Universality of Weyl Unitaries
aa r X i v : . [ m a t h . OA ] J a n UNIVERSALITY OF WEYL UNITARIES
DOUGLAS FARENICK , OLUWATOBI RUTH OJO , AND SARAH PLOSKER A BSTRACT . Weyl’s unitary matrices, which were introduced in Weyl’s 1927 paper[11] on group theory and quantum mechanics, are p × p unitary matrices givenby the diagonal matrix whose entries are the p -th roots of unity and the cyclicshift matrix. Weyl’s unitaries, which we denote by u and v , satisfy u p = v p = p (the p × p identity matrix) and the commutation relation uv = ζ vu , where ζ is aprimitive p -th root of unity. We prove that Weyl’s unitary matrices are universalin the following sense: if u and v are any d × d unitary matrices such that u p = v p = d and uv = ζvu , then there exists a unital completely positive linear map φ : M p ( C ) → M d ( C ) such that φ (u) = u and φ (v) = v . We also show,moreover, that any two pairs of p -th order unitary matrices that satisfy the Weylcommutation relation are completely order equivalent.When p =
2, the Weyl matrices are two of the three Pauli matrices from quan-tum mechanics. It was recently shown in [7] that g -tuples of Pauli-Weyl-Brauerunitaries are universal for all g -tuples of anticommuting selfadjoint unitary ma-trices; however, we show here that the analogous result fails for positive integers p >
1. I
NTRODUCTION
With respect to a positive integer p > p -th root of unity ζ , apair of d × d unitary matrices u and v satisfy the Weyl relations if(1) u p = v p = d (the d × d identity matrix) and uv = ζvu .The relation uv = ζvu is referred to as a Weyl commutation relation . The mostimmediate example of unitary matrices satisfying the Weyl relations comes fromH. Weyl’s 1927 paper on quantum mechanics and group theory [11, p. 32], in which d = p and u and v are the p × p unitary matrices denoted herein by u and v ,respectively, and are defined by(2) u = ζ ζ . . . ζ p − and v = . Date : January 5, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Weyl unitary, Pauli matrix, operator system, complete order equivalence,matrix range, extreme points. Department of Mathematics and Statistics, University of Regina, Regina SK S4S 0A2, Canada. Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9,Canada.
In the case where p =
2, the Weyl unitaries u and v are two (namely, σ Z and σ X ) ofthe three Pauli matrices: σ X = (cid:20) (cid:21) , σ Y = (cid:20) − ii 0 (cid:21) , σ Z = (cid:20) − (cid:21) .For this reason, the Weyl unitaries are often viewed as a generalised form of thePauli matrices. (In particular, Weyl’s paper [11] introduces his p × p unitaries aftera fulsome discussion of the Pauli matrices.)Weyl’s unitary matrices u and v in (2) are not just examples of unitary matricessatisfying the relations (1); they are, in fact, universal matrices for all such pairs ofunitary matrices. That is, as we shall prove herein, if u and v are d × d unitary ma-trices such that u p = v p = d , and uv = ζvu , then there exists a unital completelypositive linear map Φ : M p ( C ) → M d ( C ) such that Φ (u) = u and Φ (v) = v , where M n ( C ) denotes the algebra of complex n × n matrices.Returning to the case p =
2, the Weyl matrices u and v are the Pauli matrices σ Z and σ X , respectively. The third Pauli matrix, σ Y , is obtained from the othertwo via the equation σ Y = iσ X σ Z . Any unitary u ∈ M d ( C ) for which u = d must be selfadjoint; and the condition uv = ζvu is equivalent, in the case p =
2, to uv = − vu . Therefore, the relations (1) for pairs of unitaries extend easilyto g -tuples of unitaries, thereby describing g -tuples of anticommuting selfadjointunitary matrices.However, for p >
2, extending the relations in (1) from two unitaries to a highernumber of unitaries, say g , has a number of additional considerations. The case of g = u and v unitaries that satisfythe relations in (1). The commutation relation uv = ζvu implies that vu = ζ − uv ;it is only in the case where p = ζ − = ζ , and so one is required, for p >
2, toaccount for the change in the scalar if one swaps the order of the unitary product.If ζ is a primitive p -th root of unity, then ζ k is also a primitive p -th root of unityfor all k ∈ N that are not divisible by p —or, stated more conveniently, for anynonzero k ∈ Z p . Thus, one might have commutation relations for unitaries thatinvolve ζ and some its powers.Such considerations lead to the following definition. Definition 1.1.
