Quantum channels are functions that map density matrices to density matrices. They can be used to describe a vast array of system environment interactions for quantum mechanical systems. The focus in quantum information theory lies on the study of the information content -- the quantum entropy -- of input and output system. The property of how much information a channel can convey is called capacity. When two quantum channels are combined they act on a bigger, possibly entangled state in parallel. The effects of entanglement on the capacity of a pair of channels have been widely studied. Still, recent work showed unexpected behavior. Hastings found that the capacities of two channels to transmit classical information does not simply add, but can exceed the sum of the individual quantum capacities, a property called superadditivity. We study Hastings' work and find estimates for how large the dimensions of such channels would have to be. The question of capacity relates to the behavior of quantum channels in extremal cases, when the output entropy is minimized. It is natural to wonder about the "standard" behavior of quantum channels and consider their average output entropy. We have studied this behavior in particular in cases where many relatively simple quantum channels act in parallel. When combining quantum channels in parallel, the parts of the system evolve independently of each other -- up to entanglement -- and only interact with the environment. Expanding on this idea one can think of a system where the parts not only interact with the environment but also directly with each other, a quantum channel network. Interesting cases of such systems arise naturally in the study of large biological molecules. In this case, information is often transmitted via energy pulses. Thus, the study of entropy and the study of energy overlap. Of particular interest to us are antennas in photo-active systems, they absorb light energy and transports it to a chemical reaction site. Recently, it has been discovered in the FMO molecule that this process evolves coherently, i.e. shows quantum oscillatory behavior. We study such quantum channel networks and how they can be approximated by kinetic networks. When considering the transport efficiency, this helps separating possible coherent effects from incoherent hopping.
Quantum channels are functions that map density matrices to density matrices. They can be used to describe a vast array of system environment interactions for quantum mechanical systems. The focus in quantum information theory lies on the study of the information content -- the quantum entropy -- of input and output system. The property of how much information a channel can convey is called capacity. When two quantum channels are combined they act on a bigger, possibly entangled state in parallel. The effects of entanglement on the capacity of a pair of channels have been widely studied. Still, recent work showed unexpected behavior. Hastings found that the capacities of two channels to transmit classical information does not simply add, but can exceed the sum of the individual quantum capacities, a property called superadditivity. We study Hastings' work and find estimates for how large the dimensions of such channels would have to be. The question of capacity relates to the behavior of quantum channels in extremal cases, when the output entropy is minimized. It is natural to wonder about the "standard" behavior of quantum channels and consider their average output entropy. We have studied this behavior in particular in cases where many relatively simple quantum channels act in parallel. When combining quantum channels in parallel, the parts of the system evolve independently of each other -- up to entanglement -- and only interact with the environment. Expanding on this idea one can think of a system where the parts not only interact with the environment but also directly with each other, a quantum channel network. Interesting cases of such systems arise naturally in the study of large biological molecules. In this case, information is often transmitted via energy pulses. Thus, the study of entropy and the study of energy overlap. Of particular interest to us are antennas in photo-active systems, they absorb light energy and transports it to a chemical reaction site. Recently, it has been discovered in the FMO molecule that this process evolves coherently, i.e. shows quantum oscillatory behavior. We study such quantum channel networks and how they can be approximated by kinetic networks. When considering the transport efficiency, this helps separating possible coherent effects from incoherent hopping.
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.
In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered. These phenomena are purely quantum mechanical and completely unimaginable in classical systems. Recently, the most important discovery in Quantum Information Theory was the possibility of transmitting quantum information over zero-capacity quantum channels. In this work we prove that the possibility of superactivation of quantum channel capacities is determined by the mathematical properties of the quantum relative entropy function.
Historically at the core of thermodynamics and information theory, entropy's use in quantum information extends to diverse topics including high-energy physics and operator algebras. Entropy can gauge the extent to which a quantum system departs from classicality, including by measuring entanglement and coherence, and in the form of entropic uncertainty relations between incompatible measurements. The theme of this dissertation is the quantum nature of entropy, and how exposure to a noisy environment limits and degrades non-classical features. An especially useful and general form of entropy is the quantum relative entropy, of which special cases include the von Neumann and Shannon entropies, coherent and mutual information, and a broad range of resource-theoretic measures. We use mathematical results on relative entropy to connect and unify features that distinguish quantum from classical information. We present generalizations of the strong subadditivity inequality and uncertainty-like entropy inequalities to subalgebras of operators on quantum systems for which usual independence assumptions fail. We construct new measures of non-classicality that simultaneously quantify entanglement and uncertainty, leading to a new resource theory of operations under which these forms of non-classicalty become interchangeable. Physically, our results deepen our understanding of how quantum entanglement relates to quantum uncertainty. We show how properties of entanglement limit the advantages of quantum superadditivity for information transmission through channels with high but detectable loss. Our method, based on the monogamy and faithfulness of the squashed entanglement, suggests a broader paradigm for bounding non-classical effects in lossy processes. We also propose an experiment to demonstrate superadditivity. Finally, we estimate decay rates in the form of modified logarithmic Sobolev inequalities for a variety of quantum channels, and in many cases we obtain the stronger, tensor-stable form known as a complete logarithmic Sobolev inequality. We compare these with our earlier results that bound relative entropy of the outputs of a particular class of quantum channels.
The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and it has been used to compute the capacity of quant um channel for quantum communication process; quantum input-quantum output. Recently, a correlated state in quantum syatems, so-called quantum entangled state or quantum entanglement, are used to study quntum information, in part icular, quantum computation, quantum teleportation, quantum cryptography. In this paper, we mainly discuss three things below: (1) We point out the di fference between the capacity of quantum channel and that of classical-quant um-classical channel. (2) So far the entangled state is merely defined as a non-separable state, we give a wider definition of the entangled state and c lassify the entangled states into three categories. (3) The quantum mutual e ntropy for an entangled state is discussed. The above (2) and (3) are a join t work with Belavkin.
In communication processes, a channel has an activity to communicate information of input system to output system. The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983, and he extended it to general quantum systems by means of the relative entropy of Araki and Uhlmann. The capacity shows the ability of the information transmission of the channel, which is used as a measure for construction of channels. The fully quantum capacity is formulated by taking the supremum of Ohya mutual entropy with respect to a certain subset of the initial state space. One of the most important theorems in quantum communication theory is coding theorem. The quantum coding theorems are discussed by using the mutual entropy — type measures introduced in the studies of quantum information.In this paper, we compare with these mutual entropy‐type measures in order to obtain most suitable one for discussing the quantum coding theorem. We concretely calculate the quantum capacity for quantum channels.
In most communication scenarios, sending a symbol encoded in a quantum state requires spending resources such as energy, which can be quantified by a cost of communication. A standard approach in this context is to quantify the performance of communication protocol by classical capacity, quantifying the maximal amount of information that can be transmitted through a quantum channel per single use of the channel. However, different figures of merit are also possible and a particularly well-suited one is the classical capacity per unit cost, which quantifies the maximal amount of information that can be transmitted per unit cost. I generalize this concept to account for the quantum nature of the information carriers and communication channels and show that if there exists a state with cost equal to zero, e.g., a vacuum state, the capacity per unit cost can be expressed by a simple formula containing maximization of the relative entropy between two quantum states. This enables me to analyze the behavior of photon information efficiency for general communication tasks and show simple bounds on the capacity per unit cost in terms of quantities familiar from quantum estimation theory. I calculate also the capacity per unit cost for general Gaussian quantum channels.