Schur duality Laura Mancinska University of Waterloo July 30, 2008 - PowerPoint PPT Presentation

Schur duality Laura Mancinska University of Waterloo July 30, 2008

Outline 1 Basics of representation theory 2 Schur duality 3 Applications

Basics of representation theory

Representation Definition A representation ( φ, C n ) over the vector space C n of a group G is a homomorphism φ : G → GL( n, C ) .

Representation Definition A representation ( φ, C n ) over the vector space C n of a group G is a homomorphism φ : G → GL( n, C ) . Homomorphism: φ ( g 1 g 2 ) = φ ( g 1 ) φ ( g 2 ) for all g 1 , g 2 ∈ G GL( n, C ) : n × n invertible complex matrices

Representation Definition A representation ( φ, C n ) over the vector space C n of a group G is a homomorphism φ : G → GL( n, C ) . Example Every group has trivial representation ( φ triv , C ) : φ triv ( g ) = 1 .

Representation Definition A representation ( φ, C n ) over the vector space C n of a group G is a homomorphism φ : G → GL( n, C ) . Example Every group has trivial representation ( φ triv , C ) : φ triv ( g ) = 1 . Example S n has representation ( φ sgn , C ) given by φ sgn ( π ) = sgn( π ) .

Representation Definition A representation ( φ, C n ) over the vector space C n of a group G is a homomorphism φ : G → GL( n, C ) . Example Every group has trivial representation ( φ triv , C ) : φ triv ( g ) = 1 . Example S n has representation ( φ sgn , C ) given by φ sgn ( π ) = sgn( π ) . Example Representations of U ( d ) include: � C d � ⊗ n ) given by φ ( U ) = U ⊗ n ( φ,

Representation Definition A representation ( φ, C n ) over the vector space C n of a group G is a homomorphism φ : G → GL( n, C ) . Example Every group has trivial representation ( φ triv , C ) : φ triv ( g ) = 1 . Example S n has representation ( φ sgn , C ) given by φ sgn ( π ) = sgn( π ) . Example Representations of U ( d ) include: � C d � ⊗ n ) given by φ ( U ) = U ⊗ n ( φ, ( φ det , C ) given by φ det ( U ) = det( U )

Direct sum and tensor product Definition Let ( φ 1 , V 1 ) and ( φ 2 , V 2 ) be representations of G . Then representations ( φ 1 ⊕ φ 2 , V 1 ⊕ V 2 ) and ( φ 1 ⊗ φ 2 , V 1 ⊗ V 2 ) of G are their direct sum and tensor product, respectively.

Direct sum and tensor product Definition Let ( φ 1 , V 1 ) and ( φ 2 , V 2 ) be representations of G . Then representations ( φ 1 ⊕ φ 2 , V 1 ⊕ V 2 ) and ( φ 1 ⊗ φ 2 , V 1 ⊗ V 2 ) of G are their direct sum and tensor product, respectively. Example Let ( φ 1 , C 2 ) , ( φ 2 , C ) be representations of U (2) such that φ 1 ( U ) = U φ 2 ( U ) = 1

Direct sum and tensor product Definition Let ( φ 1 , V 1 ) and ( φ 2 , V 2 ) be representations of G . Then representations ( φ 1 ⊕ φ 2 , V 1 ⊕ V 2 ) and ( φ 1 ⊗ φ 2 , V 1 ⊗ V 2 ) of G are their direct sum and tensor product, respectively. Example Let ( φ 1 , C 2 ) , ( φ 2 , C ) be representations of U (2) such that φ 1 ( U ) = U φ 2 ( U ) = 1 Then ( φ 1 ⊕ φ 2 , C 3 ) is their direct sum and ( φ 1 ⊗ φ 2 , C 2 ) is their tensor product. � U � 0 ( φ 1 ⊕ φ 2 )( U ) = U ⊕ 1 = ( φ 1 ⊗ φ 2 )( U ) = U ⊗ 1 = U 0 1

Irreducible representations Definition We say that a representation ( φ, V ) of group G is irreducible if it is not a direct sum of at least two other representations.