Suppose that ζ ∈ C is a primitive p -th root of unity. The Weyl com-mutation relations for a g -tuple u = ( u , . . . , u g ) of d × d unitary matrices that satisfy u pk = d , for each k , are given by the equations (3) u k u ℓ = ζ c kℓ u ℓ u k , for all k , ℓ ∈ {
1, . . . , g } , for some skew-symmetric matrix C = [ c kℓ ] gk , ℓ = ∈ M g ( Z p ) . If c kℓ = whenever k < ℓ ,then the commutation relations in (3) are said to be simple . Motivated by the fact that the Pauli matrix σ Y is a scalar multiple of the product σ X σ Z , one may mimic the construction by considering w = λuv , where λ ∈ C and u and v are d × d unitary matrices such that u p = v p = d and uv = ζvu . Thecondition w p = λ . Indeed, as vu = ( /ζ ) uv , ( uv ) p = u ( vu ) p − v = ( /ζ ) p − u ( vu ) p − v = ( /ζ ) p − ( /ζ ) p − u ( vu ) p − v = · · · = ( /ζ ) + + ··· +( p − ) u p v p = ( /ζ ) ( p − ) p NIVERSALITY OF WEYL UNITARIES 3
Thus, the condition w p = λ p = ζ ( p − ) p , and so λ is uniquely deter-mined: λ = ζ p − . In this case, we have the following commutation relations (inaddition to uv = ζvu ): uw = ζwu and vw = ζ − wv .Thus, if ( u , u , u ) were given by ( u , v , w ) , then the matrix C for the Weyl com-mutation relations would be C = − − − .On the other hand, the Weyl commutation relations for ( u , u , u ) = ( u , w , v ) aresimple, as in this case the matrix C is C = − − − . Definition 1.2. If p > , then the simple Weyl unitary matrices are the three p × p unitary matrices ω a , ω b , and ω c defined by ω a = u , ω b = ζ p − uv , ω c = v . The triple
W = ( ω a , ω b , ω c ) is called the simple Weyl triple . In this paper, we establish a result that is stronger than the assertion concern-ing the universality of the Weyl matrices u and v . Specifically, we show that anytwo unitary matrices with the relations (1) are universal – not just the Weyl pair.This result is best phrased in terms of complete order equivalence, which is essen-tially a special form of isomorphism in the operator system category (see [8] for anoverview of the theory of operator systems and completely positive linear maps).To explain complete order equivalence, suppose that x = ( x , . . . , x g ) is a g -tupleof complex d × d matrices x k . The operator system of x is the linear space O x of d × d matrices defined by O x = Span C { d , x k , x ∗ k | k =
1, . . . , g } .If x = ( x , . . . , x g ) and y = ( y , . . . , y g ) are g -tuples of matrices x j ∈ M d ( C ) and y ℓ ∈ M d ( C ) , then x and y are said to be completely order equivalent , denoted by x ≃ ord y , if there exists a linear isomorphism φ : O x → O y such thata) φ and φ − are unital completely positive (ucp) linear maps, andb) φ ( u k ) = y k , for each k =
1, . . . , g .If a = ( a , . . . , a g ) is a g -tuple of d × d matrices, and if γ : C n → C d is a lineartransformation, then γ ∗ aγ denotes the g -tuple of n × n matrices given by γ ∗ aγ = ( γ ∗ a γ , . . . , γ ∗ a g γ ) .If n = d and if b = γ ∗ aγ for a unitary γ , then this situation is denoted by a ≃ u b and we say that b is unitarily equivalent to a .While it is clear that x ≃ u y = ⇒ x ≃ ord y , D. FARENICK, O. OJO, AND S. PLOSKER the converse does not hold in general, for in the complete order equivalence prob-lem for g -tuples x and y of matrices, the matrix dimensions d and d need not beequal. Hence, complete order equivalence is weaker than unitary equivalence.An answer to the question of when two operator g -tuples x and y are com-pletely order equivalent has been given by Davidson, Dor-On, Shalit, and Solel[3]: x ≃ ord y if and only if W ( x ) = W ( y ) , where, for a g -tuple z = ( z , . . . , z g ) ofoperators z k , W ( z ) denotes the matrix range of z , which is the sequence W ( z ) = ( W n ( z )) n ∈ N of subsets W n ( z ) in M n ( C ) × · · · × M n ( C ) ( g copies) defined by(4) W n ( z ) = { ( ψ ( z ) , . . . , ψ ( z g )) | ψ : O z → M n ( C ) is a ucp map } .This result of Davidson, Dor-On, Shalit, and Solel is especially useful in caseswhere the matrix ranges can be described; however, such descriptions are not al-ways available.We shall find it convenient to use the following definition. Definition 1.3.
