Study of Equivariant maps and applications to Quantum Information Theory

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.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.

quant-pharXiv:2601.02282Jan 6, 20260 citations

Characterization of Equivariant Maps and Application to Entanglement Detection

Ivan Bardet, Benoı̂t Collins, Gunjan Sapra

arXiv:1811.08193Annales Henri PoincaréAug 17, 202013 citations

Learning equivariant maps with variational quantum circuits

Zachary P. Bradshaw, Ethan N. Evans, Matthew Cook +1 more

Geometric quantum machine learning uses the symmetries inherent in data to design tailored machine learning tasks with reduced search space dimension. The field has been well studied recently in an effort to avoid barren plateau issues while improving the accuracy of quantum machine learning models. This work explores the related problem of learning an equivariant map given two unitary representations of a finite group, which in turn allows the symmetric embedding of the data to be learned rather than simply required. Moreover, this procedure allows the learning of covariant quantum channels, which are an essential tool in quantum information theory. We demonstrate the feasibility of this task and give examples to illustrate the procedure.Geometric quantum machine learning uses the symmetries inherent in data to design tailored machine learning tasks with reduced search space dimension. The field has been well studied recently in an effort to avoid barren plateau issues while improving the accuracy of quantum machine learning models. This work explores the related problem of learning an equivariant map given two unitary representations of a finite group, which in turn allows the symmetric embedding of the data to be learned rather than simply required. Moreover, this procedure allows the learning of covariant quantum channels, which are an essential tool in quantum information theory. We demonstrate the feasibility of this task and give examples to illustrate the procedure.

arXiv:2412.11889Physical Review AppliedApr 3, 20253 citations

Some Results in Quantum Information Theory

Debashree Chakraborty

Jan 1, 20150 citations

Reverse Map Projections as Equivariant Quantum Embeddings

Max Arnott, Dimitri Papaioannou, Kieran McDowall +2 more

We introduce the novel class $(E_α)_{α\in [-\infty,1)}$ of reverse map projection embeddings, each one defining a unique new method of encoding classical data into quantum states. Inspired by well-known map projections from the unit sphere onto its tangent planes, used in practice in cartography, these embeddings address the common drawback of the amplitude embedding method, wherein scalar multiples of data points are identified and information about the norm of data is lost. We show how reverse map projections can be utilised as equivariant embeddings for quantum machine learning. Using these methods, we can leverage symmetries in classical datasets to significantly strengthen performance on quantum machine learning tasks. Finally, we select four values of $α$ with which to perform a simple classification task, taking $E_α$ as the embedding and experimenting with both equivariant and non-equivariant setups. We compare their results alongside those of standard amplitude embedding.We introduce the novel class $(E_α)_{α\in [-\infty,1)}$ of reverse map projection embeddings, each one defining a unique new method of encoding classical data into quantum states. Inspired by well-known map projections from the unit sphere onto its tangent planes, used in practice in cartography, these embeddings address the common drawback of the amplitude embedding method, wherein scalar multiples of data points are identified and information about the norm of data is lost. We show how reverse map projections can be utilised as equivariant embeddings for quantum machine learning. Using these methods, we can leverage symmetries in classical datasets to significantly strengthen performance on quantum machine learning tasks. Finally, we select four values of $α$ with which to perform a simple classification task, taking $E_α$ as the embedding and experimenting with both equivariant and non-equivariant setups. We compare their results alongside those of standard amplitude embedding.

arXiv:2407.19906Jul 29, 20240 citations

Studies in quantum information theory

Nicolas C. Menicucci

Jan 1, 20080 citations

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.

We introduce the novel class

(E_α)_{α\in [-\infty,1)}

of reverse map projection embeddings, each one defining a unique new method of encoding classical data into quantum states. Inspired by well-known map projections from the unit sphere onto its tangent planes, used in practice in cartography, these embeddings address the common drawback of the amplitude embedding method, wherein scalar multiples of data points are identified and information about the norm of data is lost. We show how reverse map projections can be utilised as equivariant embeddings for quantum machine learning. Using these methods, we can leverage symmetries in classical datasets to significantly strengthen performance on quantum machine learning tasks. Finally, we select four values of

α

with which to perform a simple classification task, taking

E_α

as the embedding and experimenting with both equivariant and non-equivariant setups. We compare their results alongside those of standard amplitude embedding.