Characterization of equivariant maps and application to entanglement detection
aa r X i v : . [ m a t h - ph ] J u l Characterization of equivariant maps and application toentanglement detection
Ivan Bardet, Benoˆıt Collins, Gunjan SapraJuly 9, 2019
Abstract
We study equivariant linear maps between finite-dimensional matrix algebras, as introduced in [1].These maps satisfy an algebraic property which makes it easy to study their positivity or k -positivity.They are therefore particularly suitable for applications to entanglement detection in quantum informationtheory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call( a, b )-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitaryconjugation as studied by Bhat in [2] and as a generalization of the equivariant maps studied in [1].Using representation theory, we fully compute them and study their graphical representation, and showthat they are basically enough to study all equivariant maps. We finally apply them to the problem ofentanglement detection and prove that they form a sufficient (infinite) family of positive maps to detectall k -entangled density matrices. Due to their crucial role in numerous tasks in quantum processing and quantum computation, it is of greatimportance to decide whether a certain density matrix on a bipartite system is entangled or not [3, 4].However this problem, referred to as entanglement detection , is known to be a computationally hard one inquantum information theory [5, 6]. In the last two decades, lots of effort have been accomplished in order todetermine necessary and sufficient conditions for a density matrix to be entangled. For instance, one suchcriterion is the k -extendibility hierarchy [7], which provides a sequence of tests to check that the densitymatrix is separable, that ultimately detects all entangled states.Another appealing method is the positive map criterion [8], which gives an operational interpretationof the Hahn–Banach theorem applied to the convex set of separable density matrices. The Horodecki’sTheorem thus states that a density matrix ρ is entangled if and only if there exists a positive map Φ suchthat ( i ⊗ Φ)( ρ ) is not positive semi-definite, where Φ only acts on one of the two subsystems. Necessarily,this positive map is not completely positive.The most well-known example of such map is the transpose map, and it leads to the positive partialtranspose (PPT) criterion [9]. However, because of the complex geometrical structure of the set of separabledensity matrices, an infinite number of maps that one does not know how to describe efficiently would benecessary to detect all entangled states (see for instance [10]). The goal of this article is to propose a familyof maps, with increasing complexity, which suffice to detect any entanglement. Their main interest lies inthat it is rather easy to check if they are positive or not.Compared to the k -extendibility hierarchy, the Horodecki’s Theorem can be generalized to test thedegree of entanglement of a density matrix quantified by its Schmidt number. A density matrix ρ has Schmidtnumber less than or equal to t if and only if ( i ⊗ Φ)( ρ ) ≥ t -positive maps.Similarly to positivity, we propose a sufficient family of t -positive maps for the problem of t -positivitydetection. Again, it is rather easy to check if they are t -positive or not.Indeed, proving that a map is completely positive is easy: it is enough to check that its Choi matrix ispositive semi-definite [11]. Such a criterion does not exist in general to check that a given map is t -positive,which makes it difficult to find interesting examples of positive but not completely positive maps. Choi [12]1ave in 1973 the first example of a linear map on M n ( C ) which is ( n − n -positive. Onedecade later in 1983, Takasaki and Tomiyama [13] gave a method to construct any number of linear mapson finite-dimensional matrix algebras, which are ( k − k -positive. Interestingly, all theseexamples fall in the class of maps introduced by Collins et al. in [1], called equivariant maps . In the samearticle, they proved that if a linear map happens to be equivariant, its k -positivity depends upon the positivityof a k -blocks submatrix of the corresponding Choi matrix. In a sense, it means that it is as easy to check thatan equivariant map is k -positive, as it is to check that it is completely positive. They subsequently studied aparametric family of equivariant linear maps on M ( C ) with values in M ( C ) ⊗ . In this article, we analysein more depth equivariant maps from M n ( C ) to M n ( C ) ⊗ k for all k, n ≥
1, and characterize a large classof them. Note however that not all known examples of positive maps are equivariant (see for instance [14, 15]).More precisely, we are concerned with two different objectives. The first one is to get a fullunderstanding of equivariant maps. We only get sparse results in this direction. As a first insight, we givea characterization of equivariant maps in terms of their Choi matrices in Theorem 3.1. We also define asubclass of them, the unitarily equivariant maps, which are more tracktable objects. Corollary 3.3 is oneof the main results of this paper, where we prove that unitarily equivariant maps - were the equivarianceproperty is given in terms of a unitary representation - are in fact covariant maps with respect to a unitaryrepresentation of the unitary group.The second objective is to fully compute a subclass of such maps, the ( a, b )-unitarily equivariant maps,generalizing the examples in [1] and the characterization in [2] of linear maps invariant under conjugation.We then focus on the application to entanglement detection of such maps. We prove that any t -entangled density matrix - that is with Schmidt Number strictly lower than t - can be detected using a t -positive unitarily equivariant map. Combined with previous result, this gives an explicit family of t -positivemaps that are sufficient of t -entanglement detection.This article is organized as follows. In Section 2, we introduce the equivariant maps, list some of theirproperties and give some examples, among which the one of Choi [12], Takasaki and Tomiyama [13]. Insection 3, we give different characterizations mentioned above. We study the graphical representations ofthe Choi matrices of ( a, b )-unitarily equivariant maps in Section 4. We focus on entanglement detection inSection 5. In this section, we present definitions of equivariant linear maps and give an explanation as to why it isimportant to study these maps.
For a positive integer n , M n ( C ) is the set of square matrices, with entries from C of size n , with canonicalorthonormal basis ( e ij ) ≤ i,j ≤ n . The unitary group on C n is denoted by U n . We denote by n the identitymatrix in M n ( C ) and by i n the identity map acting on M n ( C ). We write B n = P ni =1 | e i i ⊗ | e i i , thenon-normalized maximally entangled Bell vector in ( C n ⊗ C n ). The rank-one projection on B n is denotedby B n = B n B ∗ n . Finally, A t and Tr( A ) denote the transpose and (non-normalized) trace of a matrix A ∈ M n ( C ) respectively. θ n denotes the transpose map A A t on M n ( C ).If H and K are two complex Hilbert spaces, B ( H ) denotes the space of all bounded linear operatorson H and B ( H , K ) the space of all bounded linear maps from H to K . A self-adjoint map Φ ∈ B ( H , K ) issuch that Φ( X ∗ ) = Φ( X ) ∗ for all X ∈ H . In the following, we will assume that all linear maps are self-adjoint.We are now ready to give the main definitions of this article. Definition 2.1.