A unitary matrix u is of p -th order if u p is the identity matrix. Lastly, some notation: if S is a nonempty subset of M d ( C ) , then Alg (S) denotesthe complex associative subalgebra of M d ( C ) generated by S . In particular, it isa classical result that M p ( C ) = Alg ( S ) , if S is the set consisting of the two Weylunitary matrices u and v [9].2. U NIVERSALITY OF THE W EYL U NITARY M ATRICES u AND v The eigenvalues of the Weyl unitary matrices u and v are precisely the p -rootsof unity, which is the maximal spectrum of any p -th order unitary matrix. It sohappens that this spectral property is a consequence of the Weyl commutationrelation. Lemma 2.1. If u and v are p -th order d × d unitary matrices in which uv = ζvu , then(1) p divides d ,(2) Tr ( u ) = Tr ( v ) = Tr ( uv ) = , and(3) every p -root of unity is an eigenvalue of u and an eigenvalue of v .Proof. Applying the determinant to the relation uv = ζvu leads todet ( u ) det ( v ) = det ( uv ) = ζ d det ( vu ) = ζ d det ( u ) det ( v ) ,which implies that ζ d =
1. Because ζ is primitive, the positive integer p divides d .The relation uv = ζvu is equivalent to v ∗ uv = ζu . Thus, by applying the trace,we have that Tr ( u ) = ζ Tr ( u ) , which is true only if Tr ( u ) =
0. Analogous reasoningleads to Tr ( v ) =
0. Finally, Tr ( vu ) = Tr ( uv ) = ζ Tr ( vu ) ,which leads to ζ = ( uv ) =
0. As the former cannot hold, it must be that thelatter does.We next show that every p -th root of unity is an eigenvalue of u . Because thespectrum of a matrix is invariant under unitary similarity, the relation v ∗ uv = ζu implies that ζλ is an eigenvalue of u whenever λ is an eigenvalue of u . Applyingthis observation to the eigenvalue ζλ , we see that ζ λ is an eigenvalue of u . Hence, NIVERSALITY OF WEYL UNITARIES 5 by iteration of the argument, ζ k λ is an eigenvalue of u for every k =
1, . . . , p . Be-cause the map α αλ is a bijection of C onto itself, the spectrum of u must containat least p elements. But on the other hand, the spectrum of u cannot contain morethan p elements; hence, the spectrum of u must coincide with the set of p -th rootsof unity. A similar argument applies to v . (cid:3) The next lemma determines the set of all p -th order unitaries v , given a unitary u with the spectral structure described in Lemma 2.1, for which uv = ζvu . Lemma 2.2. If d = pn and if u , v ∈ M d ( C ) are p -th order unitary matrices suchthat uv = ζvu , then there exist a unitary matrix y ∈ M d ( C ) and unitary matrices v , . . . , v p ∈ M n ( C ) such that y ∗ uy = ˜ u and y ∗ vy = ˜ v , where (5) ˜ u = n ζ n ζ n . . . ζ p − n and ˜ v = · · · v v . . . ... v ... . . . . . . ... · · · v p , where n denotes the n × n identity matrix, and where v is given by v = v ∗ v ∗ · · · v ∗ p .Proof. Lemma 2.1 indicates that every p -th root of unity is a spectral element of u and Tr ( u ) =
0. Therefore, using d = pn , each eigenvalue ζ k of u must appear withmultiplicity n . Hence, u admits a diagonalisation such as that given in (5).Now let ˜ v = y ∗ vy ; in accordance with the structure of ˜ u , the operator ˜ v canbe expressed as a p × p matrix of n × n matrices v ij . The relation ˜ u ˜ v = ζ c ˜ v ˜ u holds only if the operators v ij = ( i , j ) { ( p ) , ( k , k − ) | k =
2, . . . , p } .Thus, denote v p by v and v k , k − by v k . Because ˜ v ∗ ˜ v = ˜ v ˜ v ∗ = d , each v k satisfies v ∗ k v k = v k v ∗ k = n . Furthermore, ˜ v p = d implies that v p v p − . . . v v = n , and so v = v ∗ v ∗ · · · v ∗ p . (cid:3) As a consequence of the result above, it is possible to construct a path-connectedset of p -th order unitaries v such that uv = ζvu from a certain p -th order unitarymatrix u . Corollary 2.3. If u ∈ M d ( C ) is a p -th order unitary matrix such that every p -th rootof unity is an eigenvalue of u of multiplicity d/p , then the set of all p -th order unitarymatrices v ∈ M d ( C ) for which uv = ζvu is homeomorphic to the Cartesian product of p − copies of the unitary group U dp . The next result is crucial for establishing an explicit complete order equivalencein our main result, Theorem 2.10 below.
Lemma 2.4. If d = pn and if u , v ∈ M d ( C ) are unitaries of the form (5), for someunitaries v , . . . , v p ∈ M n ( C ) , then a p × p matrix Z = [ z ij ] pi , j = of n × n matrices z ij isan element of Alg ( { u , v } ) if and only if there exist p scalars λ ij such that: (6) z ii = λ ii n , for all i ; z p = λ p v and z k , k − = λ k , k − v k , for all k =
2, . . . , p ; z p = λ p v ∗ and z k , k + = λ k , k + v ∗ k + , for all k =
1, . . . , p − z ij = λ ij ( v i · · · v j + ) , for all i > j with | i − j | > z ij = λ ij ( v ∗ i + · · · v ∗ j ) , for all i < j with | i − j | > D. FARENICK, O. OJO, AND S. PLOSKER
Proof.
