We define a notion of entanglement-symmetry of channels (completely positive trace-preserving maps) between finite-dimensional (f.d.) $C^*$-algebras. Channels related by an entanglement-symmetry may be reversibly transformed into each other using an entangled resource. For example, the standard quantum teleportation and dense coding schemes witness that the noiseless channel on a $d$-dimensional quantum system and the noiseless channel on a $d^2$-level classical system are related by an entanglement-symmetry. Entanglement-symmetries arise from covariance of a channel; they relate $G$-covariant channels to $G'$-covariant channels, where $G$ and $G'$ are monoidally equivalent compact (quantum) groups related by a unitary pseudonatural transformation.

§ INTRODUCTION

§.§ Overview

Entanglement-assisted channel coding schemes. We begin by reviewing a basic setup for channel coding. Alice and Bob can communicate through a channel $N: A \to B$, where $A$ is some input system belonging to Alice, and $B$ is some output system belonging to Bob. They want to communicate through a channel $T: X \to Y$, where $X$ is some input system belonging to Alice and $Y$ is some output system belonging to Bob. A channel coding scheme for $T$ from $N$ is defined by an encoding channel $E: X \to A$, performed by Alice, and a decoding channel $D: B \to Y$, performed by Bob, such that $D \circ N \circ E = T$:

D ∘N ∘E

The existence of a channel coding scheme for $T$ from $N$ means that Alice and Bob can communicate through the channel $T$ even through they only share the channel $N$.
More loosely, we can say that the channel $N$ is transformed into the channel $T$ by the encoding and decoding channels $(E,D)$.

We can introduce entanglement-assistance into the picture as follows. Let $H_e$ be a Hilbert space, and let $\Psi: \mathbb{C} \to B(H_e) \otimes B(H_e)$ be the channel initialising a maximally entangled state

\begin{equation}\label{eq:entstatepuredef}
\ket{\psi}:= \frac{1}{\sqrt{\dim(H_e)}}\sum_{i=1}^{\dim(H_e)}\ket{i}\otimes \ket{i} \in H_e \otimes H_e.
\end{equation}

Operationally, we assume that this state has been initialised, and that Alice is in possession of one of the entangled systems and Bob the other. To perform her encoding map, Alice may use her half of the maximally entangled state, giving the type $E: X \otimes B(H_e) \to A$ for her encoding operation; to perform his decoding map, Bob may use his half of the maximally entangled state, giving the type $D: B \otimes B(H_e) \to Y$ for his decoding operation. We say that $(E,D,H_e)$ is an entanglement-assisted channel coding scheme for $T$ from $N$ if the following equation is obeyed (from now on, we read diagrams from bottom to top):

D ∘(N ⊗𝕀_B(H_e)) ∘(E ⊗𝕀_B(H_e)) ∘(𝕀_X ⊗Ψ)

The existence of an entanglement-assisted channel coding scheme $(E,D,H_e)$ for $T$ from $N$ means that Alice and Bob can communicate through the channel $T$ even though they only share the channel $N$ and and a maximally entangled state of two quantum systems of dimension $\dim(H_e)$.

It is not hard to show that the relation
$(N_1: A_1 \to B_1) \geq (N_2: A_2 \to B_2)$ iff there exists an entanglement-assisted channel coding scheme for $N_2$ from $N_1$

defines a partial order on any set of channels. Our entanglement-symmetries will relate channels which are equal under this partial order; that is, which can simulate each other using an entangled resource.

We observe that any pair of entanglement-equivalent channels have the same entanglement-assisted capacities, since one can interconvert the channels prior to use. We will return to this application later in this introduction when discussing entanglement-equivalences of quantum-to-quantum with classical-to-classical channels.

Tight teleportation and dense coding. Here is a well-known example. The names teleportation and dense coding are given to the following entanglement-assisted channel coding scenarios.

* Teleportation. The channel $N:A \to B$ shared by Alice and Bob is the noiseless (i.e. identity) classical channel

\id_{\mathbb{C}^{\oplus d^2}}: \mathbb{C}^{\oplus d^2} \to{} \mathbb{C}^{\oplus d^2}

on a $d^2$-level classical system. (Here $\mathbb{C}^{\oplus d^2}$ is the $d^2$-dimensional commutative $C^*$-algebra.)

The channel $T: X \to Y$ that they wish to perform is the noiseless quantum channel

\id_{M_d(\mathbb{C})}: M_d(\mathbb{C}) \to M_d(\mathbb{C})$$

on the qudit $C^*$-algebra $M_d(\mathbb{C})$.

* Dense coding. The channel $N: A \to B$ shared by Alice and Bob is the noiseless quantum channel $\id_{M_d(\mathbb{C})}.$

The channel $T: X \to Y$ they wish to perform is the noiseless classical channel
\id_{\mathbb{C}^{\oplus d^2}}$.

The standard teleportation and dense coding schemes are an entanglement-symmetry relating the channels $\id_{\mathbb{C}^{\oplus d^2}}$ and $\id_{M_d(\mathbb{C})}$.

Our construction. Entanglement-symmetries arise from covariance of a channel. Covariance for compact group actions is frequently used in quantum information theory (see e.g. [14] and references therein). We work in the broader setting of compact quantum groups, which generalise compact groups [29]. Working in this broader setting is not an arbitrary choice — it is forced on us by our consideration of Tannaka-Krein-Woronowicz duality, as we shall explain shortly.

We recall that an action of a compact group $G$ on a finite-dimensional $C^*$-algebra $A$ is a continuous homomorphism $\tau: G \to \Aut(A)$, where $\Aut(A)$ is the group of $*$-automorphisms of $A$. We call $(A,\tau)$ a finite-dimensional $G$-$C^*$-algebra.[In operator algebra a $C^*$-algebra with a continuous action of a locally compact group is often called a $C^*$-dynamical system. One benefit of the alternative `$G$-$C^*$-algebra' terminology is that, when working with multiple compact (quantum) groups, it makes clear which group is acting.] All our $G$-$C^*$-algebras are finite dimensional and equipped with a canonical  trace which is invariant under the $G$-action. A covariant channel $(A,\tau_A) \to (B,\tau_B)$ is a channel (i.e. a completely positive trace-preserving map) $N: A \to B$ such that $\tau_B(g) \circ N = N \circ \tau_A(g)$ for all $g \in G$. More generally, an action of a compact quantum group, and a corresponding notion of covariance for channels[For some quantum groups there may exist no invariant trace; in this case we demand preservation of the canonical invariant functional.], may be defined. For any compact quantum group $G$, the $G$-$C^*$-algebras and covariant channels form a category $\Chan(G)$.

For any compact quantum group $G$, we may consider its category of finite-dimensional continuous unitary representations $\Rep(G)$. A fibre functor is a faithful unitary $\mathbb{C}$-linear monoidal functor $\Rep(G) \to \Hilb$, where $\Hilb$ is the category of finite-dimensional Hilbert spaces and linear maps. The category $\Rep(G)$ has a canonical fibre functor $F$, given by taking a representation to its underlying Hilbert space and an intertwiner to its underlying linear map. However, there may be other fibre functors $F'$ not naturally isomorphic to the canonical fibre functor. By Tannaka-Krein-Woronowicz duality, every fibre functor $F'$ is associated to a compact quantum group $G_{F'}$. In general, $G_{F'}$ may be a quantum group even if $G$ is an ordinary compact group; this is why it is necessary for us to work with quantum groups.

In [26] we introduced and classified unitary maps between fibre functors, called unitary pseudonatural transformations (UPTs). Every UPT has an associated Hilbert space $H_e$.

A black-box description of our construction is as follows. Let $G$ be a compact quantum group, let $F$ be the canonical fibre functor on $\Rep(G)$, let $F'$ be some other fibre functor accessible from $F$ by a UPT, let $G'=G_{F'}$ be the compact quantum group associated to $F'$, and let $(\alpha,H_e): F \to F'$ be a UPT. Then we construct an equivalence of categories $\Chan(G) \to \Chan(G')$ (Theorem <ref>). In particular:

* For every $G$-$C^*$-algebra $(A,\tau)$, we construct a $G'$-$C^*$-algebra $(A',\tau')$.

* For every $G$-covariant channel $f: (A,\tau_A) \to (B,\tau_B)$, we construct a $G'$-covariant channel $f': (A',\tau_A') \to (B',\tau_B')$.

This equivalence has the following additional structure. For every $G$-$C^*$-algebra $(A,\tau)$, we construct channels

\begin{align*}
u_{(A,\tau)}: A \otimes B(H_e) \to A' && v_{(A,\tau)}: A' \otimes B(H_e) \to A
\end{align*}

such that for every $G$-covariant channel $f: (A,\tau_A) \to (B,\tau_B)$, the following equations are obeyed:

Here $\Psi: \mathbb{C} \to B(H_e) \otimes B(H_e)$ is the channel initialising the maximally entangled state (<ref>).
We say that $f$ and $f'$ are related by an entanglement-symmetry.

The essential idea of the construction is to use the canonical fibre functor $F$ to associate every $G$-$C^*$-algebra $(A,\tau)$ with an `abstract $C^*$-algebra' in the category $\Rep(G)$ (concretely, a standard special Frobenius algebra). The fibre functor $F'$ maps abstract $C^*$-algebras in $\Rep(G)$ to $G'$-$C^*$-algebras (Proposition <ref>), whereby we obtain the new $G'$-$C^*$-algebra $(A',\tau')$. Channels $f:(A,\tau_A) \to (B,\tau_B)$ can likewise be associated with `abstract channels' in $\Rep(G)$ (concretely, counit-preserving morphisms between Frobenius algebras obeying the CP condition (<ref>)), which the fibre functor $F'$ maps to channels $f': (A',\tau_A') \to (B,\tau_B')$ (Proposition <ref>). The channels $u_{(A,\tau)}$ and $v_{(A,\tau)}$ are obtained straightforwardly from the unitary pseudonatural transformation $(\alpha,H_e): F \to F'$ (Theorem <ref>).

§.§ Related work

Quantum bijections. The channels $u_{(A,\tau)}: A \otimes B(H_e) \to A'$ and $v_{(A,\tau)}: A' \otimes B(H_e) \to A$ realising the entanglement-symmetry are instances of the quantum bijections already studied in the context of noncommutative combinatorics <cit.> and entanglement-invertibility of channels [25]. These reduce to $*$-isomorphisms when the auxiliary Hilbert space is trivial. In fact, it was shown in <cit.> that quantum bijections from an f.d. $C^*$-algebra $A$ are precisely unitary pseudonatural transformations from the canonical fibre functor on the category of representations of the quantum permutation group [28] of $A$. This generalises the obvious correspondence between $*$-automorphisms of $A$ and unitary monoidal natural automorphisms of the canonical fibre functor on $\Rep(\Aut(A))$.

Quantum graph isomorphisms. In <cit.> the notion of a quantum graph $(A,\Gamma)$ was introduced. This is defined by an adjacency matrix $\Gamma$ on a finite-dimensional $C^*$-algebra $A$; the adjacency matrix is a certain CP map $\Gamma: A \to A$. A finite-dimensional (f.d.) quantum isomorphism of quantum graphs <cit.> is a quantum bijection intertwining the adjacency matrices. The construction of entanglement-symmetries detailed above holds for general CP maps as well as channels, and it follows from [26, 16] that a finite-dimensional quantum isomorphism is precisely an entanglement-symmetry of the CP map $\Gamma$. (Here $\Gamma$ is covariant for the natural action of the quantum automorphism group of $(A,\Gamma)$ on $A$.)

Quantum correlations and infinite-dimensional generalisations. An obvious question arising from this work is: can we realise entanglement-symmetries corresponding to fibre functors not unitarily pseudonaturally isomorphic to the canonical fibre functor? We showed in <cit.> that unitary pseudonatural transformations  between fibre functors correspond to finite-dimensional $*$-representations of Hopf-Galois objects. One would therefore expect the infinite-dimensional generalisation to correspond to infinite-dimensional $*$-representations, or, more restrictively, tracial states of Hopf-Galois objects.

There is a notion of quantum isomorphism of quantum graphs making use of an infinite-dimensional entangled resource, defined in terms of a commuting operator strategy for a nonlocal game satisfying a certain symmetry condition <cit.>. The connection between these commuting operator quantum isomorphisms and tracial states of Hopf-Galois objects was already made in <cit.>. In fact, in [23, 6] a general notion of quantum nonlocal game was introduced, which can accept inputs and outputs in an arbitrary f.d. $C^*$-algebra. We suspect that commuting operator strategies for quantum nonlocal games satisfying an analogue of the aforementioned symmetry condition will yield the required infinite-dimensional generalisation of quantum bijection.

Categorical quantum mechanics. This work makes extensive use of results obtained in the program of categorical quantum mechanics initiated in [1], particularly those relating to the Frobenius-algebraic formulation of finite-dimensional $C^*$-algebras and completely positive maps between them [27, 7, 11]. We have also made use of the more general Frobenius-algebraic characterisation of finite-dimensional $C^*$-algebras with a compact quantum group action in [18].

§.§ Contents

In Section <ref> we recall well-known material on rigid $C^*$-tensor categories, compact quantum groups and Tannaka-Krein-Woronowicz duality. In Section <ref> we recall well-known material on  the Frobenius algebraic formulation of finite-dimensional $G$-covariant physics; we then put this theory within the framework of Tannaka-Krein-Woronowicz duality. In Section <ref> we treat the construction of entanglement-symmetries we have discussed in this introduction. In Section <ref> we present a simple example, and apply this to compute the entanglement-assisted capacity of certain quantum channels. In Appendix <ref> we prove some facts about existence of invariant traces on $G$-$C^*$-algebras, extending results on $G$-invariant functionals in [18].

§.§ Acknowledgements

We thank David Reutter and Jamie Vicary for useful discussions. We are grateful to Makoto Yamashita for comments and suggestions regarding invariant traces on $G$-$C^*$-algebras which led to the inclusion of Appendix <ref>. The work was supported by EPSRC.

§ NOTATION AND CONVENTIONS

All Frobenius algebras considered in this work are dagger, so we omit the qualifier. When referring to Frobenius algebras in some rigid $C^*$-tensor category we write a square-bracketed tuple $[A,m,u]$ where $A$ is the carrier object of the algebra and $m: A \otimes A \to A$ and $u: \mathbbm{1} \to A$ are the multiplication and unit morphisms.

