The decoupling approach to quantum information theory

TL;DR

This work presents a unified decoupling framework that recasts quantum information tasks as decorrelation problems, enabling broad one-shot and asymptotic coding results for arbitrary channels. Central to the approach is a general decoupling theorem, from which state merging, FQSW, and many quantum coding theorems follow, including new results for channels with side information and quantum broadcast channels. A key contribution is showing that entanglement-assisted capacities often admit single-letter formulas and align with classical counterparts, while the framework also yields a novel locking protocol with a trace-distance security criterion. The results significantly advance the understanding of quantum channel capacities, protocol design, and cryptographic security in the presence of quantum memories, offering a powerful, versatile toolkit for quantum information theory.

Abstract

Quantum information theory studies the fundamental limits that physical laws impose on information processing tasks such as data compression and data transmission on noisy channels. This thesis presents general techniques that allow one to solve many fundamental problems of quantum information theory in a unified framework. The central theorem of this thesis proves the existence of a protocol that transmits quantum data that is partially known to the receiver through a single use of an arbitrary noisy quantum channel. In addition to the intrinsic interest of this problem, this theorem has as immediate corollaries several central theorems of quantum information theory. The following chapters use this theorem to prove the existence of new protocols for two other types of quantum channels, namely quantum broadcast channels and quantum channels with side information at the transmitter. These protocols also involve sending quantum information partially known by the receiver with a single use of the channel, and have as corollaries entanglement-assisted and unassisted asymptotic coding theorems. The entanglement-assisted asymptotic versions can, in both cases, be considered as quantum versions of the best coding theorems known for the classical versions of these problems. The last chapter deals with a purely quantum phenomenon called locking. We demonstrate that it is possible to encode a classical message into a quantum state such that, by removing a subsystem of logarithmic size with respect to its total size, no measurement can have significant correlations with the message. The message is therefore "locked" by a logarithmic-size key. This thesis presents the first locking protocol for which the success criterion is that the trace distance between the joint distribution of the message and the measurement result and the product of their marginals be sufficiently small.

Figures (5)

• Figure 1: Diagram illustrating Theorem \ref{['thm:vanilla-channels-oneshot']}. Each line represents a quantum system, boxes represent isometries, and the horizontal axis represents the passage of time. Lines joined together at either end of the diagram represent pure states. Alice used $V$ to encode her message $A$ into the input to the channel $A'$, and Bob uses the channel output $C$ together with the $B$ that he had since the beginning to decode $A$ (and $B$) back. The decoder also produces a system $F$ that purifies the environment.
• Figure 2: Diagram illustrating the states $\sigma$ and $\omega$ in Theorem \ref{['thm:vanilla-channels-oneshot']}. Each line represents a quantum system, boxes represent isometries, and the horizontal axis represents the passage of time. Lines joined together at either end of the diagram represent pure states.
• Figure 3: Diagram illustrating Theorem \ref{['thm:side-info-channels-oneshot']}, with encoder, channel and decoder purified. Each line represents a quantum system, boxes represent isometries, and the horizontal axis represents the passage of time. Lines joined together at either end of the diagram represent pure states. $V$ represents Alice's encoder: she uses the side information $S'$ to encode the message $A$ into the channel input $A'$ and discards a system $D$. The decoder $D$ takes the channel output $C$ together with Bob's initial system $B$ and produces $A$ and $B$ as output; the result being close to the initial state $\psi$.
• Figure 4: Diagram illustrating the states $\omega$ and $\sigma$ which define the input distribution in Theorem \ref{['thm:side-info-channels-oneshot']}. Each line represents a quantum system, boxes represent isometries, and the horizontal axis represents the passage of time. Lines joined together at either end of the diagram represent pure states.
• Figure 5: Diagram illustrating Theorem \ref{['thm:broadcast-1shot']}, with encoder, channel and decoders purified. Each line represents a quantum system, boxes represent isometries, and the horizontal axis represents the passage of time. Lines joined together at either end of the diagram represent pure states. $W$ represents Alice's encoder: she encodes the messages $A_1$ and $A_2$ into the channel input $A'$ and discards a system $D$. The decoders $D_1$ and $D_2$ take the channel outputs $C_1$ and $C_2$ together with Bob 1 and 2's initial systems $B_1$ and $B_2$ and produce $A_1 B_1$ and $A_2B_2$ as output; the result being close to the initial state $\psi$.

Theorems & Definitions (129)

• Theorem 2.1: Helstrom's theorem helstrom
• proof
• Definition 2.1: Fidelity
• Definition 2.2: Fidelity distance
• Lemma 2.2: Fuchs-van de Graaf inequalities
• Definition 2.3: Diamond norm
• Definition 2.4: von Neumann entropy vonneumann-grundlagen
• Definition 2.5: Conditional von Neumann entropy
• Definition 2.6: Quantum mutual information
• Definition 2.7: Conditional quantum mutual information