Distinguishability and Accessible Information in Quantum Theory

Abstract

This document focuses on translating various information-theoretic measures of distinguishability for probability distributions into measures of distin- guishability for quantum states. These measures should have important appli- cations in quantum cryptography and quantum computation theory. The results reported include the following. An exact expression for the quantum fidelity between two mixed states is derived. The optimal measurement that gives rise to it is studied in detail. Several upper and lower bounds on the quantum mutual information are derived via similar techniques and compared to each other. Of note is a simple derivation of the important upper bound first proved by Holevo and an explicit expression for another (tighter) upper bound that appears implicitly in the same derivation. Several upper and lower bounds to the quan- tum Kullback relative information are derived. The measures developed are also applied to ferreting out the extent to which quantum systems must be disturbed by information gathering measurements. This is tackled in two ways. The first is in setting up a general formalism for describing the tradeoff between inference and disturbance. The main point of this is that it gives a way of expressing the problem so that it appears as algebraic as that of the problem of finding quantum distinguishability measures. The second result on this theme is a theorem that prohibits "broadcasting" an unknown (mixed) quantum state. That is to say, there is no way to replicate an unknown quantum state onto two separate quantum systems when each system is considered without regard to the other. This includes the possibility of correlation or quantum entanglement between the systems. This result is a significant extension and generalization of the standard "no-cloning" theorem for pure states.

Ph. D. Dissertation, University of New Mexico, prepared in LaTeX, 174 pages in single-space format, 640 equations, 528 references, 11 PostScript figures, requires epsfig.sty

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• Choice of Measurement Sets in Qubit Tomography
• The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography
• General conditions for approximate quantum error correction and near-optimal recovery channels
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• Adversarial quantum circuit learning for pure state approximation
• On Asymptotic Quantum Statistical Inference
• On Variational Expressions for Quantum Relative Entropies
• Fast parallel circuits for the quantum Fourier transform
• Fidelity Between Partial States as Signature of Quantum Phase Transitions
• Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map
• Statistical speed of quantum states: Generalized quantum Fisher information and Schatten speed
• Hilbert's projective metric in quantum information theory
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• Non-linear operations in quantum information theory
• Transition Probability (Fidelity) and Its Relatives
• Optimal State Discrimination in General Probabilistic Theories
• Versatile microwave-driven trapped ion spin system for quantum information processing
• Macroscopic Distinguishability Between Quantum States Defining Different Phases of Matter: Fidelity and the Uhlmann Geometric Phase
• Locking classical information
• Estimating distinguishability measures on quantum computers
• Cryptographic Distinguishability Measures for Quantum Mechanical States
• Quantum-Bayesian Coherence
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• Quantifying dynamical coherence with dynamical entanglement
• Second law of thermodynamics for relativistic fluids formulated with relative entropy
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• Assessments of macroscopicity for quantum optical states
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• Subtleties of witnessing quantum coherence in non-isolated systems
• Decrease of Fisher information and the information geometry of evolution equations for quantum mechanical probability amplitudes
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• Quantum Public-Key Encryption with Information Theoretic Security
• Optimizing entropy relative to a channel or a subalgebra
• Cryptography in a Quantum World
• A relational approach to quantum reference frames for spins
• Optimal Observables for Minimum-Error State Discrimination in General Probabilistic Theories
• Norms and Cones in the Theory of Quantum Entanglement
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• Compatibility of quantum states
• Generating functions for generalized binomial distributions
• Fluctuation Theorems for multitime processes
• Complementarity and Entanglement in Quantum Information Theory
• Just Two Nonorthogonal Quantum States
• Reverse test and quantum analogue of classical fidelity and generalized fidelity
• Quantum Complexity: restrictions on algorithms and architectures
• Lowering Helstrom Bound with non-standard coherent states
• Towards optimal experimental tests on the reality of the quantum state
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• On conclusive eavesdropping and measures of mutual information in quantum key distribution
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• Non-monotonicity of trace distance under tensor products
• Distances between quantum states in the tomographic-probability representation
• Microfabrication techniques for trapped ion quantum information processing
• Topics in quantum information and the theory of open quantum systems
• Distinguishing n Hamiltonians on C^n by a single measurement
• Supercomputer simulations of transmon quantum computers
• How much does it cost to teleport?
• Quantum Chernoff bound as a measure of efficiency of quantum cloning for mixed states
• Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction
• Tighter Lower Bounds on Quantum Annealing Times
• Identification of a natural fieldlike entanglement resource in trapped-ion chains
• Fast, High-Fidelity, Quantum Non-demolition Readout of a Superconducting Qubit Using a Transverse Coupling
• Power of unentangled measurements on two antiparallel spins
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• Some properties of partial fidelities
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• Characterizing the Depolarizing Quantum Channel in Terms of Riemannian Geometry
• Nonextensive thermodynamic functions in the Schrödinger-Gibbs ensemble
• Proof of the Orthogonal Measurement Conjecture for Qubit States
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• Optimal Bounds on Functions of Quantum States under Quantum Channels
• Decoherence without decoherence