Let $[A_1,m_1,u_1]$, $[A_2,m_2,u_2]$ be special Frobenius algebras in some rigid $C^*$-tensor category. We show that a morphism $f:A_1 \to A_2$ is CP by writing it as $f: [A_1,m_1,u_1] \to{} [A_2,m_2,u_2]$.

When referring to a unitary corepresentation of a compact quantum group algebra $A_G$ we write a tuple $(V,\rho)$, where $V$ is a Hilbert space and $\rho: V \to V \otimes A_G$ is the corepresentation map. For two unitary corepresentations $(V,\rho_V), (W,\rho_W)$ we show that a linear map $f: V \to W$ is an intertwiner by writing it as $f:(V,\rho_V) \to (W,\rho_W)$.

When referring to a finite-dimensional $G$-$C^*$-algebra for a compact quantum group $G$ of Kac type we write a tuple $([A,m,u],\rho)$ where $[A,m,u]$ is a special symmetric Frobenius algebra in the category of finite-dimensional Hilbert spaces and linear maps and $\rho: A \to A \otimes A_G$ is an $A_G$-coaction.

Let $([A_1,m_1,u_1],\rho_1)$, $([A_2,m_2,u_2],\rho_2)$ be finite-dimensional $G$-$C^*$-algebras. We show that a channel $f:[A_1,m_1,u_1] \to{} [A_2,m_2,u_2]$ is covariant by writing it as $f:([A_1,m_1,u_1],\rho_1) \to{} ([A_2,m_2,u_2],\rho_2)$.

§ BACKGROUND

§.§ Diagrammatic calculus for rigid $C^*$-tensor categories

A rigid $C^*$-tensor category is in particular a pivotal (in fact, a spherical) dagger category <cit.> in which the Hom-sets are Banach spaces. For the exact definition, see e.g. <cit.>. We will here only highlight some aspects.

The adjective `tensor' indicates that a rigid $C^*$-tensor category is monoidal. We use the well-known diagrammatic calculus for monoidal categories [11, 21].

We read diagrams from bottom to top. Objects are drawn as wires, while morphisms are drawn as boxes whose type corresponds to their input and output wires. Composition of morphisms is represented by vertical juxtaposition, while monoidal product is represented by horizontal juxtaposition. For example, two morphisms $f:X\to Y $ and $g: Y \to Z$ can be composed as follows:

g ∘f: X →Z

f ⊗g: X ⊗Y →Y ⊗Z

The wire for the monoidal unit $\mathbbm{1}$, and the identity morphism $\id_X$ for any object $X$, are invisible in the diagram. Two diagrams which are planar isotopic represent the same morphism.

`Rigid'. The adjective `rigid' indicates that a rigid $C^*$-tensor category has duals. Here and throughout when we talk about the duals of a rigid $C^*$-tensor category we mean the standard solutions to the conjugate equations <cit.>.

Let $X$ be an object in a monoidal category. A dual $[X^*,\eta,\epsilon]$ for $X$ is:

* An object $X^*$ (the dual object). To represent dual objects in the graphical calculus, we draw an upward-facing arrow on the $X$-wire and a downward-facing arrow on the $X^*$-wire.

* Morphisms $\eta_R: \mathbbm{1} \to X^* \otimes X$, $\eta_L: \mathbbm{1} \to X \otimes X^*$, $\epsilon_R: X \otimes X^* \to \mathbbm{1}$, $\epsilon_L: X^* \otimes X \to \mathbbm{1}$, which we draw as cups and caps:

\begin{align*}
\includegraphics[scale=1]{Figures/svg/2cats/dagduallcapflip.pdf}\qquad
\includegraphics[scale=1]{Figures/svg/2cats/dagdualrcup.pdf}\qquad
\includegraphics[scale=1]{Figures/svg/2cats/dagdualrcupflip.pdf}\qquad
\includegraphics[scale=1]{Figures/svg/2cats/dagduallcap.pdf}\qquad
\\
\eta_R:\mathbbm{1} \to X^* \otimes X
\eta_L: \mathbbm{1} \to X \otimes X^*
\epsilon_R: X \otimes X^* \to \mathbbm{1}
\epsilon_L: X^* \otimes X \to \mathbbm{1}
\end{align*}

* These morphisms must satisfy the following snake equations:

In a rigid $C^*$-tensor category, the duals allow us to define the dimension of an object and the trace of an endomorphism. Let $X$ be some object and let $f \in \End(X):=\Hom(X,X)$ be a morphism; then the trace of $f$ is defined as follows:

\begin{align}\label{eq:dim}
\includegraphics{Figures/svg/2cats/lefttrace.pdf}
\includegraphics{Figures/svg/2cats/righttrace.pdf}
\end{align}

The fact that the left and right trace are equal here is due to sphericality of a rigid $C^*$-tensor category. The dimension of an object $X$ is defined by $\dim(X):= \Tr(\id_X)$.

`C*'. The adjective $`C^*{}-$' indicates that a rigid $C^*$-tensor category is a dagger category whose Hom-sets are Banach spaces. Moreover, for any morphism $f: X \to Y$ we have $||f^{\dagger} \circ f|| = ||f||^2$; in particular, for any object $X$ the set $\End(X)$ is a $C^*$-algebra.

In the graphical calculus, we give the morphism boxes an offset edge, and represent the dagger of a morphism by reflection in a horizontal axis, preserving the direction of any arrows:

Let $\mathcal{C}$ be a dagger category.  We say that a morphism $\alpha: X \to Y$ is an isometry if $\alpha^{\dagger} \circ \alpha = \id_X$. We say that it is a coisometry if $ \alpha \circ \alpha^{\dagger}  = \id_Y$. We say that it is unitary if it is both an isometry and a coisometry.

In a rigid $C^*$-tensor category, the left cup and cap are the dagger of the right cup and cap:

§.§.§ Example: the category $\Hilb$

A simple example of a pivotal dagger category is the category $\Hilb$. The objects of the monoidal category $\Hilb$ are finite-dimensional Hilbert spaces, and the morphisms are linear maps between them; composition of morphisms is composition of linear maps. The monoidal product is given on objects by the tensor product of Hilbert spaces, and on morphisms by the tensor product of linear maps; the unit object is the 1-dimensional Hilbert space $\mathbb{C}$.

For any object $H$, its right dual object can be chosen to be $H$ itself; every object is self-dual. For this reason we do not generally draw arrows on Hilbert space wires in this work. Indeed, any orthonormal basis $\{\ket{i}\}$ for $H$ defines a cup and cap:

(0,0) to [out=up, in=up, looseness=2.5] (2,0);
[dimension, left] at (2.0,0) $H$;
[dimension, right] at (0,0) $H$;
[zxvertex=,zxdown] at (1,1.5);

(0,0) to [out=up, in=up, looseness=2.5] (2,0);
[dimension, right] at (2.05,0) $H$;
[dimension, left] at (0,0) $H$;
[zxvertex=,zxdown] at (1,1.4);

|i⟩ ⊗|j⟩↦δ_ij

1↦∑_i |i⟩ ⊗|i⟩

We draw the cup and cap on $H$ with a white vertex, as in the above diagram. In what follows, we assume that all Hilbert spaces we consider come equipped with some chosen orthonormal basis $\{\ket{i}\}$, defining a self-duality as above. The dagger structure is given by Hermitian adjunction of linear maps. Dimension and trace in the rigid $C^*$-tensor category $\Hilb$ reduce to the usual notions of dimension of a Hilbert space and trace of a linear map.

The category $\Hilb$ is additionally symmetric — for any $V,W$ there is a swap map $\sigma_{V,W}: V \otimes W \to W \otimes V$. We draw this as an intersection:

|v⟩ ⊗|w⟩ ↦|w⟩ ⊗|v⟩

The four-dimensional calculus allows us to untangle arbitrary diagrams and remove any twists, as exemplified by the following equations:

§.§ Fibre functors and unitary pseudonatural transformations

§.§.§ Fibre functors

Let $\mathcal{C}$ be a rigid $C^*$-tensor category. We call a faithful unitary $\mathbb{C}$-linear monoidal functor $F: \mathcal{C} \to \Hilb$ a fibre functor.

To depict fibre functors, we use a graphical calculus of functorial boxes [13]. In this calculus, we represent the effect of a fibre functor $F: \mathcal{C} \to \Hilb$ by drawing a shaded box around objects and morphisms in $\mathcal{C}$. For example, let $X,Y$ be objects and  $f: X \to Y$ a morphism in $\mathcal{C}$. Then the morphism $F(f): F(X) \to F(Y)$ in $\Hilb$ is represented as:

We represent the multiplicators $\mu_{X,Y}:F(X) \otimes F(Y) \to F(X \otimes Y)$ and their daggers as follows:

μ_X,Y: F(X) ⊗F(Y) →F(X ⊗Y)     μ_X,Y^†: F(X ⊗Y) →F(X) ⊗F(Y)

We represent the unitor $\upsilon: \mathbb{C} \to F(\mathbbm{1}_C)$ and its dagger as follows (recall that the monoidal unit is invisible in the diagrammatic calculus):

υ: ℂ →F(1_C)     υ^†: F(1_C)  →ℂ

§.§.§ Unitary pseudonatural transformations

Let $\mathcal{C}$ be a rigid $C^*$-tensor category and let $F_1,F_2: \mathcal{C} \to \Hilb$ be fibre functors. (We colour the functorial boxes blue and red respectively.) A unitary pseudonatural transformation $(\alpha,H): F_1 \to F_2$ is defined by the following data:

* An Hilbert space $H$ (drawn as a green wire).

* For every object $X$ of $\mathcal{C}$, a unitary $\alpha_X: F_1(X) \otimes H \to H \otimes F_2(X)$ (drawn as a white vertex):

The unitaries $\{\alpha_X\}$ must satisfy the following conditions:

* Naturality. For every morphism $f:X \to Y$ in $\mathcal{C}$:

* Monoidality.

* For every pair of objects  $X,Y$ of $\mathcal{C}$:

* $\alpha_{\mathbbm{1}}$ is defined as follows:

Unitary pseudonatural transformations generalise unitary monoidal natural isomorphisms, which we recover when $H \cong \mathbb{C}$.

Since the components of the UPT are unitary, it follows that $\dim(F_1(X)) = \dim(F_2(X))$ for any object $X$ in $\mathcal{C}$.

The diagrammatic calculus shows that pseudonatural transformation is a planar notion. The $H$-wire forms a boundary between two regions of the $\mathcal{D}$-plane, one in the image of $F$ and the other in the image of $G$. By pulling through the $H$-wire, morphisms from $\mathcal{C}$ can move between the two regions (<ref>).

UPTs $(\alpha,H): F_1 \to F_2$ and $(\beta,H'): F_2 \to F_3$ can be composed associatively to obtain a UPT $(\alpha \circ \beta,H \otimes H'): F_1 \to F_3$ whose components $(\alpha \circ \beta)_X$ are as follows (we colour the $H'$-wire orange, and the $F_3$-box brown):

Let $(\alpha,H): F_1 \to F_2$ be a UPT. Then the dual of $\alpha$ is a UPT $(\alpha^*,H): F_2 \to F_1$ whose components $(\alpha^*)_X$ are defined as follows:

It can be shown <cit.> that $\alpha$ and $\alpha^*$ obey the following pull-through equations:

UPTs from the canonical fibre functor on a rigid $C^*$-tensor category, as well as the fibre functors themselves, were classified in terms of Hopf-Galois theory in [26].

§.§ Compact quantum groups and Tannaka-Krein-Woronowicz reconstruction

§.§.§ Compact quantum groups

A very general example of a rigid $C^*$-tensor category is the category of representations of a compact quantum group.

A unital $*$-algebra $A$ equipped with a unital $*$-homomorphism $\Delta: A \to A \otimes A$ (the comultiplication) is called a Hopf-$*$-algebra if $(\Delta \otimes \id_A) \circ \Delta = (\id_A \otimes \Delta) \circ \Delta$ and there exist linear maps $\epsilon: A \to \mathbb{C}$ (the counit) and $S: A \to A$ (the antipode) such that

(ϵ⊗𝕀_A) ∘Δ= 𝕀_A = (𝕀_A ⊗ϵ) ∘Δ
m ∘(S ⊗𝕀_A) ∘Δ= u ∘ϵ= m ∘(𝕀_A ⊗S) ∘Δ
where $m: A \otimes A \to A$ is the multiplication and $u: \mathbb{C} \to A$ the unit of the algebra $A$.

Let $G$ be a compact group, and let $C(G)$ be the $\mathbb{C}^*$-algebra of continuous complex functions on $G$. For any finite-dimensional $*$-representation $\pi: G \to B(H)$, each matrix entry $\pi_{ij}: G \to \mathbbm{C}$ is an element of $C(G)$. Let $\mathbb{C}[G] \subset C(G)$ be the linear span of matrix coefficients of finite-dimensional representations of $G$. This naturally possesses the structure of a Hopf $*$-algebra (see <cit.>).

A corepresentation $(H,\delta)$ of a Hopf $*$-algebra $A$ on a vector space $H$ is defined by a linear map $\delta: H \to H \otimes A$ such that

(δ⊗𝕀_A) ∘δ= (𝕀_H ⊗Δ) ∘δ
(𝕀_H ⊗ϵ) ∘δ= 𝕀_H

Suppose that $H$ is a Hilbert space and let $\{\ket{i}\}$ be an orthonormal basis. We call the elements $\delta_{ij}:=(\bra{i} \otimes \id_A) \circ \delta (\ket{j})$ of $A$ the matrix coefficients of $(H,\delta)$.
We then say that the corepresentation $(H,\delta)$ is unitary if the matrix coefficients obey the following equation:

\sum_{j} \delta_{ji}^* \delta_{jk} = 1_A = \sum_{j} \delta_{ij} \delta_{kj}^*

For $(H_1,\delta_1), (H_2,\delta_2)$ corepresentations of a Hopf-$*$-algebra $A$, we say that a linear map $f: H_1 \to H_2$ is an intertwiner $f:(H_1,\delta_1) \to (H_2,\delta_2)$ if $\delta_2 \circ f = (f \otimes \id_A) \circ \delta_1$.

We say that a Hopf-$*$-algebra is a compact quantum group algebra if it is generated as an algebra by matrix coefficients of its finite-dimensional unitary corepresentations.

