This note will introduce some notation and definitions for information theoretic quantities in the context of quantum systems, such as (conditional) entropy and (conditional) mutual information. We will employ the natural C*-algebra formalism, and it turns out that one has an allover dualism of language: we can define everything for (compatible) observables, but also for (compatible) C*-subalgebras. The two approaches are unified in the formalism of quantum operations, and they are connected by a very satisfying inequality, generalizing the well known Holevo bound. Then we turn to communication via (discrete memoryless) quantum channels: we formulate the Fano inequality, bound the capacity region of quantum multiway channels, and comment on the quantum broadcast channel.
This note will introduce some notation and definitions for information theoretic quantities in the context of quantum systems, such as (conditional) entropy and (conditional) mutual information. We will employ the natural C*-algebra formalism, and it turns out that one has an allover dualism of language: we can define everything for (compatible) observables, but also for (compatible) C*-subalgebras. The two approaches are unified in the formalism of quantum operations, and they are connected by a very satisfying inequality, generalizing the well known Holevo bound. Then we turn to communication via (discrete memoryless) quantum channels: we formulate the Fano inequality, bound the capacity region of quantum multiway channels, and comment on the quantum broadcast channel.
Coding theorems and (strong) converses for memoryless quantum communication channels and quantum sources are proved: for the quantum source the coding theorem is reviewed, and the strong converse proven. For classical information transmission via quantum channels we give a new proof of the coding theorem, and prove the strong converse, even under the extended model of nonstationary channels. As a by-product we obtain a new proof of the famous Holevo bound. Then multi-user systems are investigated, and the capacity region for the quantum multiple access channel is determined. The last chapter contains a preliminary discussion of some models of compression of correlated quantum sources, and a proposal for a program to obtain operational meaning for quantum conditional entropy. An appendix features the introduction of a notation and calculus of entropy in quantum systems.
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.
We define classical quantum multiway channels for transmission of classical information, after the previous work by Allahverdyan and Saakian (see Quantum Computing and Quantum Communications (Lecture Notes in Computer Science). Berlin, Germany: Springer-Verlag, vol.1509, 1999). Bounds on the capacity region are derived in a uniform way, which are analogous to the classically known ones, simply replacing Shannon (1961) entropy with von Neumann (1955) entropy. For the single receiver case (multiple-access channel) the elect capacity region is determined. These results are applied to the case of noisy channels, with arbitrary input signal states. A second issue of this work is the presentation of a calculus of quantum information quantities, based on the algebraic formulation of quantum theory.
We define classical quantum multiway channels for transmission of classical information, after recent work by Allahverdyan and Saakian. Bounds on the capacity region are derived in a uniform way, which are analogous to the classically known ones, simply replacing Shannon entropy with von Neumann entropy. For the single receiver case (multiple-access channel) the exact capacity region is determined. These results are applied to the case of noisy channels, with arbitrary input signal states. A second issue of this work is the presentation of a calculus of quantum information quantities, based on the algebraic formulation of quantum theory.
Quantum channels, also called quantum operations, are linear, trace preserving and completely positive transformations in the space of quantum states. Such operations describe discrete time evolution of an open quantum system interacting with an environment. The thesis contains an analysis of properties of quantum channels and different entropies used to quantify the decoherence introduced into the system by a given operation. Part I of the thesis provides a general introduction to the subject. In Part II, the action of a quantum channel is treated as a process of preparation of a quantum ensemble. The Holevo information associated with this ensemble is shown to be bounded by the entropy exchanged during the preparation process between the initial state and the environment. A relation between the Holevo information and the entropy of an auxiliary matrix consisting of square root fidelities between the elements of the ensemble is proved in some special cases. Weaker bounds on the Holevo information are also established. The entropy of a channel, also called the map entropy, is defined as the entropy of the state corresponding to the channel by the Jamiolkowski isomorphism. In Part III of the thesis, the additivity of the entropy of a channel is proved. The minimal output entropy, which is difficult to compute, is estimated by an entropy of a channel which is much easier to obtain. A class of quantum channels is specified, for which additivity of channel capacity is conjectured. The last part of the thesis contains characterization of Davies channels, which correspond to an interaction of a state with a thermal reservoir in the week coupling limit, under the condition of quantum detailed balance and independence of rotational and dissipative evolutions. The Davies channels are characterized for one-qubit and one-qutrit systems.
An elementary algebraic approach to unified quantum information theory is given. The operational meaning of entanglement as specifically quantum encoding is disclosed. General relative entropy as information divergence is introduced, and three most important types of relative information, namely, the Araki-Umegaki type (A-type), the Belavkin-Staszewski type (B-type), and the thermodynamical (C-type) are discussed. It is shown that true quantum entanglement-assisted entropy is greater than semiclassical (von Neumann) quantum entropy, and the proper positive quantum conditional entropy is introduced. The general quantum mutual information via entanglement is defined, and the corresponding types of quantum channel capacities as a supremum via the generalized encodings are formulated. The additivity problem for quantum logarithmic capacities for products of arbitrary quantum channels under appropriate constraints on encodings is discussed. It is proved that true quantum capacity, which is achieved on the standard entanglement as an optimal quantum encoding, retains the additivity property of the logarithmic quantum channel entanglement-assisted capacities on the products of quantum input states. This result for quantum logarithmic information of A-type, which was obtained earlier by the author, is extended to any type of quantum information.