Programmability of covariant quantum channels

A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.


Introduction
Programmable quantum processors are devices which can apply desired quantum operations, specified by the user via program states, to arbitrary input states. This is convenient because one machine can be used to implement several operations and it resembles very much the way classical computers work based on classical programs and input data. Therefore, quantum processors have been studied since the early days of quantum information theory. In 1997, Nielsen and Chuang proved the No-Programming Theorem which states that it is not possible to implement infinitely many unitary channels exactly with finitedimensional program register [32], i.e. exact universal programmable quantum processors are impossible. However, exact programmable quantum processors were studied for special families of quantum operations [21] and approximate programmable quantum processors were considered in several contexts [4,13,20,22,24,36,37,39]. The question of optimal program dimension (the dimension of the program register) for an approximate universal quantum processor was only recently answered. The works [27] and [47] provided new upper and lower bounds on the program dimension applying methods from Banach space theory and quantum entropies, respectively. In this work, we consider a special class of channels, the covariant channels, i.e. a programmable quantum processor that implements all channels which are covariant with respect to representations U , V of a compact Lie group. Note that, in particular, all results also hold for finite groups. Symmetries are of fundamental importance in physics, since they give rise to conserved quantities via Noether's theorem [33]. In open systems, these symmetries arise as covariant quantum channels and are studied using tools from quantum information theory [11,29]. From a practical point of view, symmetries often simplify problems and break the curse of dimensionality, thus making them amenable to rigorous analysis. For these reasons, covariant quantum channels appear in many different settings, such as channel discrimination, capacities and communication tasks (see Ref. [30] and the references therein). Since this is a special set, the question arises whether those channels can be implemented exactly by a programmable quantum processor. In this article, we show that an exact implementation is possible if the group acts irreducibly on the channel input. While we prove upper bounds on the program dimension for general representations U , we focus on the case in which U is irreducible. As any representation can be decomposed into irreducible ones, it is natural to start studying this scenario before investigating more general representations in future work. In Section 2, we present some preliminaries and our notation. We consider exact programmability of group-covariant channels in Section 3, where we first look at a method based on extreme points in Subsection 3.1. We show that processors have a particularly simple measure-and-prepare form if and only if the commutant of the tensor representation is abelian. This yields a program dimension equal to the number of irreducible representations occurring in the direct sum decomposition of the tensor representation in Corollary 19. Subsection 3.2 discusses the structure of the commutant of the tensor representation in more detail. We give a different construction of covariant programmable quantum processors based on teleportation in Subsection 3.3. This construction is subsequently concatenated with a compression map which allows us to utilize the special structure of the Choi-Jamiołkowski states corresponding to the covariant channels. This leads to Theorem 25, where we show that we obtain a program dimension of at most the sum of the dimensions of the blocks occurring in the structure of the Choi-Jamiołkowski states. After the analysis of exact programmability, we consider an approximate version thereof (see Section 4). First, we provide approximate upper bounds in the case of arbitrary representations U , V in Proposition 29. They are in general worse than the exact bounds in Theorem 25, but they apply more generally. This result is the only one in which we consider arbitrary representations U instead of irreducible ones. In Theorem 31, we provide lower bounds on the program dimension of approximate covariant quantum processors. In particular, this shows that the construction in Theorem 25 is optimal for the exact case.