Irreducible representations Definition We say that a representation ( φ, V ) of group G is irreducible if it is not a direct sum of at least two other representations. Example If the representation space V of representation ( φ, V ) is 1-dimensional, then ( φ, V ) is irreducible.

Irreducible representations Definition We say that a representation ( φ, V ) of group G is irreducible if it is not a direct sum of at least two other representations. Example If the representation space V of representation ( φ, V ) is 1-dimensional, then ( φ, V ) is irreducible. Theorem Every representation ( φ, V ) of G is isomorphic to a direct sum of irreducible representations of G : � φ ( g ) ∼ = λ ( g ) ⊗ I n λ λ ∈ ˆ G

Schur duality

Representations of U ( d ) and S n Consider representations � C d � ⊗ n � � Q , of U ( d ) , where Q ( U ) | i 1 i 2 . . . i n � = U | i 1 � U | i 2 � . . . U | i n �

Representations of U ( d ) and S n Consider representations � C d � ⊗ n � � Q , of U ( d ) , where Q ( U ) | i 1 i 2 . . . i n � = U | i 1 � U | i 2 � . . . U | i n � � C d � ⊗ n � � P , of S n , where � � � � � � P ( π ) | i 1 i 2 . . . i n � = � i π − 1 (1) � i π − 1 (2) . . . � i π − 1 ( n )

Representations of U ( d ) and S n Consider representations � C d � ⊗ n � � Q , of U ( d ) , where Q ( U ) | i 1 i 2 . . . i n � = U | i 1 � U | i 2 � . . . U | i n � � C d � ⊗ n � � P , of S n , where � � � � � � P ( π ) | i 1 i 2 . . . i n � = � i π − 1 (1) � i π − 1 (2) . . . � i π − 1 ( n ) � C d � ⊗ n � � We can consider representation QP , of U ( d ) × S n , given by QP ( U, π ) = Q ( U ) P ( π )

Representations of U ( d ) and S n Consider representations � C d � ⊗ n � � Q , of U ( d ) , where Q ( U ) | i 1 i 2 . . . i n � = U | i 1 � U | i 2 � . . . U | i n � � C d � ⊗ n � � P , of S n , where � � � � � � P ( π ) | i 1 i 2 . . . i n � = � i π − 1 (1) � i π − 1 (2) . . . � i π − 1 ( n ) � C d � ⊗ n � � We can consider representation QP , of U ( d ) × S n , given by QP ( U, π ) = Q ( U ) P ( π ) = P ( π ) Q ( U )

Schur duality Theorem. (Schur duality) There exist a basis (Schur basis) in which representation � C d � ⊗ n � � QP , of U ( d ) × S n decomposes into irreducible representations q λ and p λ of U ( d ) and S n respectively: � QP ( U, π ) ∼ = q λ ( U ) ⊗ p λ ( π ) λ ∈ Par( n,d )

Schur duality Theorem. (Schur duality) There exist a basis (Schur basis) in which representation � C d � ⊗ n � � QP , of U ( d ) × S n decomposes into irreducible representations q λ and p λ of U ( d ) and S n respectively: � QP ( U, π ) ∼ = q λ ( U ) ⊗ p λ ( π ) λ ∈ Par( n,d ) Definition Schur transform U sch is unitary transformation implementing the base change from standard basis to Schur basis: � U sch = | sch i � � i | i

Schur duality � QP ( U, π ) ∼ = q λ ( U ) ⊗ p λ ( π ) λ ∈ Par( n,d )

Schur duality � QP ( U, π ) ∼ = q λ ( U ) ⊗ p λ ( π ) λ ∈ Par( n,d ) Example C 2 � ⊗ 2 we get � In case of 2 qubits, i.e.,