First note that if a matrix b commutes with a unitary w , then b also com-mutes with w − = w ∗ , implying that b ∗ commutes with w ; hence, the set { w } ′ ofmatrices commuting with w is closed under the adjoint ∗ . Second, as the inverse w − of a unitary matrix w is a polynomial in w , then algebra generated by one ormore unitary matrices is ∗ -closed. Hence, by von Neumann’s Double CommutantTheorem, Alg ( { u , v } ) = { u , v } ′′ ,where S ′′ denotes the double commutant ( S ′ ) ′ of a set S . We begin, therefore, byshowing that(7) { u , v } ′ = (cid:14) p − M k = ( v p · · · v k + ) ∗ x ( v p · · · v k + ) ! M x | x ∈ M n ( C ) (cid:15) .To this end, suppose that X = [ x ij ] pi , j = ∈ M d ( C ) , where x ij ∈ M n ( C ) for all i and j , commutes with u and v . Then Xu = uX implies that ζ i − x ij = ζ j − x ij for all i , j , and so x ij = i and j with j = i . The equation Xv = vX yields x v = v x pp and v k x k − k − = x kk v k for k =
2, . . . , p . Thus,(8) x pp = v ∗ x v , x = v ∗ x v , x = v ∗ x v , · · · , x p − p − = v ∗ p x pp v p ,which yields(9) x kk = ( v k · · · v ) x pp ( v ∗ · · · v ∗ k ) , for all k =
1, . . . , p .Therefore, once x pp is specified, all other diagonal entries of X are determined.Suppose next that Z = [ z ij ] pi , j = ∈ M d ( C ) , where z ij ∈ M n ( C ) for all i and j , commutes with every X ∈ { u , v } ′ . Write the diagonal entries of X ∈ { u , v } ′ as x , . . . , x p , and recall from equation (9) that x k = ( v k · · · v ) x p ( v ∗ · · · v ∗ k ) , for all k =
1, . . . , p .The equation XZ = ZX implies that x i z ij = z ij x j for all i , j . For i = j , theserelations imply that z ii commutes with every n × n matrix; hence, z ii = λ ii n , forsome λ ii ∈ C .If k ∈ {
2, . . . , p } , then x k z k , k − = z k , k − x k − is written, using the equations (8),as x k z k , k − = z k , k − ( v ∗ k x k v k ) ,and so x k ( z k , k − v ∗ k ) = ( z k , k − v ∗ k ) x k . As this holds for all x k ∈ M n ( C ) , z k , k − v ∗ k = λ k , k − n , for some λ k , k − ∈ C ; that is, z k , k − = v k , for k =
2, . . . , p . The same typeof calculation leads to z p = λ p v .Similarly, if k ∈ {
1, . . . , p − } , then x k z k , k + = z k , k + x k + is, using the equations(8), equivalent to the equation x k + ( v k + z k , k + ) = ( v k + z k , k + ) x k + ,which implies that z k , k + = λ k , k + v ∗ k + , as the equation above must hold for all x k + . Likewise, z p = v ∗ .Consider now the entries of Z for which | i − j | >
2. Assume first the cases where i > j . By equations (8), x i = ( v i · · · v j + )( v j · · · v ) x p ( v ∗ · · · v ∗ j )( v ∗ j + · · · v ∗ i )= ( v i · · · v j + ) x j ( v ∗ j + · · · v ∗ i ) . NIVERSALITY OF WEYL UNITARIES 7
Thus, the equation x i z ij = z ij x j is x i z ij = z ij ( v i · · · v j + ) ∗ x i ( v i · · · v j + ) ,which is equivalent to x i ( z ij ( v i · · · v j + ) ∗ ) = ( z ij ( v i · · · v j + ) ∗ ) x i .As x i ∈ M n ( C ) can be arbitrary, z ij = λ ij n for some λ ij ∈ C , which implies that z ij = λ ij ( v i · · · v j + ) , for all i > j with | i − j | > | i − j | > i < j , the same type of arguments lead to z ij = λ ij ( v ∗ i + · · · v ∗ j ) , for all i < j with | i − j | > (cid:3) As an example of the matrix structure indicated Lemma 2.4, consider the casewhere p =
5. Select any n ∈ N and unitary matrices v , . . . , v ∈ M n ( C ) , and set v = ( v v v v ) ∗ . Thus, u = n ζ n ζ n ζ n ζ n and v = v v v v v .By Lemma 2.4, z ∈ Alg ( { u , v } ) if and only if z = λ n λ v ∗ λ ( v ∗ v ∗ ) λ ( v ∗ v ∗ v ∗ ) λ v λ v λ n λ v ∗ λ ( v ∗ v ∗ ) λ ( v ∗ v ∗ v ∗ ) λ ( v v ) λ v λ n λ v ∗ λ ( v ∗ v ∗ ) λ ( v v v ) λ ( v v ) λ v λ n λ v ∗ λ ( v v v v ) λ ( v v v ) λ ( v v ) λ v λ n ,for some λ ij ∈ C .Let us also make note of the following useful consequence of Lemma 2.4. Corollary 2.5. If u , v ∈ M p ( C ) are unitaries of the form (5), for some complex numbers v , . . . , v p , then { u , v } ′ = { λ p | λ ∈ C } and Alg ( { u , v } ) = M p ( C ) . The structure of Alg ( { u , v } ) provided by Lemma 2.4 gives an explicit ∗ -isomorphismof matrix algebras: Proposition 2.6. If d = pn and if u , v ∈ M d ( C ) are unitaries of the form (5), for someunitaries v , . . . , v p ∈ M n ( C ) , then the function ρ : M p ( C ) → M d ( C ) defined by ρ (cid:16) [ λ ij ] pi , j = (cid:17) = [ z ij ] pi , j , where the matrices z ij ∈ M n ( C ) are given as in equations (6), is a unital ∗ -isomorphismof M p ( C ) and Alg ( { u , v } ) .Proof. By Lemma 2.4, the function ρ maps M p ( C ) onto C ∗ ( u , v ) . Furthermore, ρ is plainly linear and ker ρ = { } , and so ρ is a linear isomorphism. It remains toshow that ρ is a ∗ -homomorphism. Equations (6) indicate that ρ ( Λ ∗ ) = ρ ( Λ ) ∗ , forall Λ ∈ M p ( C ) , and so the multiplicativity of ρ is the only property left to confirm. D. FARENICK, O. OJO, AND S. PLOSKER
To this end, let A = [ α ij ] i , j and B = [ β ij ] i , j be elements of M p ( C ) . We shallcompare the entries of ρ ( A ) ρ ( B ) with those of ρ ( AB ) . If, for example, i > j and | i − j | >
2, then the entries of row i of ρ ( A ) are α i ( v i · · · v ) , α i ( v i · · · v ) , · · · , α ii n , α i , i + v ∗ i + , · · · , α ip ( v ∗ i + · · · v ∗ p ) ,while the entries of column j of ρ ( B ) are β j ( v ∗ · · · v ∗ j ) , β j ( v ∗ · · · v ∗ j ) , · · · , β jj n , β j + j v p , · · · , β pj ( v p · · · v j + ) .Therefore, the ( i , j ) -entry of ρ ( A ) ρ ( B ) is p X k = α ik β kj ( v i · · · v j + ) , which is the ( i , j ) -entry of ρ ( AB ) . The arguments for all other choices of i and j are similar. (cid:3) Theorem 2.7.