Let n, N ≥ be two natural numbers. A linear map Φ : M n ( C ) → M N ( C ) is called:(i) Equivariant , if for every unitary matrix U ∈ M n ( C ) there exists V = V ( U ) ∈ M N ( C ) such that Φ( U XU ∗ ) = V ( U ) Φ( X ) V ( U ) ∗ ∀ X ∈ M n ( C ); (1)2 ii) Unitarily equivariant , if furthermore the operator V ( U ) in the previous definition can be taken unitary.(iii) ( a, b )-unitarily equivariant , if there are a, b natural numbers such that N = n a + b and M N ( C ) ≡ M n ( C ) ⊗ a ⊗ M n ( C ) ⊗ b , and such that for every unitary U ∈ M n ( C ) , Φ( U XU ∗ ) = ( U ⊗ a ⊗ U ⊗ b ) Φ( X ) ( U ⊗ a ⊗ U ⊗ b ) ∗ ∀ X ∈ M n ( C ) . (2)Thus, ( a, b )-unitarily equivariant maps are a subfamily of unitarily equivariant maps, that is itselfa subclass of equivariant maps. We prove in Corollary 3.3 that every unitarily equivariant map where U V ( U ) is a unitary representation can be seen as a corner of a sum of ( a, b )-unitarily equivariant maps.In the following, we give a family of equivariant and not unitariy equivariant maps. Example 2.2.
Let A ∈ M n ( C ) be a non-unitary invertible matrix. Define: Φ A : M n ( C ) → M n ( C ) X AXA ∗ Let U ∈ M n ( C ) be a unitary. Then for every X ∈ M n ( C ) , Φ( U XU ∗ ) = AU XU ∗ A ∗ = ( AU A − )( AXA ∗ )( A ∗ ) − U ∗ A ∗ = ( AU A − )Φ( X )( AU A − ) ∗ Clearly,
AU A − may not be a unitary as A is not unitary. Therefore, Φ A is equivariant and notunitarily equivariant. Example 2.3.
Bhat characterized in [2] all (0 , -unitarily equivariant maps (that is, for a = 0 and b = 1 )on B ( H ) for some Hilbert space H , not necessarily finite dimensional. More precisely, he proved that a linearmap Φ acting on B ( H ) satisfies Φ( U XU ∗ ) = U Φ( X ) U ∗ for all X ∈ B ( H ) iff there exist α, β ∈ C such that Φ( X ) = αX + β Tr [ X ] I H . Directly from this, we get that any (1 , -unitarily equivariant map Ψ on M n ( C ) is of the form Ψ( X ) = αθ n ( X ) + β Tr [ X ] I H , where θ n is the transpose map. Indeed, we can check that θ n ◦ Ψ is (0 , -unitarilyequivariant and apply Bhat’s result. We shall similarly characterize all ( a, b ) -unitarily equivariant maps inTheorem 3.4, thus generalizing these two case, when H is finite dimensional. Establishing k -positivity of a linear map is a difficult task, even on low dimensional matrix algebras.In this regard, a criterion which is a necessary and sufficient condition for an equivariant map to be k -positivewas given in [1]. Theorem 2.4. [1, Theorem 2.2] Let
Φ : M n ( C ) → M N ( C ) be an equivariant map. Then, for k ≤ min { n, N } , Φ is k -positive if and only if the block matrix [Φ( e ij )] ki,j =1 is positive semi-definite, where ( e ij ) ≤ i,j ≤ n are the matrix units in M n ( C ) . Incidentally, some well-known examples of k -positive but not completely positive linear maps areactually examples of unitarily equivariant maps, even though this was not explicitly stated when they wereintroduced. We recall these examples and give alternative proof of their k -positivity, based on Theorem 2.4.(i) Every ∗ -homomorphism or ∗ -anti-homomorphism on a finite-dimensional matrix algebras is equivariant.Such maps are always completely positive or co-completely positive.(ii) Choi [12, Theorem 1] gave the first example of a linear map on M n ( C ) which is ( n − n -positive, given by Φ : M n ( C ) → M n ( C ) A
7→ { ( n − A ) } n − A .
The above map is unitarily equivariant (take V ( U ) = U ). We apply Theorem 2.4 to prove that Φ is( n − e ij )] n − i,j =1 are 0 with multiplicity 1 and ( n −
1) with multiplicities( n ( n − − , which are positive. Hence, Φ is ( n − k -positivity. The map is defined by:Ψ : M n ( C ) → M n ( C ) A λn Tr( A ) n + (1 − λ ) A .
This map is k -positive for any 1 ≤ k ≤ n if and only if 0 ≤ λ ≤ nk − . We give an alternative proofbased on Theorem 2.4. Indeed, Ψ is unitarily equivariant (take V ( U ) = U ) and to find the conditions of k -positivity on Ψ, we only need to find the values of λ such that [Ψ( e ij )] ki,j =1 is positive. It can be easilyseen that λn and λn + (1 − λ ) k are two distinct eigenvalues of [Ψ( e ij )] ki,j =1 with different multiplicities.Therefore, the map Ψ is k -positive if and only if λ ≥ λ ≤ nk − .(iv) Collins et.al [1] gave the following family of parametric linear maps which are (1 , α and β be two real numbers and n ≥ . Then the family of maps,Φ α,β,n : M n ( C ) → M n ( C ) ⊗ M n ( C ) A A t ⊗ n + n ⊗ A + Tr( A )( α n + βB n ) . (3)There are values of parameters α and β for which the family of maps Φ α,β,n is positive and notcompletely positive, more detail can be found in [1]. In this section, we study some basic properties of equivariant linear maps, which will help us to give acharacterization of these maps.
Lemma 2.5.
