This dissertation deals with entropic inequalities for bosonic channels in quantum information theory. Such inequalities have applications in foundational questions of quantum communication and can be used to make statements about the convergence of quantum dynamical semigroups. We prove, among other results, an inequality for the output entropy of a quantum channel which models additive classical noise and apply this inequality to derive bounds on the classical capacity of a general family of non-Gaussian quantum channels.
The speed of convergence of controlled dissipative dynamics to the desired state or set of states is critical in many tasks in experimental physics and quantum information processing. Here we focus on quantum Markovian master equations and, by exploiting the linear character of the dynamical generator, we derive an alternative characterization of the semigroups stabilizing a target subspace, as well as the asymptotic speed of convergence to the desired set. Our analysis highlights the effect of the Hamiltonian parameters (which can be considered as our controls) on the convergence speed, and suggests two basic principles for their design.
Quantum information theory is the study of the achievable limits of information processing within quantum mechanics. Many different types of information can be accommodated within quantum mechanics, including classical information, coherent quantum information, and entanglement. Exploring the rich variety of capabilities allowed by these types of information is the subject of quantum information theory, and of this Dissertation. In particular, I demonstrate several novel limits to the information processing ability of quantum mechanics. Results of especial interest include: the demonstration of limitations to the class of measurements which may be performed in quantum mechanics; a capacity theorem giving achievable limits to the transmission of classical information through a two-way noiseless quantum channel; resource bounds on distributed quantum computation; a new proof of the quantum noiseless channel coding theorem; an information-theoretic characterization of the conditions under which quantum error-correction may be achieved; an analysis of the thermodynamic limits to quantum error-correction, and new bounds on channel capacity for noisy quantum channels.
This dissertation investigates relative entropies, also called generalized divergences, and how they can be used to characterize information-theoretic tasks in quantum information theory. The main goal is to further refine characterizations of the optimal rates for quantum source coding, state redistribution, and measurement compression with quantum side information via second order asymptotic expansions and strong converse theorems. The dissertation consists of a mathematical and an information-theoretic part. In the mathematical part, we focus on the
α-sandwiched Rényi divergence (
α-SRD). We first investigate the limit
α→0 to determine whether this recovers the well-known
0-Rényi relative divergence. We then prove various new results for entropic quantities derived from the
α-SRD, including dimension bounds and useful bounds in terms of the fidelity between two quantum states. Furthermore, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality (viz. monotonicity under quantum operations) for the
α-SRD, and give applications to entropic bounds. In the information-theoretic part, we first derive the second order asymptotics of visible quantum source coding using a mixed source. For the achievability part, we develop universal quantum source codes achieving a given second order rate for a memoryless source. As a corollary of the main result, we obtain the second order asymptotics of quantum source coding using a single memoryless source. We then prove strong converse theorems for state redistribution (with or without feedback) and measurement compression with quantum side information. The key ingredients in proving these theorems are the aforementioned fidelity bounds on Rényi entropic quantities derived from the
α-SRD.
We study the information transmission capacities of quantum Markov semigroups acting on finite-dimensional quantum systems. We show that, in the limit of infinite time, the capacities can be efficiently computed in terms of the structure of the peripheral space of the semigroup, are strongly additive, and satisfy the strong converse property. We also establish convergence bounds to show that the infinite-time capacities are reached after time scaling quadratically with the system dimension. From a data storage perspective, our analysis provides tight bounds on the number of bits or qubits that can be reliably stored for long times in a quantum memory experiencing Markovian noise. From a practical standpoint, we show that typically, a quantum memory with Markovian noise acting independently and identically on all qubits and a fixed time-independent global error correction mechanism becomes useless for storage after time scaling exponentially with the number of qubits. In contrast, if the error correction is local, the memory becomes useless much more quickly, i.e., after time scaling logarithmically with the number of qubits. In the setting of point-to-point communication between two spatially separated parties, our analysis provides efficiently computable bounds on the optimal rate at which bits or qubits can be reliably transmitted via long Markovian communication channels, both in the finite block-length and asymptotic regimes.
Given a quantum Markov semigroup (T t ) t≥o on B (h), with a faithful normal invariant state ρ, we associate to it the semigroup (T t ) t≥o on Hilbert-Schmidt operators on h (the L 2 (ρ) space) defined by (T t (ρ 1/4xρ 1/4) = ρ 1/4 T t (x)ρ 1/4. This allows us to use spectral theory to study the infinitesimal generator of (T t ) t≥o and deduce information on the speed of convergence to equilibrium of the given semigroup. We apply this idea to show that some quantum Markov semigroups related to birth-and-death processes converge to equilibrium exponentially rapidly in L 2 (p).