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Information gain in quantum theory

Mohammad F. Maghrebi

Published Jan 1, 2008• 0 citations
Publisher

Abstract

In this thesis I address the fundamental question that how the information gain is possible in the realm of quantum mechanics where a single measurement alters the state of the system. I study an ensemble of particles in some unknown (but product) state in detail and suggest an optimal way of gaining the maximum information and also quantify the corresponding information exactly. We find a rather novel result which is quite different from other well-known definitions of the information gain in quantum theory.

DOI
10.14288/1.0066769

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In this thesis I address the fundamental question that how the information gain is possible in the realm of quantum mechanics where a single measurement alters the state of the system. I study an ensemble of particles in some unknown (but product) state in detail and suggest an optimal way of gaining the maximum information and also quantify the corresponding information exactly. We find a rather novel result which is quite different from other well-known definitions of the information gain in quantum theory.
In quantum information the fundamental information-containing system is the qubit. A measurement of a single qubit can at most yield one classical bit. However, a dichotomous measurement of an unknown qubit will yield much less information about the qubit state. We use Bayesian inference to compute how much information one progressively gets by making sucessive, individual measurements on an ensemble of identically prepared qubits. Perhaps surprisingly, even if the measurements are arranged so that each measurement yields one classical bit, that is, the two possible measurement outcomes are a priori equiprobable, it takes almost a handful of measurements before one has gained one bit of information about the gradually concentrated qubit probability density. We also show that by following a strategy that reaps the maximum information per measurement, we are led to a mutually unbiased basis as our measurement bases. This is a pleasing, although not entirely surprising, result.
We calculate the single-outcome information gain of quantum measurements on a completely unknown qubit state randomly chosen from different state ensembles. It is shown that the information gain decreases when the ensemble size increases. We obtain analytic single-outcome information gain formulas for continuous equatorial and spherical state ensembles, which show that the states in the former ensemble are relatively easier to identify. The squared fidelity between post-measurement state and pre-measurement state is also calculated, and the qualitative tradeoff relations between information gain and fidelity are illustrated by our results.
We determine the optimal measurement that maximizes the average information gain about the state of a qubit system. The qubit is prepared in one of two known states with known prior probabilities. To treat the problem analytically we employ the formalism developed for the maximum confidence quantum state discrimination strategy and obtain the POVM which optimizes the information gain for the entire parameter space of the system. We show that the optimal measurement coincides exactly with the minimum-error quantum measurement only for two pure states, or when the two states have the same Bloch radius or they are on the same diagonal of the Bloch disk.
We address the problem of extracting information from a single quantum system whose state is known to be in one of several possible states. In the generic case, it is notoriously difficult to find the optimal measurement, that is the measurement that provides the most possible information about the system's state. We consider a simple general prescription for a measurement that is typically not optimal but appears to be quite good. It seems to be particularly good when the states to be distinguished are equally likely and almost orthogonal.
This thesis is a study of how information theory can be used to analyse and interpret quantum mechanical systems, specifically, systems that are being used for information processing and communication tasks. Quantum information science is a relatively new field, and it has only been recently that we have started to look at using quantum mechanical systems to accomplish information processing tasks that are not possible using classical systems. As this method of information processing is a relatively new paradigm, there are many possibilities for new algorithms and discoveries in this area of computing. The application of information theory towards the analysis and description of quantum mechanical systems yields new interesting results - it can even be argued that information theory provides a natural and intuitive way to intepret quantum mechanics, as correlations and measurement outcomes in quantum processes can be described using information theory. Chapter 1 is an introduction to the strange new world of quantum mechanics and takes a brief look at one of the applications of quantum information science in quantum cryptography. A mathematical overview of the postulates and framework that quantum mechanics is built on is presented in Chapter 2. Chapter 3 is a brief study of classical information theory and some of the fundamental concepts and ideas that form the crux of the subject. In Chapter 4, we present the main results of the thesis by taking a look at an optimal measurement scheme that improves on previous schemes by allowing us to extract the maximal amount of information from states in a quantum channel of specific geometry. The findings have an impact on various other areas of the field, including security analysis of quantum cryptographic protocols and building better quantum cloning machines.