Geometric constructions over $\mathbb{C}$ and $\mathbb{F}_2$ for Quantum InformationGeometric constructions over $\mathbb{C}$ and $\mathbb{F}_2$ for Quantum Information

Frédéric Holweck

In this review paper I present two geometric constructions of distinguished nature, one is over the field of complex numbers $\mathbb{C}$ and the other one is over the two elements field $\mathbb{F}_2$ . Both constructions have been employed in the past fifteen years to describe two quantum paradoxes or two resources of quantum information: entanglement of pure multipartite systems on one side and contextuality on the other. Both geometric constructions are linked to representation of semi-simple Lie groups/algebras. To emphasize this aspect one explains on one hand how well-known results in representation theory allows one to see all the classification of entanglement classes of various tripartite quantum systems ( $3$ qubits, $3$ fermions, $3$ bosonic qubits...) in a unified picture. On the other hand, one also shows how some weight diagrams of simple Lie groups are encapsulated in the geometry which deals with the commutation relations of the generalized $N$ -Pauli group.In this review paper I present two geometric constructions of distinguished nature, one is over the field of complex numbers $\mathbb{C}$ and the other one is over the two elements field $\mathbb{F}_2$. Both constructions have been employed in the past fifteen years to describe two quantum paradoxes or two resources of quantum information: entanglement of pure multipartite systems on one side and contextuality on the other. Both geometric constructions are linked to representation of semi-simple Lie groups/algebras. To emphasize this aspect one explains on one hand how well-known results in representation theory allows one to see all the classification of entanglement classes of various tripartite quantum systems ($3$ qubits, $3$ fermions, $3$ bosonic qubits...) in a unified picture. On the other hand, one also shows how some weight diagrams of simple Lie groups are encapsulated in the geometry which deals with the commutation relations of the generalized $N$-Pauli group.

arXiv:1810.04258Oct 9, 20180 citations

Geometric Approach to Digital Quantum Information

Chad T. Rigetti, Rémy Mosseri, Michel Devoret

arXiv:quant-ph/0312196Quantum Information ProcessingDec 1, 200413 citations

Resolution of the Black Hole Information Paradox via Unified Relational Projection Dynamics (URPD): A Minimal Information-Theoretic Framework

A.I. Trinity Lab

This paper presents a novel resolution of the black hole information paradox using Unified Relational Projection Dynamics (URPD), a minimal information-theoretic framework grounded in the geometry of the complex projective state space CPN−1\mathbb{C}P^{N-1}CPN−1.URPD treats physical states not as absolute objects but as relational frames, whose overlaps ∣⟨F1∣F2⟩∣|\langle F_1 | F_2 \rangle|∣⟨F1∣F2⟩∣ never vanish due to the compactness of the Fubini–Study geometry.As a result, information is never destroyed during black-hole evaporation; instead, it is continuously compressed into the residual overlaps between interior and exterior frames.This mechanism naturally reproduces the Page curve and preserves global unitarity without relying on holography, replica wormholes, island formulas, or any additional quantum-gravity assumptions.The work provides the first fully relational and geometrically minimal explanation of information preservation across the entire evaporation process.This paper presents a novel resolution of the black hole information paradox using Unified Relational Projection Dynamics (URPD), a minimal information-theoretic framework grounded in the geometry of the complex projective state space CPN−1\mathbb{C}P^{N-1}CPN−1.URPD treats physical states not as absolute objects but as relational frames, whose overlaps ∣⟨F1∣F2⟩∣|\langle F_1 | F_2 \rangle|∣⟨F1∣F2⟩∣ never vanish due to the compactness of the Fubini–Study geometry.As a result, information is never destroyed during black-hole evaporation; instead, it is continuously compressed into the residual overlaps between interior and exterior frames.This mechanism naturally reproduces the Page curve and preserves global unitarity without relying on holography, replica wormholes, island formulas, or any additional quantum-gravity assumptions.The work provides the first fully relational and geometrically minimal explanation of information preservation across the entire evaporation process.

Nov 22, 20250 citations

New asymmetric quantum codes over $\textbf{F}_{q^2}$ New asymmetric quantum codes over $$\textbf{F}_{q^2}$$

Yuena Ma, Yang Liu, Liangdong Lü +1 more

Quantum Information ProcessingApr 16, 20230 citations

On Quantum Codes Obtained From Cyclic Codes Over $\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$ On Quantum Codes Obtained From Cyclic Codes Over $\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$

Sukhamoy Pattanayak, Abhay Kumar Singh, Pratyush Kumar

Let $R=\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$ be a non-chain finite commutative ring, where $u^3=u$ . In this paper, we mainly study the construction of quantum codes from cyclic codes over $R$ . We obtained self-orthogonal codes over $\mathbb{F}_2$ as gray images of linear and cyclic codes over $R$ . The parameters of quantum codes which are obtained from cyclic code over $R$ are discussed.Let $R=\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$ be a non-chain finite commutative ring, where $u^3=u$. In this paper, we mainly study the construction of quantum codes from cyclic codes over $R$. We obtained self-orthogonal codes over $\mathbb{F}_2$ as gray images of linear and cyclic codes over $R$. The parameters of quantum codes which are obtained from cyclic code over $R$ are discussed.

arXiv:1601.02328Jan 11, 20160 citations

Geometric Constructions over ℂ

{\mathbb {C}}

and 𝔽 2

{\mathbb {F}}_2

for Quantum Information

Geometric constructions over

\mathbb{C}

and

\mathbb{F}_2

for Quantum Information

In this review paper I present two geometric constructions of distinguished nature, one is over the field of complex numbers

\mathbb{C}

and the other one is over the two elements field

\mathbb{F}_2

. Both constructions have been employed in the past fifteen years to describe two quantum paradoxes or two resources of quantum information: entanglement of pure multipartite systems on one side and contextuality on the other. Both geometric constructions are linked to representation of semi-simple Lie groups/algebras. To emphasize this aspect one explains on one hand how well-known results in representation theory allows one to see all the classification of entanglement classes of various tripartite quantum systems (

3

qubits,

3

fermions,

3

bosonic qubits...) in a unified picture. On the other hand, one also shows how some weight diagrams of simple Lie groups are encapsulated in the geometry which deals with the commutation relations of the generalized

N

-Pauli group.

New asymmetric quantum codes over

\textbf{F}_{q^2}

On Quantum Codes Obtained From Cyclic Codes Over

\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2

Let

R=\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2

be a non-chain finite commutative ring, where

u^3=u

. In this paper, we mainly study the construction of quantum codes from cyclic codes over

R

. We obtained self-orthogonal codes over

\mathbb{F}_2

as gray images of linear and cyclic codes over

R

. The parameters of quantum codes which are obtained from cyclic code over

R

are discussed.