Let a, b ∈ N . Then,(1) The set of all ( a, b ) -unitarily equivariant maps is a vector space.(2) The set of all unitarily equivariant maps is not a vector space. Neither is the set of equivariant maps.Proof. It is an easy calculation to show that the set of all ( a, b )-unitarily equivariant maps is closed underaddition and scalar multiplication. Hence, this set is a vector space.We give an example of two unitarily equivariant maps such that their sum is not even equivariant, which isenough to conclude that both sets are not vector spaces.Let Φ , Φ : M ( C ) → M ( C ) be given by Φ = i and Φ = θ . It is straightforward that both mapsare unitarily equivariant. We prove that the map (Φ + Φ ) is not equivariant, that is, there exists a unitary U ∈ M ( C ) such that there is no V ∈ M ( C ) with(Φ + Φ )( U XU ∗ ) = V (Φ + Φ )( X ) V ∗ ∀ X ∈ M ( C ) . Let U = " √ √ − ι √ ι √ . Take X = e , where e is a matrix unit in M ( C ) and ι = −
1. We prove that there is no V ∈ M ( C ) suchthat, (Φ + Φ )( U e U ∗ ) = V (Φ + Φ )( e ) V ∗ . (4)(Φ + Φ )( U e U ∗ ) = (cid:20) (cid:21) . Therefore, rank ((Φ + Φ )( U e U ∗ )) = 2.Since, rank (Φ + Φ )( e ) = 1. Therefore there exists no V ∈ M ( C ) such that,(Φ + Φ )( U e U ∗ ) = V (Φ + Φ )( e ) V ∗ . (5)4 xample 2.6. To conclude this section, we give an example of a completely positive linear map on M ( C )which is not equivariant. Consider the linear mapΦ : M ( C ) → M ( C ) A A − A t + Tr( A ) . The map Φ is completely positive, but it is not equivariant.Let U = " √ √ − ι √ ι √ and X = − ιe + ιe Where ι = −
1. Then Φ(
U XU ∗ ) = (cid:20) − (cid:21) and Φ( X ) = (cid:20) (cid:21) . Therefore, there exists no V ∈ M ( C ) such that Φ( U XU ∗ ) = V Φ( X ) V ∗ for unitary U and X = − ιe + ιe . Hence, Φ is not equivariant.
In this section, we tackle the problem of characterizing equivariant maps defined on finite-dimensional matrixalgebras. We first characterize general equivariant linear maps in terms of their Choi matrices in Theorem 3.1.Then, we prove that every unitarily equivariant map where U V ( U ) is a unitary representation can berealized as an ⊕ i ∈ I ( a i , b i )-unitarily equivariant map for a finite set I and for some values of a i and b i . Finally,we compute explicitly all ( a, b )-unitarily equivariant maps in Theorem 3.4. We recall that the Choi matrix C Φ of a linear map Φ ∈ B ( M n ( C ) , M N ( C )) is defined by C Φ := ( i n ⊗ Φ)( B n ) (6)where B n is the unnormalized Bell density matrix. Equation (6) defines a one-to-one linear correspondencebetween matrices C ∈ M nN ( C ) and maps Φ ∈ B ( M n ( C ) , M N ( C )). The following theorem is applicable to allequivariant maps. Theorem 3.1.
Let
Φ : M n ( C ) → M N ( C ) be a linear map. Then the following two assertions are equivalent:(i) Φ is an equivariant map.(ii) For all unitary matrices U in M n ( C ) , there exists a matrix V = V ( U ) ∈ M N ( C ) such that [ C Φ , U ⊗ V ] = 0 . Proof.
To prove ( i ) = ⇒ ( ii ), we use the fact that ( U ⊗ U ) commutes with B n for every unitary matrix U ∈ M n ( C ). We have the following C Φ = ( i n ⊗ Φ)( B n )= ( i n ⊗ Φ)( U ⊗ U )( X i,j e ij ⊗ e ij )( U ⊗ U ) ∗ = ( X i,j U e ij U ∗ ) ⊗ Φ( U e ij U ∗ ) (7)Since Φ is equivariant, for every unitary U ∈ M n ( C ) there exists V ∈ M N ( C ) such that Φ satisfies Equation(1). From Equation (7), C Φ becomes, C Φ = X i,j U e ij U ∗ ⊗ V Φ( e ij ) V ∗ = ( U ⊗ V ) C Φ ( U ⊗ V ) ∗ . C Φ commutes with ( U ⊗ V ).We now prove the converse implication ( ii ) = ⇒ ( i ). Assume that assertion ( ii ) holds. Then the previouscomputation shows that ( U ⊗ V ) C Φ ( U ⊗ V ) ∗ is the Choi matrix of the map Y V Φ( U ∗ Y U ) V ∗ . By unicityof the Choi matrix, we get that for all Y ∈ M n ( C ),Φ( Y ) = V Φ( U ∗ Y U ) V ∗ . Taking X = U ∗ Y U yields the result. Hence, the proof is completed.
The definition of equivariant maps given by Equation (1) is quite general in the sense that we do not put anyconstraint on the operator V ( U ). It turns out that in the case of unitary invariance, the following theoremimplies that U → V ( U ) might be assumed to be a group morphism: Theorem 3.2.
Let
Φ : M n ( C ) → M N ( C ) be a unitarily equivariant linear map. Then there exists a unitaryrepresentation U V ( U ) of the unitary group U n on C N such that for all X ∈ M n ( C ) and all unitary U ∈ U n , Φ( U XU ∗ ) = V ( U ) Φ( X ) V ( U ) ∗ . Proof.
We denote by A (Φ) the C ∗ -algebra generated by the image of Φ: it is the subalgebra of M N ( C )spanned by the identity operator N and products of Φ( X ) , Φ( X ) ∗ for X ∈ M n ( C ). Given U ∈ U n , we define on A (Φ) an endomorphism as follows.We fix a basis of A (Φ) as a vector space, (Φ( X j ) · · · Φ( X j l )) where the j i s run in some finite set J .The endomorphism L U is defined by extending by linearity the formula L U (Φ( X j ) · · · Φ( X j l )) = Φ( U X j U ∗ ) · · · Φ( U X j l U ∗ ) . It can be noted that thanks to the unitarity assumption, this equation becomes L U (Φ( X j ) · · · Φ( X j l )) = V Φ( X j ) · · · Φ( X j l ) V ∗ , therefore for any Y ∈ A (Φ), the map is actually L U : Y ∈ A (Φ) V ( U ) Y V ( U ) ∗ , therefore this is actually a ∗ -automorphism of A (Φ).Let us record two other important properties of L : U n → Aut ( A (Φ)) , U L U . First of all, L iscontinuous, as any linear map between finite dimensional spaces is continuous and so is the multiplicationof operators. Secondly, L is actually a group morphism. Indeed, for any U , U ∈ U n and any X , ..., X k ∈ M n ( C ), L U U (Φ( X ) · · · Φ( X k )) = V ( U ) Φ( U X U ∗ ) · · · Φ( U X k U ∗ ) V ( U ) ∗ = L U ( Φ( U X U ∗ ) · · · Φ( U X k U ∗ ) )= L U ◦ L U (Φ( X ) · · · Φ( X k )) . So, L : U n → Aut ( A (Φ)) , U L U is a continuous group morphism.It follows from the classification of automorphisms of a finite-dimensional C ∗ -algebra [24, Proposition2.2.6] that there exists W = W ( U ) in M N ( C ) that normalizes A (Φ) such that L U ( X ) = W XW ∗ and this W is unique up to a central element in the algebra generated by A (Φ) and the normalizer of the commutant of A (Φ) (which we will call NC A (Φ) for the purpose of this proof).So, the map L : U n → Aut ( A (Φ)) , U L U gives rise to a continous application U n →U N / Z NC A (Φ) , U W