The category $\Corep(A)$ whose objects are finite-dimensional unitary corepresentations of a compact quantum group algebra $A$ and whose morphisms are intertwiners is a rigid $C^*$-tensor category <cit.>.

Let $G$ be a compact group. A continuous function $f:G \to \mathbb{C}$ is called representative if there exists a continuous representation $\pi$ of $G$ on some finite-dimensional complex vector space $H$ and elements $v \in H$, $\phi \in H^*$ such that $f(x) = \phi(\pi(x)v)$ for all $x \in G$. The algebra of representative functions $R(G)$ has the structure of a Hopf-$*$-algebra where the comultiplication, counit and antipode are defined as follows:

\begin{align*}
\Delta(f)(x,y) = f(xy)
\epsilon(f) = f(e)
(S(f))(x) = f(x^{-1})
\end{align*}

The Hopf-$*$-algebra $R(G)$ is generated by the coefficients of its finite-dimensional unitary corepresentations; in other words, it is a compact quantum group algebra.

There is an isomorphism of categories between the category of continuous unitary representations of $G$ and the category of unitary corepresentations of $R(G)$ (see <cit.> for the bijection on objects; the extension to morphisms is obvious). Roughly, for any continuous unitary representation $\pi$ of $G$ on $H$ we may identify $\End(H)$ with $M_n(\mathbb{C})$ by some orthonormal basis $\{\ket{i}\}$ and consider $\pi$ as an $R(G)$-valued matrix $(\pi_{ij})$; this defines a unitary corepresentation $\pi^c: H \to H \otimes R(G)$ whose matrix coefficients are $(\bra{i} \otimes \id_{R(G)}) \pi^c \ket{j} = \pi_{ij}$.

In the spirit of Example <ref>, a compact quantum group algebra $A$ is considered as the algebra of matrix coefficents of continuous unitary representations of some `compact quantum group' $G$, such that $\Rep(G) = \Corep(A)$. We will refer to compact quantum groups $G$, and write $\Rep(G)$, in order to emphasise the similarity with the representation theory of compact groups. However, the compact quantum group algebra is the concrete object, at least as far as this work is concerned. We write $A_G$ for the compact quantum group algebra corresponding to a compact quantum group $G$.

For a f.d. unitary representation $X$ of a compact quantum group $G$ we call the dimension (<ref>) of $X$ as an object in the rigid $C^*$-tensor category $\Rep(G)$ its quantum dimension $\dim_q(X)$. We call the dimension of the underlying Hilbert space of $X$ the ordinary dimension $\dim_c(X)$.  We say that a compact quantum group $G$ is of Kac type if $\dim_q(X) = \dim_c(X)$ for every f.d. unitary representation $X$ of $G$.

§.§.§ Tannaka-Krein-Woronowicz duality

The category $\Rep(G)$ has a canonical fibre functor which takes each corepresentation to its underlying Hilbert space and each intertwiner to its underlying linear map.

Let $\mathcal{C}$ be a rigid $C^*$-tensor category and let $U: \mathcal{C} \to \Hilb$ be a fibre functor.
There exists a compact quantum group $G$ and a unitary monoidal equivalence $E: \mathcal{C} \to \Rep(G)$ such that the following diagram of functors commutes[In <cit.> it is only stated that the diagram commutes up to unitary monoidal natural isomorphism. In fact for the construction given in <cit.>, it commutes on-the-nose.], where $F: \Rep(G) \to \Hilb$ is the canonical fibre functor:

\begin{equation}\label{diag:tannaka}
\begin{diagram}
\mathcal{C}	&	\rTo^{E}	&	\Rep(G)		\\
&	\rdTo_{U}	&	\dTo_{F}	\\
&				&	\Hilb		\\
\end{diagram}
\end{equation}

In particular, for any object $V$ of $\mathcal{C}$ one can define  a unitary corepresentation $\rho_{U,V}: U(V) \to U(V) \otimes A_G$. These unitary corepresentations $\{\rho_{U,V}\}_{V \in \Obj(\mathcal{C})}$ have the following properties:

* Naturality. For any $f: V \to W$ in $\mathcal{C}$, the morphism $U(f): (U(V),\rho_{U,V}) \to (U(W),\rho_{U,W})$ is an intertwiner, i.e.:

\begin{equation}\label{eq:rhonat}
\rho_{U,W} \circ U(f) = (U(f) \otimes \id_{A_G}) \circ \rho_{U,V}
\end{equation}

* Monoidality. We have the following equations, where the black vertex represents the multiplication and $u_{A_G}$ is the unit of the Hopf-$*$-algebra $A_G$:

* For any objects $V,W$ in $\mathcal{C}$:

* For the monoidal unit object $\mathbbm{1}$ in $\mathcal{C}$:

\rho_{U,\mathbbm{1}} = \id_{U(\mathbbm{1})} \otimes u_{A_G}

The unitary monoidal equivalence $E: \mathcal{C} \to \Rep(G)$ is then defined on objects by $X \mapsto (U(X),\rho_{U,X})$ and on morphisms by $f \mapsto U(f)$. It is clear that the diagram (<ref>) commutes on-the-nose.

§ FROBENIUS ALGEBRAS AND $G$-$C^*$-ALGEBRAS

In this section we will reformulate finite-dimensional $G$-$C^*$-algebra theory in terms of Frobenius algebras. Essentially everything in this section is well-known [12, 27, 4, 7, 18, 11]; we have just brought it together in one place and made the presentation as self-contained as possible. To our knowledge the formulation of $G$-$C^*$-algebra theory in terms of Tannaka duality in Section <ref> is new, but this is a very straightforward consequence of the results in <cit.>.

§.§ Frobenius algebras

First, we recall basic definitions regarding Frobenius algebras.

Let $\mathcal{C}$ be a rigid $C^*$-tensor category. An algebra $[A,m,u]$ in $\mathcal{C}$ is an object $A$ with multiplication and unit morphisms, depicted as follows:

(A) at (0,0);
(0.75,1) to (0.75,2);

(A) at (0.75,2);

m:A⊗A →A    u: 1 →A

These morphisms satisfy the following associativity and unitality equations:

Analogously, a coalgebra $[A,\delta,\epsilon]$ is an object $A$ with a  comultiplication $\delta: A \to A\otimes A$ and a counit $\epsilon:A\to \mathbb{C}$ obeying the following coassociativity and counitality equations:

The dagger of an algebra $[A,m,u]$ is a coalgebra $[A,m^{\dagger},u^{\dagger}]$.  A algebra $[A,m,u]$ in $\mathcal{C}$ is called Frobenius if the algebra and adjoint coalgebra structures are related by the following Frobenius equation:

Frobenius algebras are canonically self-dual. Indeed, it is easy to see that for any Frobenius algebra the following cup and cap fulfil the snake equations (<ref>):

We say that the Frobenius algebra is standard if, for any $f \in \End(A)$, we have the following equality:

\begin{align*}
\includegraphics[scale=1]{Figures/svg/frobalgs/frobstandard1.pdf}
\includegraphics[scale=1]{Figures/svg/frobalgs/frobstandard2.pdf}
\end{align*}

A Frobenius algebra is special[Here we have followed the definition of speciality of a Frobenius algebra common in the categorical quantum mechanics literature (e.g. [10, 11, 8, 7, 27]), namely $mm^{\dagger} = \id$. In the compact quantum groups literature another use of the term speciality is common, where it is only required for $m m^{\dagger}$ to be scalar (see e.g. <cit.>).
] if the following equation is satisfied:

If $\mathcal{C}$ is additionally symmetric (for instance, $\mathcal{C}=\Hilb$) a Frobenius algebra is symmetric or furthermore commutative if the following equations are satisfied, where symmetry is the left equation and commutativity the right:

Let $[A,m_A,u_A],[B,m_B,u_B]$ be Frobenius algebras in a rigid $C^*$-tensor category. We say that a morphism $f: A \to B$ is a $*$-homomorphism precisely when it satisfies the following equations:

We say that a morphism $f: A \to B$ is a $*$-cohomomorphism precisely when it satisfies the following equations:

A unitary $*$-isomorphism $[A,m_A,u_A] \to{} [B,m_B,u_B]$ is a unitary $*$-homomorphism or $*$-cohomomorphism. (It is easy to check that unitarity of $f$ means that either of (<ref>) or (<ref>) implies the other.)

There is a weaker relevant notion of isomorphism: an  $*$-isomorphism of Frobenius algebras is an invertible $*$-homomorphism. A $*$-isomorphism of Frobenius algebras can equivalently be defined as an invertible morphism $f: A \to B$ obeying the first two equations of (<ref>) and such that $f^{\dagger}\circ f$ is a left $A$-module map (see <cit.>).

A $*$-isomorphism of Frobenius algebras is a unitary $*$-isomorphism precisely when it preserves the counit (the second equation of (<ref>)).

In a symmetric rigid $C^*$-tensor category such as $\Hilb$, we may define the tensor product of Frobenius algebras.

Let $[A,m_A,u_A]$ and $[B,m_B,u_B]$ be Frobenius algebras in a symmetric monoidal dagger category. Then their tensor product is a Frobenius algebra $[A \otimes B,m_{A \otimes B},u_{A \otimes B}]$, where $m_{A \otimes B}$ and $u_{A \otimes B}$ are defined as follows:

It is straightforward to show that the tensor product preserves speciality and symmetry.

§.§ Finite-dimensional $C^*$-algebras and channels

We will now recall that Frobenius algebras in the category $\Hilb$ correspond precisely to finite-dimensional $C^*$-algebras equipped with a faithful positive linear functional. For one direction, we observe that any finite-dimensional $C^*$-algebra $A$ equipped with a faithful positive linear functional $\phi: A \to \mathbb{C}$ may be given the structure of a Hilbert space using the inner product $\braket{x|y} = \phi(x^*y)$. We thereby obtain an algebra $[A,m,u]$ in $\Hilb$. In fact, this algebra is Frobenius.

The algebra $[A,m,u]$ in $\Hilb$ corresponding to a finite-dimensional $C^*$-algebra $A$ with faithful positive linear functional $\phi: A \to \mathbb{C}$ is Frobenius. Moreover, we have $\phi = u^{\dagger}$ and the involution may be expressed as follows:

We first observe that the involution takes the form (<ref>). Indeed, we observe that for any $a,b \in A$ we have $\braket{ab | 1} = \phi((ab)^*) = \phi(b^*a^*)= \braket{b|a^*}$. Depicting this graphically, we obtain the following equation for any $a ∈A$:
\begin{calign}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarinvolpf1.pdf}~~=~~\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarinvolpf2.pdf}\qquad \qquad \qquad \forall~ b \in A
\end{calign}
Since the equation holds for all $b ∈A$, it follows that the involution takes the form~\eqref{eq:frobinvol}.

We now show that $A$ is a Frobenius algebra. We observe the following implication of the equality $⟨1|b^*a|=⟩ϕ(b^*a)= ⟨b|a|$⟩:

∀ a,b∈A

We also have the following equation, following from the equation $\braket{c|a^*b} = \phi(c^*a^*b) = \phi((ac)^*b) = \braket{ac|b}$:
\begin{calign}\label{eq:frobpfcomult}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpfcomult11.pdf}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpfcomult12.pdf}
\qquad \forall~a,b,c\in A
\Rightarrow~~~
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpfcomult21.pdf}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpfcomult22.pdf}
\end{calign}
From this we obtain the following additional equation:
\begin{calign}\label{eq:frobpfmult}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpfmult1.pdf}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpfmult2.pdf}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpfmult3.pdf}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpfmult4.pdf}
\end{calign}
Then for any elements $|a⟩,|b⟩,|c⟩,|d⟩ ∈A$ and some orthonormal basis ${|k⟩}$ of $A$:
\begin{calign}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpf1new.pdf}
\sum_{k}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpf2new.pdf}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpf3new.pdf}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpf4new.pdf}
\includegraphics[scale=.6]{Figures/svg/catqm/frobhstarfrobpf5new.pdf}
\end{calign}
Here the first equality is by insertion of a resolution of the identity $∑_k |k⟩⟨k|$; the second equality is by~\eqref{eq:frobpfcomult} and~\eqref{eq:frobpfmult}; the third equality is by removal of the resolution of the identity; and the fourth equality is by the dagger of~\eqref{eq:frobpfsnake}. The Frobenius equation is therefore shown.

