[Paper Review] Quantum mappings and designs

Grzegorz Rajchel-Mieldzioć|arXiv (Cornell University)|Apr 27, 2022

graph theory and CDMA systems4 citations

TL;DR

This thesis introduces novel quantum mappings and designs, focusing on unistochasticity in bistochastic matrices and entangling power in multipartite systems. It presents an analytical formula for the average entangling power of tripartite orthogonal gates and constructs the first known absolutely maximally entangled (AME) state of four quhexes (AME(4,6)), revealing a block-like structure with 112 non-zero terms and high symmetry, advancing quantum combinatorics and quantum information theory.

ABSTRACT

In order to use quantum devices for computations, it is necessary to understand the intricacies of the theoretical description. To this end, we provide several novel constructions useful for the comprehension of quantum mechanics from the perspective of mappings and designs. The unistochasticity problem, which relates the classical and the quantum domain, is solved in specific cases, e.g. for all matrices of dimension 4. Furthermore, we provide an explicit formula for entangling power in the multipartite case. Most importantly, the thesis presents and elaborates on the path that lead to the recent construction of absolutely maximally entangled state of four subsystems six levels each. Finally, we study cardinality as a measure of "quantumness" of quantum Latin squares and quantum Sudoku (SudoQ). Arrays of the highest cardinality yield families of quantum measurements of special properties. A connection between SudoQ designs and mutually unbiased bases is demonstrated.

Motivation & Objective

• To develop new theoretical tools for understanding quantum mechanics through mappings and designs.
• To solve the unistochasticity problem for 4×4 bistochastic matrices and extend it to rays and counter-rays in arbitrary dimensions using robust Hadamard matrices.
• To derive an analytical formula for the average entangling power of tripartite orthogonal quantum gates.
• To construct and characterize the first known absolutely maximally entangled (AME) state of four quhexes (AME(4,6)) with high symmetry and 112 non-zero terms.
• To generalize quantum Sudoku (SudoQ) designs by introducing cardinality as a measure of 'quantumness' and linking them to mutually unbiased bases.

Proposed method

• Developed an algorithm to determine unistochasticity of 4×4 bistochastic matrices using the bracelet condition.
• Proved that the bracelet condition is sufficient for unistochasticity of circulant 4×4 matrices.
• Established that rays and counter-rays in the Birkhoff polytope are unistochastic if a robust Hadamard matrix of dimension N exists.
• Derived an explicit analytical expression for the average entangling power of a tripartite orthogonal gate using unitary invariance and integration over the orthogonal group.
• Introduced the Hessian of the entangling power and the average singular entropy to assess extremality of unitary matrices.
• Defined the cardinality of a SudoQ design as the number of distinct quantum states and linked it to quantum Latin squares and mutually unbiased bases.

Experimental results

Research questions

• RQ1 Is every 4×4 bistochastic matrix unistochastic, and can the bracelet condition fully characterize unistochastic circulant matrices of this size?
• RQ2 Are rays and counter-rays in the Birkhoff polytope of dimension N unistochastic when a robust Hadamard matrix of size N exists?
• RQ3 What is the analytical form of the average entangling power for tripartite orthogonal quantum gates?
• RQ4 Can a genuinely quantum AME(4,6) state be constructed, and does it exhibit a block-like structure with high symmetry and non-zero term count?
• RQ5 What is the full range of admissible cardinalities for quantum Latin squares, and how does cardinality relate to the 'quantumness' of SudoQ designs?

Key findings

• The bracelet condition is sufficient to determine unistochasticity of 4×4 circulant bistochastic matrices.
• Rays and counter-rays in the Birkhoff polytope are unistochastic if a robust Hadamard matrix of dimension N exists.
• An explicit analytical formula for the average entangling power of a tripartite orthogonal gate was derived.
• The first known AME(4,6) state was constructed with 112 non-zero terms and a distinct block-like structure, suggesting deeper symmetry.
• The Hessian of the entangling power and average singular entropy were computed, enabling extremality assessment of unitary matrices.
• A maximal cardinality SudoQ design exists for any N², and cardinality serves as a quantitative measure of 'quantumness' in quantum Latin squares.