Classical and quantum information theory are simply explained. To be more specific it is clarified why Shannon entropy is used as measure of classical information and after a brief review of quantum mechanics it is possible to demonstrate why the density matrix is the main tool of quantum information theory. Then von Neumann entropy is introduced and with its help a great difference between classical and quantum information theory is presented: quantum entanglement. Moreover an information theoretic interpretation of quantum measurement is discussed. Data compression, error correction and noisy channel transmission are simply demonstrated for both classical and quantum cases. Finally using the above theory quantum cryptography is reviewed and the possibility of a commercial device realizing it is explored.
Classical and quantum information theory are simply explained. To be more specific it is clarified why Shannon entropy is used as measure of classical information and after a brief review of quantum mechanics it is possible to demonstrate why the density matrix is the main tool of quantum information theory. Then von Neumann entropy is introduced and with its help a great difference between classical and quantum information theory is presented: quantum entanglement. Moreover an information theoretic interpretation of quantum measurement is discussed. Data compression, error correction and noisy channel transmission are simply demonstrated for both classical and quantum cases. Finally using the above theory quantum cryptography is reviewed and the possibility of a commercial device realizing it is explored.
Abstract This book deals with the basic concept of information, its importance, and the similarities and differences between classical and quantum information. First, classical information theory and its application to communication are discussed. Quantum mechanics and communication based on the laws of quantum mechanics are then considered. Quantum entanglement, a key resource in quantum communication, is analysed both from the fundamental perspective and from the perspective of its information-processing capability. This book also introduces quantum computation (in the context of the search and factorisation algorithms) and the basics of quantum error correction. Throughout this book, the connections between information theory, thermodynamics, and (quantum) physics are emphasised and discussed. Other topics covered in the book are data compression, related measures of information such as relative entropy, joint entropy, conditional entropy, and mutual information, along with the capacity of a noisy channel.
I find that a great part of the information I have was acquired by looking up something and finding something else on the way. Franklin P. Adams The topic of this chapter is quantum information , i.e., information in the quantum world. The material that is presented here lies at the core of a variety of areas within the compass of quantum computation, such as quantum cryptography, described in Chapter 9, and quantum data compression. As quantum information extends and modifies concepts developed in classical information theory, a brief review of the mathematical notion of information is in order. In Section 10.1, we recall the basics of classical information theory. Section 10.2 generalizes this work to get quantum information theory. Section 10.3 discusses classical and quantum data compression. We conclude with a small section on errorcorrecting codes. CLASSICAL INFORMATION AND SHANNON ENTROPY What is information really? In the mid-forties, an American mathematician named Claude Shannon set out to establish a mathematical theory of information on a firm basis. To see this, we use Alice and Bob. Alice and Bob are exchanging messages. Let us say that Alice can only send one of four different messages coded by the letters A, B, C , and D .
In this Thesis, several results in quantum information theory are collected, most of which use entropy as the main mathematical tool. *While a direct generalization of the Shannon entropy to density matrices, the von Neumann entropy behaves differently. A long-standing open question is, whether there are quantum analogues of unconstrained non-Shannon type inequalities. Here, a new constrained non-von-Neumann type inequality is proven, a step towards a conjectured unconstrained inequality by Linden and Winter. *IID quantum state merging can be optimally achieved using the decoupling technique. The one-shot results by Berta et al. and Anshu at al., however, had to bring in additional mathematical machinery. We introduce a natural generalized decoupling paradigm, catalytic decoupling, that can reproduce the aforementioned results when used analogously to the application of standard decoupling in the asymptotic case. *Port based teleportation, a variant of standard quantum teleportation protocol, cannot be implemented perfectly. We prove several lower bounds on the necessary number of output ports N to achieve port based teleportation for given error and input dimension, showing that N diverges uniformly in the dimension of the teleported quantum system, for vanishing error. As a byproduct, a new lower bound for the size of the program register for an approximate universal programmable quantum processor is derived. *In the last part, we give a new definition for information-theoretic quantum non-malleability, strengthening the previous definition by Ambainis et al. We show that quantum non-malleability implies secrecy, analogous to quantum authentication. Furthermore, non-malleable encryption schemes can be used as a primitive to build authenticating encryption schemes. We also show that the strong notion of authentication recently proposed by Garg et al. can be fulfilled using 2-designs.
This chapter sets the basis of quantum information theory (QIT) . The central purpose of QIT is to qualify the transmission of either classical or quantum information over quantum channels . The starting point of the QIT description is von Neumann entropy , S (ρ), which represents the quantum counterpart of Shannon's classical entropy, H ( X ). Such a definition rests on that of the density operator (or density matrix ) of a quantum system, ρ, which plays a role similar to that of the random-events source X in Shannon's theory. As we shall see, there also exists an elegant and one-to-one correspondence between the quantum and classical definitions of the entropy variants relative entropy , joint entropy , conditional entropy , and mutual information . But such a similarity is only apparent. Indeed, one becomes rapidly convinced from a systematic analysis of the entropy's additivity rules that fundamental differences separate the two worlds. The classical notion of information correlation between two event sources for quantum states shall be referred to as quantum entanglement . We then define a quantum communication channel, which encodes and decodes classical information into or from quantum states. The analysis shows that the mutual information H ( X ; Y ) between originator and recipient in this communication channel cannot exceed a quantity χ, called the Holevo bound , which itself satisfies χ ≤ H ( X ), where H ( X ) is the entropy of the originator's classical information source.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy (or simply quantum entropy ). It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state.1 The quantum entropy gives meaning to the notion of an information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes on a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations. We then define several other quantum information measures, such as quantum mutual information, that bear similar definitions as in the classical world, but with Shannon entropies replaced with quantum entropies. This replacement may seem to make quantum entropy somewhat trivial on the surface, but a simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information (recall that the largest the mutual information can be in the classical world is one bit for the case of two maximally correlated bits). We then discuss several entropy inequalities that play an important role in quantum information processing: the monotonicity of quantum relative entropy, strong subadditivity, the quantum data-processing inequalities, and continuity of quantum entropy.
Abstract This chapter discusses the principles of quantum information. The quantum mechanical equivalent of the Shannon noisy-channel communication theorem is obtained, along with some profound statements about the behavior of quantum information during generalised quantum measurements. The latter are very important in studies of quantum entanglement, but also in exploring the connections between thermodynamics, information theory, and quantum physics. This chapter also discusses equalities and inequalities related to entropy, the Holevo bound, capacity of a bosonic channel, information gained through measurements, relative entropy and thermodynamics, entropy increase due to erasure, and Landauer's erasure and data compression.