Finally, we must show that $ϕ= u^†$. For this we need only observe that for all $a ∈A$, $⟨1|a|=⟩ ϕ(1^*a) = ϕ(a)$.
\end{proof}
\noindent
The converse also holds.
\begin{proposition}[{\cite[Lem. 2.2]{Neshveyev2018}}]\label{prop:frobtocstar}
Let $[A,m,u]$ be a Frobenius algebra in $\Hilb$. Then there is a unique involution on $A$, defined as in~\eqref{eq:frobinvol}, such that $A$ becomes a $C^*$-algebra with faithful positive linear functional $\phi=u^{\dagger}: A \to \mathbb{C}$ such that $\braket{x|y} = \phi(x^{*}y)$.
\end{proposition}
\noindent
We have therefore identified Frobenius algebras in $$ with f.d. $C^*$-algebras equipped with a faithful positive linear functional. The maps between Frobenius algebras we have already considered in Definition~\ref{def:starhomcohom} correspond to their equivalent notions for finite-dimensional $C^*$-algebras.
\begin{proposition}\label{prop:starhomscstaralg}
In $\Hilb$:
\begin{itemize}
\item A $*$-homomorphism of Frobenius algebras is a $*$-homomorphism of $C^*$-algebras.
\item A $*$-isomorphism of Frobenius algebras is a $*$-isomorphism of $C^*$-algebras.
\item A unitary $*$-isomorphism of Frobenius algebras is a $*$-isomorphism of $C^*$-algebras preserving the faithful positive linear functional $u^{\dagger}$.
\end{itemize}
\end{proposition}
\noindent
We have shown a correspondence between Frobenius algebras in $$ and pairs of an f.d. $C^*$-algebras and a faithful positive linear functional. We now consider a Frobenius-algebraic formulation of channels between these $C^*$-algebras. We will consider a channel to be a completely positive linear map preserving the chosen functional. Of course, it is common in physics to restrict to the case where the functional is a trace (see Remark~\ref{rem:trace}).
\begin{definition}
Let $A,B$ be Frobenius algebras in a rigid $C^*$-tensor category $\mathcal{C}$ and let $f: A \to B$ be a morphism. We say that $f$ satisfies the \emph{CP condition}, or is a \emph{CP morphism}, when there exists an object $X$ and a morphism $g: A \otimes B \to X$ such that the following equation holds:
\begin{calign}\label{eq:cpcond}
\includegraphics[scale=.8]{Figures/svg/catqm/cpdef1.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/cpdef2.pdf}
\end{calign}
In other words, the morphism on the LHS of~\eqref{eq:cpcond} is a positive element of the $C^*$-algebra $\End(A \otimes B)$.
\end{definition}
\begin{theorem}[{\cite[Thm. 7.18]{Heunen2019}}]\label{thm:cpmap}
Let $A,B$ be Frobenius algebras in $\Hilb$. A morphism $N: A \to B$ is completely positive as a map between $C^*$-algebras iff it satisfies the CP condition.
\end{theorem}
\noindent
We therefore obtain the following definition of a channel.
\begin{definition}\label{def:chan}
Let $A, B$ be Frobenius algebras and $f: A \to B$ a morphism in a rigid $C^*$-tensor category. We say that $f$ is a \emph{channel} when it satisfies the CP condition~\eqref{eq:cpcond} and preserves the counit:
\begin{calign}\label{eq:tpcond}
\includegraphics[scale=.8]{Figures/svg/catqm/tracepres1.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/tracepres2.pdf}
\end{calign}
\end{definition}
\begin{corollary}\label{cor:chan}
Let $A,B$ be Frobenius algebras in $\Hilb$. A linear map $N: A \to B$ is a completely positive functional-preserving map between $C^*$-algebras precisely when it is a channel in the sense of Definition~\ref{def:chan}.
\end{corollary}
\noindent
Using this description, we can quickly show that $*$-cohomomorphisms between special Frobenius algebras are channels.
\begin{lemma}\label{lem:starcohom}
Let $A,B$ be special Frobenius algebras in $\Hilb$. Any $*$-cohomomorphism $f: A \to B$ is a channel.
\end{lemma}
\begin{proof}
The functional-preservation condition~\eqref{eq:tpcond} is immediately satisfied, by~\eqref{eq:cohomo}. We show the CP condition:
\begin{calign}
\includegraphics[scale=.8]{Figures/svg/catqm/starcohomischannel1.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/starcohomischannel2.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/starcohomischannel3.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/starcohomischannel4.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/starcohomischannel5.pdf}
\end{calign}
Here the first equality is by the Frobenius equation~
\eqref{eq:Frobenius} and counitality; the second equality is by the first $*$-cohomomorphism condition~\eqref{eq:cohomo}; the third equality is by the third $*$-cohomomorphism condition~\eqref{eq:cohomo}; and the final equality is by speciality of the Frobenius algebra and~\eqref{eq:Frobenius}.
\end{proof}

\begin{example}[The algebra $B(H)$ and the maximally entangled state]\label{ex:bh}
Let $H$ be a Hilbert space of dimension $d$.
We define a special Frobenius algebra on the Hilbert space $H \otimes H$ with the following multiplication $m: (H \otimes H) \otimes (H \otimes H) \to H \otimes H$ and unit $u: \mathbb{C} \to H \otimes H$:
\begin{align*}
\frac{1}{\sqrt{d}}~\includegraphics[scale=1]{Figures/svg/frobalgs/endomult.pdf}
\sqrt{d}~\includegraphics[scale=1]{Figures/svg/frobalgs/endounit.pdf}
\end{align*}
By Proposition~\ref{prop:frobtocstar}, this is a finite-dimensional $C^*$-algebra with involution~\eqref{eq:frobinvol} and faithful trace $u^{\dagger}$. We will now show that there is a trace-preserving $*$-isomorphism between this $C^*$-algebra and the algebra $B(H)$ equipped with the \emph{special trace} $d \Tr: B(H) \to \mathbb{C}$, where $\Tr$ is the matrix trace.

We claim that the following linear map is the $*$-isomorphism:
\begin{align}\label{eq:endhiso}
f: B(H) &\to H \otimes H \\
X &\mapsto \sqrt{d}(X \otimes \mathbbm{1})(\sum_i \ket{i} \otimes \ket{i})
\end{align}
Multiplicativity is seen as follows:
\begin{calign}
\sqrt{d}
\includegraphics[scale=.8]{Figures/svg/frobalgs/bhhommult1.pdf}
\sqrt{d}
\includegraphics[scale=.8]{Figures/svg/frobalgs/bhhommult2.pdf}
\end{calign}
That the unit is preserved is clear from the definition. To see that the involution is preserved:
\begin{calign}
\sqrt{d}
\includegraphics[scale=.8]{Figures/svg/frobalgs/bhhominvol1.pdf}
\sqrt{d}
\includegraphics[scale=.8]{Figures/svg/frobalgs/bhhominvol2.pdf}
\end{calign}
To see that it is an injection, observe that any element of $B(H)$ can be recovered from its image:
\begin{calign}
\sqrt{d}
\includegraphics[scale=.8]{Figures/svg/frobalgs/bhhominject1.pdf}
\sqrt{d}
\includegraphics[scale=.8]{Figures/svg/frobalgs/bhhominject2.pdf}
\end{calign}
To see that it is a surjection, observe that any element of $H \otimes H$ can be written as $\sum_{ij} \lambda_{ij} \ket{i} \otimes \ket{j}$. This is the image of the element $\frac{1}{\sqrt{d}}\sum_{ij}\lambda_{ij} \ket{i}\bra{j}$ of $B(H)$.
To see that the trace is preserved:
\begin{calign}
d \Tr(X)
\end{calign}
We now consider the representation of the maximally entangled state of $H \otimes H$. Let us consider the $C^*$-algebra $B(H \otimes H)$ with the special trace, defined as above as a Frobenius algebra on $(H \otimes H) \otimes (H \otimes H)$. Since the maximally entangled state of $H \otimes H$ is just a normalised version of the cup of the self-duality of $H$, the maximally entangled state of $B(H \otimes H)$ has the following form as an element of the Frobenius algebra:
\begin{align*}
\frac{1}{d^2}
\includegraphics[scale=.8]{Figures/svg/frobalgs/entstate11.pdf}
\frac{1}{d^2}
\includegraphics[scale=.8]{Figures/svg/frobalgs/entstate12.pdf}
\end{align*}
Using the canonical unitary $*$-isomorphism $B(H \otimes H) \cong B(H) \otimes B(H)$, we obtain the following expression for the maximally entangled state as an element of the Frobenius algebra $B(H) \otimes B(H)$:
\begin{align}
\label{eq:maxentstate}
\frac{1}{d^2}\includegraphics[scale=.8]{Figures/svg/frobalgs/entstate22.pdf}
\end{align}
\end{example}

\subsection{Finite-dimensional $G$-$C^*$-algebras and covariant channels}
\label{sec:gcstaralg}
We now generalise to the case where operations are covariant for the action of a compact quantum group $G$.
We first recall from~\cite[\S{}2.3]{Neshveyev2018} the following completely general characterisation of a finite-dimensional $G$-$C^*$-algebra $A$ equipped with a $G$-invariant positive linear functional.
\begin{definition}\label{def:generalgcstaralg}
Let $[A,m,u]$ be a Frobenius algebra in $\Hilb$ and let $A_G$ be a compact quantum group algebra. We say that a linear map $\rho: A \to A \otimes A_G$ is a \emph{coaction} of $A_G$ on $A$ if the pair $(A,\rho)$ is a unitary corepresentation (Definition~\ref{def:unitarycorep}) and $m:A \otimes A \to A, u: \mathbb{C} \to A$ are intertwiners, i.e.:
\begin{calign}
\includegraphics[scale=.8]{Figures/svg/catqm/gcstaralgdefmultpullthrough1.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/gcstaralgdefmultpullthrough2.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/gcstaralgdefunitpullthrough1.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/gcstaralgdefunitpullthrough2.pdf}
\end{calign}
Here the black vertex represents the multiplication and unit of the Hopf $*$-algebra $A_G$. We call $([A,m,u],\rho)$ a \emph{$G$-$C^*$-algebra with functional}. (Here the $G$-invariant faithful positive linear functional is the counit $u^{\dagger}: A \to \mathbb{C}$.)

We say that a linear map between $G$-$C^*$-algebras with functional is \emph{covariant} if it is an intertwiner of corepresentations (Definition~\ref{def:unitarycorep}).
\end{definition}
\noindent
Definition~\ref{def:generalgcstaralg} is just a concrete way of saying that a $G$-$C^*$-algebra equipped with a $G$-invariant faithful positive linear functional is a Frobenius algebra in $(G)$. Following Proposition~\ref{prop:starhomscstaralg} we can identify $*$-homomorphisms,  $*$-isomorphisms, and unitary $*$-isomorphisms of Frobenius algebras in $(G)$ with covariant $*$-homomorphisms, covariant $*$-isomorphisms, and functional-preserving covariant $*$-isomorphisms of $G$-$C^*$-algebras respectively.

A $G$-$C^*$-algebra therefore corresponds to a $*$-isomorphism class of Frobenius algebras in $(G)$. In what follows we will use a result from~[18] to uniquely fix a functional for each $G$-$C^*$-algebra.
\begin{lemma}\label{lem:uniquespecialfunctional}
Every $G$-$C^*$-algebra admits a unique $G$-invariant faithful positive linear functional such that the corresponding Frobenius algebra in $\Rep(G)$ is special, standard and satisfies $u^{\dagger} u = \dim_q(A)$.
\end{lemma}
\begin{proof}
This is just~\cite[Thm. 2.11]{Neshveyev2018} with a different normalisation. There it was shown that there is a unique $G$-invariant state on any $G$-$C^*$-algebra such that the corresponding Frobenius algebra $A$ is standard and satisfies $u^{\dagger} u= 1$ and  $mm^{\dagger}=\dim_q(A)\id_A$. Using a scalar $*$-isomorphism of Frobenius algebras ($u \mapsto \sqrt{\dim_q(A)} u$ and $m \mapsto \frac{1}{\sqrt{\dim_q(A)}} m$) we obtain the desired unique $G$-invariant faithful positive linear functional.
\end{proof}
\noindent
We call the unique $G$-invariant faithful positive linear functional of Lemma~\ref{lem:uniquespecialfunctional} the \emph{canonical $G$-invariant functional} on $A$. This is just a different normalisation of the canonical invariant state of~\cite[Thm. 2.11]{Neshveyev2018}.
\begin{remark}\label{rem:trace}
It is common in physics to assume that the functional is a trace. In the case of a general compact quantum group action there may be no $G$-invariant trace on a $G$-$C^*$-algebra. We discuss this in Appendix~\ref{sec:app}. In particular, we show that, whenever a $G$-invariant trace exists, the canonical invariant functional of Lemma~\ref{lem:uniquespecialfunctional} is tracial. We also provide a necessary and sufficient condition for the existence of a $G$-invariant trace; namely, that the quantum dimension $\dim_q(A)$ should be equal to the ordinary dimension $\dim(A)$. In this case we show in that the canonical functional is precisely the \emph{special trace} on the underlying $C^*$-algebra~[27]. In particular, this implies that the canonical invariant functional is always tracial whenever the compact quantum group is of \emph{Kac type}~\cite[Prop. 1.7.9]{Neshveyev2013} (this includes all ordinary compact groups).
\end{remark}
\noindent
Let $𝒞$ be a rigid $C^*$-tensor category. Recall that, by Tannaka duality (Theorem~\ref{thm:tannaka}), when $U: 𝒞 →$ is a fibre functor there is a compact quantum group $G$ associated to the pair $(𝒞,U)$ such that $𝒞 ≃(G)$. One therefore expects that $G$-$C^*$-algebras can be identified with special standard Frobenius algebras ($$s) in $𝒞$; in what follows we make this precise.
\begin{proposition}\label{prop:imisgcalg}
Let $\mathcal{C}$ be a rigid $C^*$-tensor category and $U: \mathcal{C} \to \Hilb$ be a fibre functor. Let $[A,m,u]$ be an $\F$ in $\mathcal{C}$. Then $[U(A),U(m) \circ \mu_{A,A},U(u) \circ \upsilon]$ is a special Frobenius algebra in $\Hilb$:
\begin{calign}
\includegraphics[scale=.8]{Figures/svg/catqm/fadef1.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/fadef2.pdf}
\\ \nonumber
U(m) \circ \mu_{A,A}
U(u) \circ \upsilon
\end{calign}
Let $G$ be the CQG obtained by Tannaka reconstruction from the pair $(\mathcal{C},U)$, and let $\rho_{U,A}: U(A) \to U(A) \otimes A_G$ be the corresponding corepresentation of $A_G$ on $U(A)$. Then $([U(A),U(m) \circ \mu_{A,A},U(u) \circ \upsilon],\rho_{U,A})$ is a $G$-$C^*$-algebra with functional.
\end{proposition}
\begin{proof}
For the first statement, to show that $[U(A),U(m) \circ \mu_{A,A},U(u) \circ \upsilon]$ is a \F{} it is easy to check that the equations of a special Frobenius algebra are preserved under a unitary monoidal functor. For example, speciality follows from unitarity of the fibre functor and speciality of $[A,m,u]$:
\begin{calign}
\includegraphics[scale=.8]{Figures/svg/catqm/faspeciality1.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/faspeciality2.pdf}
\includegraphics[scale=.8]{Figures/svg/catqm/faspeciality3.pdf}
\end{calign}
For the second statement, we already know that $\rho_{U,A}: U(A) \to U(A) \otimes A_G$ is a unitary corepresentation, so we need only show that $U(m)\circ \mu_{A,A}$ and $U(u) \circ \upsilon$ are intertwiners. This follows from the properties of the representations $\{\rho_{U,X}\}_{X \in \Obj(\mathcal{C})}$ listed in Section~\ref{sec:tannaka}. To see that $U(m) \circ \mu_{A,A}$ is an intertwiner:
\begin{calign}
\includegraphics[scale=.9]{Figures/svg/catqm/farhomultpullthrough1.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/farhomultpullthrough2.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/farhomultpullthrough3.pdf}
\end{calign}
Here the first equality is by naturality~\eqref{eq:rhonat} of $\{\rho_{U,X}\}$ and the second is by monoidality~\eqref{eq:rhomon} of $\{\rho_{U,X}\}$.