analogously for d 2 . The set of all pure states is denoted by D P . A quantum channel is a completely positive trace-preserving map T : We review some basic results from representation theory which we need in our analysis. For further details we refer to Ref. [38]. Throughout the paper we shortly write compact group instead of compact Lie groups.
Definition 1 (Unitary representation). Let G be a compact group. A unitary representation of G is a continuous homomorphism from G to the unitary operators Since all representations in this paper will be unitary, we will refer to them as representations for brevity. Any representation can be decomposed into irreducible representations, which are its fundamental building blocks because those cannot be decomposed any further. The image of the representation, i.e. the set of unitaries U g , g ∈ G, generates a matrix algebra (i.e. a finite-dimensional unital * -algebra of linear operators). To study symmetries in the next chapter, we define the commutant of this algebra as: Note that the commutant is again a matrix algebra. If the algebra has a special structure, its commutant also has a corresponding structure which directly follows from the structure of A and the definition of a commutant.
. . , K}, are irreps of G. Furthermore, let A(U ) be the operator algebra generated by the {U g } g∈G and A (U ) the corresponding commutant. Then, The special structure of the commutant is illustrated in Figure 1 for a concrete example. A useful tool in the analysis of direct sum decompositions, irreps and their multiplicities is the notion of a character which is a complex number associated to each group element and related to a unitary representation through the trace operation. It is defined as: [38, p. 41]). Let U : G → GL(H 1 ) be a representation such that g → U g . The character χ : G → C maps every element of G to a complex number according to g → tr(U g ).
In the case of a one-dimensional representation, the character is equal to the one-dimensional representation. The following theorem shows how characters can be used to study representations. [38,Theorem VII.9.5]). For all α, β irreps,
The multiplicities of irreps in the direct sum decomposition of a representation can be calculated as follows: Corollary 9 (Multiplicity relation [38, Corollary VII.9.6.]). Let U be a representation of a compact Lie group G and χ U the corresponding character. Let χ α be the character of an irrep α. Then, is the multiplicity of α in the direct sum decomposition of U . Note that the n α are uniquely determined by U .
The set of all U V -covariant channels is represented by It is standard in quantum information theory to identify quantum channels with positive matrices via the Choi-Jamiołkowski isomorphism [10,25]. We refer to Ref. [19] for a good book on quantum information theory. Let |i ⊗ |i be a maximally entangled state, where {|i } i∈{1,...,d} is an orthonormal basis on H 1 and H 2 , respectively. We define the set of all Choi-Jamiołkowski states corresponding to quantum channels T ∈ T U V as The U V -covariance property on channel level has a well-known correspondance on statespace level which is stated in the following lemma. Lemma 11. The covariance property of a channel T ∈ T U V w.r.t. the unitary representations U , V of a group G is equivalent to the condition that the corresponding Choi- Proof. Using (Ū g ⊗ U g )|Ω Ω|(Ū g ⊗ U g ) * = |Ω Ω|, we can compute that Since the Choi-Jamiołkowski map T → c T is an isomorphism, it follows that that the U V -covariance property of T is equivalent to the corresponding Choi-Jamiołkowski state commuting withŪ g ⊗ V g .
The No-Programming Theorem states that it is not possible to build a device which can implement all unitary channels, or in fact any infinite set of unitaries, exactly and with a finite-dimensional program register. In this work, we consider a setting where the No-Pogramming Theorem is not applicable because we do not want our processor to implement all unitary channels but a family with a certain symmetry consisting of possibly noisy quantum operations. Therefore, we study exact programmability first before considering an approximate version.

