A Resource Framework for Quantum Shannon Theory

Quantum Shannon theory is loosely defined as a collection of coding theorems, such as classical and quantum source compression, noisy channel coding theorems, entanglement distillation, etc., which characterize asymptotic properties of quantum and classical channels and states. In this paper we advocate a unified approach to an important class of problems in quantum Shannon theory, consisting of those that are bipartite, unidirectional and memoryless. We formalize two principles that have long been tacitly understood. First, we describe how the Church of the larger Hilbert space allows us to move flexibly between states, channels, ensembles and their purifications. Second, we introduce finite and asymptotic (quantum) information processing resources as the basic objects of quantum Shannon theory and recast the protocols used in direct coding theorems as inequalities between resources. We develop the rules of a resource calculus which allows us to manipulate and combine resource inequalities. This framework simplifies many coding theorem proofs and provides structural insights into the logical dependencies among coding theorems. We review the above-mentioned basic coding results and show how a subset of them can be unified into a family of related resource inequalities. Finally, we use this family to find optimal trade-off curves for all protocols involving one noisy quantum resource and two noiseless ones.Comment: 60 page

Distillation and simulation in quantum information

University of Technology Sydney. Faculty of Engineering and Information Technology.We use the techniques of convex optimization, especially semidefinite programming, to study two kinds of fundamental tasks, i.e., distillation and simulation in quantum information theory. We investigate these tasks in a unified framework of resource theory and focus on their computation and characterization with finite resources. Particularly we study the tradeoff among relevant parameters such as the number of resource copies, resource transformation rate, error tolerance and success probability. In the first part, we study the task of distillation for two different resources, maximally entangled state and maximally coherent state, representing nonlocal and local “quantumness” respectively. For entanglement distillation, we derive an efficiently computable second-order estimation of the distillation rate for general quantum states, which are tight for quantum states of practical interest. Our study overcomes the limitation of conventional research either focusing on the asymptotic rate or ignoring the computability. For the coherence distillation, we perform finite analysis for both deterministic and probabilistic scenarios. Our results unveil several new features of coherence from a resource theoretic viewpoint and contribute to an increased understanding of the fundamental properties of different sets of free operations. In the second part, we investigate the resource cost of simulating a quantum channel via quantum coherence or another quantum channel. We introduce the channel’s analogs of max-relative entropy, logarithmic robustness and max-information of quantum states, providing their operational interpretation with the channel simulation cost via different resources. Particularly, we establish the asymptotic equipartition property of the channel’s max-information, that is, it converges to the quantum mutual information of the channel in the independent and identically distributed asymptotic limit. As applications, this asymptotic equipartition property implies the quantum reverse Shannon theorem in the presence of non-signalling correlations. From the perspective of resource theory, the worth of a resource can usually be characterized by the minimum distance to a set of useless resources under a proper distance measure. We give such characterization for all the tasks studied in this thesis, and find that the distance measure for the distillation and simulation process naturally corresponds to the quantum hypothesis testing relative entropy and the max-relative entropy, respectively

Toward physical realizations of thermodynamic resource theories

Conventional statistical mechanics describes large systems and averages over many particles or over many trials. But work, heat, and entropy impact the small scales that experimentalists can increasingly control, e.g., in single-molecule experiments. The statistical mechanics of small scales has been quantified with two toolkits developed in quantum information theory: resource theories and one-shot information theory. The field has boomed recently, but the theorems amassed have hardly impacted experiments. Can thermodynamic resource theories be realized experimentally? Via what steps can we shift the theory toward physical realizations? Should we care? I present eleven opportunities in physically realizing thermodynamic resource theories.Comment: Publication information added. Cosmetic change

Asymptotic Compressibility of Entanglement and Classical Communication in Distributed Quantum Computation

We consider implementations of a bipartite unitary on many pairs of unknown input states by local operation and classical communication assisted by shared entanglement. We investigate to what extent the entanglement cost and the classical communication cost can be compressed by allowing nonzero but vanishing error in the asymptotic limit of infinite pairs. We show that a lower bound on the minimal entanglement cost, the forward classical communication cost, and the backward classical communication cost per pair is given by the Schmidt strength of the unitary. We also prove that an upper bound on these three kinds of the cost is given by the amount of randomness that is required to partially decouple a tripartite quantum state associated with the unitary. In the proof, we construct a protocol in which quantum state merging is used. For generalized Clifford operators, we show that the lower bound and the upper bound coincide. We then apply our result to the problem of distributed compression of tripartite quantum states, and derive a lower and an upper bound on the optimal quantum communication rate required therein.Comment: Section II and VIII adde

The apex of the family tree of protocols: Optimal rates and resource inequalities

We establish bounds on the maximum entanglement gain and minimum quantum communication cost of the Fully Quantum Slepian-Wolf protocol in the one-shot regime, which is considered to be at the apex of the existing family tree in Quantum Information Theory. These quantities, which are expressed in terms of smooth min- and max-entropies, reduce to the known rates of quantum communication cost and entanglement gain in the asymptotic i.i.d. scenario. We also provide an explicit proof of the optimality of these asymptotic rates. We introduce a resource inequality for the one-shot FQSW protocol, which in conjunction with our results, yields achievable one-shot rates of its children protocols. In particular, it yields bounds on the one-shot quantum capacity of a noisy channel in terms of a single entropic quantity, unlike previously bounds. We also obtain an explicit expression for the achievable rate for one-shot state redistribution.Comment: 31 pages, 2 figures. Published versio

Superadditivity in trade-off capacities of quantum channels

In this article, we investigate the additivity phenomenon in the dynamic capacity of a quantum channel for trading classical communication, quantum communication and entanglement. Understanding such additivity property is important if we want to optimally use a quantum channel for general communication purpose. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity, and it is largely unknown if this could lead to an superadditive double or triple resource capacity. For example, if a channel has an additive classical and quantum capacity, can the classical-quantum capacity be superadditive? In this work, we answer such questions affirmatively. We give proof-of-principle requirements for these channels to exist. In most cases, we can provide an explicit construction of these quantum channels. The existence of these superadditive phenomena is surprising in contrast to the result that the additivity of both classical-entanglement and classical-quantum capacity regions imply the additivity of the triple capacity region.Comment: 15 pages. v2: typo correcte

Towards Quantum Cybernetics

Classical cybernetics is a successful meta-theory to model the regulation of complex systems from an abstract information-theoretic viewpoint, regardless of the properties of the system under scrutiny. Fundamental limits to the controllability of an open system can be formalized in terms of the law of requisite variety, which is derived from the second law of thermodynamics. These concepts are briefly reviewed, and the chances, challenges and potential gains arising from the generalisation of such a framework to the quantum domain are discussed.Comment: Perspective article, close to published versio

Enhanced communication with the assistance of indefinite causal order

In quantum Shannon theory, the way information is encoded and decoded takes advantage of the laws of quantum mechanics, while the way communication channels are interlinked is assumed to be classical. In this Letter we relax the assumption that quantum channels are combined classically, showing that a quantum communication network where quantum channels are combined in a superposition of different orders can achieve tasks that are impossible in conventional quantum Shannon theory. In particular, we show that two identical copies of a completely depolarizing channel become able to transmit information when they are combined in a quantum superposition of two alternative orders. This finding runs counter to the intuition that if two communication channels are identical, using them in different orders should not make any difference. The failure of such intuition stems from the fact that a single noisy channel can be a random mixture of elementary, non-commuting processes, whose order (or lack thereof) can affect the ability to transmit information