If two p -th order unitary matrices u and v satisfy the Weyl commutationrelation uv = ζvu , then ( u , v ) ≃ ord (u , v) .Proof. If u , v ∈ M d ( C ) , then Lemma 2.1 asserts that p divides d and that there is aunitary matrix y ∈ M d ( C ) such that y ∗ uy and y ∗ vy are the matrices in (5). As themap x y ∗ xy is a ∗ -automorphism of M d ( C ) , it is enough to assume that u and v are in this form already. In that regard, the isomorphism ρ : M p ( C ) → M d ( C ) in Theorem 2.6 satisfies ρ (u) = u and ρ (v) = v . Furthermore, as ρ is a unital ∗ -isomorphism, its restriction φ to the operator system generated by the Weyl ma-trices u and v has the property that both φ and φ − are completely positive. (cid:3) Corollary 2.8.
The Weyl unitary matrices u and v are universal for the commutationrelation uv = ζvu .Proof. Suppose that u and v are p -th order d × d unitary matrices that satisfy theWeyl commutation relation uv = ζvu . By Theorem 2.10, the linear isomorphism φ : O (u , v) → O ( u , v ) is completely positive. Viewing φ as a ucp from the operatorsystem O (u , v) into the matrix algebra M d ( C ) , the map φ admits a ucp extension Φ to M p ( C ) , by the Arveson Extension Theorem [1, 8]. Clearly the map Φ sends u to u and v to v . (cid:3) Corollary 2.9.
If two p -th order unitary matrices u and v satisfy the Weyl commutationrelation uv = ζvu , then the pair ( u , v ) is universal for all p -th order unitary matrices thatsatisfy the Weyl commutation relation. Our final observation is that if the Weyl commutation relations are satisfied by p -th order p × p unitaries, then these unitaries must be unitarily equivalent to theWeyl unitaries. Corollary 2.10.
If two p -th order p × p unitary matrices u and v satisfy the Weyl com-mutation relation uv = ζvu , then ( u , v ) ≃ u (u , v) .Proof. By hypothesis, there is a unital completely positive bijection φ : O (u , v) → O ( u , v ) in which φ (u) = u , φ (v) = v , and φ − is completely positive. Let Φ and Φ be extensions of φ and φ − , respectively, to ucp maps M p ( C ) → M p ( C ) . The ucpmap Φ ◦ Φ fixes every element of the irreducible operator system O (u , v) , and so byArveson’s Boundary Theorem [2, 6], Φ ◦ Φ is the identity map of M p ( C ) . Hence, Φ is a ucp map of M p ( C ) with a completely positive inverse, which by Wigner’sTheorem implies that Φ is a unitary similarity transformation x w ∗ xw for someunitary w ∈ M p ( C ) . (cid:3) NIVERSALITY OF WEYL UNITARIES 9
3. W
EYL -B RAUER U NITARIES
Consider the Weyl triple
W = ( ω a , ω a , ω c ) of p × p unitary matrices, where ω a = u , ω b = ζ p − uv , ω c = v ,and where u and v are the Weyl unitary matrices. Recall that the triple W =( ω a , ω a , ω c ) satisfies the simple Weyl commutation relations.Observe that v = ζ − p u p − w , and so v is in the associative algebra generated by ω a and ω b . Because M p ( C ) is generated as an algebra by u and v , this means thatthe algebra generated ω a and ω b is also M p ( C ) . Hence,Alg ( Q − ) = Alg ( Q − ) = M p ( C ) ,where Q = { ω a , ω b , ω c } and Q − = { ω a , ω b } .As in [10, Definition 6.63] and by adapting the method of the proof of Theorem4.3 in [7], we shall invoke an iteration whereby we produce, from m invertiblematrices x , . . . , x m , a set of m + x ⊗ p , . . . , x m − ⊗ p , x m ⊗ ω a , x m ⊗ ω b , x m ⊗ ω c .Specifically, in taking x , x , and x to be ω a , ω b , and ω c , respectively, the iterationyields a set Q ⊂ M p ( C ) ⊗ M p ( C ) of 5 elements: Q = { ω a ⊗ ω b ⊗ ω c ⊗ ω a , ω c ⊗ ω b , ω c ⊗ ω c } = Q − ∪ { ω c ⊗ ω c } ,where