Exact and Approximate Programmability
We consider a programmable quantum processor that implements all U V -covariant channels for a unitary representation U of a compact group G. We define a programmable quantum processor with input ρ ∈ D(H 1 ) that implements all covariant channels T ∈ T U V with program states π T ∈ D(H P ) of dimension d P . This is schematically illustrated in Figure 2. Mathematically, we define this processor as follows: Definition 12 ( -PQP C ). Let H 1 and H 2 be separable Hilbert spaces. Then we call P ∈ CPTP(H 1 ⊗ H P , H 2 ), with finite-dimensional H P , an -programmable quantum processor for a set C ⊂ CPTP(H 1 , H 2 ) of channels ( -PQP C ), if for every quantum channel T ∈ C there exists a state π T ∈ D(H P ) such that To address the Hilbert spaces H 1 , H 2 and H P , we refer to the former two as the input and output registers and to the latter as program register. We say that the processor P -implements the class C of channels; for = 0 we say that P exactly implements the class C, and address it as a PQP C .
In particular, when C = T U V , we write CPQP U V for the covariant programmable quantum processor and -CPQP U V for the approximate version thereof. To allow for potentially mixed states in the program register is natural, since the set T U V is convex, whereas the set of pure states is not.

Remark 13.
In the literature, where the set of channels C often consists of isometries or even unitaries, it is customary to impose that the program state π T is pure. While this does not affect the upper bounds on the program dimension (allowing mixed states cannot make it more difficult to program a processor compared to pure states), it is in particular an essential assumption for some lower bounds when unitaries are implemented [27,47]. On the other hand, it does not affect our lower bounds based on the properties of the Holevo information.
It is however always possible to modify a given processor P ∈ CPTP(H 1 ⊗ H P , H 2 ) to a new processor P ∈ CPTP(H 1 ⊗ H P , H 2 ) such that every channel implemented by P, i.e. T = P(·⊗π T ), is implemented by P using a pure program state, i.e. T = P (·⊗|ψ T ψ T |) for |ψ T ∈ H P . The idea is to purify π T on a larger system, which gives rise to two upper bounds on d P = dim H P . To obtain the first, every state π T on H P can be purified to a pure state |ψ T ∈ H P ⊗H P =: H P and we can let P : The second upper bound is a refinement of the first, starting from the observation that we can decompose a mixed program state into a convex combination of pure states, , which can be seen from the Choi-Jamiołkowski isomorphism, hence by Carathéodory's theorem T can be written as a convex combination of not more than D : which has rank ≤ r := min{D, d P }, implements T as well: T = P(· ⊗ π T ). But now we only need an r-dimensional Hilbert space H R to purify π T = tr H R |ψ T ψ T |, |ψ T ∈ H P ⊗H R . As before, we can let the processor be P : We conclude by noting that the first method is good if d P is already small, whereas the second is preferred if d P is large. The former is the case if we consider unitarily covariant channels (see Example 20). The latter is the case if C = T U V and the symmetries we consider are trivial, as d P is exponentially large in the dimension d = d 1 = d 2 [27,47].
We start with the observation that the processor can without loss of generality be chosen to be (U ⊗ 1 d P )V -covariant as well: Proof. We can construct the desired processor via twirling, i.e.
where µ is the Haar measure on G. We compute for T ∈ T U V and T a quantum channel such that for T ∈ T U V by covariance, it holds that as well. This shows that P is also an -CPQP U V with the same program dimension.
Using the invariance of the Haar measure, it can be verified that for any g ∈ G, which shows that P is (U ⊗ 1 d P )V -covariant as desired.

Recall Lemma 11 which states that
Due to this correspondence, we consider representations of the formŪ ⊗ V (which are isomorphic to the adjoint representation of G if V = U ) with U g ∈ U 1 , g ∈ G and the commutant Note that since H 1 and H 2 are finite dimensional, only K ≤ d 1 d 2 irreps can appear in the commutant with multiplicity n k > 0, where n k = dim(H k ). We identify these elements with an index k ∈ {1, . . . , K} motivated by the fact that we want to relate the irreps occuring in the direct sum decomposition ofŪ ⊗ V with the number of extreme points of J U V , for instance. While it is not true in general that all states in K are Choi-Jamiołkowski states of a quantum channel, this is true if U is an irrep. This is proven in the following lemma which aligns with results in Refs. [15, p. 6] and [1, p. 7].

Lemma 15. Let K be as defined above and let U be an irrep of a compact group G on
Since all elements of J U V are Choi-Jamiołkowski states corresponding to U V -covariant channels, they satisfy c T ≥ 0 and tr(c T ) = 1 by definition. Hence, c T ∈ D(H 1 ⊗ H 2 ). According to Lemma 11, T ∈ T U V corresponds to [c T ,Ū g ⊗ V g ] = 0 for all g ∈ G and thus, c T ∈ K ∩ D(H 1 ⊗ H 2 ). "⊆" Let us refer to H 1 as system A and to H 2 as system B. If we intersect K with the set of states D(H 1 ⊗ H 2 ), then every ρ AB ∈ K ∩ D(H 1 ⊗ H 2 ) satisfies tr(ρ AB ) = 1 and To obtain ρ AB ∈ J U V , we additionally have to show the required property tr B (ρ AB ) = for any g ∈ G which is equal to Since U is an irrep if and only ifŪ is, we infer due to Schur's Lemma: Taking the trace on both sides results in for all c T ∈ J U V , which implies that T is unital.
In Ref. [21], the authors derived that channels implemented by a processor that is covariant with respect to the special unitary group SU (H 1 ) are unital using a similar argument. We will now consider how to construct covariant programmable quantum processors in the case where K is abelian.

