[Paper Review] Applications of coherent classical communication and the Schur transform to quantum information theory

Isaac L. Chuang|ArXiv.org|Dec 30, 2005

Quantum Information and Cryptography159 references73 citations

TL;DR

This thesis introduces two foundational quantum information tools: coherent classical communication, a unitary framework for transmitting classical information via quantum operations, and an efficient polynomial-time quantum circuit for the Schur transform. The key contribution is a constructive, efficient implementation of the Schur transform using Clebsch-Gordan transforms, enabling new quantum algorithms and unifying quantum Fourier transforms on symmetric groups with Schur-Weyl duality.

ABSTRACT

Quantum mechanics has led not only to new physical theories, but also a new understanding of information and computation. Quantum information began by yielding new methods for achieving classical tasks such as factoring and key distribution but also suggests a completely new set of quantum problems, such as sending quantum information over quantum channels or efficiently performing particular basis changes on a quantum computer. This thesis contributes two new, purely quantum, tools to quantum information theory--coherent classical communication in the first half and an efficient quantum circuit for the Schur transform in the second half.

Motivation & Objective

• To develop a formal framework for quantum Shannon theory that unifies classical and quantum communication tasks using unitary interactions.
• To introduce and formalize the concept of coherent classical communication as a unitary alternative to standard classical communication in quantum protocols.
• To construct a polynomial-time quantum circuit for the Schur transform, resolving a long-standing open problem in quantum information theory.
• To establish algorithmic connections between the Schur transform and the quantum Fourier transform on the symmetric group $\mathcal{S}_n$.
• To demonstrate how the Schur transform enables efficient implementation of quantum algorithms involving symmetric and unitary group symmetries.

Proposed method

• Formalize quantum Shannon theory using resource inequalities to describe communication tasks in terms of entanglement and classical communication resources.
• Introduce coherent classical communication as a nearly unitary process that replaces classical communication in quantum protocols, preserving quantum coherence.
• Develop a quantum circuit for the Schur transform using Clebsch-Gordan transforms to decompose tensor product spaces into irreducible representations of $\mathcal{S}_n$ and $\mathcal{U}_d$.
• Embed the group algebra $\mathbb{C}[\mathcal{S}_n]$ into the $1^n$ weight space of $(\mathbb{C}^n)^{\otimes n}$ to map permutations to quantum states.
• Use the Schur transform to perform a Fourier transform on $\mathcal{S}_n$ by mapping $\mathbb{C}[\mathcal{S}_n]$ to $\bigoplus_{\lambda} \mathcal{P}_\lambda^* \otimes \mathcal{P}_\lambda$, where $\mathcal{P}_\lambda$ are irreducible representations.
• Establish isomorphisms between representation matrices of $\mathcal{S}_n$ and the Schur basis, showing that the GZ basis of the Schur space corresponds to the GZ basis of the irreducible representations up to phase.

Experimental results

Research questions

• RQ1 Can classical communication in quantum protocols be replaced by a coherent, unitary process that preserves quantum coherence and enables new protocol designs?
• RQ2 What is the optimal trade-off between entanglement and classical communication in quantum communication tasks, and how can it be derived using coherent communication?
• RQ3 Is there an efficient quantum circuit for implementing the Schur transform that runs in polynomial time in $n$ and $\log(1/\epsilon)$?
• RQ4 How can the Schur transform be used to implement the quantum Fourier transform on the symmetric group $\mathcal{S}_n$?
• RQ5 What is the precise relationship between the Schur transform and the quantum Fourier transform on $\mathcal{S}_n$, and how do their representation matrices compare?

Key findings

• A new formalism for quantum Shannon theory is developed, expressing all known coding theorems as resource inequalities involving entanglement and classical communication.
• Coherent classical communication is introduced as a unitary alternative to classical communication, enabling new protocols and unifying existing ones under a common framework.
• An efficient polynomial-time quantum circuit for the Schur transform is constructed using Clebsch-Gordan transforms, achieving runtime $\operatorname{poly}(n, \log 1/\epsilon)$.
• The Schur transform enables a quantum Fourier transform on $\mathcal{S}_n$ by mapping $\mathbb{C}[\mathcal{S}_n]$ to $\bigoplus_{\lambda} \mathcal{P}_\lambda^* \otimes \mathcal{P}_\lambda$ via the isomorphism $\mathbb{C}[\mathcal{S}_n] \cong \bigoplus_{\lambda} \mathcal{Q}_\lambda^n(1^n) \otimes \mathcal{P}_\lambda$.
• The representation matrices of $\mathcal{S}_n$ under the Schur transform are shown to be isomorphic to those of the irreducible representations $\mathcal{P}_\lambda$, differing only by an arbitrary phase per basis vector, which remains an open question to fully characterize.