The coherent information of quantum channels plays a important role in quantum information theory as it can be used to calculate the quantum capacity of a channel. However, it is a non-linear, non-differentiable optimization problem. This thesis discusses that by restricting to the space of positive definite density matrices and restricting the class of quantum channels to be strictly positive, the coherent information becomes differentiable. This allows the computation of the Riemannian gradient and Hessian of the coherent information. It will be shown that the maximally mixed state is a critical point for the $n$-shot coherent information of the Pauli, dephrasure and Pauli-erasure channels. In addition, the classification of the maximally mixed state as a local maxima/minima and saddle-point will be solved for the one shot coherent information. The hope of this work is to provide a new avenue to explore the quantum capacity problem.
The coherent information of quantum channels plays a important role in quantum information theory as it can be used to calculate the quantum capacity of a channel. However, it is a non-linear, non-differentiable optimization problem. This thesis discusses that by restricting to the space of positive definite density matrices and restricting the class of quantum channels to be strictly positive, the coherent information becomes differentiable. This allows the computation of the Riemannian gradient and Hessian of the coherent information. It will be shown that the maximally mixed state is a critical point for the n-shot coherent information of the Pauli, dephrasure and Pauli-erasure channels. In addition, the classification of the maximally mixed state as a local maxima/minima and saddle-point will be solved for the one shot coherent information. The hope of this work is to provide a new avenue to explore the quantum capacity problem.
This paper will explore the restriction of the coherent information to the positive definite density matrices in the special case where the quantum channels are strictly positive linear maps. The space of positive definite density matrices is equipped with an embedded submanifold structure of the real vector space of Hermitian matrices. These ensure that the n-shot coherent information is differentiable and allows for the computation of its gradient and Hessian. We show that any tensor products of critical points preserve being a critical point of the coherent information. Furthermore, we show that for any positive integer n, the maximally mixed state is always a critical point for the class of mixed unitary quantum channels with orthogonal, unitary Kraus operators. We determine when the maximally mixed state is a local maximum/minimum or saddle point, including its eigenvectors, for the class of Pauli-erasure channels when n is equal to 1. This class includes the dephrasure channel and Pauli channel and refines potential regions where super-additivity is thought to occur. These techniques can be used to study other optimization problems over density matrices and allow the use of manifold optimization algorithms and a better understanding of the quantum capacity problem by utilizing the first and second order geometry.
Quantum capacity quantifies the amount of quantum information that can be transmitted by a quantum channel with an arbitrary small probability of error. Mathematically, the quantum capacity is given by an asymptotic formula involving the one-shot quantum capacity of the associated channel, which, due to purely quantum effects such as superadditivity of one-shot quantum capacity, is rarely computable. The one-shot quantum capacity is mathematically characterized in terms of optimizing an entropic quantity referred to as coherent information over all possible input states of a channel, the computation of which also tends to be intractable due to the difficulty of optimizing the coherent information. In this work, we develop perturbative methods for analyzing the behavior of coherent information of a quantum channel with respect to small perturbations of the input state. By doing so, we are able to derive three general criteria for determining whether an input state yields suboptimal coherent information. We then show how our criteria yield sufficient conditions for superadditivity of one-shot quantum capacity, and also for detecting a positive gap between one-shot private capacity and one-shot quantum capacity. The utility of our criteria is illustrated through examples, which yield new results regarding the one-shot quantum capacity of qubit depolarizing channels, Pauli channels and dephrasure channels.
This thesis consists of three parts. We begin our investigation with a brief introduction into open quantum systems. Then we explain different measures of distinguishability of density operators, especially the Uhlmann Fidelity, which will be the basis for the model function we investigate in this work. \nWe continue by explaining the time evolution of open quantum systems in more detail, and introducing the important notion of quantum channels as a concept in quantum information theory. This allows us to state a central problem of quantum information theory: the characterization of the quantum information capacity of a given quantum channel. The major challenge is the maximization of the coherent information, a function defined on a n-fold tensor product of a Hilbert space, which is non-additive and thus has to be considered in the limit as n→∞. \n \nTo gain an insight into this unsolved problem, we study the channel fidelity of a quantum channel, which is a simpler function, also defined on n-fold tensor product spaces. It shares an essential feature with the coherent information in being non-additive. However, in contrast to the coherent information it is mathematically accessible. We establish the channel fidelity as a model for the coherent information and study its properties. \n \nIn the second part, a short introduction to the representation theory of symmetric and unitary groups is followed by concrete instructions for Collins' and \\'{S}niady's formula for the integration of functions of matrix elements of unitary groups with respect to the Haar measure. This exposition culminates in a simplification of the general formula that is optimal for investigating the channel fidelity. \n \nIn the third part, we calculate channel fidelity moments for arbitrary n. In order to obtain more concrete results, we restrict ourselves to the study of Pauli channels. For these we discuss the transition of the average and variance from small n to the limit n→∞ and give an explicit formula for both in this limit. In particular, we find that for a large number n of Pauli channels, the channel fidelity distribution is peaked very strongly. Additionally, under certain restrictions, the simplified formula from part two also allows us to give concrete expressions for higher moments in the limit n→∞. \n \nWe conclude by comparing our new results with results from a former work, where, in the search for maximizing states of the channel fidelity, we found states that maximize the channel fidelity of Pauli channels, at least locally. Because these local maxima have a much higher fidelity than the average of the very strongly peaked distribution, we infer that these states would not be found by a standard numerical maximization procedure. If the channel fidelity models the coherent information accurately in this regard, its maximization thus poses a very hard problem.