Covariant programmable quantum processors from extreme points
In this section, we consider a special class of CPQP U V , the measure-and-prepare CPQP U V . We will show that T U V can be implemented by a measure-and-prepare CPQP U V if and only if the commutant K is abelian. Before we state the main result, we present a lemma required for its proof which is a correspondence between the commutant and the affiliated state space which we require in order to show our first main result for exact programmability.

Lemma 16.
Let U be an irrep of a compact group G, and let V be another representation of G. Then, the following are equivalent: Proof. "i) ⇒ ii)" This statement is mentioned in Ref. [40] without proof. Let B ∈ K and let K be an abelian matrix algebra. This implies that n k = 1 for all k ∈ {1, . . . , K}. Furthermore, let K be the number of irreps appearing in the direct sum decomposition of U ⊗ V . We obtain . , K} such that the corresponding block is of dimension n k > 1. Let us consider elements of the form . These elements are extreme points of J U V by Lemma 15. Thus, there are infinitely many extreme points of J U V . Hence, the set cannot be isomorphic to a polytope which has finitely many extreme points by definition. The last implication "ii) ⇒ iii)" is obvious.
Let us now state the definition of the class of processors we consider in this section.
Definition 17 (Measure-and-prepare P QP C ). Let H 1 and H 2 be separable Hilbert spaces. Then, we call a PQP C P with finite-dimensional H P a measure-and-prepare PQP C for a set of quantum channels C if there exists a K ∈ N, a POVM {E i } i∈{1,...,K} ⊂ D(H P ) and a set of quantum channels T k ∈ C, k ∈ {1, . . . , K}, such that Proof. We fix a quantum channel T ∈ C with its corresponding Choi-Jamiołkowski state c T ∈ J C . Since J C is isomorphic to a polytope, it is spanned by K extreme points c T k and can therefore be written as convex combination of these where x k ∈ [0, 1] and K k=1 x k = 1. By the Choi-Jamiołkowski isomorphism, there is a channel T k ∈ T U V corresponding to each of the extreme points c T k of J C , i.e. T can be linearly decomposed with extreme points T k (·). We encode the {x k } k∈{1,...,K} in the program state as follows with an arbitrary orthonormal basis {|k } k∈{1,...,K} on H P . The following processor implements T ∈ C exactly with a program register of dimension d P = K: and extended by linearity. We verify that this is indeed a measure-and-prepare PQP C : where, E k = |k k| for all k ∈ {1, . . . , K}. Thus, we showed that if J C is isomorphic to a polytope, there is a measure-and-prepare processor P that implements all T ∈ C exactly with program dimension dim(H P ) = d P = K. Conversely, let P be a measure-and-prepare PQP C with a K-outcome POVM. Then, C is the convex hull of at most K extreme points, since by definition for any T ∈ C (p 1 , . . . , p K ) is a probability distribution. Thus, the extreme points of C are a subset of {T 1 , . . . , T K } and hence C is a polytope.
The following corollary assures that if we want to check whether channels T ∈ T U V are programmable exactly by a measure-and-prepare CPQP U V with finite-dimensional program register, we can consider the specific structure of the commutant K.
with extreme points |Ω Ω| and 1 d 2 −1 (1 − |Ω Ω|). Thus, T has the form T (·) =α tr(·) 1 d + (1 −α)id [26,40]. Every c T can be written as convex combination of the two extreme points and the set of all convex combinations is isomorphic to a 1-simplex. Thus, the measure-and-prepare CPQP U V can be implemented with program dimension d P = 2 using the construction in Proposition 18.

