A Mini-Introduction To Information Theory

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Some basic facts about QM systems . . . . . . . . . . . . . . . . . . . . . . 2 3 The partial trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 Classical Shannon entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 Sundry formulas for the Shannon entropy . . . . . . . . . . . . . . . . . . . 4 6 Von Neumann Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7 Sundry formulas for the Von Neumann entropy . . . . . . . . . . . . . . . . 5 8 H(A) versus S(A)? What is dierent? . . . . . . . . . . . . . . . . . . . . 6 9 Case I (Classical) Independent qubits . . . . . . . . . . . . . . . . . . . . . 6 10 Case II (Classical) Correlated qubits . . . . . . . . . . . . . . . . . . . . . 7 11 Case III (Nonclassical--Purely Quantum Mechanical) Entangled (superc

2001

Abstract Quantum information theory provides a foundation for such topics as quantum cryptography, quantum error-correction and quantum teleportation. This paper seeks to provide an introduction to quantum information theory for non-physicists at an undergraduate level. It covers basic concepts in quantum mechanics as well as in information theory, and proceeds to explore some results such as Von Neumann entropy, Schumacher coding and quantum error-correction.

Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences and distinguishability and is formalized using the distinctions (" dits ") of a partition (a pair of points distinguished by the partition). All the definitions of simple, joint, conditional and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates (cohered together in the pure superposition state being measured) that are distinguished by the measurement (decohered in the post-measurement mixed state). Both the classical and quantum versions of logical entropy have simple interpretations as " two-draw " probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states.

Physics Letters A, 1991

Contrary to the widespread belief, Shannon's information and entropy are not in general equivalent. The purpose of this Letter is to discuss their conceptual difference and to pinpoint to mathematical reason for this. This fact is further illustrated through a toy model consisting of a harmonic oscillator in a coherent state, showing explicitly the dependence of Shannon's information on the class of quantum states the system is in. The interrelation between Shannon's information and entropy is reestablished after a theorem telling which class of states maximizes the (missing) information is proved. We shall conclude that entropy is the maximum amount of missing information.

Arxiv preprint quant-ph/0411172, 2004

We look at certain thought experiments based upon the 'delayed choice' and 'quantum eraser' interference experiments, which present a complementarity between information gathered from a quantum measurement and interference effects. It has been argued that these experiments show the Bohm interpretation of quantum theory is untenable. We demonstrate that these experiments depend critically upon the assumption that a quantum optics device can operate as a measuring device, and show that, in the context of these experiments, it cannot be consistently understood in this way. By contrast, we then show how the notion of 'active information' in the Bohm interpretation provides a coherent explanation of the phenomena shown in these experiments. We then examine the relationship between information and entropy. The thought experiment connecting these two quantities is the Szilard Engine version of Maxwell's Demon, and it has been suggested that quantum measurement plays a key role in this. We provide the first complete description of the operation of the Szilard Engine as a quantum system. This enables us to demonstrate that the role of quantum measurement suggested is incorrect, and further, that the use of information theory to resolve Szilard's paradox is both unnecessary and insufficient. Finally we show that, if the concept of 'active information' is extended to cover thermal density matrices, then many of the conceptual problems raised by this paradox appear to be resolved.

Journal of Mathematical Physics, 2000

A method of representing probabilistic aspects of quantum systems is introduced by means of a density function on the space of pure quantum states. In particular, a maximum entropy argument allows us to obtain a natural density function that only reflects the information provided by the density matrix. This result is applied to derive the Shannon entropy of a quantum state. The information theoretic quantum entropy thereby obtained is shown to have the desired concavity property, and to differ from the the conventional von Neumann entropy. This is illustrated explicitly for a two-state system.

2021

The information content of a source is defined in terms of the minimum number of bits needed to store the output of the source in a perfectly recoverable way. A similar definition can be given in the case of quantum sources, with qubits replacing bits. In the mentioned cases the information content can be quantified through Shannon’s and von Neumann’s entropy, respectively. Here we extend the definition of information content to operational probabilistic theories, and prove relevant properties as the subadditivity, and the relation between purity and information content of a state. We prove the consistency of the present notion of information content when applied to the classical and the quantum case. Finally, the relation with one of the notions of entropy that can be introduced in general probabilistic theories, the maximum accessible information, is given in terms of a lower bound.

2015

Mark M. Wilde, Assistant Professor at Louisiana State University, has improved this theorem in a way that allows for understanding how quantum measurements can be approximately reversed under certain circumstances. The new results allow for understanding how quantum information that has been lost during a measurement can be nearly recovered, which has potential implications for a variety of quantum technologies. [9]

2021

Logical entropy gives a measure, in the sense of measure theory, of the distinctions of a given partition of a set, an idea that can be naturally generalized to classical probability distributions. Here, we analyze how fundamental concepts of this entropy and other related definitions can be applied to the study of quantum systems, leading to the introduction of the quantum logical entropy. Moreover, we prove several properties of this entropy for generic density matrices that may be relevant to various areas of quantum mechanics and quantum information. Furthermore, we extend the notion of quantum logical entropy to post-selected systems.

Physical Review A, 2002

The theory of noncommutative dynamical entropy and quantum symbolic dynamics for quantum dynamical systems is analised from the point of view of quantum information theory. Using a general quantum dynamical system as a communication channel one can define different classical capacities depending on the character of resources applied for encoding and decoding procedures and on the type of information sources. It is shown that for Bernoulli sources the entanglement-assisted classical capacity, which is the largest one, is bounded from above by the quantum dynamical entropy defined in terms of operational partitions of unity. Stronger results are proved for the particular class of quantum dynamical systems-quantum Bernoulli shifts. Different classical capacities are exactly computed and the entanglementassisted one is equal to the dynamical entropy in this case.