Q − = Q \ { ω c ⊗ ω c } . The matrices in Q satisfy the Weyl commutationrelations ˜ u ˜ v = ζ ˜ v ˜ u when ˜ u is selected before ˜ v in Q and the set Q is consideredas an ordered list.Another iteration of the construction generates a set Q consisting of 7 elements: Q = Q − ∪ { ω c ⊗ ω c ⊗ ω c } ,where Q − = { ω a ⊗ ⊗ ω b ⊗ ⊗ ω c ⊗ ω a ⊗ ω c ⊗ ω b ⊗ ω c ⊗ ω c ⊗ ω a , ω c ⊗ ω c ⊗ ω b } .Once again, the matrices in Q satisfy the Weyl commutation relations ˜ u ˜ v = ζ ˜ v ˜ u when ˜ u is selected before ˜ v in Q and the set Q is considered as an ordered list.Repeated iteration produces, for each positive integer k , a set Q k , − of cardinality2 k and a set Q k with one additional element, namely Q k = Q k , − ∪ (cid:14) k O ω c (cid:15) ,such that the matrices in Q k satisfy the Weyl commutation relations ˜ u ˜ v = ζ ˜ v ˜ u when ˜ u is selected before ˜ v in Q k and the set Q k is considered as an ordered list.The 2 k elements of Q k , − consist of k pairs such that, in the order given by theiterative construction, the product of each pair is a product tensor in which allfactors are the identity matrix and one tensor factor is ω a ω b . More specifically, if Q k , − = { z , z , z , z , . . . , z k − , z k } ⊂ k O M p ( C ) , then z z = ( ω a ω b ) ⊗ ⊗ · · · ⊗ = ζ − p ( ω c ⊗ ⊗ · · · ⊗ ) z z = ⊗ ( ω a ω b ) ⊗ · · · ⊗ = ζ − p ( ⊗ ω c ⊗ · · · ⊗ ) ... = ... z k − z k = ⊗ ⊗ · · · ⊗ ( ω a ω b ) = ζ − p ( ⊗ ⊗ ⊗ · · · ⊗ ω c ) .Hence, k O ω c = ( ζ − p ) − k k Y j = w j − w j ∈ Alg ( Q k , − ) ,which shows that Alg ( Q k , − ) = Alg ( Q k ) ,for every k ∈ N .We now show that Alg ( Q k , − ) = Alg ( Q k ) = k O M p ( C ) . Of course, it is sufficientto show this for Alg ( Q k , − ) . The claim holds for k = Q − = { ω a , ω b } generates M p ( C ) . Looking at the case k = Q − = { ω a ⊗ ω b ⊗ ω c ⊗ ω a , ω c ⊗ ω b } .The algebra generated by { ω a ⊗ ω b ⊗ } consists of all matrices of the form s ⊗ s ∈ M p ( C ) , whereas the algebra generated by { ω c ⊗ ω a , ω c ⊗ ω a } consists ofall matrices of the form ω c ⊗ t , for t ∈ M p ( C ) . Because the set of all products ofmatrices of these two types is the set of all elementary tensors in M p ( C ) ⊗ M p ( C ) ,we see that the claims holds for k =
2. In general, using induction, if we considermatrices of the form given by the construction x ⊗ p , . . . , x m − ⊗ p , x m ⊗ ω a , x m ⊗ ω b , x m ⊗ ω c ,where the algebra generated by invertible matrices x , . . . , x m − is a full matrixalgebra M d ( C ) , then our argument here shows that x ⊗ p , . . . , x m − ⊗ p generatematrices of the form s ⊗ p while x m ⊗ ω a and x m ⊗ ω b generate all matrices ofthe form x m ⊗ t . Thus, collectively, these matrices generate M d ( C ) ⊗ M p ( C ) .The construction above proves the following theorem. Theorem 3.1.
For every positive integer k there exist p -th order unitaries u , . . . , u k + that satisfy the simple Weyl commutation relations and are such that Alg ( { u , . . . , u k } ) = Alg ( { u , . . . , u k , u k + } ) = k O M p ( C ) .As mentioned in [7], it is often the case that spin systems arise in quantumtheory; for this reason, the following modification of the construction above isworth a brief mention. Theorem 3.2. If H is an infinite-dimensional complex Hilbert space, then there exists acountable sequence { u n } n ∈ N of p -th order unitary operators u n such that u k u ℓ = ζu ℓ u k whenever k < ℓ and such that the norm-closed algebra generated by the sequence { u n } n ∈ N is isomorphic to the C ∗ -algebra ∞ O M p ( C ) . NIVERSALITY OF WEYL UNITARIES 11
Proof.