Structure of the commutant of the tensor representation
Based on the previous section, one could argue that in the case where the direct sum decomposition ofŪ ⊗ V consists of at least one one-dimensional irrep with multiplicity n k > 1, it would not be possible to implement the corresponding U V -covariant channel exactly in this case. It is instructive to see why this situation never arises, which we will discuss in this section. The argument is the following: Assume d 1 = d 2 = d and that there is a k ∈ {1, . . . , K} such that n k > 1. Then, there would be elements in J U V of the form is a rank-one projector and dim(0) = d 2 − n k . By the Choi-Jamiołkowski isomorphism, there is a corresponding channel T ∈ T U V . We consider a Kraus representation of the channel T . Since T is a completely positive map and rank(B ϕ ) = 1, the channel T can be written as T (·) = X(·)X * with one Kraus operator X ∈ B(H 2 ). Since, additionally, T is trace-preserving, we know that XX * = 1 = X * X, i.e. the Kraus operator is a unitary and T is a unitary channel Since there are infinitely many pure states, the processor would have to implement infinitely many corresponding unitary channels. This contradicts the No-Programming Theorem according to which there is no processor that implements infinitely many unitary channels exactly with program dimension d P < ∞. This shows that a processor P with d P < ∞ cannot exist if b k = 1 and n k > 1 for some k ∈ {1, . . . , K}. Note that the above argument fails for b k > 1, since then there might be no rank-1 elements in K. However, from a representation-theoretic perspective this situation does not arise because a one-dimensional irrep in the direct sum decomposition of U ⊗V always has multiplicity ≤ 1 as the following proposition shows.

Proposition 21.
Let U be an irrep on H 1 of a compact group G. Let V be another representation of G on H 2 with dimension d 2 ≤ d 1 . In a direct sum decomposition of U ⊗ V , the one-dimensional irreps λ appear with multiplicity n λ ≤ 1.
Proof. Let χ U be the character of the irrep U , χ V the character of V , and λ the character of the one-dimensional irrep λ. We use the following scalar product (see Lemma 8) where µ is the Haar measure on G. Here, ψ is the character of an arbitrary representation of G. Note that if U is an irrep, then this scalar product of the corresponding characters gives the multiplicity of U in the representation corresponding to ψ (see Corollary 9). We want to show that the multiplicity of λ inŪ ⊗ V is ≤ 1. Note that the character ofŪ ⊗ V isχ U · χ V . LetĜ be the set of irreps of G and let be the decomposition of V into irreducible representations on the level of characters. Then, Since λ is the character of a one-dimensional irrep, it is equal to the representation itself and |λ(g)| 2 = 1 for all g ∈ G. Thus, λ·χ U , λ·χ U = 1 and the representation corresponding to λ · χ U is again irreducible. Note that the representation corresponding to λ · χ U has the same dimension as U .
The scalar product λ · χ U , χ α thus gives the multiplicity of λ · χ U in χ α . Let α be such that n α > 0. If the dimension of the representation corresponding to χ α is smaller than d 2 , then λ · χ U , χ α = 0 by Lemma 8. Thus, λ,χ U · χ V = 0. If the dimension is d 2 , then χ V = χ α for dimensional reasons and V is irreducible. The multiplicity of an irrep in another irrep can be at most equal to 1 which proves the assertion.

Remark 22. We can also give a direct algebraic argument for the impossibility of a onedimensional irrep of G to occur inŪ ⊗ V with multiplicity larger than one, if U is an irrep and V is an irrep, too, and has the same dimension as U . Namely, observe that all pure states in the multiplicity space K of a one-dimensional irrep have to be maximally entangled due to Schur's Lemma (their reduced states on both factors have to be maximally mixed). However, there is no two-or higher-dimensional subspace in d × d that merely consists of maximally entangled states (see, for instance Ref. [12, Proposition 6]).
After this discussion, we give a different construction of covariant programmable quantum processors which is more widely applicable.