which implements L U (the continuity follows from the fact that a continuousbijection between two compact sets is bi-continuous).In order to complete the proof, we have to prove that this map can be lifted continuously to a map U n → U N , U W that generates the same L U . This follows from the fact that, by construction, W can betaken as an element of NC A (Φ), so, given a minimal projection P in Z NC A (Φ), it is enough to show thatthere exists a continuous choice for P W .There exists θ ( U , U ) ∈ R such that P W ( U ) W ( U ) = e ιθ ( U ,U ) P W ( U U ) (8)6ote that it follows from the above that the phase e ιθ ( U ,U ) is a continuous function of U , U . Letus prove that P W can undergo a phase modification that will turn it into a special unitary representation.First of all, without loss of generality, we may assume that θ ( I n , I n ) = 0, at the possible expense of replacing P W by e ιθ ( I n ,I n ) P W . Then, by continuity looking in a vicinity of the identity, taking the determinant, weget that det
P W ( U ) det P W ( U ) = e N ιθ ( U ,U ) det P W ( U U ) , in other words e ιθ ( U ,U ) = (det P W ( U ) det P W ( U ) / det P W ( U U )) /N , where the N -th root is obtained with the prinicipal branch of the logarithm, which is well defined in aneighbourhood of 0.So, we may rewrite Equation (8) as P W ( U ) P W ( U ) = (det P W ( U ) det P W ( U ) / det P W ( U U )) /N P W ( U U ) , or equivalently, P W ( U ) det P W ( U ) − /N P W ( U ) det P W ( U ) − /N = P W ( U U )(det P W ( U U )) − /N Therefore, setting P ˜ W ( U ) = P W ( U ) det W ( U ) − /N , and doing the same for each minimal projection P in Z NC A (Φ) we see that, in a neighbourhood of 0,˜ W ( U ) ˜ W ( U ) = ˜ W ( U U ) . It is known that a finite dimensional representation of a Lie group is in one to one correspondance with thatof its Lie algebra, and therefore that it suffices to define it on a neighbourhood of identity, in the simplyconnected case. So, this proves that ˜ W extends on S U n to a representation of S U n (note that the above is amodification of Bargmann’s results [17, 18]). We can therefore decompose π into irreducible representations: C N = M λ E λ ⊗ C N λ , where E λ are irreducible representations of S U n appearing in the unitary representation π with multiplicity N λ . We denote by π λ the unitary representation of S U n on E λ induced by π .By [19, Proposition 22.2], corresponding to every irreducible representation π λ of S U n , there exists anirreducible representation ˜ π λ of U n such that ˜ π λ | S U n = π λ . Hence, ˜ π = L λ N λ ˜ π λ is a unitary representationof U n . This complete the proof.Let us note that the last part of the proof can be slightly simplified in case A (Φ) = M N ( C ). Indeed, inthis case it is enough to replace U N / Z NC A (Φ) , U W by PU N (the projective unitary group) and we canrefer to Bargmann’s theorem. It is also interesting to note that no continuity is needed in the assumption, itis granted automatically by the theorem (together with rationality and analyticity) when one replaces V by W . The next corollary builds a bridge between unitarily equivariant maps and ( a, b )-unitarily equivariantmaps. It justifies the computation of the latter in the next section. Corollary 3.3.
Let Φ ∈ B ( M n ( C ) , M N ( C )) be a unitarily equivariant map. Then there exist a finite sequenceof pair ( a i , b i ) of natural integers, a partial isometric map W : C N → ⊕ i ( C n ) ⊗ a i ⊗ ( C n ) ⊗ b i andan ⊕ i ( a i , b i ) -unitarily equivariant map Ψ on M n ( C ) such that for all X ∈ M n ( C ) , W Φ( X ) W ∗ = Ψ( X ) . (9) Proof.
This is a direct consequence of the representation theory of the unitary group. Indeed, for anyrepresentation, there exist a finite sequence of pairs ( a i , b i ) of natural integers such that this representationappears in the unitary representation (see [20] for instance) U
7→ ⊕ i U ⊗ a i ⊗ U ⊗ b i . It means that there exists a partial isometry W from C N to ⊕ i ( C n ) ⊗ a i ⊗ ( C n ) ⊗ b i such that for all U ∈ U n , W V ( U ) W ∗ = P ⊕ i ( U ⊗ a i ⊗ U ⊗ b i ) P , where P is the orthogonal projection on some supspace of ⊕ i (( C n ) ⊗ a i ⊗ ( C n ) ⊗ b i ) invariant by ⊕ i ( U ⊗ a i ⊗ U ⊗ b i )for all U ∈ U n . We can then define the map Ψ : X W Φ( X ) W ∗ and it can be readily checked that it is an ⊕ i ( a i , b i )-unitarily equivariant map. 7 .3 Characterization of ( a, b ) -unitarily equivariant maps We now deal with the question of characterizing ( a, b )-unitarily equivariant maps. More explicitly, let a, b ∈ N and n ≥
2. Then, what are all the linear mapsΦ : M n ( C ) → M n ( C ) ⊗ a ⊗ M n ( C ) ⊗ b such that for every unitary U ∈ M n ( C ), Φ satisfy Equation (2). By Lemma 2.5(1) the set of all ( a, b )-unitarilyequivariant maps is a vector space. Characterizing this vector space is then equivalent to exhibiting oneof its basis. In order to do that, we need some basic results from group representation theory and moreprecisely the Schur-Weyl duality Theorem [21, Theorem 8.2.10]. More detail can be found in [21, Chapters7,8]. Let us define two unitary representations σ k and ρ k on ( C n ) ⊗ k , with a + b = k , of the symmetric group S k and the unitary group U n respectively. For all v , ..., v k ∈ C n , π ∈ S k and U ∈ U n , they are defined as σ k ( π ) ( v ⊗ · · · ⊗ v k ) = ( v π − (1) ⊗ · · · ⊗ v π − ( k ) ) ,ρ k ( U ) ( v ⊗ · · · ⊗ v k ) = ( U v ⊗ · · · ⊗ U v k ) . We denote by σ k ( C [ S k ]) (resp. ρ k ( C [ U n ]))the ∗ -algebra generated by the representation σ k (resp. ρ k ). That is, σ k ( C [ S k ]) = { σ k ( π ) ; π ∈ S k } ′′ = ( X π ∈ S k f ( π ) σ k ( π ) ; f : S k → C ) ,ρ k ( C [ U n ]) = { ρ k ( U ) ; U ∈ U n } ′′ , where the prime symbol denotes the commutant of a set (we will not need the explicit formula for the elementsof ρ k ( U n )). Then Schur-Weyl duality Theorem asserts that σ k ( C [ S k ]) and ρ k ( C [ U n ])) are the commutant ofeach other.In other words, for any operator C ∈ M n ( C ) ⊗ k ,[ U ⊗ k , C ] = 0 ∀ U ∈ U n iff C ∈ σ k ( S k ) . (10)We refer to [21, Theorem 8.2.8] for instance for a presentation of this Theorem. With this result in hand, wecan now prove one of the main results of this paper. Theorem 3.4.
