[Paper Review] Distinguishability and Accessible Information in Quantum Theory

Christopher A. Fuchs|ArXiv.org|Jan 23, 1996

Quantum Information and CryptographyComputer Science456 references204 citations

TL;DR

This dissertation establishes a rigorous information-theoretic framework for quantifying the distinguishability of nonorthogonal quantum states using three key measures: fidelity, Kullback-Leibler divergence, and mutual information. It derives exact expressions and optimal measurements for these quantities, with key contributions including a simplified proof of Holevo's upper bound on quantum mutual information and a novel method for generating increasingly tight lower bounds via nonlinear matrix equations.

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.

Motivation & Objective

• To formalize the operational distinguishability of nonorthogonal quantum states using classical information-theoretic measures as a foundation.
• To address the fundamental problem that nonorthogonal quantum states cannot be perfectly distinguished, necessitating statistical measures of distinguishability.
• To derive quantum versions of classical distinguishability measures—fidelity, Kullback-Leibler divergence, and mutual information—by optimizing over quantum measurements.
• To establish a formalism linking inference and disturbance in quantum measurements, culminating in a proof of the no-broadcasting theorem for unknown quantum states.
• To provide exact expressions and tight bounds for quantum distinguishability measures, with applications in quantum cryptography, computation, and communication.

Proposed method

• Uses classical information-theoretic measures (fidelity, Kullback-Leibler divergence, mutual information) as starting points for defining quantum distinguishability.
• Applies optimization over positive operator-valued measures (POVMs) to define the 'quantum distinguishability' as the optimal value of each measure.
• Employs advanced matrix analysis techniques, including solutions to nonlinear matrix equations, to derive progressively tighter lower bounds on quantum Kullback information.
• Utilizes trace inequalities and spectral theory to derive bounds on quantum mutual information, including a simplified derivation of Holevo's 1973 upper bound.
• Applies the purification map and sesquilinear form techniques to analyze quantum state distinguishability and measurement disturbance.
• Develops an algebraic formalism to express the trade-off between inference (information gain) and disturbance (state disturbance) in quantum measurements.

Experimental results

Research questions

• RQ1 How can classical distinguishability measures be adapted to quantify the distinguishability of nonorthogonal quantum states?
• RQ2 What is the optimal quantum measurement that maximizes or minimizes each of the three distinguishability measures: fidelity, Kullback-Leibler divergence, and mutual information?
• RQ3 What are the tightest known upper and lower bounds on quantum mutual information, and how can they be derived using matrix analysis?
• RQ4 Can a systematic method be developed to generate increasingly tighter lower bounds on quantum Kullback-Leibler information?
• RQ5 To what extent is it possible to infer the identity of a quantum state without disturbing it, and what fundamental limits exist?

Key findings

• An exact expression for the quantum fidelity is derived, with the optimal measurement for fidelity maximization fully characterized.
• A simplified derivation of Holevo's 1973 upper bound on quantum mutual information is provided, clarifying its structure and assumptions.
• An explicit, tighter upper bound on quantum mutual information is identified, which appears implicitly in the same derivation as the simplified Holevo bound.
• A systematic method for generating successively tighter lower bounds on quantum Kullback-Leibler information is developed, relying only on solving higher-order nonlinear matrix equations.
• The no-broadcasting theorem is proven: it is impossible to create two copies of an unknown quantum state on separate systems without correlation, even if the copies are not individually pure.
• The formalism for trade-off between inference and disturbance is established, showing that the problem can be cast algebraically, analogous to distinguishability measure optimization.