Let H = ∞ O C p , which is the direct limit of the finite-dimensional Hilbertspaces H k = k O C p . On each H k construction the Weyl-Brauer unitaries, and thenform the tensor product of these unitaries with infinitely many copies of the p × p identity matrix so as to produce unitary operators u , . . . , u k on H of order p thatsatisfy the simple Weyl commutation relations. This construction also shows thatthe algebra A k generated by u , . . . , u k is isomorphic to k O M p ( C ) and that A k isa unital subalgebra of A k + . Hence, the norm-closed algebra generated by { u n } n ∈ N coincides with the norm-closure of [ k ∈ N A k , which is precisely ∞ O M p ( C ) . (cid:3)
4. W
EYL -B RAUER U NITARIES A RE N OT U NIVERSAL
Although the Pauli-Weyl-Brauer matrices are universal for selfadjoint anticom-muting unitaries [7], the analogous result fails for p > Proposition 4.1.
Assume that p > and let W = ( ω a , ω b , ω c ) denote the triple ofsimple Weyl-Brauer matrices. Let x , y , z ∈ M p ( C ) be the p -th order unitary matricesgiven by x = ω a , z = ω c , and (10) y = ζ ζ . . . . . . ζ p − ζ p − . . . ζ ( − p ) / . The triple ( x , y , z ) satisfies the simple Weyl commutation relations, but there does not existany unital completely positive linear map φ : M p ( C ) → M p ( C ) in which φ ( ω a ) = x , φ ( ω b ) = y , and φ ( ω c ) = z .Proof. If λ p denotes the ( p , p − ) -entry of y , then the condition y p = p impliesthat λ p = (cid:0) · ζ · ζ · · · ζ p − (cid:1) − = ζ − P p − k = k = ζ ( − p ) / ;furthermore, matrix multiplication confirms that the triple ( x , y , z ) satisfies thesimple Weyl commutation relations. Hence, ( x , y , z ) is a simple Weyl triple.Assume that a unital completely positive linear map φ : M p ( C ) → M p ( C ) inwhich φ ( ω a ) = x , φ ( ω b ) = y , and φ ( ω c ) = z does exist. The fixed point set F φ = { s ∈ M d ( C ) | φ ( s ) = s } of φ is an operator system that contains the Weylunitaries ω a = u and ω c = v ; because { u , v } ′ = C d , the operator system F φ isirreducible.Let ψ : F φ → M p ( C ) be the ucp map ψ ( s ) = s , for all s ∈ F φ . Thus, φ is a ucpextension of ψ from F φ to M p ( C ) . However, as F φ is irreducible, ψ has uniquecompletely positive extension to M p ( C ) , by Arveson’s Boundary Theorem [2, 6].Therefore, φ can only be the identity map, as the identity map on M d ( C ) is one ucp extension of ψ . However, as φ ( ω a ) = x = ω a , φ is not the identity map. Hence,this contradiction leads us to conclude that φ is not completely positive. (cid:3) Corollary 4.2.
The Weyl-Brauer unitaries are not universal for g -tuples (where g > )of p -th order unitaries that satisfy the simple Weyl commutation relations.Proof. If universality were to hold for some g >
3, then it would need to hold for g =
3. However, Proposition 4.1 indicates that universality fails for g = (cid:3)
5. T HE M ATRIX R ANGE OF THE W EYL U NITARIES
The work of Arveson [2] and Davidson, Dor-On, Shalit, and Solel [3] demon-strate the role of the matrix range in questions such as those we have consideredherein. For this reason, it is of interest to consider the matrix range of the Weylunitaries, especially in connection with the geometry of the matrix range from theperspective of free convexity [4].
Definition 5.1.
A sequence K = ( K n ) n ∈ N of subsets K n in the Cartesian product M n ( C ) g of g copies of M n ( C ) is matrix convex if (11) m X j = γ ∗ j a j γ j ∈ K n for all m ∈ N , all a j ∈ K n j , and all linear transformations γ j : C n → C n j for which (12) m X j = γ ∗ j γ j = n . Linear transformations γ j that satisfy (12) are called matrix convex coefficients andelements of the form (11) are called matrix convex combinations of the elements a j . We are interested in the following notions [4] of extremal element in the contextof matrix convexity.
Definition 5.2. If K = ( K n ) n ∈ N is matrix convex, where K n ⊆ M n ( C ) g for each n , thenan element b ∈ K n is:(1) an absolute extreme point of K if whenever b is a matrix convex combination(11) of elements a j ∈ K n j such that each matrix convex coefficient γ j is nonzero,then, for each j , either (i) n j = n and a j ≃ u b or (ii) n j > n and there exists a c j such that a j ≃ u b ⊕ c j ;(2) a matrix extreme point of K if whenever b is a matrix convex combination (11)of elements a j ∈ K n j such that each matrix convex coefficient γ j is surjective,then, for each j , n j = n and a j ≃ u b . Theorem 5.3.
