Applications of coherent classical communication and the Schur transform to quantum information theory

TL;DR

The thesis develops two central, purely quantum tools: coherent classical communication, which elevates classical messages to coherently transmitted resources via unitary channels, and an efficient quantum circuit for the Schur transform based on Schur-Weyl duality. It frames quantum information tasks within a resource-inequality formalism, deriving optimal trade-offs for using noisy channels or states to generate or consume cbits, ebits, and qubits, including entanglement-assisted and bidirectional scenarios. The coherent communication framework unifies and simplifies known protocols (teleportation, super-dense coding, remote state preparation, HSW coding) and yields new capacity results for unitary gates, revealing deep links between classical and quantum resources and enabling reversible protocols. The Schur transform chapter provides a polynomial-time quantum circuit via Clebsch-Gordan decompositions, connecting the transform to Schur duality and highlighting algorithmic ties to the quantum Fourier transform on the symmetric group. Together, these contributions advance the understanding of fundamental limits in quantum information processing and offer practical primitives for efficient quantum communication and computation.

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.

Figures (18)

• Figure 1: Dependencies between different chapters of this thesis. The solid lines indicate that one chapter depends on another, while the dashed lines mean a partial dependence: Section \ref{['sec:schur-qit-apps']} has references to some of the protocols in Section \ref{['sec:known']} and Chapter \ref{['chap:ccc']} is motivated by and extends the results of Chapter \ref{['chap:unitary']}.
• Figure 2: The sliding protocol. We would like to simulate ${\bf P}$, which uses ${\cal N}_1, \ldots, {\cal N}_\ell$ consecutively, but we are only given ${\cal N}_1 \otimes \ldots {\cal N}_\ell$. The horizontal blocks represent uses of ${\cal N}_1 \otimes \ldots \otimes {\cal N}_\ell$ and stacking them vertically indicates how we perform them consecutively with the output of one block becoming the input of the block above it (i.e. time flows from the bottom to the top). Thus $m+l-1$ consecutive uses of ${\cal N}_1 \otimes \ldots \otimes {\cal N}_\ell$ can simulate $m$ copies of ${\bf P}$.
• Figure 3: Example of a possible achievable rate region $\mathop{\mathrm{CC}}\nolimits(U)$, with the limiting capacities of $C_\rightarrow, C_\leftarrow$ and $C_+$ indicated.
• Figure 4: Schematic diagrams for ${\cal P}_n$ and ${\cal P}_n'$. (a) A given protocol ${\cal P}_n$ for two-way classical communication. The output is a superposition (over all $a',b'$) of the depicted states, with most of the weight in the $(a',b')=(a,b)$ term. The unlabeled output systems in the state $|\gamma_{a' \!, b'}^{a,b}\rangle$ are $\mathop{\mathrm{A}}\nolimits_2,\mathop{\mathrm{B}}\nolimits_2$. (b) The same protocol with the inputs copied to local ancillas $\mathop{\mathrm{A}}\nolimits_0, \mathop{\mathrm{B}}\nolimits_0$ before ${\cal P}_n$. If $|\gamma_{a,b}^{a,b}\rangle$ is independent of $a,b$, two-way coherent classical communication is achieved. (c) The five steps of ${\cal P}_n'$. Steps (1)-(4) are shown in solid lines. Again, the inputs are copied to local ancillas, but ${\cal P}_n$ is used on messages encrypted by a coherent one-time-pad (the input $|a\rangle_{\mathop{\mathrm{A}}\nolimits_1}$ is encrypted by the coherent version of the key $|x\rangle_{\mathop{\mathrm{A}}\nolimits_3}$ and the output $|a' \space \oplus x\rangle_{\mathop{\mathrm{B}}\nolimits_1}$ is decrypted by $|x\rangle_{\mathop{\mathrm{B}}\nolimits_3}$; similarly, $|b\rangle_{\mathop{\mathrm{B}}\nolimits_1}$ is encrypted by $|y\rangle_{\mathop{\mathrm{B}}\nolimits_4}$ and $|b' \space \oplus y\rangle_{\mathop{\mathrm{A}}\nolimits_1}$ decrypted by $|y\rangle_{\mathop{\mathrm{A}}\nolimits_4}$. The intermediate state is shown in the diagram. Step (5), shown in dotted lines, decouples the messages in $\mathop{\mathrm{A}}\nolimits_{0,1},\mathop{\mathrm{B}}\nolimits_{0,1}$ from $\mathop{\mathrm{A}}\nolimits_{2,3,4},\mathop{\mathrm{B}}\nolimits_{2,3,4}$, which is in the joint state very close to $|\Gamma_{00}\rangle$.
• Figure 5: A general protocol for noisy super-dense coding.

Theorems & Definitions (116)

• Lemma 1.1: Fannes
• Lemma 1.2
• Lemma 1.3
• proof
• Lemma 1.4
• proof
• Definition 1.5: Finite resources
• Definition 1.6: Depth-$\ell$ resources
• Definition 1.7: Elementary protocols
• Definition 1.8: Standard protocol