To see that $U(u) \circ \upsilon$ is an intertwiner:
\begin{calign}
\includegraphics[scale=.9]{Figures/svg/catqm/farhounitpullthrough1.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/farhounitpullthrough2.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/farhounitpullthrough3.pdf}
\end{calign}
Here the first equality is by naturality~\eqref{eq:rhonat} and the second equality is by monoidality~\eqref{eq:rhomonunit}.
\end{proof}
\begin{notation}
For concision, for any $\F$ $[A,m,u]$ in $\mathcal{C}$ we write $U([A,m,u])$ for the $G$-$C^*$-algebra $([U(A),U(m) \circ \mu_{A,A},U(u) \circ \upsilon],\rho_{U,A})$.
\end{notation}
\noindent
We now show that, up to covariant $*$-isomorphism, all $G$-$C^*$-algebras can be obtained in this way.
\begin{proposition}
Let $\mathcal{C}$ be a rigid $C^*$-tensor category, let $U: \mathcal{C} \to \Hilb$ be a fibre functor, and let $G$ be the compact quantum group obtained by Tannaka reconstruction for the pair $(\mathcal{C},U)$.

Let $([A,m,u],\rho)$ be a $G$-$C^*$-algebra with functional. Then there exists a $\F$ $[\bar{A},\bar{m},\bar{u}]$ in $\mathcal{C}$ such that $U([\bar{A},\bar{m},\bar{u}])$ is covariantly $*$-isomorphic to $([A,m,u],\rho)$.
\end{proposition}
\begin{proof}
Recall the commuting diagram of functors~\eqref{diag:tannaka}. We first observe that, since $(A,\rho)$ is a unitary corepresentation of $G$ and $E$ is a unitary equivalence, there exists an object $\tilde{A}$ of $\mathcal{C}$ and a unitary intertwiner $f: (U(\tilde{A}),\rho_{U,\tilde{A}}) \to (A,\rho)$. We pull back the algebra structure of $[A,m,u]$ along $f$ to obtain a $G$-$C^*$-algebra $([U(\tilde{A}),f^{\dagger} \circ m \circ (f \otimes f),f^{\dagger} \circ u],\rho_{U,\tilde{A}})$, so that $f$ is a covariant $*$-isomorphism. It is easy to show by fullness and unitarity of the monoidal equivalence $E$ that there exist morphisms $\tilde{m}: \tilde{A} \otimes \tilde{A} \to \tilde{A}$ and $\tilde{u}: \mathbbm{1} \to \tilde{A}$ in $\mathcal{C}$ such that $([U(\tilde{A}),f^{\dagger} \circ m \circ (f \otimes f),f^{\dagger} \circ u],\rho_{U,\tilde{A}}) = U([\tilde{A},\tilde{m},\tilde{u}])$. Finally, by Lemma~\ref{lem:uniquespecialfunctional} there is a standard special Frobenius algebra $[\bar{A},\bar{m},\bar{u}]$ in $\mathcal{C}$ which is covariantly $*$-isomorphic to $[\tilde{A},\tilde{m},\tilde{u}]$.
\end{proof}
\noindent
Having identified $G$-$C^*$-algebras with $$s in $𝒞$, we now turn to covariant channels.
\begin{definition}
Let $([A_1,m_1,u_1],\rho_1)$, $([A_2,m_2,u_2],\rho_2)$ be $G$-$C^*$-algebras with functional. We say that a CP morphism $f: [A_1,m_1,u_1] \to{} [A_2,m_2,u_2]$ is \emph{covariant} if it is also an intertwiner $(A_1,\rho_1) \to (A_2,\rho_2)$.
\end{definition}
\noindent
We first show that a CP morphism between $$s in $𝒞$ induces a covariant CP morphism between the corresponding $G$-$C^*$-algebras.
\begin{proposition}\label{prop:imiscovchan}
Let $\mathcal{C}$ be a rigid $C^*$-tensor category, and let $U: \mathcal{C} \to \Hilb$ be a fibre functor. Let $[A_1,m_1,u_1]$, $[A_2,m_2,u_2]$ be $\F$s in $\mathcal{C}$ and let $f: [A_1,m_1,u_1] \to{} [A_2,m_2,u_2]$ be a CP morphism in $\mathcal{C}$. Then $$U(f): U([A_1,m_1,u_1]) \to{} U([A_2,m_2,u_2])$$ is a covariant CP morphism.
If $f$ is trace-preserving then $U(f)$ is a covariant channel.
\end{proposition}
\begin{proof}
We know that $U(f)$ is an intertwiner by naturality~\eqref{eq:rhonat} of $\{\rho_{U,X}\}_{X \in \Obj(\mathcal{C})}$.
We show that $U(f)$ is a CP map. Indeed, the CP condition~\eqref{eq:cpcond} is preserved under the unitary monoidal functor $U$:
\begin{calign}
\includegraphics[scale=.9]{Figures/svg/catqm/cppreserved1.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/cppreserved2.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/cppreserved3.pdf}
\end{calign}
The map $U(f)$ is therefore covariant CP.
Finally, the trace-preservation condition~\eqref{eq:tpcond} is preserved under the functor $U$:
\begin{calign}
\includegraphics[scale=.9]{Figures/svg/catqm/unitpreserved1.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/unitpreserved2.pdf}
\end{calign}
\end{proof}
\noindent
We now show that all covariant CP maps and channels are obtained as the image of CP morphisms in $𝒞$.
\ignore{
\begin{lemma}\label{lem:pos*hom}
Let $A,B$ be f.d. $C^*$-algebras and let $\pi:A \to B$ be a $*$-homomorphism. Then a element $\pi(a)$ is self-adjoint and positive in $B$ iff $a$ is self-adjoint and positive in $A$.
\end{lemma}
\begin{proof}
The self-adjoint part is clear. The positivity part follows from the characterisation of positivity of a self-adjoint element in  terms of positivity of the spectrum~\cite[Thm. 1.6.5]{Putnam}. In one direction, if $a$ is positive in $A$ --- i.e. $a = b b^*$ for some $b \in A$ --- then $f(a) = f(b) f(b)^*$ and so $f(a)$ is positive in $B$. In the other direction, if $f(a)$ is positive in $B$, then $\lambda >0$ for all $\lambda$ s.t. $\lambda1_B-f(a)$ is invertible in $B$. Consider $\lambda 1_A - a \in A$. If this is invertible then $f(\lambda 1_A-a)=\lambda 1_B - f(a)$ is invertible in $B$: thus $\lambda>0$.
\end{proof}}

\begin{proposition}
Let $\mathcal{C}$ be a rigid $C^*$-tensor category, and let $U: \mathcal{C} \to \Hilb$ be a fibre functor.
Let $[A_1,m_1,u_1]$,$[A_2,m_2,u_2]$ be $\F$s in $\mathcal{C}$.
Then for every covariant CP map $$f: U([A_1,m_1,u_1]) \to U([A_2,m_2,u_2])$$ there is a unique CP morphism $\tilde{f}: [A_1,m_1,u_1] \to{} [A_2,m_2,u_2]$ in $\mathcal{C}$ such that $U(\tilde{f}) = f$.

If $f$ is trace-preserving then $\tilde{f}$ is trace-preserving also.
\end{proposition}
\begin{proof}
Recall the commuting triangle~\eqref{diag:tannaka}. By covariance of $f$, there is a unique morphism $\hat{f}$ in $\Rep(G)$ such that $F(\hat{f}) = f$, where $F$ is the canonical fibre functor. Now since $U=F \circ E$ and $E$ is an equivalence, there exists a unique morphism $\tilde{f}$ in $\mathcal{C}$ such that $U(\tilde{f}) = f$.

We now need to show that $\tilde{f}$ obeys the CP condition in $\mathcal{C}$. We know that $f$ satisfies the CP condition~\eqref{eq:cpcond} in $\Hilb$; this is precisely to say that the morphism
\begin{calign}
\includegraphics[scale=.9]{Figures/svg/catqm/cppullback1.pdf}
\end{calign}
is a positive element in the $C^*$-algebra $\End_{\Hilb}(U(A) \otimes U(B))$.
This in turn implies (by pre- and post-composition with $\mu_{A,B}^{\dagger}$ and $\mu_{A,B}$ respectively) that the following morphism is a positive element in $\End_{\Hilb}U(A \otimes B)$:
\begin{calign}
\includegraphics[scale=.9]{Figures/svg/catqm/cppullback2.pdf}
\end{calign}
But the fibre functor $U$ is faithful and unitary; in particular, the induced map $\End_{\mathcal{C}}(A \otimes B) \to \End_{\Hilb}(U(A \otimes B))$ is an injective $*$-homomorphism. It follows from spectral permanence~\cite[Thm. 1.5.3]{Putnam} that the following is a positive element of $\End_{\mathcal{C}}(A \otimes B)$:
\begin{calign}
\includegraphics[scale=.9]{Figures/svg/catqm/cppullback3.pdf}
\end{calign}
This is precisely to say that $\tilde{f}$ is a CP morphism in $\mathcal{C}$.

Finally, we consider the trace-preservation condition. If the covariant CP map $f$ is trace-preserving, then we have the following sequence of implications:
\begin{calign}
\includegraphics[scale=.9]{Figures/svg/catqm/tracepullback11.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/tracepullback12.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/tracepullback21.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/tracepullback22.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/tracepullback31.pdf}
\includegraphics[scale=.9]{Figures/svg/catqm/tracepullback32.pdf}
\end{calign}
Here the first implication is by postcomposition with $\upsilon$, and the second is by faithfulness of the functor $U$.
The last equality precisely states that $\tilde{f}: [A_1,m_1,u_1] \to{} [A_2,m_2,u_2]$ is trace-preserving.
\end{proof}
\noindent
Altogether, what we have proven is an equivalence of categories. We formulate this equivalence now. Let $𝒞$ be a rigid $C^*$-tensor category, let $U: 𝒞 →$ be a fibre functor, and let $G$ be the compact quantum group recovered from the pair $(𝒞,U)$ by Tannaka duality. Let $(𝒞)$ be the category whose objects are standard special Frobenius algebras in $𝒞$ and whose morphisms are CP morphisms between them. Let $(𝒞)$ be the subcategory whose morphisms additionally preserve the counit. Let $(G)$ be the category whose objects are f.d. $G$-$C^*$-algebras and whose morphisms are covariant CP maps. Let $(G)$ be the subcategory whose morphisms additionally preserve the canonical $G$-invariant functional.
\begin{theorem}\label{thm:equivalence}
The functors $\CP(\mathcal{C}) \to \CP(G)$ and $\Chan(\mathcal{C}) \to \Chan(G)$ defined on objects by $[A,m,u] \mapsto U([A,m,u])$ and on morphisms by $f \mapsto U(f)$ are equivalences of categories.
\end{theorem}

\section{Entanglement-symmetries}

\subsection{Theorem}
\label{sec:thm}
We can finally prove the construction of entanglement-symmetries discussed in the introduction.

For the remainder of this section we fix a compact quantum group $G$ and let $F: (G) →$ be the canonical fibre functor. Let $F': (G) →$ be some other fibre functor , and let $G'$ be the compact quantum group associated to it by Tannaka duality. Let $(α,H_e): F →F'$ be a UPT, and let $d:=(H_e)$.

Recall the definition of a $*$-cohomomorphism (Definition~\ref{def:starhomcohom}) between Frobenius algebras, and that a $*$-cohomomorphism between special Frobenius algebras in $$ is in particular a channel (Lemma~\ref{lem:starcohom}). Recall also the Frobenius algebra structure of $B(H_e) ≅H ⊗H$ (Example~\ref{ex:bh}).
\begin{theorem}\label{thm:starcohoms}
Let $[A,m,u]$ be an $\F$ in $\Rep(G)$. Then we obtain:
\begin{itemize}
\item A $*$-cohomomorphism
$$u_{[A,m,u]}: F([A,m,u]) \otimes B(H_e) \to{} F'([A,m,u])$$
defined as follows:
\begin{calign}\label{eq:udef}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomdef1.pdf}
\sqrt{d}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomdef2.pdf}
\end{calign}
\item A $*$-cohomomorphism
$$v_{[A,m,u]}: F'([A,m,u]) \otimes B(H_e) \to{} F([A,m,u])$$
defined as follows:
\begin{calign}\label{eq:vdef}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomdualdef1.pdf}
\sqrt{d}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomdualdef2.pdf}
\end{calign}
\end{itemize}
These $*$-cohomomorphisms obey the following \emph{entanglement-invertibility} equations with respect to the maximally entangled state $\Psi: \mathbb{C} \to B(H) \otimes B(H)$:
\begin{calign}\label{eq:entinvertibility}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymm11.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymm12.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymm21.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymm22.pdf}
\end{calign}
These $*$-cohomomorphisms are also \emph{natural} for CP morphisms $f: [A_1,m_1,u_1] \to{} [A_2,m_2,u_2]$:
\begin{calign}\label{eq:starcohomsnat}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomnatural11.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomnatural12.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomnatural21.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomnatural22.pdf}
\end{calign}
\end{theorem}
\begin{proof}
We begin by showing that the maps $\{u_{[A,m,u]}\}$ and $\{v_{[A,m,u]}\}$ are $*$-cohomomorphisms. Recall the definition of the multiplication and unit of $B(H)$ (Example~\ref{ex:bh}) and the tensor product of $\F$s (Definition~\ref{def:tensorprodfrob}). We show the proof for $\{u_{[A,m,u]}\}$; the proof for $\{v_{[A,m,u]}\}$ is similar. For the first equation of~\eqref{eq:cohomo}:
\begin{calign}
\sqrt{d}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf11.pdf}
\sqrt{d}~
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf12.pdf}
\sqrt{d}~~
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf13.pdf}
\end{calign}
Here the first equality is by a snake equation and monoidality of the UPT $\alpha$~\eqref{eq:pntmonmon}; the second equation is by naturality of $\alpha$~\eqref{eq:pntmonnat}. For the second equation of~\eqref{eq:cohomo}:
\begin{calign}
\sqrt{d}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf21.pdf}
\sqrt{d}~~
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf22.pdf}
\sqrt{d}~~
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf23.pdf}
\end{calign}
Here the first equality is by naturality of the UPT $\alpha$~\eqref{eq:pntmonnat} and the second equality is by monoidality of $\alpha$~\eqref{eq:pntmonmonunit} and unitarity of the functor $F'$.
For the third equation of~\eqref{eq:cohomo}:
\begin{calign}
\sqrt{d}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf31.pdf}
\sqrt{d}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf32.pdf}
\sqrt{d}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf33.pdf}
\\~~=~~
\sqrt{d}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf34.pdf}
\sqrt{d}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomcohompf35.pdf}
\end{calign}
Here the first equality is by the `horizontal flip' calculus of the dagger; the second equality is by a snake equation; the third equality is by the definition of the dual UPT~\eqref{eq:dualpnt};\footnote{We have simplified the proof by assuming that the chosen dual of $A$ in the pivotal dagger category $\Rep(G)$ is $A$ itself under the duality corresponding to the Frobenius algebra $[A,m,u]$. The proof can easily be made fully general using uniqueness of duals up to isomorphism.} and the fourth equality is by~\eqref{eq:untangling}. In the final diagram we recognise the cup~\eqref{eq:cupcapfrob} of $F([A,m,u]) \otimes B(H_e)$ and the cap~\eqref{eq:cupcapfrob} of $F'([A,m,u])$.