Covariant programmable quantum processors from teleportation
If K is not abelian, we need a different method to show exact programmability. This case appears for example for the finite group A 4 [16, p. 20], which is outside the scope of Ref. [30]. The group has four irreps ϕ (0) , ϕ (1) , ϕ (2) , and ϑ. The first three, ϕ (0) , ϕ (1) , ϕ (2) , are onedimensional, while ϑ is three-dimensional [16,Section 2.3]. Indeed, with the third root of unity ζ = e 2πi/3 , we have ϕ (j) (123) k K 4 = ζ jk . On the other hand, ϑ has a very intuitive form as the real SO(3) symmetries of a regular tetrahedron in three-space: concretely, let H = (1, 1, 1, 1) ⊥ = (a 1 , a 2 , a 3 , a 4 ) : −1, −1, 3) of the tetrahedron. The group action of ϑ is simply by permutation of the coordinate axes, and so all its matrices are real; as a consequence, ϑ = ϑ. Consider now the set of ϑϑ-covariant channels. Note that they form a semigroup (since they are closed under composition), and at the same time a closed convex set, isomorphic to the 3 × 3-states invariant under ϑ ⊗ ϑ. The latter representation decomposes as Then, the unitary channels T j (ρ) = V j ρV * j (j = 0, 1, 2) are ϑϑ-covariant. For instance, V can be given as where T denotes the transpose, whose Choi-Jamiołkowski state α is the normalized projector onto the 3 × 3-antisymmetric subspace. It isŪ U -covariant for arbitrary U ∈ SU(3); in particular it is ϑϑ-covariant if we recall that ϑ is real, but it is much more symmetric than that. It has the curious property that it is extremal in the set of all CPTP maps, hence in particular in the set of covariant ones. Moreover, we note that the channels T j • W • T k (j, k = 0, 1, 2) are all ϑϑ-covariant by the semigroup property, giving further extreme points. Their Choi-Jamiołkowski states, which are (V k ⊗ V j )α(V k ⊗ V j ) * , generate the qubit algebra A of the multiplicity space of ϑ in Eq. (3.1). In particular, we will find subsequently elements from the algebra which are a representation of the Pauli matrices σ X , σ Y , σ Z . This was inspired by the approach taken in [14] for a different problem. Let Using the form of α, which is proportional to the projector onto the antisymmetric subspace, we infer that P F ∈ A, where F is the flip (or swap) operator on C 3 ⊗ C 3 . Furthermore, we can verify that [P, 1 3 ⊗ V ] = 0 and [P, F ] = 0. Combining this with the previous finding that the Choi-Jamiołkowski states of T j • W • T k are in A, we obtain four linearly independent elements of A, namely P , P F , P (V ⊗ V * ) and P F (V ⊗ V * ). Since P F is traceless, P is orthogonal to all elements except P (V ⊗ V * ) and P F is orthogonal to all elements except P F (V ⊗V * ). Thus, we can identify P 1, V X := P F σ X and make
We can construct an exact quantum processor for covariant channels based on teleportation simulation, i.e. the simulation of quantum channels by quantum teleportation. In the case of the Pauli group this goes back to the Refs. [6,7]. Important developments concerning the teleportation of covariant channels can be found in Refs. [9,23,31]. See also the very recent Ref. [41]. We know from Refs. [35, p. 58] and [45, Proposition 2] that it is always possible to simulate U V -covariant channels exactly using the corresponding Choi-Jamiołkowski state. This can easily be formulated as a processor which uses the Choi-Jamiołkowski state as program state and performs the teleportation protocol. Therefore, the dimension of the program register is d 1 d 2 . We will make this precise in the next proposition already present in Refs. [35,45], which we have included here for convenience.