In quantum information theory, in order for two (or more) parties to communicate, there must be a communication channel. The two central quantities that must be considered are quantum channel capacity and the quantum coherent information. The first describes the capability for a given channel to communicate the required quantum information and the latter describes the amount of information that has been successfully transmitted through the corresponding quantum channel. The capacity of a quantum channel is actually a function of the coherent information, and so the basic step for getting the channel capacity is to calculate its coherent information which serves as a lower bound on the capacity. Generally speaking, it is hard to find the capacity of a given quantum channel, this is because of regularization which is to introduce additional process to solve an ill-conditioned problem. Comparing to general quantum channels, the capacity is easily achieved for the two fundamental classes of quantum channels, degradable and antidegradable, in such a way that degradable channels have capacity given exactly by the coherent information and antidegradable channels have zero coherent information. However, this is not true for general quantum channels, by all means that, under certain restrictions, it is possible to find a non-Pauli quantum channel such that its coherent information is zero but its capacity is non-zero.
The quantum capacity captures the value of a quantum channel for transmitting quantum information, establishing the fundamental limits on quantum communication. In spite of its central role in quantum information theory, the quantum capacity of most channels is unknown, with wide gaps between the best upper and lower bounds. Even deciding whether a channel has nonzero capacity -- finding its capacity threshold -- is difficult. In this paper we report significant increases in the capacity thresholds of two prototypical noise models: the depolarizing channel and Pauli channels. In the case of the depolarizing channel, this is the first improvement in 18 years, giving a bigger increase beyond the hashing bound than all previous improvements combined. Our starting point is the representation theoretic framework recently proposed by Bhalerao and Leditzky (2025) to compute coherent information for special permutation invariant states. We generalize their framework to the full symmetric subspace, which allow us to optimize coherent information over rank two states in that space. A representation theoretic calculation shows that exponentially many Kraus operators of the channel annihilate the symmetric space, corresponding to a massive decrease in environment entropy for states on the symmetric space compared to the maximally mixed state. This explains the enhanced coherent information as a manifestation of degeneracy for the resulting codes.
Determining the ultimate limits of quantum communication, such as the quantum capacity of a channel and the distillable entanglement of a shared state, remains a central challenge in quantum information theory, primarily due to the phenomenon of superadditivity. This work develops Riemannian optimization methods to establish significantly tighter, computable two-sided bounds on these fundamental quantities. For upper bounds, our method systematically searches for state and channel extensions that minimize known information-theoretic bounds. We achieve this by parameterizing the space of all possible extensions as a Stiefel manifold, enabling a universal search that overcomes the limitations of ad-hoc constructions. Combined with an improved upper bound on the one-way distillable entanglement based on a refined continuity bound on quantum conditional entropy, our approach yields new state-of-the-art upper bounds on the quantum capacity of the qubit depolarizing channel for large values of the depolarizing parameter, strictly improving the previously best-known bounds. For lower bounds, we introduce Riemannian optimization methods to compute multi-shot coherent information. We establish lower bounds on the one-way distillable entanglement by parameterizing quantum instruments on the unitary manifold, and on the quantum capacity by parameterizing code states with a product of unitary manifolds. Numerical results for noisy entangled states and different channels demonstrate that our methods successfully unlock superadditive gains, improving previous results. Together, these findings establish Riemannian optimization as a principled and powerful tool for navigating the complex landscape of quantum communication limits. Furthermore, we prove that amortization does not enhance the channel coherent information, thereby closing a potential avenue for improving capacity lower bounds in general.