We now show the entanglement-invertibility equations~\eqref{eq:entinvertibility}. Recall the expression~\eqref{eq:maxentstate} for the channel  $\Psi: \mathbb{C} \to B(H) \otimes B(H)$ initialising the maximally entangled state. We prove the first equation of~\eqref{eq:entinvertibility}:
\begin{calign}
\frac{1}{d}~~
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmpf1.pdf}
\frac{1}{d}~~
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmpf2.pdf}
\frac{1}{d}~~
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmpf3.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmpf4.pdf}
\end{calign}
Here the first equality is by~\eqref{eq:untangling}; the second equality is by the pull-through equations for the UPT and its dual~\eqref{eq:cupcapmodsdualpnt}; and the third equality is by evaluation of $\dim(H_e)=d$.
The second equation of~\eqref{eq:entinvertibility} is proven similarly.

The naturality equations~\eqref{eq:starcohomsnat} follow immediately from naturality of the UPTs $\alpha$ and $\alpha^*$~\eqref{eq:pntmonnat}.
\end{proof}
\begin{corollary}\label{cor:entequivs}
Let $f: [A_1,m_1,u_1] \to{} [A_2,m_2,u_2]$ be a channel between standard special Frobenius algebras in $\Rep(G)$, and let
\begin{align*}
F(f): F([A_1,m_1,u_1]) &\to F([A_2,m_2,u_2])\\
F'(f): F'([A_1,m_1,u_1]) &\to F'([A_2,m_2,u_2])
\end{align*}
be the corresponding channels in $\Chan(G)$ and $\Chan(G')$ respectively. Then:
\begin{itemize}
\item $(u_{[A_1,m_1,u_1]},v_{[A_2,m_2,u_2]},H_e)$ is an entanglement-assisted channel coding scheme for $F(f)$ from $F'(f)$.
\item $(v_{[A_1,m_1,u_1]},u_{[A_2,m_2,u_2]},H_e)$ is an entanglement-assisted channel coding scheme for $F'(f)$ from $F(f)$.
\end{itemize}
That is, the following equations hold:
\begin{calign}\label{eq:starcohomentsymms}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmeqs11.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmeqs12.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmeqs21.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmeqs22.pdf}
\end{calign}
\end{corollary}
\begin{proof}
This follows straightforwardly from Theorem~\ref{thm:starcohoms}. For the first equation of~\eqref{eq:starcohomentsymms}:
\begin{calign}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmeqs11.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmeqspf.pdf}
\includegraphics[scale=0.9]{Figures/svg/entanglementsymms/starcohomentsymmeqs12.pdf}
\end{calign}
Here the first equality is by naturality~\eqref{eq:starcohomsnat} and the second equality is by entanglement-invertibility~\eqref{eq:entinvertibility}. The second equation of~\eqref{eq:starcohomentsymms} is proven similarly.
\end{proof}
\noindent
Taken together, these entanglement-symmetries give rise to an equivalence of categories.
\begin{definition}\label{def:equivalence}
We define a map from objects and morphisms of $\CP(G)$ (resp. $\Chan(G)$) to objects and morphisms of $\CP(G')$ (resp. $\Chan(G')$) as follows.

For each $G$-$C^*$-algebra $([A,m,u],\rho)$, choose a covariant unitary $*$-isomorphism $\iota_{([A,m,u],\rho)}: ([A,m,u],\rho) \to F([\tilde{A},\tilde{m},\tilde{u}])$ for some $[\tilde{A},\tilde{m},\tilde{u}]$ in $\Rep(G)$. (Such a $*$-isomorphism exists by Theorem~\ref{thm:equivalence}.)

The mapping is defined as follows:
\begin{itemize}
\item \emph{On objects}.
([A,m,u],\rho) \mapsto F'([\tilde{A},\tilde{m},\tilde{u}]).
\item \emph{On morphisms.} We map any covariant CP map (resp. channel) $f: ([A_1,m_1,u_1],\rho_1) \to ([A_2,m_2,u_2],\rho_2)$ to the following morphism $F'([\tilde{A}_1,\tilde{m}_1,\tilde{u}_1]) \to F'([\tilde{A}_2,\tilde{m}_2,\tilde{u}_2])$:
\begin{calign}
\includegraphics[scale=1]{Figures/svg/entanglementsymms/equivalenceonmorphs.pdf}
\end{calign}
\end{itemize}
\end{definition}
\begin{theorem}\label{thm:starcohomequiv}
The map of Definition~\ref{def:equivalence} is an equivalence of categories $\CP(G) \cong \CP(G')$ (resp. $\Chan(G) \cong \Chan(G')$).
\end{theorem}
\begin{proof}
This follows straightforwardly from Theorem~\ref{thm:equivalence}; indeed, the map is precisely the composition of a weak inverse of the equivalence $\CP(\Rep(G)) \to \CP(G)$ (resp. $\Chan(\Rep(G)) \to \Chan(G)$) with the equivalence $\CP(\Rep(G)) \to \CP(G')$ (resp. $\Chan(\Rep(G)) \to \Chan(G')$).
\end{proof}

\subsection{Example and application}\label{sec:ex}
We now present a brief example and application of our construction in the case where the $G$-action corresponds to a finite group grading.

\subsubsection{Example: finite group grading}
Let $G$ be a finite group. Consider the rigid $C^*$-tensor category $(G)$ of $G$-graded Hilbert spaces and grading-preserving linear maps. This category is generated by simple objects ${[g]  |  g ∈G}$, with tensor product $[g_1] ⊗[g_2] = [g_1g_2]$; the canonical fibre functor takes all these simple objects to the one-dimensional Hilbert space $ℂ$. The compact quantum group associated to $(G)$ is the Hopf $*$-algebra $C(G)$ of complex functions on $G$. The category $(C(G))$ is therefore the category of $G$-graded f.d. $C^*$-algebras (where the grading is orthogonal with respect to the inner product arising from the special trace) and grading-preserving CP maps.

We now recall the classification of fibre functors on $(G)$.  First we define an equivalence relation on pairs $(L,[ψ])$ of a subgroup $L<G$ and a 2-cohomology class $[ψ] ∈H^2(L, U(1))$, as follows:
\begin{align}\nonumber
(L_1,[\psi_1]) \sim (L_2,[\psi_2]) \qquad \Leftrightarrow \qquad L_2 = g L_1 g^{-1} &\textrm{~and~} \psi_1 \textrm{~is cohomologous to~}
\\\label{eq:ctequivrel}
&\psi_2^g(x,y):= \psi_2(g x g^{-1}, gyg^{-1}) \textrm{~for some~}g \in G
\end{align}
An equivalence class of rank-one module categories over $(G)$ is determined by a 2-cohomology class $[ψ] ∈H^2(G,U(1))$, up to the equivalence relation~\eqref{eq:ctequivrel}~\cite[Ex. 7.4.10]{Etingof2016}. Such a module category has a single object, with associativity constraint specified in the obvious way by the 2-cocycle $ψ$. We obtain the following characterisation of the associated fibre functor $F_ψ$.
\begin{lemma}
Let $\psi \in Z^2(G,U(1))$ be some 2-cocycle. Then $F_{\psi} \cong F \circ E_{\psi}$, where $F: \Hilb(G) \to \Hilb$ is the canonical fibre functor and $E_{\psi}$ is an autoequivalence of $\Hilb(G)$, specifically the identity functor equipped with the following generally nontrivial multiplicator:
\begin{align*}
m_{[g_1],[g_2]} = \overline{\psi(g_1,g_2)} ~\id_{[g_1g_2]}: E_{\psi}([g_1]) \otimes E_{\psi}([g_2]) \to E_{\psi}([g_1 g_2])
\end{align*}
\end{lemma}
\noindent
We see therefore that the compact quantum group associated to any fibre functor is just $C(G)$ again. We will compute the autoequivalence of $(C(G))$ associated to the fibre functor $F_ψ$.

We first recall the classification of $G$-graded f.d. $C^*$-algebras (where the grading is orthogonal w.r.t. the inner product defined by the special trace). This is equivalent to the classification of $$s in $(G)$. Every $$ is a direct sum of indecomposable $$s, so we need only classify the indecomposable $$s. For any subgroup $L<G$ and 2-cocycle $ϕ∈Z^2(L, U(1))$, let $A(L,ϕ)$ be the $ϕ$-twisted group algebra for $L$; this is an $|L|$-dimensional algebra whose underlying Hilbert space has orthonormal basis ${g̅ | g ∈L}$, whose multiplication is specified by $g̅ ·h̅ = ϕ(g,h) gh$ and whose unit is $e̅$. Suitably normalised, this is an $$ in $(G)$, where the element $g̅$ is graded by $g$ for all $g ∈L$. Then, by~\cite[Prop. 3.1]{Ostrik2003}, indecomposable $$s in $(G)$ are all of the form $V ⊗A(H,ψ) ⊗V^*$ for some pair $(L,ψ)$ and some object $V$ of $(G)$, where the multiplication and unit of the $$ are defined using the multiplication and unit of $A(H,ϕ)$ together with the cup and cap of the duality on $V$. The following proposition is then clear.
\begin{proposition}
Let $\psi \in Z^2(G,U(1))$ be a 2-cocycle. The autoequivalence of $\CP(C(G))$ associated to the fibre functor $F_{\psi}$ is defined as follows on indecomposable objects:
\begin{itemize}
\item On objects: the indecomposable $\F$ $V \otimes A(L,\phi) \otimes V^*$ is taken to the indecomposable $\F$ $V \otimes A(L,\overline{\psi} \phi) \otimes V^*$.
\item On morphisms: a CP map $V_1 \otimes A(L_1,\phi_1) \otimes V_1^* \to V_2 \otimes A(L_2,\phi_2) \otimes V_2^*$ is taken to the identical CP map (i.e. identical as a grading-preserving map on the underlying graded Hilbert spaces) $V_1 \otimes A(L_1,\overline{\psi}\phi_1) \otimes V_1^* \to V_2 \otimes A(L_2,\overline{\psi}\phi_2) \otimes V_2^*$.
\end{itemize}
This transformation is extended to all $\F$s and CP maps in the obvious way.
\end{proposition}
\begin{example}[Transformations on classical channels]\label{ex:gradedchans}
Let $G$ be an finite abelian group (this will make calculation easier). Fibre functors on $\Hilb(G)$ correspond to 2-cohomology classes $[\psi] \in H^2(G,U(1))$.

Consider two indecomposable commutative $G$-graded f.d. $C^*$-algebras. By what has already been said, these will be twisted group algebras $A(L_1,1)$, $A(L_2,1)$ , for some abelian subgroups $L_1,L_2<G$. There are two orthonormal bases of interest for these algebras. The first is the graded basis $\{\bar{g}~|~g \in L_i\}$ by which we defined the twisted group algebra. The second is the factor basis; that is,  the basis realising the isomorphism $A(L_i,1) \cong \oplus_{i=1}^{|L_i|}\mathbb{C}$. Explicitly, the factor basis is the basis of characters $\{\bar{\chi}~|~ \chi \in L_i^*\}$, where $\bar{\chi}:= \frac{1}{|L_i|} \sum_{g \in L_i} \chi(g) \bar{g}$. Let $r_i: G^* \to L_i^*$ be the restriction homomorphism; we obtain an action of $G^*$ on $A(L_i,1)$ by
\begin{align}\label{eq:charaction}
\xi \cdot \bar{\chi} = \overline{r_i(\xi)\chi}
\end{align}
for all $\xi \in G^*$. In the factor basis, a grading-preserving channel $f: A(L_1,1) \to A(L_2,1)$ is a stochastic matrix $(f_{\bar{\chi}_2,\bar{\chi}_1})_{\chi_i \in L_i^*}$ which is covariant with respect to these actions.

We now compute the transformation. Let $\psi \in Z^2(G,U(1))$ be a 2-cocycle. The new algebras will be $A(L_1,\bar{\psi}|_{L_1})$, $A(L_2,\bar{\psi}|_{L_2})$. Choose a grading-preserving channel $A(L_1,1) \to A(L_2,1)$, and let $f$ be its stochastic matrix. To determine the transformed channel we will express $f$ in the graded basis. Let $\mu_i: A(L_i,1) \to A(L_i,1)$ be the Fourier transform matrix which takes the factor basis onto the graded basis. The basis-changed matrix $f^{\mu}:= \mu_2 f \mu_1^{\dagger}$ is diagonal. In the graded bases of the new algebras, the new channel $A(L_1,\psi|_{L_1}) \to A(L_2,\psi|_{L_2})$ has exactly the same matrix $f^{\mu}$.