Proposition 24. Let G be a compact group and let U be an irreducible representation on
Proof. Let ρ ∈ D(H 1 ) be the state to be teleported. In this proof, we identify the input space as system A, H 1 H A , and the program as a composite system with parts A and is isomorphic to the output space of T . We will write d 1 for the dimension of A. Note that dim A = dim A = dim A = d 1 . The Choi-Jamiołkowski states corresponding to the channels that are simulated serve as program states of the processor running the following protocol: i) the processor measures according to the POVM Due to Schur's Lemma this is a POVM. Note that this POVM can potentially be continuous.
ii) On outcome g, apply V * g (·)V g to the outcome of the protocol.
This construction of a processor implements U V -covariant channels T ∈ T U V with the Choi-Jamiołkowski state as program state. The map is defined as and extended by linearity. Let us verify that indeed In the following, we insert the definition of the Choi-Jamiołkowski state Here, A is again a system isomorphic to A. We calculate Applying the unitary conjugation as in Eq. (3.2) and integrating yields Eq. (3.3). This shows that P is indeed a CPQP U V with the desired program dimension.
Using the structure of the commutant K, we can reduce the program requirements further: Theorem 25. Let U be an irrep on H 1 of a compact group G, V another representation of G on H 2 , and let K be of the form Then, there exists a CPQP U V with program dimension d P = K k=1 n k , where n k = dim(H k ).
Proof. Since we have a programmable quantum processor P, constructed in Proposition 24, available, we combine its protocol, i.e. the teleportation simulation part, with a compression map which reduces the program dimension. Instead of K we consider a simpler matrix algebra with all multiplicities removed. These matrix algebras are isomorphic. Let be this isomorphism and let H P = K k=1 H k , which has dimension d P = K k=1 n k . Thus, D * is a unital completely positive map with unital completely positive inverse C * [8, Example II. 6 and it follows that P (ρ ⊗ C(c T )) = T (ρ).

Remark 26.
Comparing Proposition 24 to Theorem 25, we see that the bound in the theorem is strictly better as soon as there is one index k such that b k > 1. Recall that the b k are the dimensions of the irreps appearing inŪ ⊗ V . Thus, the improvement of Theorem 25 over Proposition 24 is larger, the more different irreps of dimension strictly larger than 1 appear inŪ ⊗ V and the larger the dimension of these irreps is.

Bounds for Approximate Programmability
Since exact universal programmable quantum processors are impossible, there is a great interest in approximate versions of such devices. After having considered exact programmability in the last section, we now want to approximate the output such that the programmed output is close to the ideal output. Therefore, we need the concept of an -CPQP U V from Definition 12, which is a processor that implements a channel T instead of the exact result T . However, it should be -close to the ideal one in diamond norm. Having recalled this, the question of program requirements arises and hence, we show that the program register requirements for an -CPQP U V are not much lower than for an exact CPQP U V . Note that we can still benefit from the covariance property and the corresponding structure of the commutant K.

Upper bounds on the program dimension
We construct generic upper bounds for the dimension of the program register d P . We seek to establish an -net on the set of covariant channels T U V . Therefore, we need the following result about -nets from Ref. [28].
Lemma 27 ( -nets in R n [28,Lemma 9.5]). Let ∈ (0, 1) and let · be any norm on R n . There is an -net S on the unit sphere S n−1 · of (R n , · ) of cardinality That means, for all x ∈ S n−1 · , there is a y ∈ S such that x − y ≤ .
Due to the Choi-Jamiołkowski isomorphism, there is a c T ∈ J U V corresponding to each T ∈ T U V . According to Lemma 11, we know that J U V ⊆ K ∩ D(H 1 ⊗ H 2 ). Thus, we can make use of the special block-diagonal structure of K.

Proposition 28. Let
Proof. Since J U V ⊆ K by Lemma 11, the real vector space Lin R J U V generated by J U V has dimension at most d n . Using the Choi-Jamiołkowski isomorphism, we infer that Lin R T U V is a real subspace of B(B(H 1 ), B(H 2 )) of dimension at most d n . The restriction of the diamond norm turns (Lin R T U V , · ) into a real normed space which is isometrically isomorphic to (R dn , · ) for some induced norm · . Moreover, T = 1 for all T ∈ T U V . Lemma 27 thus ensures the existence of an -net S on the unit sphere of (Lin R T U V , · ) and the unit sphere contains T U V . To obtain S U V , we repeat the following steps. Take Φ ∈ S. If Φ ∈ T U V , keep it and proceed to the next element. If there is no T ∈ T U V such that Φ − T ≤ , remove Φ from the set and proceed to the next element. If there is a T ∈ T U V such that Φ − T ≤ , exchange Φ by T and proceed to the next element. This algorithm constructs S U V with the desired properties. Indeed, for any T ∈ T U V , there is a Φ ∈ S and a T ∈ S U V such that The cardinality bound follows since by construction |S U V | ≤ |S|.
The previous proposition allows us to construct an -CPQP U V .