The Weyl pair (u , v) is a matrix extreme point of its matrix range.Proof. Let b = (u , v) and suppose that b = m X j = γ ∗ j a j γ j for some surjective matrixconvex coefficients γ j and matrix pairs a j = ( a j , a j ) ∈ W n j ( b ) . For each j thereis a ucp map ψ j : O (u , v) → M n j ( C ) such that a j = ψ j (u) and a j = ψ j (v) . Let Ψ j NIVERSALITY OF WEYL UNITARIES 13 be a ucp extension of ψ j to a ucp map Ψ j : M p ( C ) → M n j ( C ) , for each j , and let Φ : M p ( C ) → M p ( C ) be given by Φ = m X j = γ ∗ j Ψ j γ j .Note that Φ |O (u , v) is the identity map on O (u , v) ; hence, in considering the iden-tity map in O (u , v) as a ucp map from O (u , v) into M p ( C ) , Φ is a ucp extension ofthat map. Because the operator system O (u , v) is irreducible, Arveson’s BoundaryTheorem [2, 6] implies that Φ is the identity map on M p ( C ) . That is,id M p ( C ) = m X j = γ ∗ j Ψ j γ j .Now because the identity map of M p ( C ) is a pure matrix state of M p ( C ) , it is alsoa matrix extreme point of its matrix state space [5]. Hence, n j = p for every j andthere are unitaries w j ∈ M p ( C ) such that Ψ j ( x ) = w ∗ j xw j for every x ∈ M p ( C ) . Inparticular, u = w ∗ j a j w j and v = w ∗ j a j w j for all j , which yields a j ≃ u (u , v) = b for all j . (cid:3) Theorem 5.4.
The Weyl pair (u , v) is an absolute extreme point of its matrix range.Proof. Let b = (u , v) and suppose that, for some ℓ ∈ N , the pair ( a , a ) = (cid:18)(cid:20) u r s t (cid:21) , (cid:20) v r s t (cid:21)(cid:19) ∈ W p + ℓ ( b ) ,for some matrices r i , s i , t i of appropriate sizes. We claim that the matrix pair above ( a , a ) can be an element of W p + ℓ ( b ) only if the off-diagonal matrices r i and s i are zero, for i =
1, 2. The reason for this is straightforward. Because each a i is aucp image of a Weyl unitary, k a i k i =
1, 2. Therefore, no row or columnin a i can have norm (in C p + ℓ ) exceeding 1. However, because each row of u and v has exactly one nonzero entry and this entry is of modulus 1, a nonzero entry in r or r would cause a row in a or a to have norm exceeding 1, which we notedcannot happen. Thus, r and r are zero matrices. Using a similar argument forthe columns, we deduce that s and s are also zero matrices.In the language of matrix convexity, the previous paragraph proves that thepair (u , v) is an Arveson extreme point of its matrix range W ( b ) . By [4, Theorem1.1(3)], if an Arveson extreme point of a matrix convex set is irreducible, then it isan absolute extreme point. Since the commutant { u , v } ′ is 1-dimensional, the Weylpair is irreducible and, hence, an absolute extreme point of its matrix range. (cid:3) The proofs of Theorems 5.3 and 5.4 extend beyond Weyl pairs to all Weyl-Brauerunitaries, once it has been shown that the Weyl-Brauer matrices generate irre-ducible operator systems. The details are left to the reader (using, if one wishes,the method of proof in [7] that showed the irreducibility of the operator systemgenerated by the Pauli-Weyl-Brauer unitaries). However, at the very least, westate below the version of this theorem for the three basic Weyl-Brauer unitaries ω a , ω b , and ω c . Theorem 5.5. If W = ( ω a , ω b , ω c ) is the Weyl-Brauer triple, then W is a matrixextreme point and an absolute extreme point of its matrix range. A CKNOWLEDGEMENTS
This work was supported, in part, by the NSERC Discovery Grant program, theCanada Foundation for Innovation, and the Canada Research Chairs program.R
EFERENCES1. William Arveson,
Subalgebras of C ∗ -algebras , Acta Math. (1969), 141–224. MR MR0253059 (40 Subalgebras of C ∗ -algebras. II , Acta Math. (1972), no. 3-4, 271–308. MR MR0394232 (52 Dilations, inclusions ofmatrix convex sets, and completely positive maps , Int. Math. Res. Not. IMRN (2017), no. 13, 4069–4130.MR 36715114. Eric Evert, J. William Helton, Igor Klep, and Scott McCullough,
Extreme points of matrix convex sets,free spectrahedra, and dilation theory , J. Geom. Anal. (2018), no. 2, 1373–1408. MR 37905045. Douglas Farenick, Extremal matrix states on operator systems , J. London Math. Soc. (2) (2000), no. 3,885–892. MR 1766112 (2001e:46103)6. , Arveson’s criterion for unitary similarity , Linear Algebra Appl. (2011), no. 4, 769–777.MR 2807232 (2012d:15002)7. Douglas Farenick, Farrah Huntinghawk, Adili Masanika, and Sarah Plosker,
Complete order equiva-lence of spin unitaries , Linear Algebra Appl. (2021), 1–28.8. Vern Paulsen,
Completely bounded maps and operator algebras , Cambridge Studies in Advanced Math-ematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR MR1976867 (2004c:46118)9. Julian Schwinger,
Unitary operator bases , Proc. Nat. Acad. Sci. U.S.A. (1960), 570–579. MR 11564810. John Watrous, The theory of quantum information , Cambridge University Press, Cambridge, 2018.11. H. Weyl,
Quantenmechanik und Gruppentheorie , Z. Physik A46