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The individual qudit basis states are here denoted by | 0 ⟩, …, | d − 1 ⟩
Toth G Guhne O 2009 Phys. Rev. Lett. 102 170503 10.1103/PhysRevLett.102.170503
The action of the permutation operator P ^ σ on the computational basis states | i 1, …, i N ⟩ ( i k = 0, …, d − 1 for k = 1, …, N ) is standardly defined by P ^ σ | i 1, …, i N ⟩ = | i σ − 1 ( 1 ), …, i σ − 1 ( N ) ⟩
Ceccherini-Silberstein T Scarabotti F Tolli F 2010 Representation Theory of the Symmetric Groups, The Okounkov-Vershik Approach, Character Formulas and Partition Algebras Cambridge University Press
The product of two PI operators is PI and so is the Hermitian conjugate of a PI operator: the commutant L S N ( H ) is a ∗ -algebra of operators on H
As a reminder, the Hermitian conjugate of a superoperator L is the unique superoperator L † such that Tr ( A ^ † L † [ B ^ ] ) = Tr ( L [ A ^ ] † B ^ ), for all A ^, B ^ ∈ L ( H )
Within the formalism of the superoperators of permutation P σ, a PI operator A ^ PI is an operator that satisfies P σ [ A ^ PI ] = A ^ PI, ∀ σ
This means that each superoperator P σ defines a ∗ -isomorphism on the Liouville space L ( H )
Indeed, ∀ σ, n, and local operator X ^, we have P σ [ X ^ ( n ) ] = X ^ ( σ ( n ) ), which implies { P σ [ X ^ ( n ) ] } = { X ^ ( σ ( n ) ) } = { X ^ ( n ) } and P σ [ X ^ c ] = X ^ c
The commutant L S N ( H ) is the subspace of PI operators, i.e., of operators A ^ PI that satisfy P σ [ A ^ PI ] = A ^ PI, ∀ σ [68]. It is therefore nothing but the symmetric subspace of ( L ( H d ) ) ⊗ N ≅ L ( H d ⊗ N ) = L ( H )
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The irreducible representations of the symmetric group S N are indexed by the partitions ν of N . A partition ν of an integer N ≥ 0 is a sequence of integers ( ν 1, …, ν l ), with ν 1 ≥ ν 2 ≥ … ≥ ν l > 0 and ∑ i = 1 l ν i = N . We write in this case ν ⊢ N . The numbers ν i are called the parts of ν and the number l of parts is the length of ν, denoted by l ( ν ) . The weight of ν is the sum of its parts, N, also denoted by | ν | . A partition of an integer N of at most d parts ( d > 0 ) is a partition ν with l ( ν ) ≤ d . We write in this case ν ⊢ ( N, d ) . The case N = 0 is particular and only counts the so-called empty partition of length 0. The irreducible representations of the general linear group GL(d) are indexed by so-called highest weights ν ≡ ( ν 1, …, ν d ), with ν 1 ≥ ⋯ ≥ ν d and ν i ( i = 1, …, d ) positive or negative integers. In the context of the Schur-Weyl duality, only GL(d) irreps of highest weights ν with positive parts play a role, in which case ν identifies to a partition of at most d parts (including the empty partition of length 0). The irreducible representations of the unitary group U(d) are indexed similarly.
The dimension of the irreducible representation S ν is given by the elegant hook length formula: f ν = N ! / ∏ ( i, j ) ∈ ν h ( i, j ), where the product runs over the hook length h ( i, j ) of each box (i, j) of the diagram ν [103]. The historical Frobenius-Young determinantal formula [104, 105] can also be used instead: for ν ≡ ( ν 1, …, ν l ), f ν = N ! ∏ i < j ( l i − l j ) / ∏ i l i ! with l i = ν i + l ( ν ) − i and l ( ν ) the length of partition ν. In particular, for ν = ( N ), f ν = 1, and for ν ≡ ( ν 1, ν 2 ) ⊢ ( N, 2 ), f ν = ( ν 1 − ν 2 + 1 ) ( N ν 2 ) / ( ν 1 + 1 ) . This also allows one to express the ratio [100] r ν ν − τ ≡ N f ν − τ / f ν in the form r ν ν − τ = l τ ∏ i ≠ τ ( l i − l τ + 1 ) / ( l i − l τ )
The dimension of the irreducible representation U ν ( d ) reads f ν ( d ) = ∏ 1 ⩽ i < j ⩽ d ( ν i − ν j + j − i ) / ( j − i ), with ν i ≡ 0, ∀ i > l ( ν i ) (Weyl dimension formula). In particular, f ( . ) ( d ) = 1 [(.) is the empty partition of length 0 and U ( . ) ( d ) is the trivial representation], f ( 1 ) ( d ) = d, and f ( N ) ( d ) = ( N + d − 1 N ), ∀ N > 0 . The number f ν ( d ) can also be expressed in the form f ν ( d ) = s ν ( 1, …, 1 ), where ( 1, …, 1 ) is a d-uple and s ν ( x 1, …, x d ) is the Schur’s polynomial in the d variables x 1, …, x d associated to partition ν (an homogeneous symmetric polynomial of degree | ν |, with | ν | the weight of ν) [65]