Let a, b ∈ N and Φ : M n ( C ) → M n ( C ) ⊗ a ⊗ M n ( C ) ⊗ b . Then the following assertions areequivalent.(i) Φ is ( a, b ) -unitarily equivariant.(ii) [ C Φ , U ⊗ a +1 ⊗ U ⊗ b ] = 0 for all U ∈ U n , where C Φ is the Choi matrix of Φ .(iii) There exists f : S k +1 → C such that C Φ = X π ∈ S k +1 f ( π ) (cid:0) θ ⊗ a +1 n ⊗ i ⊗ bn (cid:1) [ σ k +1 ( π )] , (11) where θ n is the transpose map on M n ( C ) .Proof. The equivalence of ( i ) and ( ii ) is clear from Theorem 3.1.( ii ) ⇒ ( iii ).Assume that C Φ commutes with ( U ⊗ a +1 ⊗ U ⊗ b ) for all U ∈ U n , that is,( U ⊗ a +1 ⊗ U ⊗ b ) C Φ ( U ⊗ a +1 ⊗ U ⊗ b ) ∗ = C Φ (12)Remark that the transpose map θ n ∈ B ( M n ( C )) is (1 , U ∈ U n and all C ∈ M n ( C ) ⊗ a + b +1 ,( θ ⊗ a +1 n ⊗ i ⊗ bn )( U ⊗ a +1 ⊗ U ⊗ b C ( U ⊗ a +1 ⊗ U ⊗ b ) ∗ ) = U ⊗ k +1 [( θ ⊗ a +1 n ⊗ i ⊗ bn ) C ] U ∗⊗ k +1 . (13)8here k = a + b . Applying Equations (12) and (13), we get:( θ ⊗ a +1 n ⊗ i ⊗ bn )[ C Φ ] = U ⊗ k +1 ( θ ⊗ a +1 n ⊗ i ⊗ bn )[ C Φ ] U ∗⊗ k +1 . Consequently, ( θ ⊗ a +1 n ⊗ i ⊗ bn )[ C Φ ] ∈ ρ k ( U n ) ′ . By the Schur-Weyl duality given in Equation (10), we obtainthat ( θ ⊗ a +1 n ⊗ i ⊗ bn )[ C Φ ] ∈ σ k +1 ( C [ S k +1 ]), which implies (iii).( iii ) ⇒ ( ii ) is straightforward using the converse part of Schur-Weyl duality and the previous computation.Clearly by point (iii) in Theorem 3.4, the set of ( a, b )-unitarily equivariant maps is isomorphic as avector space to the space σ k +1 ( C [ S k +1 ]). Therefore we get the straightforward corollary: Corollary 3.5.
A basis of ( a, b ) -unitarily equivariant maps is given by the maps { Φ π : π ∈ S k +1 } , where k = a + b , with the corresponding Choi matrices: C Φ π = ( θ ⊗ a +1 n ⊗ i ⊗ bn )[ σ k +1 ( π )] , (14) where the matrix σ k +1 ( π ) acts on vectors ( v ⊗ · · · ⊗ v k +1 ) as σ k +1 ( π ) ( v ⊗ · · · ⊗ v k +1 ) = v π − (1) ⊗ · · · ⊗ v π − ( k +1) . (15) In particular, when n ≥ k + 1 , then the dimension of this vector space is ( k + 1)! . We discuss the graphical representation of this basis in the next section. ( a, b ) -unitarilyequivariant maps In this section we study the graphical representation of ( a, b )-unitarily equivariant maps for any a, b ∈ N .This gives a visual and very convenient method to compute them. We illustrate this with the (1 , raphical representationVectors/Operators | v i ∈ C n | v ih v | ∈ ( C n ) ∗ h v | T ∈ B ( H ) TT ∈ B ( H ⊗ H ) T H H H H AB ∈ M n ( C ) A BA t ∈ M n ( C ) A Figure 1: Graphical representations of vectors and operatorsLet W a,b denote the vector space of all ( a, b )-unitarily equivariant maps. Recall from Section 3.3 thatthe maps C Φ π = ( θ ⊗ a +1 n ⊗ i ⊗ bn )( σ k +1 ( π )) form a basis of W a,b with π running through the permutation group S k +1 , where σ k +1 is the unitary representation of the permutation group defined in Equation (15). Basedon the graphical representations given in Table 1, we give the graphical representations of the C Φ π . We firstconsider the case of a = 1 and b = 1 and make some observations. We discuss the general case in Theorem4.1. We compute the graphical representation of the Choi matrices C Φ π with π ∈ S , by first computingthe one of σ ( π ). We start with the representation of σ (123) as an example. This map is defined as follows: σ (123) : ( C n ) ⊗ → ( C n ) ⊗ v ⊗ v ⊗ v v ⊗ v ⊗ v . Its graphical representation is given on the left in Figure 2. By Equation (14), C Φ (123) = ( θ ⊗ n ⊗ i n ) σ (123).Using the graphical representation of the transpose map, we obtain that C Φ (123) is represented by the rightfigure in Figure 2. θ ⊗ n ⊗ i n ( · ) −→ Figure 2: Graphical representation of σ (123) (on the left) and C Φ (123) (on the right)We take the following convention in order to represent the operators σ k +1 ( π ): input on the left arerepresented by black (full) dots while output on the right are represented by white (empty) dots. Thegraphical representation of the operator σ k +1 ( π ) then corresponds to tracing a wire from the i th (black) doton the left to the π − ( i ) (white) dot on the right. Using graphical computation, it can be directly check that σ k +1 ( π ) and U ⊗ k +1 commute for all unitary operator U ∈ U n .Then, applying the transpose map to the first ( a + 1) tensors corresponds graphically to interchange the blackand the white dot at the first ( a + 1) rows (starting from the highest). Again, it can be directly checked10 UU U t U t U ∗ =Figure 3: Graph of C Φ (123) . Each black dot is connected to a white dot in a one-to-one way, according to thepermutation (123).from a graphical computation that C Φ( π ) and ( U ⊗ a +1 ⊗ U ⊗ b ) commute. This is illustrated in the case of thepermutation (123) in Figure 3.We can obtain similarly the graphical representations of the Choi matrices C Φ π corresponding to all π ∈ S . There are 3! = 6 different possibilities to trace a wire between black and white dots in a one-to-oneway. Each of them corresponds to a different permutation π ∈ S . This is illustrated in Figure 4. πC Φ π (23) (12) (13) (123) (132)Figure 4: Graphical representations of the C Φ π for each π ∈ S Figure 4 gives the basis elements of the vector space W , . Theorem 4.1 generalizes this fact to any( a, b )-unitarily equivariant map. We omit the proof, as it follows exactly the same idea as the example above. Theorem 4.1.
