Distinguishability and Accessible Information in Quantum Theory

TL;DR

The work develops a comprehensive framework for translating classical information-theoretic distinguishability into quantum-state terms, delivering exact and bound expressions for key quantum measures such as fidelity, mutual information (accessible information), and quantum KL information. It establishes optimal quantum measurements (POVMs) that realize these quantities, derives a simplified Holevo bound, and introduces ensemble-dependent bounds that sharpen our understanding beyond mean-state quantities. The dissertation also connects distinguishability to fundamental limits on information gathering, notably proving a general no-broadcasting theorem for mixed states and formulating an algebraic tradeoff between inference and disturbance. Collectively, these results illuminate the limits of quantum state discrimination and have direct implications for quantum cryptography and quantum communication theory. The methods combine operator inequalities, purification techniques, and Lyapunov-operator tools to yield both exact formulas and practically computable bounds.

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.

Figures (11)

• Figure 1: Binary Question Schemes
• Figure 2: Coding Tree Imbedded in a Full Tree
• Figure 3: Probability of error in guessing a photon's polarization that is either horizontal or $45^\circ$ from the horizontal. Error probability is plotted here as a function of measurement (i.e., radians from the horizontal) and number of measurement repetitions $M$ before the guess is made.
• Figure 4: The Holevo upper bound $S(t)$, the upper bound $L(t)$, the information $I(t)$ extractable by optimal orthogonal projection-valued measurement (found numerically), the lower bound $M(t)$, and the Jozsa-Robb-Wootters lower bound $Q(t)$, all for the case that $\hat{\rho}_0$ is pure ($a=1$), $\hat{\rho}_1$ is mixed with $b=2/3$, and the angle between the two Bloch vectors is $\pi/3$.
• Figure 5: All the bounds to accessible information studied here, for the case that $\hat{\rho}_0$ is pure ($a=1$), $\hat{\rho}_1$ is mixed with $b=2/3$, and the angle between the two Bloch vectors is $\pi/4$.

Theorems & Definitions (14)

• Theorem 2.1: Chernoff
• Theorem 2.2: Chernoff
• Theorem 2.3: Aczél
• Lemma 2.1
• Theorem 2.4
• Theorem 2.5
• Corollary 2.1
• Lemma 2.2
• Theorem 2.6
• Theorem 2.7: Daróczy