The irreps of the subgroups Si ( 1 ⩽ i < N ) a Schur basis vector | ν, T ν, W ν ⟩ belongs to are given by the shapes of the SYT Tν restricted to only boxes 1 to i. Similarly, the irreps of the subgroups U(k) ( 1 ⩽ k < d ) the vector | ν, T ν, W ν ⟩ belongs to are given by the shapes of the SWT Wν restricted to only boxes 0 to k − 1
Vilenkin N J Klimyk A U 1992 Representation of Lie Groups and Special Functions vol 1-3 Kluwer Academic Publishers
f ν P σ [ F ^ ν ( W ν, W ν ′ ) ] = ∑ λ, T λ, W λ ∑ λ ′, T λ ′ ′, W λ ′ ′ ∑ T ν | λ, T λ, W λ ⟩ ⟨ λ, T λ, W λ | P ^ σ | ν, T ν, W ν ⟩ ⟨ ν, T ν, W ν ′ | P ^ σ † | λ ′, T λ ′ ′, W λ ′ ′ ⟩ ⟨ λ ′, T λ ′ ′, W λ ′ ′ | = ∑ ν, T ν, T ν ′, T ν | ν, T ν, W ν ⟩ ⟨ T ν | σ ^ | T ν ⟩ ⟨ T ν | σ ^ † | T ν ′ ⟩ ⟨ ν, T ν ′, W ν ′ | = f ν F ^ ν ( W ν, W ν ′ ), with σ ^ and | T ν ⟩ the representation operators and GT-basis states in the S ν -irrep of the symmetric group SN, respectively
Hence, replacing F ^ ν ( W ν, W ν ′ ) and F ^ ν ( W ν ′, W ν ) by ( F ^ ν ( W ν, W ν ′ ) + F ^ ν ( W ν ′, W ν ) ) / 2 and i ( F ^ ν ( W ν, W ν ′ ) − F ^ ν ( W ν ′, W ν ) ) / 2, ∀ ν ⊢ ( N, d ), W ν, W ν ′ ∈ W ν : W ν ≠ W ν ′, yields together with the operators F ^ ν ( W ν, W ν ) an orthonormal basis of PI Hermitian operators in the commutant L S N ( H ) . In addition, having Tr [ F ^ ν ( W ν, W ν ′ ) ] = f ν δ W ν, W ν ′, an orthogonal basis in the subspace of traceless PI Hermitian operators is straightfowardly obtained by further replacing all but one operators F ^ ν ( W ν, W ν ) by traceless linear combinations of them
The structure constant of the commutant operator algebra follows immediately: F ^ λ ( W λ, W λ ′ ) F ^ μ ( W μ, W μ ′ ) = ∑ ν, W ν, W ν ′ c λ, W λ, W λ ′; μ, W μ, W μ ′ ν, W ν, W ν ′ F ^ ν ( W ν, W ν ′ ), with f λ c λ, W λ, W λ ′; μ, W μ, W μ ′ ν, W ν, W ν ′ = δ λ, μ δ λ, ν δ W λ ′, W μ δ W λ, W ν δ W μ ′, W ν ′
The product of two ν-type operators is a ν-type operator and so is the Hermitian conjugate of a ν-type operator: each operator subspace L ν ( H ) is a ∗ -algebra of operators on H and a subalgebra of the commutant L S N ( H )
The diagram or shape of a partition ν ≡ ( ν 1, …, ν l ) ⊢ N is an array of N boxes arranged on l left-justified rows, with row i ( 1 ⩽ i ⩽ l ) containing νi boxes. The shape of a partition ν is usually denoted by the same symbol ν. An inner corner of a shape ν is a box ∈ ν whose removal leaves us with a valid partition shape. An outer corner of ν is a box ∉ ν whose addition produces a valid partition shape
Having ∑ λ − f λ − = f λ, we get in particular K 1 ^ ( λ, W λ, W ~ λ ) = N δ W λ, W ~ λ, so that equation (32) yields as expected from definition K 1 ^, 1 ^ [ F ^ ν ( W ν, W ν ′ ) ] = N F ^ ν ( W ν, W ν ′ ) . A similar expression as equation (32) is also directly obtained for the operator K 1 ^, Y ^ [ F ^ ν ( W ν, W ν ′ ) ] using the equality K 1 ^, Y ^ [ F ^ ν ( W ν, W ν ′ ) ] = K Y ^, 1 ^ [ F ^ ν ( W ν ′, W ν ) ] †
For any operator L ^, D L ^ † [ ρ ^ ] = L ^ † ρ ^ L ^ − 1 2 L ^ † L ^ ρ ^ − 1 2 ρ ^ L ^ † L ^, so that if L ^ is Hermitian, then D L ^ † = D L ^
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For any partition ν, ν ∓ τ denotes the partition obtained by the removal [addition] of the inner [outer] corner of ν at row τ. We have μ = ν ± τ ⇔ ν = μ ∓ τ, so that μ ∈ { ν ± } ⇔ ν ∈ { μ ∓ }
For any partition ν ∓, the row at which the removal [addition] of the inner [outer] corner of ν occurs is denoted by τ ν − / ν [ τ ν + / ν ]. We have τ ν ± / ν = τ ν / ν ±
The trace is invariant under cyclic permutations, so that Tr ( P σ [ A ^ ] ) = Tr ( A ^ ) . As a result, we get Tr ( A ^ PI † B ^ ) = Tr ( P σ [ A ^ PI † B ^ ] ) = Tr ( P σ [ A ^ PI † ] P σ [ B ^ ] ) = Tr ( A ^ PI † P σ [ B ^ ] ), ∀ σ . Alternatively, we can also write Tr ( A ^ PI † P σ [ B ^ ] ) = Tr ( P σ † [ A ^ PI ] † B ^ ) = Tr ( P σ − 1 [ A ^ PI ] † B ^ ) = Tr ( A ^ PI † B ^ )
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