The graphical representation of the Choi matrices of the basis elements C Φ( π ) of the vectorspace W a,b is given by the following rule. Being a matrix on ( C n ) ⊗ k +1 , its graphical representative has ( k + 1) input (on the left) and ( k + 1) output (on the right). The first ( a + 1) inputs are symbolized by black (full)dots, the remaining b by white (empty) dots. In the same way, the first ( a + 1) outputs are symbolized bywhite (empty) dots, the remaining b by black (full) dots. Then each element of this basis corresponds to apossible wiring between black and white dot, in a one-to-one way. Theorem 4.1 gives a graphical representation of the Choi matrices of the basis elements of W a,b ,which completes the characterization of the vector space W a,b of all ( a, b )-unitarily equivariant maps. As anillustration, we study the graphical representation of the Choi matrix of the family of linear maps Φ α,β, studied in [1] and given by Equation (3). It is given as follows:+ + α + β Figure 5: Graphical representation of C Φ α,β,n Remark that it corresponds to taking linear combinations of the graphical representations of thepermutations (12), (13), (1) and (23) respectively and from left to right. A more general 3-parameters family11an be defined by adding the permutations (123) and (132) and it gives:Φ α,β,γ,n : M n ( C ) → M n ( C ) ⊗ M n ( C ) A A t ⊗ n + n ⊗ A + Tr( A )( α n + βB n )+ γ ( B n ( n ⊗ A ) + ( n ⊗ A ) B n ) . (16)We leave the study of this more general map to future work. One application of positive maps that are not completely positive is entanglement detection in quantuminformation theory. We recall that a density matrix in M n ( C ) for some integer n ≥ C n with trace equal to one. We denote by S ( n ) the set of density matrices on C n and by S ( m × n )the set of density matrices on ( C m ⊗ C n ). Then a density matrix on ( C m ⊗ C n ) is called separable if it isin the convex hull of tensor-product density matrices. We denote by SEP( m, n ) the set of separable densitymatrices on ( C m ⊗ C n ), that is,SEP( m, n ) = (X i λ i ρ iA ⊗ ρ iB ; ρ iA ∈ S ( m ) , ρ iB ∈ S ( n ) , λ i ≥ , X i λ i = 1 ) . We are interested in the complement of this set in S ( m × n ). If a density matrix ρ is called S ( m × n ) \ SEP( n, m ), it is called entangled . The first operational characterization of entanglement/separabilitywas proved in [8]: a density matrix ρ ∈ S ( m × n ) is entangled if and only if there exists a positive map φ : M n ( C ) → M m ( C ) such that ( i m ⊗ φ )( ρ ) is not positive semi-definite. Remark that if φ is completelypositive, then ( i m ⊗ φ )( ρ ) is necessarily positive semi-definite. This means that positive – but not completelypositive - maps lead to entanglement detectors . The most well-known example of positive maps leading toentanglement detection is the partial transpose, and the density matrices that remain positive under itsaction are the so-called PPT density matrices . More generally, given a positive map φ : M n ( C ) → M N ( C )for any natural number N >
0, we can define S m ( φ ) = { ρ ∈ S ( m × n ) ; ( i m ⊗ φ )( ρ ) ≥ } . (17)Thus the set of PPT density matrices on C m ⊗ C n is S m ( θ n ), where θ n is the transposition on M n ( C ).Apart from ( m, n ) = (2 ,
2) or ( m, n ) = (2 , S m ( θ n ) is strictly different from SEP( m, n ).In fact, deciding whether a density matrix is separable or not is known as a hard computational problem[5, 6]. It means in particular that an infinite number of positive maps are needed to detect all entangleddensity matrices. It is thus a central question in quantum information theory to find families of maps thatare sufficient for this task.More generally, it is also interesting to detect the amount of entanglement contains in a bipartitedensity matrix. The Schmidt number represents in this respect a good integer quantifier. We recall itsdefinition. Recall that any norm-one vector ψ ∈ ( C m ⊗ C n ) admits a unique Schmidt decomposition as ψ = t X i =1 p λ i ψ mi ⊗ ψ ni , where ( ψ mi ) i =1 ,...,t (resp. ( ψ ni ) i =1 ,...,t ) is an orthogonal family on C m (resp. on C n ) with consequently1 ≤ t ≤ min { m, n } , and where P ti =1 λ i = 1. Then its Schmidt rank is defined as SR( ψ ) := t . The Schmidtnumber of a density matrix ρ ∈ S ( m × n ) is defined subsequently asSN( ρ ) := min max j SR( ψ j ) ; ρ = X j ρ j | ψ j ih ψ j | , where the minimum is over all possible decompositions of ρ as a linear combination of pure states.We denote by SN t ( m, n ) the set of density matrices with Schmidt number smaller than t ≥
1. Weshall also refer to elements of this set as t -separable density matrices and elements of its complement S ( m × n ) \ SN t ( m, n ) as t -entangled density matrices. Remark in particular that the separable densitymatrices are exactly the ones with Schmidt number equal to one: SEP( m, n ) = SN ( m, n ). In the same way12hat separability (or 1-separability) can be checked using positive maps, t -separability can be checked using t -positive maps. That is, a density matrix ρ ∈ S ( m × n ) is t -separable if and only if ( i m ⊗ φ )( ρ ) ≥ t -positive maps φ : M n ( C ) → M n ( C ).We are now in position to state the main result of this article. Theorem 5.1.