We remark that coboundaries cannot be ignored. For instance, suppose that $\psi|_{L_1}, \psi|_{L_2}$ are both coboundaries, so the new systems are classical again. Let $i_{i}: A(L_i,\psi|_{L_i}) \to A(L_i,1)$ be the obvious $*$-isomorphism; in the graded basis this is a diagonal matrix. Then in the graded basis the transformed channel is $i_{2} \circ f^{\mu} \circ i_{1}^{\dagger}$. This is in general a different channel.
\end{example}
\begin{remark}
We will briefly explain how to obtain the UPTs realising these entanglement-symmetries. Since $C(G)$ is a finite compact quantum group, every fibre functor is accessible from the canonical fibre functor by a unitary pseudonatural transformation~\cite[Cor. 3.17]{Verdon2020}. For a 2-cocycle $\psi \in Z^2(G,U(1))$, a UPT $F \to F_{\psi}$ can be straightforwardly calculated using~\cite[Thm. 4.12]{Verdon2020}. Explicitly, let $\pi: G \to B(H)$ be an irreducible projective representation of $G$ with cocycle $\psi$.  Let $[g]$ be a simple object of $\Hilb[G]$; we observe that $F(V) = F_{\psi}(V) = \mathbb{C}$. We then define a UPT $(\alpha,H): F \to F_{\psi}$ with components $\alpha_{[g]}: H \to H$ by $\alpha_{[g]}:= \pi(g)$. The $*$-cohomomorphism $F(A) \otimes B(H) \to F_{\psi}(A)$ can then be computed for any f.d. $C^*$-algebra $A$ using~\eqref{eq:udef}.
\end{remark}
\ignore{
For the following proposition, which is an immediate application of Theorem~\ref{}, we fix some notation. By Theorem~\ref{}, for any system $\tilde{A}$ in $\CP(C(G))$ the algebras $A = \tilde{F}(\tilde{A})$ and $\tilde{F}_{\psi}(\tilde{A})$ are identical as graded Hilbert spaces; the only difference is the twist in the multiplication. Let $A_g$ be the homogeneous subspace of $A$ for the group element $g \in G$. There are embeddings $i_g: A_g \to\tilde{F}(\tilde{A})$ and $i_g': A_g \to \tilde{F}_{\psi}(\tilde{A})$. Let $I_g := i_g' \circ i_g^{\dagger}: \tilde{F}(\tilde{A}) \to \tilde{F}_{\psi}(\tilde{A})$.
\begin{proposition}
Let $\tilde{A}$ be a system in $\CP(C(G))$. Let $\tilde{F}$ be the canonical fibre functor on $\CP(C(G))$, and let $\tilde{F}_{\psi}$ be the fibre functor associated to some 2-cocycle $\psi \in Z^2(G,U(1))$.

Then the entanglement-invertible channel $(M_{\tilde{A}},H): \tilde{F}(\tilde{A}) \to \tilde{F}_{\psi}(\tilde{A})$ and its inverse $(N_{\tilde{A}},H): \tilde{F}_{\psi}(\tilde{A}) \to \tilde{F}(\tilde{A})$ are defined as follows:
\begin{align*}
M_{\tilde{A}}(\rho \otimes \sigma) = \frac{1}{\sqrt{\dim(H)}} \sum_{g \in G} \Tr[\pi(g)\sigma] I_g (\rho)  &&
N_{\tilde{A}}(\rho \otimes \sigma) = \frac{1}{\sqrt{\dim(H)}} \sum_{g \in G} \Tr[\pi(g)^* \sigma] I_g^{\dagger} (\rho)
\end{align*}
\end{proposition}
\begin{remark}
These are not the only entanglement-symmetries associated to a finite group. Indeed, rather than the category $\Rep(C(G))$ of $G$-graded Hilbert spaces one can consider the category $\Rep(G)$ directly; now the systems possess a $G$-action rather than a $G$-grading. In this case the interesting fibre functors arise when $G$ is a \emph{group of central type}. Again, all fibre functors are accessible by a unitary pseudonatural transformation. We will not go into details here as the description of the fibre functors is more complicated.
\end{remark}

\ignore{
This is already enough to compute $\hat{E}$ as far as we need to for our purposes:
\begin{itemize}
\item The $\F$ $A(H,\psi)$ is mapped to the $\F$ $A(H,\overline{\omega}\psi)$.
\item The right $A(H,\psi)$ module $V \otimes A(H,\psi)$ is mapped to the right $A(H,\overline{\omega}\psi)$-module $V \otimes A(H,\overline{\omega}\psi)$.
\item A module homomorphism $f: V \otimes A(H,\psi) \to W \otimes A(H,\psi)$ is mapped to itself, considered as a module homomorphism $f: V \otimes A(H,\overline{\omega}\psi) \to W \otimes A(H,\overline{\omega}\psi)$.
\end{itemize}
The first bullet point specifies the equivalence $\tilde{E}$ on objects of $\CP(G)$.

The second and third bullet points can be used to compute the effect of $\tilde{E}$ on morphisms. The Choi theorem~\eqref{} gives a bijective correspondence between CP morphisms $V \otimes A(H_1,\psi_1) \otimes  V^* \to W \otimes A(H_2,\psi_2) \otimes W^*$ and positive elements of $\End_{A(H_2,\psi_2)-\Mod-A(H_1,\psi_1)}[A(H_2,\psi_2) \otimes W^* \otimes V \otimes A(H_1,\psi_1)]$. The equivalence $\hat{E}$ maps these elements to $\End_{A(H_2,\overline{\omega}\psi_2)-\Mod-A(H_1,\overline{\omega}\psi_1)}[A(H_2,\overline{\omega}\psi_2) \otimes W^* \otimes V \otimes A(H_1,\overline{\omega}\psi_1)]$ in the obvious way. It may of course happen that $\bar{\omega} \psi_i$ is not the chosen representative of its cohomology class; in this case there is an obvious isomorphism $A(H_i,\bar{\omega}\psi_i) \cong A(H_i,\underline{\bar{\omega}\psi_i})$, where $\underline{\bar{\omega}\psi_i}$ is the chosen representative.
\ignore{
\noindent
We will now present an example where classical channels are transformed into quantum channels. For this we introduce the following notion.
\begin{definition}
Let $G$ be a group. We say that a 2-cocycle $\psi \in Z^2(G,U(1))$ is \emph{nondegenerate} if the twisted group algebra $A(G,\psi)$  is a matrix algebra; or equivalently, if $G$ has, up to isomorphism, a single projective representation $H_{\psi}$ (of dimension $\sqrt{|G|}$) with 2-cocycle $\psi$, defining a $*$-isomorphism $A(G,\psi) \cong B(H_{\psi})$.

Nondegeneracy is preserved under multiplication by a coboundary; we call a cohomology class $[\psi] \in H^2(G,U(1))$ of nondegenerate 2-cocycles a \emph{nondegenerate cohomology class}.

A group possessing a nondegenerate cohomology class is called a \emph{group of central type.} We will sometimes find it convenient to abuse language by referring to a pair $(G,[\psi])$, where $G$ is a group of central type and $[\psi] \in H^2(G,U(1))$ is a nondegenerate 2-cohomology class, as a group of central type.
\end{definition}
\noindent
An abelian group $A$ is of central type when it is of \emph{symmetric type}, i.e. when there exists some group $S$ such that $A \cong S \times S$. Nonabelian groups $G$ of central type do not admit such an easy classification; see~\cite{} for a list of such groups up to order 121.
\ignore{
\begin{example}[Transformations from a classical system]
We consider all transformations of a single classical system arising from a finite group grading. By the classification~\eqref{}, all indecomposable graded commutative f.d. $C^*$-algebras are twisted group algebras $A(G,1)$, for some abelian group $G$. Possible transformations therefore correspond to 2-cohomology classes $[\psi] \in H^2(G,U(1))$.

Since $G$ is abelian, it is convenient to use the obvious isomorphism $\Hilb(G) \cong \Rep(G^*)$. We thereby consider the twisted group algebra $A(G,1)$ as the algebra $C(G^*)$ of functions on the group $G^*$, with basis $\{\delta_{\chi}~|~\chi \in G^*\}$, and an action of $G^*$ by $\chi' \cdot \delta_{\chi} =  \delta_{\chi' \chi}$. A grading-preserving channel $f$ on $A(G,1)$ is simply one covariant with respect to this action, i.e. satisfying $\chi \cdot f(\bar{\chi} \cdot \delta_{y}) = f(\delta_{y})$ for all $y \in G^*$. Such channels therefore correspond to stochastic matrices $(f_{x,y})_{x, y \in G^*}$, where $f(\delta_{y}) = \sum_{x \in G^*} f_{x,y} \delta_x$; the covariance constraint implies that $f_{x,y} = f_{\chi \cdot x, \bar{\chi} \cdot y}$, so the channel is determined by a choice of $|G^*|$ probabilities $\{f_{x,e}\}_{x \in G^*}$ satisfying $\sum_{x \in G^*} f_{x,e} = 1$.

To compute the transformation we move from the basis $\{\delta_{\chi}\}$ to the basis $\{\bar{g}\}$. This is a Fourier transform, since $\bar{g} = \sum_{\chi \in G^*} \chi(g) \delta_{\chi}$. Let $(\mu_{g,\chi})_{g \in G, \chi \in G^*}$ be the matrix of this Fourier transform, i.e. $\mu_{\chi,g} = \chi(g)$. Then $f^{\mu}:= \mu f \mu^{\dagger}$ is a diagonal matrix with diagonal coefficients $(f^\mu)_{g,g} = \sum $.  Now the transformation associated to some cohomology class $[\bar{\psi}]$ can straightforwardly be computed. The new algebra is $A(G,\psi)$. In the basis $\{\bar{g}\}$ of the new algebra, the transformed channel has exactly the same matrix $f^{\mu}$.
\end{example}
\begin{example}[Transformations on multiple classical systems]
When more than one system is considered, one must take into account the relationship between subgroups of a larger group. Fix some group $G$ and let $H_1,H_2 < G$ be abelian subgroups. We consider transformations arising from the $G$-grading on the classical systems $A(H_1,1)$ and $A(H_2,1)$, which correspond to 2-cohomology classes $[\psi] \in H^2(G,U(1))$.

As before, the 2-cocycle $\bar{\psi} \in Z^2(G,U(1))$ takes $A(H_1,1)$ and $A(H_2,1)$ to $A(H_1,\psi|_{H_1})$ and $A(H_2,\psi|_{H_2})$ respectively. Endo-channels on these algebras are transformed as in Example~\ref{}. A channel $A(H_1,1) \to A(H_2,1)$ is transformed
\end{example}}

\ignore{
%\subsubsection{Example: abelian groups}

For an abelian group $L$, we recall (e.g. from~\cite{}) the correspondence between 2-cocycles $\psi \in Z^2(L)$ and alternating bimultiplicative forms $\rho: L \times L \to U(1)$, which takes a cocycle $\psi$ to the form $\rho_{\psi}$ defined by $\rho_{\psi}(x,y) := \psi(x,y) \overline{\psi(x,y)}$. It is not hard to show that this induces an isomorphism of the cohomology group $H^2(L)$ with the group $\Hom(\Lambda^2 A, U(1))$ of alternating bimultiplicative forms.
We say that a 2-cohomology class $[\psi]$ is \emph{nondegenerate} if the associated form $\rho_{\psi}$ is nondegenerate in the usual sense (i.e. $\rho(x,-): L \to U(1)$ is nontrivial for all $x \neq e$).

In fact, it is not hard to show that all 2-cocyles are induced from nondegenerate 2-cocycles on quotients. Recall that the \emph{radical} of an alternating form $\rho$ on $L$ is the subgroup $L_{\rho}:= \{x \in L~|~ \rho(x,-) = 1\}$. The following lemma is very easy to prove and is left to the reader.
\begin{lemma}
Let $\rho \in \Hom(\Lambda^2 A,U(1))$. Choose any section $\mu$ of the quotient $q: L \to L/L_{\rho}$. Define a map $\tilde{\rho}: L/L_{\rho} \times L/L_{\rho} \to U(1)$ by $\tilde{\rho}(g,h):= \rho(\mu(g),\mu(h))$.

The map $\tilde{\rho}$ does not depend on the choice of section; it is an nondegenerate alternating bimultiplicative form on $L/L_{\rho}$, such that $\rho$ is obtained by inflating $\tilde{\rho}$ using the quotient $q:L \to L/L_{\rho}$.
\end{lemma}
\ignore{\begin{proof}
To make everything concrete, let $L \to B(H)$, $g \mapsto U_g$ be some unitary projective representation of $L$ with cocycle $\psi$. Now this reduces to an ordinary linear representation of the $\alpha$-regular elements, which we call $\alpha: L_{\alpha} \to B(H)$. It is straightforwardly seen that the map $c_{\mu}(g,h): L/L_{\alpha} \times L/L_{\alpha} \to B(H)$ defined by $c_{\mu}(g,h):= \alpha(\mu(g) \mu(h) \mu(gh)^{-1})$ obeys the 2-cocycle equation $c_{\mu}(g,h)c_{\mu}(gh,i) = c_{\mu}(h,i)c_{\mu}(g,hi)$.
\ignore{; indeed, we have $\alpha(\mu(g) \mu(h) \mu(gh)^{-1}) \alpha(\mu(gh) \mu(i) \mu(ghi^{-1}) = \alpha( \mu(g) \mu(h) \mu(i) \mu(ghi)^{-1}) = \alpha(\mu(g) \mu(hi) \mu(ghi)^{-1}) \alpha(\mu(h) \mu(i) \mu(hi)^{-1})$.}

We first show that $\tilde{\psi}$ is a nondegenerate 2-cocycle on $L/L_{\alpha}$. Indeed, by multiplying the central term in the two possible orders we see that
\begin{align*}
\psi(\tilde{h},\tilde{i})\psi(\tilde{g},\tilde{hi}) c_{\mu}(h,i) c_{\mu}(g,hi) U_{\tilde{ghi}} = U_{\tilde{g}} U_{\tilde{h}} U_{\tilde{i}} =  \psi(\tilde{g},\tilde{h}) \psi(\tilde{gh},\tilde{i}) c_{\mu}(g,h) c_{\mu}(gh,i)U_{\tilde{ghi}},
\end{align*}
so using that $c_{\mu}$ is a 2-cocycle it follows that $\tilde{\psi}$ is a 2-cocycle also. Nondegeneracy is clear since $\rho_{\tilde{psi}}(g,h) = e$ implies that $\rho_{\psi}(\tilde{g},h) = e$ for all $h \in L$; thus $\tilde{g} \in L_{\alpha}$ and so $g = e$.

