• Quantum marginal problem and representations of the symmetric group
• Uncertainty, Monogamy, and Locking of Quantum Correlations
• A Generic Security Proof for Quantum Key Distribution
• Binegativity and geometry of entangled states in two qubits
• Coherent states, entanglement, and geometric invariant theory
• On the one-particle reduced density matrices of a pure three-qutrit quantum state
• Inequalities and separations among assisted capacities of quantum channels
• Computing finite-dimensional bipartite quantum separability

Cited by in corpus (17)

• Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
• Permutationally invariant state reconstruction
• Entanglement Theory and the Quantum Simulation of Many-Body Physics
• Entanglement Distillation; A Discourse on Bound Entanglement in Quantum Information Theory
• Cryptography in a Quantum World
• Complexity is not Enough for Randomness
• End-to-End Capacities of Imperfect-Repeater Quantum Networks
• Distributed Compression and Squashed Entanglement
• Symmetric functions of qubits in an unknown basis
• All Kronecker coefficients are reduced Kronecker coefficients
• The Role of Entanglement in Quantum Communication, and Analysis of the Detection Loophole
• Quantum Cryptography Protocol Based on Sending Entangled Qubit Pairs
• Max- Relative Entropy of Entanglement, alias Log Robustness
• Optimal Manipulation Of Correlations And Temperature In Quantum Thermodynamics
• The Saxl Conjecture and the Dominance Order
• Irreversibility of entanglement loss
• Entanglement and its applications in systems with many degrees of freedom