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Schwarz maps with symmetry

Alfonso García-Velo, Alberto Ibort

TL;DR

This work develops a general, representation-theoretic framework for quantum channels with symmetry, showing how $G$-equivariance imposes a block-decomposed structure on maps and Choi matrices. It achieves a complete classification of $U(n)$-equivariant maps on $M_n(\mathbb{C})$ and explicit CP and Schwarz regions, revealing Schwarz maps that are not CP. It also establishes PPT$^2$ results for several symmetry classes, including $U(n)$, $DU(n)$ (qubits and symmetric qutrits), and certain tensor-product symmetries, by exploiting the isotypic decomposition and simple composition rules. The results illuminate how symmetry controls the landscape of positive, Schwarz, and CP maps, and provide concrete, geometry-rich examples where PPT$^2$ holds, with potential extensions to broader, infinite-dimensional contexts.

Abstract

The theory of symmetry of quantum mechanical systems is applied to study the structure and properties of several classes of relevant maps in quantum information theory: CPTP, PPT and Schwarz maps. First, we develop the general structure that equivariant maps $Φ:\mathcal A \to \mathcal B$ between $C^\ast$-algebras satisfy. Then, we undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions. Schwarz maps satisfy Kadison's inequality $Φ(X^\ast X) \geq Φ(X)^\ast Φ(X)$ and form an intermediate class between positive and completely positive maps. We completely classify $U(n)$-equivariant on $M_n(\mathbb C)$ and determine those that are completely positive and Schwarz. Partial classifications are then obtained for the weaker $DU(n)$-equivariance (diagonal unitary symmetry) and for tensor-product symmetries $U(n_1) \otimes U(n_2)$. In each case, the parameter regions where $Φ$ is Schwarz or completely positive are described by explicit algebraic inequalities, and their geometry is illustrated. Finally, we further show that the $U(n)$-equivariant family satisfies $\mathrm{PPT} \iff \mathrm{EB}$, while the $DU(2)$, symmetric $DU(3)$, $U(2) \otimes U(2)$ and $U(2) \otimes U(3)$, families obey the $\mathrm{PPT}^2$ conjecture through a direct symmetry argument. These results reveal how group symmetry controls the structure of non-completely positive maps and provide new concrete examples where the $\mathrm{PPT}^2$ property holds.

Schwarz maps with symmetry

TL;DR

This work develops a general, representation-theoretic framework for quantum channels with symmetry, showing how -equivariance imposes a block-decomposed structure on maps and Choi matrices. It achieves a complete classification of -equivariant maps on and explicit CP and Schwarz regions, revealing Schwarz maps that are not CP. It also establishes PPT results for several symmetry classes, including , (qubits and symmetric qutrits), and certain tensor-product symmetries, by exploiting the isotypic decomposition and simple composition rules. The results illuminate how symmetry controls the landscape of positive, Schwarz, and CP maps, and provide concrete, geometry-rich examples where PPT holds, with potential extensions to broader, infinite-dimensional contexts.

Abstract

The theory of symmetry of quantum mechanical systems is applied to study the structure and properties of several classes of relevant maps in quantum information theory: CPTP, PPT and Schwarz maps. First, we develop the general structure that equivariant maps between -algebras satisfy. Then, we undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions. Schwarz maps satisfy Kadison's inequality and form an intermediate class between positive and completely positive maps. We completely classify -equivariant on and determine those that are completely positive and Schwarz. Partial classifications are then obtained for the weaker -equivariance (diagonal unitary symmetry) and for tensor-product symmetries . In each case, the parameter regions where is Schwarz or completely positive are described by explicit algebraic inequalities, and their geometry is illustrated. Finally, we further show that the -equivariant family satisfies , while the , symmetric , and , families obey the conjecture through a direct symmetry argument. These results reveal how group symmetry controls the structure of non-completely positive maps and provide new concrete examples where the property holds.
Paper Structure (18 sections, 33 theorems, 68 equations, 2 figures)

This paper contains 18 sections, 33 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Analyzing quantum channels
  3. Positive, Schwarz and completely positive maps
  4. Entanglement breaking channels: the PPT$^2$ conjecture
  5. Fundamentals of representation theory
  6. Groups, group actions, and symmetry
  7. Linear unitary representations of groups
  8. Equivalent representations
  9. Irreducible representations and Schur's lemma
  10. Channels with symmetry
  11. Equivariant CP and Schwarz maps
  12. Unitary equivariance
  13. Diagonal unitary equivariance
  14. Qubits: Recovering the $DU(2)$ classification
  15. Qutrits: a symmetric $DU(3)$ family
  16. ...and 3 more sections

Key Result

Lemma 3.1

Let $(G, V_1, \mu_1)$ and $(G, V_2, \mu_2)$ be two finite-dimensional irreducible representations, and let $\phi: V_1 \to V_2$ be a continuous linear map satisfying $\phi \circ \mu_1 (g) = \mu_2 (g) \circ \phi$, for all $g \in G$. Then:

Figures (2)

  • Figure 1: A diagram representing the regions of (unital) $U(n)$-equivariant CP maps (red), (unital) $U(n)$-equivariant Schwarz maps (blue), and $U(n)$-equivariant PPT maps (grey) which are equivalent to EB maps.
  • Figure 2: Symmetric $DU(3)$ family. Light: Schwarz region $|\lambda| \le \sqrt{\tfrac{1}{2}\,\min\{p,1-p\}}$. Dark: CP region $|\lambda| \le \min\{p,1-p\}$.

Theorems & Definitions (61)

  • Example 3.1
  • Lemma 3.1: Schur's Lemma
  • Theorem 4.1: Channel decomposition
  • proof
  • Lemma 4.1: Composition of $G$-equivariant channels
  • proof
  • Corollary 4.1.1
  • Theorem 4.2: Capacity decomposition
  • proof
  • Proposition 5.1: $U(n)$-equivariant maps
  • ...and 51 more