Proposition 29.
For a compact group G and representations U on H 1 , V on H 2 such that there exists an -CPQP U V with program dimension Proof. Let S U V = {T 1 , . . . , T s } be the set from Proposition 28. Then, we can define a processor by and extending by linearity. Here, {|i } i∈{1,...,s} is an orthonormal basis of H P . Choosing the program state for any T ∈ T U V to be |i i| if T − T i ≤ 2 , the map P can be checked to be an -CPQP U V using Proposition 28.

Remark 30.
Note that in the case in which U and V are the trivial representation and d 1 = d 2 = d, Proposition 29 states that This does not match the upper bounds of d P ≤ (K/ ) d 2 obtained in Ref. [27], which are also derived using -nets. The reason is that the construction in Ref. [27] only implements all unitary channels in dimension d instead of all quantum channels, which is what our construction does. In the same setting, a slight arithmetic improvement of the work [47] yields lower bounds of the form

Lower Bounds for the program dimension
We seek to provide lower bounds on the program dimension of an -CPQP U V . The main idea is that all information about the U V -covariant channel T ∈ T U V is contained in its which can be related to the corresponding Choi-Jamiołkowski states [42, eq. 3.414] Note that c T has a block-diagonal structure inherited from Since K k=1 B(H k ) and K are isomorphic as matrix algebras, extending the dual map of this isomorphism to a map C : B(H 1 ⊗ H 2 ) → B(H dc ) (as in the proof of Theorem 25), discards the multiplicity spaces and thus reduces the dimensions from d 1 d 2 to d c = K k=1 n k . Since the trace distance is contractive under quantum channels, With this trace-norm distance, we apply the Alicki-Fannes-Winter (AFW) inequality [3,34], [ , which proves the assertion.
The bound on d P we have derived increases with the block dimension n k .

Discussion
In this section, we discuss our findings and mention some potential future work. We start by pointing out that Theorem 25 is in fact optimal. Note that for = 0 we obtain exact lower bounds from Theorem 31. Combining them with the teleportation protocol in Theorem 25, the optimal program requirements d P for an CPQP U V are thus If we insisted on pure program states, we would get d P = d 2 c from Theorem 25, replacing π T by a suitable purification (for example the canonical purification √ d P (1 ⊗ π 1/2 T W )|Ω , where W is a suitable unitary). More refined constructions have been discussed in Remark 13. We leave it as an open question whether the upper bounds can be improved to match the lower ones also for pure program states. In the case where the commutant K is abelian, Corollary 19 answers the question in the affirmative. In the present paper, we have shown that the problem of finding programmable quantum processors for U V -covariant channels for irreducible U is very different from the situation concerning universal programmable quantum processors. While the former is always possible exactly with finite-dimensional program register, the latter is only possible approximately with finite-dimensional program register and requires a large program dimension. The question remains whether there are intermediate situations: Is there a group G and a reducible representation U such that there exists an exact CPQP U V with finite-dimensional program register? If so, what determines if there are exact CPQP U V with finite-dimensional program register? A major roadblock to extend our results to the case in which U is no longer irreducible is Lemma 15, since we can no longer guarantee that all elements in K ∩ D(H 1 ⊗ H 2 ) correspond to trace-preserving maps. A further question is, how upper bounds for > 0 can be constructed because thenets used in Ref. [27] and the general-purpose result Proposition 29, for instance, rely on orthogonal program states. Remark 30 already shows that the proposition is suboptimal for trivial symmetries. If U is irreducible it can be outperformed applying the teleportation protocol as we saw in Section 3.3. Hence, we have good exact upper bounds and for an approximation in the case where the program states are not orthogonal, a different way to compress the program states is required. Therefore, the question arises whether we can find approximate upper bounds which improve the exact ones from Theorem 25 for U irrep. However, Theorem 31 shows us that there is not much space for improvement using approximate processors.