Let φ : M n ( C ) → M N ( C ) be a t -positive - but not completely positive - map, with ≤ t ≤ min { m, n } . Then there exists a family of unitarily equivariant t -positive maps (Φ a ) a ≥ with Φ a : M n ( C ) → M N a ( C ) , lim a → + ∞ N a = + ∞ , such that ( S m (Φ a )) a ≥ form a decreasing family of sets and: ∩ a ≥ S m (Φ a ) ⊂ S m ( φ ) . (18) In other words, for all t -entangled density matrices ρ / ∈ S m ( φ ) , there exists an integer a ≥ such that ρ / ∈ S m (Φ a ) for all a ≥ a . We divide the proof of Theorem 5.1 in two parts: first in Section 5.1, we prove the existence of a Φ : M n ( C ) → B ( H ), for some universal infinite dimensional Hilbert space H and universal unitary representation U V ( U ) that fulfills the equivariance property (1), such that S m (Φ) ⊂ S m ( φ ). Then we prove the theoremby an application of the Peter-Weyl Theorem together with Corollary 3.3. Theorem 5.2.
Let φ : M n ( C ) → M N ( C ) be a t -positive map. Then there exists a unitarily equivariant t -positive map Φ : M n ( C ) → B ( H ) for some Hilbert space H such that: S m (Φ) ⊂ S m ( φ ) , (19) where these quantities were defined in Equation (17) . Before proving the theorem, we recall some facts about the left regular representation of the unitarygroup U n . Recall that L ( U n ) is the Hilbert space of square integrable functions on U n with respect to theHaar measure: L ( U n ) = { f : U n → C ; Z U n | f ( U ) | µ Haar ( dU ) < + ∞} . The left regular representation λ of U n on L ( U n ) is defined for all W ∈ U n as: λ W f ∈ L ( U n ) ( U f ( W ∗ U )) . (20)Now define H = L ( U n , C N ) ≈ L ( U n ) ⊗ C N , that is, H = (cid:26) ( ψ U ) U ∈U n ; ψ U ∈ C N ∀ U ∈ U n , Z U n k ψ U k µ Haar ( dU ) < + ∞ (cid:27) (21)where the integration is with respect to the Haar measure on U n . The Hilbert space H will play the role ofthe image of Φ in Theorem 5.2. The equivariance property will be given in terms of the following unitaryrepresentation of U n on H : V : W (cid:0) V ( W ) : ψ ∈ H 7→ ( ψ W ∗ U ) U ∈U n (cid:1) . (22)We can readily check that V is the left regular representation on L ( U n ) tensored with the trivialrepresentation on C N , that is, V = λ ⊗ i N .Finally, let φ ∈ B ( M n ( C ) , M N ( C )) be a linear map. We can define a map Φ ∈ B ( M n ( C ) , B ( H )) as X ∈ M n ( C ) Φ( X ) = ( φ ( U ∗ XU )) U ∈U n . (23)Thus for all X ∈ M n ( C ) and all ( ψ U ) U ∈U n ∈ H , Φ( X ) ψ = ( φ ( U ∗ XU ) ψ U ) U ∈U n . Lemma 5.3.
Let φ ∈ B ( M n ( C ) , M N ( C )) be a linear map and define the linear map Φ : M n ( C ) → B ( H ) asin Equation (23) . Then Φ is unitarily equivariant with respect to the unitary representation U V ( U ) . Iffurthermore φ is t -positive, then Φ is also t -positive. roof. We first prove that Φ is U V ( U ) equivariant. Indeed, for all W ∈ U n , X ∈ M n ( C ) and all ψ ∈ H , V ( W ) ∗ Φ( W XW ∗ ) V ( W ) ψ = V ( W ) ∗ Φ( W XW ∗ ) ( ψ W ∗ U ) U ∈U n = V ( W ∗ ) (cid:0) φ (( W ∗ U ) ∗ X ( W ∗ U )) ψ W ∗ U (cid:1) U ∈U n = Φ( X ) ψ . Secondly, we prove that Φ is t -positive. As it is equivariant, by Theorem 2.4, we only need to check that thematrix (Φ( e ij )) ≤ i,j ≤ t is positive semi-definite, where ( e ij ) ≤ i,j ≤ n are the matrix units in M n ( C ). This canbe directly checked by noticing that H ⊗ C t ≃ L ( U n , C tN ) so that(Φ( e ij )) ≤ i,j ≤ t ≃ (( φ ( U ∗ e ij U )) ≤ i,j ≤ t ) U ∈U n . If φ is t -positive, then ( φ ( U ∗ e ij U )) ≤ i,j ≤ t is positive semi-definite for all U ∈ U n , and so is (Φ( e ij )) ≤ i,j ≤ t . Proof of Theorem 5.2.
Let ρ ∈ S ( m × n ) be a t -entangled density matrix such that ( i m ⊗ φ )( ρ ) is notpositive semi-definite, i.e. ρ / ∈ S m ( φ ). We can now show that Φ “detects” ρ , that is, ( i m ⊗ Φ)( ρ ) is notpositive semi-definite. Indeed, this is true as by continuity there exist a vector ϕ ∈ ( C m ⊗ C N ) and a smallneighborhood U of I n in U n such that for all U ∈ U , h ϕ , ( i m ⊗ φ )( I m ⊗ U ρ I m ⊗ U ∗ )) ϕ i <
0. Then definingthe vector ψ ∈ ( C m ⊗ H ) as: ψ U = ϕ ∀ U ∈ U , ψ U = 0 elsewhere , we get that h ψ , ( i m ⊗ Φ)( ρ ) ψ i < i m ⊗ Φ)( ρ ) is not positive semi-definite. Going from Theorem 5.2 to Theorem 5.1 is a simple application of the Peter-Weyl Theorem applied to thecompact group U n . It states that L ( U n ) isisomorphic as a U n –space tothe closure of the direct sum of all finite dimensional irreducible unitary representations E π of U n ,each with multiplicity equal to its dimension (see [19] for instance): L ( U n ) = dM π ∈ Λ E π ⊗ F π , (24)where Λ is the countable set that indexes all irreps of U n and where F π is of dimension dim E π . Furthermore,the left-regular representation λ of U n on L ( U n ) given in Equation (20) can be decomposed according toEquation (24): λ = b ⊕ π ∈ Λ λ ( π ) ⊗ i F π , (25)where λ ( π ) is the irreducible unitary representation of U n on E π .We are now ready to prove the main theorem Proof of Theorem 5.1.