We must show that a different choice of section produces a cohomologous 2-cocycle. This is clear, since a different choice of section comes down to $\mu_{1}(g) = \mu_2(g) \phi(g)$ for some coboundary $\phi: L/L_{\alpha} \to U(1)$. Finally we must show that $[\psi]$ is obtained by inflation of $[\tilde{\psi}]$. For this observe $\mu(q(g)) = g \phi(g)$ for some coboundary $\phi: L \to U(1)$, so the result follows.
\end{proof}}
\noindent
In terms of the fibre functor $F_{\rho}$, we have the following corollary.
\begin{corollary}
Let $A$ be an abelian group, and let $\rho \in \Hom(\Lambda^2 A, U(1))$. Let $q: A \to A/A_{\rho}$ be the quotient by the radical, and let $\tilde{\rho} \in \Hom(\Lambda^2 A/A_{\rho}, U(1))$ be the nondegenerate form.

Then $F_{\rho} \cong F \circ E \circ Q$, where $Q:\Hilb(A) \to \Hilb(A/A_{\rho})$ is the quotient functor induced by $q$; $E$ is the autoequivalence of $\Hilb(A/A_{\rho})$ induced by $\tilde{\psi}$; and $F$ is the canonical fibre functor on $\Hilb(A/A_{\rho})$.
\end{corollary}
\noindent
In the abelian case we therefore restrict our attention to fibre functors coming from nondegenerate 2-cocycles $\omega: G \times G \to U(1)$; i.e. those which do not factor through a quotient. A group admitting a nondegenerate 2-cocyle is called a \emph{group of central type}; in the abelian case these are precisely the groups which can be split as $A = S \oplus S^*$.

\begin{example}
Our main example for computations will be the groups $G = Z_{p}^{2n}$ for prime $p$. These are $2n$-dimensional vector space over the finite field $\mathbb{Z}_p$, and alternating bimultiplicative forms are precisely alternating bilinear forms in the usual sense. One can pick a basis $\{e_i\}$ of $G$; the alternating forms then form a $\mathbb{Z}_p$-vector space with basis $\{e_i \wedge e_j~|~i < j\}$.  As we have seen, these correspond to pairs of a subspace $L$ and a nondegenerate form on the quotient $G/L$. The set of nondegenerate alternating forms on $G/L$ is the quotient of the general linear group by the symplectic group.

The twisted group algebra $A(H,\rho)$ corresponding to a pair of a subspace $H<G$ and an alternating form $\rho$ on $H$ may be described as follows. Let $H_{\rho}$ be the radical and pick an isomorphism $H \cong H_{\rho} \oplus H/H_{\rho}$. This induces an isomorphism $A(H,\rho) \cong A(H_{\rho},1) \otimes A(H/H_{\rho},\tilde{\rho})$.
\end{example}

\subsubsection{Application: Entanglement-assisted capacities of quantum channels}

Quantum channels have several distinct capacities, such as the classical capacity $C$ and the quantum capacity $Q$. Following~[3], we are interested in $C_E$, the \emph{entanglement-assisted classical capacity} of a quantum channel. This is a quantum channel's capacity for transmitting classical information with the help of unlimited prior entanglement between sender and receiver. (Note that $C_E$ determines the analogous entanglement-assisted quantum capacity $Q_E$, since $Q_E = C_E/2$ by teleportation and dense coding.)

It is well-known that, for classical channels, $C_E = C$; entanglement cannot increase the classical capacity. However, for quantum channels this no longer holds. The following proposition allows $C_E$ to be computed precisely for certain quantum channels.
\begin{proposition}\label{prop:computecap}
Let $G$ be a compact quantum group, and let $f: A \to B$ be a $G$-covariant channel. Let $F'$ be any fibre functor on $\Rep(G)$ which is accessible by a unitary pseudonatural transformation, and let $E: \Chan(G) \to \Chan(G')$ be the corresponding equivalence. Then $C_E[E(f)] = C_E[f]$.
\end{proposition}
\begin{proof}
By Corollary~\ref{cor:entequivs} we have entanglement-assisted channel coding schemes interchanging the two channels.
\end{proof}
\begin{example}
One straightforward way to apply Proposition~\ref{prop:computecap} is to start with some covariant classical channels whose capacity is known, and which also possess a symmetry. Let $X,Y$ be finite sets. A \emph{weakly symmetric} classical channel $f:X \to Y$ is one whose stochastic matrix $p(y|x)$ satisfies the following conditions:
\begin{itemize}
\item All rows are permutations of each other.
\item The channel is unital (i.e. it preserves the uniform distribution).
\end{itemize}
The capacity of a weakly symmetric classical channel is known to be
$$C = \log_2(|Y|) - H(\textrm{row of transition matrix})$$
where $H$ is the Shannon entropy.

Let us consider how covariance of a unital classical channel under a finite group action relates to weak symmetry.
Fix a group $G$, fix $G$-actions $\pi_X: G \to \Aut(X)$ and $\pi_Y: G \to \Aut(Y)$, and let $f: X \to Y$ be a covariant unital channel. We claim that, provided the $G$-action on $X$ is transitive, the channel $f$ is always weakly symmetric. Indeed, let $X$ have factor basis $\{\ket{1},\dots,\ket{|X|}\}$. Then for any $x \in X$:
$$f \pi_X(g) \ket{x} = \pi_Y(g) (f \ket{x}) = \pi_Y(g) (\sum_{y \in Y} p(y|x) \ket{y}) =  \sum_{y \in Y} p(y|x) \pi_Y(g) \ket{y}$$
We showed in Example~\ref{ex:gradedchans} that a channel $A(L_1,1) \to A(L_2,1)$ is grading-preserving precisely when it is covariant for the action~\eqref{eq:charaction} of $G^*$ on the systems (which is transitive by indecomposability of the twisted group algebras). Unital grading-preserving channels $A(L_1,1) \to A(L_2,1)$ are therefore weakly symmetric.

For example, let $L_1 = G$ be a group of central type (i.e. possessing a 2-cocycle $\phi$ such that $A(L_1,\phi)$ is a matrix algebra). Then for every unital classical channel $A(L_1,1) \to A(L_1,1)$ we obtain a unital quantum channel $A(L_1,\phi) \to A(L_1,\phi)$ whose entanglement-assisted capacity is identical.
\end{example}

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\section{Appendix: Faithful traces on $G$-$C^*$-algebras} \label{sec:app}

In physics, it is common to assume that the faithful positive linear functional on a $C^*$-algebra is tracial. This is precisely to demand that the corresponding Frobenius algebra should be symmetric.
In the case where there is no $G$-action, one loses no generality since there is always a tracial state on the algebra. In fact, a normalisation can be chosen for the trace that will pick it out uniquely; here we follow~[27] and demand that the Frobenius algebra should be special~\eqref{eq:frobspecial}.
\begin{proposition}\label{prop:specialtrace}
Every f.d. $C^*$-algebra $A$ admits a unique \emph{special} trace making the corresponding symmmetric Frobenius algebra $[A,m,u]$ special. Moreover, with this special trace $u^{\dagger} u = \dim(A)$.
\end{proposition}
\begin{proof}
Every f.d. $C^*$-algebra $A$ admits an orthogonal decomposition as a multimatrix algebra, i.e. $A \cong \oplus_{i = 1}^n M_{n_i}(\mathbb{C})$. Let $\{p_i\}$, $\{\iota_i\}$ be the corresponding orthogonal projections and injections onto and from the factors, and define $\Tr_i: M_{n_i}(\mathbb{C}) \overset{\iota_i}{\hookrightarrow} A  \to[\Tr] \mathbb{C}$. Then we have $\Tr(x) = \Tr(\sum_i \iota_i p_i x)= \sum_i \Tr_i(p_i x)$, so $\Tr$ is determined by the $\Tr_i: M_{n_i}(\mathbb{C}) \to \mathbb{C}$ on each factor. \ignore{Moreover the trace inner product satisfies $\Tr(a^{\dagger}b) = \sum_i \Tr( a^\dagger \iota_i p_i b) = \sum_i \Tr( (p_i a)^{\dagger} p_i b)$, implying the decomposition $A \cong \oplus_{i = 1}^n M_{n_i}(\mathbb{C})$ is orthogonal.} Since the multiplication $m: A \otimes A \to A$ has the form $m = \sum_i \iota_i\circ m_i \circ (p_i \otimes p_i)$, the adjoint comultiplication has the form $m^{\dagger} = \sum_i (\iota_i \otimes \iota_i) \circ m_i^{\dagger} \circ p_i$, and we see that $m \circ m^{\dagger} = \sum_i \iota_i \circ m_i \circ m_i^{\dagger} \circ p_i$. It follows that $\Tr$ will be special on $A$ iff the $\Tr_i$ are special on the factors.

Recall that the trace on a matrix algebra is unique up to scaling. Consider the matrix algebra $M_{n_i}(\mathbb{C})$. Under the inner product resulting from the canonical trace $\overline{\Tr_i}$ it is straightforward to show  that $m_i \circ m_i^{\dagger} = n_i \mathbbm{1}$. Scaling the trace by $n$ scales $m^{\dagger}$ by $n^{-1}$ (this is shown in~\cite[Lem. 4.4]{Vicary2011}). Therefore the unique trace on $M_{n_i}(\mathbb{C})$ yielding a special Frobenius algebra is $n_i \overline{\Tr_i}$. It follows that the the unique choice of trace on $A$ yielding speciality is:
\Tr(x) = \sum_{i} n_i \overline{\Tr_i}(p_i x)

Finally, for the last statement we observe that $u^{\dagger} u = \Tr(1^*1) = \Tr(1) = \sum_i n_i\overline{\Tr_i}(1_i) = \sum_i n_i^2 = \dim(A)$.
\end{proof}
\noindent
In the case of a $G$-$C^*$-algebra for a compact quantum group $G$ the situation becomes more interesting. Although every $G$-$C^*$-algebra admits an invariant functional (in particular, we can pick the canonical invariant functional from Lemma~\ref{lem:uniquespecialfunctional}), not every $G$-$C^*$-algebra will admit a $G$-invariant faithful trace.

We first observe that any Frobenius algebra $[A,m,u]$ in $(G)$ corresponding to a given $G$-$C^*$-algebra $A$ is $*$-isomorphic to any other, so in particular their quantum dimension is identical. It therefore makes sense to talk about the quantum dimension $_q(A)$ of a $G$-$C^*$-algebra (Definition~\ref{def:quantumdimension}). Likewise, we can consider the ordinary dimension $_c(A)$ of a $G$-$C^*$-algebra; this is the dimension of the object $F(A)$ in $$, where $[A,m,u]$ is a Frobenius algebra in $(G)$ corresponding to the $G$-$C^*$-algebra $A$ with some choice of invariant functional and $F: (G) →$ is the canonical fibre functor.

We remark that the special trace of Proposition~\ref{prop:specialtrace} is precisely the canonical $G$-invariant functional on a $G$-$C^*$-algebra for trivial $G$.
\begin{lemma}\label{lem:invspectrace}
Let $\phi$ be a $G$-invariant faithful trace on a $G$-$C^*$-algebra $A$. Then there exists a $G$-invariant faithful trace on $A$ making the corresponding Frobenius algebra in $\Rep(G)$ special.
\end{lemma}
\begin{proof}
We use~\cite[Lem. 2.8]{Neshveyev2018}, which states that any Frobenius algebra in $\Rep(G)$ is unitarily $*$-isomorphic to a direct sum of simple Frobenius algebras. We can therefore reduce to the case of a simple Frobenius algebra with $G$-invariant trace. In this case we already have $m \circ m^{\dagger} = \lambda \id$ for some $\lambda>0$, so using a scalar $*$-isomorphism ($m \mapsto \frac{1}{\sqrt{\lambda}} m$ and $u \mapsto \sqrt{\lambda} u$) we obtain a $G$-invariant faithful trace on $A$ making the corresponding Frobenius algebra special.
\end{proof}
\noindent
\begin{theorem}\label{thm:caninvtraceisspec}
A $G$-$C^*$-algebra $A$ admits a $G$-invariant faithful trace iff $\dim_q(A) = \dim(A)$. In this case the canonical $G$-invariant functional is tracial and is precisely the special trace of Proposition~\ref{prop:specialtrace}.
\end{theorem}
\begin{proof}
We first show that if $\dim_q(A)=\dim(A)$ then the canonical $G$-invariant functional is tracial. Consider the special Frobenius algebra $[A,m,u]$ in $\Rep(G)$ corresponding to $A$ with its canonical $G$-invariant functional. If we forget the $G$-action we obtain a special Frobenius algebra in $\Hilb$ satisfying $u^{\dagger}u = \dim_q(A) = \dim(A)$. But then this is precisely a finite-dimensional $C^*$-algebra equipped with its canonical functional, which was shown to be tracial in Proposition~\ref{prop:specialtrace}.

We now show that if $\dim_q(A) > \dim(A)$ (by~\cite[Cor. 2.2.20]{Neshveyev2013}, it can only be greater), then the $G$-$C^*$-algebra $A$ admits no $G$-invariant faithful trace. We will prove this by contradiction. Suppose that $A$ admits a $G$-invariant faithful trace. Then by Lemma~\ref{lem:invspectrace} it admits a $G$-invariant faithful trace making the corresponding Frobenius algebra special; we may therefore consider a special Frobenius algebra $[A,m,u]$ in $\Rep(G)$ such that the associated positive linear functional is tracial.

We observe that a special Frobenius algebra $[A,m,u]$ in $\Rep(G)$ satisfies $\dim_q(A) \leq u^{\dagger}u$, with equality iff the Frobenius algebra is standard. For this we use~\cite[Thm. 2.2.19]{Neshveyev2013}. Indeed, we note that, in the language of that theorem, $R=\overline{R}= m^{\dagger}u$ is a solution of the conjugate equations for $A$. We then have $\dim_q(A) \leq u^{\dagger}mm^{\dagger}u=u^{\dagger}u$ with equality iff $m^{\dagger}u$ is a standard solution of the conjugate equations up to a scalar factor; the scalar factor must be trivial since $R=\overline{R}$.

Forgetting the $G$-action, we obtain a special Frobenius algebra in $\Hilb$ such that the faithful positive linear functional is tracial. But in Proposition~\ref{prop:specialtrace} it was shown that the only such positive linear functional is the canonical one, satisfying $u u^{\dagger} = \dim(A)$. But since we know that $u^{\dagger} u \geq \dim_q(A) > \dim(A)$, we arrive at a contradiction. There can therefore be no $G$-invariant faithful trace on $A$.
\end{proof}
\end{document}