Schur duality � QP ( U, π ) ∼ = q λ ( U ) ⊗ p λ ( π ) λ ∈ Par( n,d ) Example C 2 � ⊗ 2 we get � In case of 2 qubits, i.e., λ =(1 , 1) λ =(2 , 0) � �� � � �� � QP ( U, π ) ∼ = ( q det ( U ) ⊗ p sgn ( π )) ⊕ ( q 3 dim ( U ) ⊗ p triv ( π ))

Schur duality � QP ( U, π ) ∼ = q λ ( U ) ⊗ p λ ( π ) λ ∈ Par( n,d ) Example C 2 � ⊗ 2 we get � In case of 2 qubits, i.e., λ =(1 , 1) λ =(2 , 0) � �� � � �� � QP ( U, π ) ∼ = ( q det ( U ) ⊗ p sgn ( π )) ⊕ ( q 3 dim ( U ) ⊗ p triv ( π )) = � det( U ) sgn( π ) � 0 = 0 q 3 dim ( U )

Schur duality � QP ( U, π ) ∼ = q λ ( U ) ⊗ p λ ( π ) λ ∈ Par( n,d ) Example C 2 � ⊗ 2 we get � In case of 2 qubits, i.e., λ =(1 , 1) λ =(2 , 0) � �� � � �� � QP ( U, π ) ∼ = ( q det ( U ) ⊗ p sgn ( π )) ⊕ ( q 3 dim ( U ) ⊗ p triv ( π )) = � det( U ) sgn( π ) � 0 | 01 � − | 10 � = 0 q 3 dim ( U ) | 00 � , | 11 � , | 01 � + | 10 �

Applications

Unitaries commuting with qubit permutations P π = QP ( I, π )

Unitaries commuting with qubit permutations � P π = QP ( I, π ) ∼ q λ ( I ) ⊗ p λ ( π ) = λ ∈ Par( n,d )

Unitaries commuting with qubit permutations � P π = QP ( I, π ) ∼ I dim( q λ ) ⊗ p λ ( π ) = λ ∈ Par( n,d )

Unitaries commuting with qubit permutations � P π = QP ( I, π ) ∼ I dim( q λ ) ⊗ p λ ( π ) = λ ∈ Par( n,d ) Example Recall Schur duality for 2 qubits: � det( U ) sgn( π ) � 0 QP ( U, π ) ∼ = 0 q 3 dim ( U )

Unitaries commuting with qubit permutations � P π = QP ( I, π ) ∼ I dim( q λ ) ⊗ p λ ( π ) = λ ∈ Par( n,d ) Example Recall Schur duality for 2 qubits: � det( I ) sgn( π ) � 0 QP ( I , π ) ∼ = 0 q 3 dim ( I )

Unitaries commuting with qubit permutations � P π = QP ( I, π ) ∼ I dim( q λ ) ⊗ p λ ( π ) = λ ∈ Par( n,d ) Example Recall Schur duality for 2 qubits: � det( I ) sgn( π ) � 0 P π = QP ( I , π ) ∼ = 0 q 3 dim ( I )

Unitaries commuting with qubit permutations � P π = QP ( I, π ) ∼ I dim( q λ ) ⊗ p λ ( π ) = λ ∈ Par( n,d ) Example Recall Schur duality for 2 qubits: � det( I ) sgn( π ) � � sgn( π ) � 0 0 P π = QP ( I , π ) ∼ = = 0 q 3 dim ( I ) 0 I 3

Unitaries commuting with qubit permutations � P π = QP ( I, π ) ∼ I dim( q λ ) ⊗ p λ ( π ) = λ ∈ Par( n,d ) Example Recall Schur duality for 2 qubits: � det( I ) sgn( π ) � � sgn( π ) � 0 0 P π = QP ( I , π ) ∼ = = 0 q 3 dim ( I ) 0 I 3 Unitaries commuting with 2-qubit permutations are given by � U (1) � 0 0 U (3)