Let φ be a t -positive map with 1 ≤ t ≤ min { m, n } . Define H and the t -positive mapΦ : M n ( C ) → B ( H ) as in Equations (21) and (23) respectively. Remark that by the Peter-Weyl Theorem asstated in Equation (24), H = L ( U n , C N ) can be decomposed as H = dM π ∈ Λ E π ⊗ ( F π ⊗ C N ) , and the unitary representation λ ⊗ i N of U n on H can be decomposed as λ ⊗ i N = b ⊕ π ∈ Λ λ ( π ) ⊗ i F π ⊗ C N . As Λ is a countable set, we can choose an increasing family of finite sets (Λ d ) d ≥ such that ∪ d ≥ Λ d = Λ. Let P d be the orthogonal projection on cL π ∈ Λ d E π ⊗ F π . Remark that H d := P d H is a finite dimensional subspaceof H as Λ d is finite. We can define on H d the restriction of Φ as Φ d := P d Φ( · ) P d . This directly defines14 t -positive map as Φ is itself t -positive by Lemma 5.3. We check that Φ d is equivariant. We define theunitary representation Π d on H d as the restriction of ( λ ⊗ i N ) to H d : Π d ( U ) := P d λ ( U ) ⊗ N P d . Using thatby Lemma 5.3 Φ is ( λ ⊗ i N )-equivariant, it is a straightforward computation to show that Φ d is Π d -equivariant.Let now ρ ∈ S ( m × n ) be a t -entangled density matrix such that ρ / ∈ S m ( φ ). By Theorem 5.2, ρ / ∈ S m (Φ), so that there exists ψ ∈ ( C m ⊗ H ) such that h ψ , ( i m ⊗ Φ)( ρ ) ψ i <
0. Define ψ d = P d ψ ∈ H d . As ψ d → ψ in H as d goes to infinity, we also have h ψ d , ( i m ⊗ Φ d )( ρ ) ψ d i → h ψ , ( i m ⊗ Φ)( ρ ) ψ i as d → + ∞ , sothere exists d ≥ d ≥ d , h ψ d , ( i m ⊗ Φ d )( ρ ) ψ d i <
0. This concludes the proof.Let us finish this section by the following corollary that might be of theoretical interest.
Corollary 5.4.
There exists a family of unitarily equivariant t -positive maps (Φ a ) a ≥ with Φ a : M n ( C ) → M N a ( C ) such that, for each m , ( S m (Φ a )) a ≥ decreases to SN t ( m, n ) as a → ∞ .Proof. For the same statement with m fixed, it is a diagonal argument applied to a dense countable subsetof the t -entangled states (given that a direct sum of l entanglement witnesses for l entangled states witnessesall l states simultaneously). For generic m this is an additional diagonal argument. Part of this work was made during IB’s visit to Kyoto University, supported by Campus France Sakuraproject. IB and BC were supported by the ANR project StoQ ANR-14-CE25-0003-01. BC was supportedby Kakenhi 15KK0162, 17H04823, 17K18734. GS would like to acknowledge FRIENDSHIP project of JapanInternational Corporation Agency (JICA) for research fellowship (D-15-90284) and kakenhi 15KK0162. GSwould like to thank Prof. B.V Raja Ram Bhat for supporting her from his JC Bose grant. All authorsare very grateful to Prof Hiroyuki Osaka for a careful reading and very helpful comments on a preliminaryversion of the paper, to Prof Yuki Arano for suggesting an improvement to the statement and the proof ofTheorem 3.2, and to Prof Masaki Izumi for sharing insight on Theorem 5.1.
References [1] B. Collins, H. Osaka, and G. Sapra, “On a family of linear maps from M n ( C ) to M n ( C ),” Linear Algebraand its Applications , vol. 555, pp. 398 – 411, 2018.[2] B. V. R. Bhat, “Linear maps respecting unitary conjugation,”
Banach J. Math. Anal , vol. 5, no. 2,pp. 1–5, 2011.[3] M. A. Nielsen and I. Chuang, “Quantum computation and quantum information,” 2002.[4] B. M. Terhal, “Detecting quantum entanglement,”
Theoretical Computer Science , vol. 287, no. 1,pp. 313–335, 2002.[5] S. Gharibian, “Strong np-hardness of the quantum separability problem,”
Quantum Info. Comput. ,vol. 10, pp. 343–360, Mar. 2010.[6] L. Gurvits, “Classical deterministic complexity of Edmonds’ Problem and quantum entanglement,” in
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing , pp. 10–19, ACM, 2003.[7] A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, “Complete family of separability criteria,”
PhysicalReview A , vol. 69, no. 2, p. 022308, 2004.[8] M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficientconditions,”
Physics Letters A , vol. 223, no. 1, pp. 1 – 8, 1996.[9] A. Peres, “Separability criterion for density matrices,”
Physical Review Letters , vol. 77, no. 8, p. 1413,1996.[10] S.-H. Kye and H. Osaka, “Classification of bi-qutrit positive partial transpose entangled edge states bytheir ranks,”
Journal of Mathematical Physics , vol. 53, no. 5, p. 052201, 2012.1511] M.-D. Choi, “Completely positive linear maps on complex matrices,”
Linear algebra and its applications ,vol. 10, no. 3, pp. 285–290, 1975.[12] M.-D. Choi, “Positive linear maps on C ∗ -algebras,” Canadian Math.J , vol. 24, no. 3, pp. 520–529, 1972.[13] T. Takasaki and J. Tomiyama, “On the geometry of positive maps in matrix algebras,”
MathematischeZeitschrift , vol. 184, pp. 101–108, Mar 1983.[14] S. J. Cho, S.-H. Kye, and S. G. Lee, “Generalized choi maps in three-dimensional matrix algebra,”
Linear algebra and its applications , vol. 171, pp. 213–224, 1992.[15] A. M¨uller-Hermes, “Decomposability of linear maps under tensor products,” arXiv preprintarXiv:1805.11570 , 2018.[16] J. Tomiyama, “On the geometry of positive maps in matrix algebras. II,”
Linear Algebra and itsApplications , vol. 69, pp. 169–177, 1985.[17] V. Bargmann, “On Unitary ray representations of continuous groups,”
Annals Math. , vol. 59, pp. 1–46,1954.[18] M. . S. Aguilar
M. International Journal of Theoretical Physics , vol. 39: 997, 2000.[19] M.Taylor, “Lectures on lie groups,”[20] M. R. Sepanski,
Compact lie groups , vol. 235. Springer Science & Business Media, 2007.[21] T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli,
Representation Theory of the Symmetric Groups:The Okounkov-Vershik Approach, Character Formulas, and Partition Algebras . Cambridge Studies inAdvanced Mathematics, Cambridge University Press, 2010.[22] C. J. Wood, J. D. Biamonte, and D. G. Cory, “Tensor networks and graphical calculus for open quantumsystems,”
Quantum Info. Comput. , vol. 15, pp. 759–811, July 2015.[23] B. Collins and I. Nechita, “Random Quantum Channels I: Graphical Calculus and the Bell StatePhenomenon,”
Communications in Mathematical Physics , vol. 297, pp. 345–370, Jul 2010.[24] Goodman, F.M. and de la Harpe, P. and Jones, V.F.R., “Coxeter Graphs and Towers ofAlgebras,”