Quantum Information Processing and Composite Quantum Fields

Some beautiful identities involving hook contents of Young diagrams have been found in the field of quantum information processing, along with a combinatorial proof. We here give a representation theoretic proof of these identities and a number of generalizations. Our proof is based on trace identities for elements belonging to a class of permutation centralizer algebras. These algebras have been found to underlie the combinatorics of composite gauge invariant operators in quantum field theory, with applications in the AdS/CFT correspondence. Based on these algebras, we discuss some analogies between quantum information processing tasks and the combinatorics of composite quantum fields and argue that this can be fruitful interface between quantum information and quantum field theory, with implications for AdS/CFT.


Identities from Quantum Information Theory
Some beautiful symmetric group identities have been found in the subject of quantum information processing [1]. A combinatoric proof has been given [2].
The key identity is where r is a Young diagram with n boxes. R is a Young diagram with n + 1 boxes, d r is the dimension of the irreducible representation (irrep) of the symmetric group S n associated with the Young diagram r, d R is the dimension of the S n+1 irrep associated with R. Finally, g(r, , R) is the Littlewood-Richardson coefficient coupling the V U r ⊗ V U with V U R , where V U r , V U , V U R are the U(N) (or GL(N)) irreps associated with the respective Young diagrams and c (R, r) is the content (difference of the column and row number) of the additional box in Young diagram R which is not contained in r.

Representation theoretic proof
Consider the tensor product of U(N) irreps.
It is an irredicible representation of U(N) × U(N). Under the diagonal subgroup, it is reducible. The decomposition is given by We also have a representation of S n × S 1 associated with (r, ). S n × S 1 is a subgroup of S n+1 . We can induce a representation of S n+1 from this representation of S n × S 1 . This is a reducible representation of S n+1 . The multiplicity of an irrep R in this reducible rep is g(r, , R) (see for example [3]). This means that (n + 1)d r = R⊢(n+1) g(r, , R)d R (2.4) On the LHS we used the fact that the dimension of the rep of S n+1 induced from the irrep V Sn On the RHS we use the decomposition in terms of irreps of S n+1 . Useful relation between Dim N r and d r is (2.6) where f r is the product over the boxes of the Young diagram of (N − c ) and c = j − i for a box at row i and column j. Similarly, Thus the ratio reads where c (R, r) is the content of the box by which R and r differ. We can also write c (R, r) =

Trace of a sum of permutations in
In the group algebra C(S n+m ), an interesting sub-algebra is formed by the subspace which is invariant under conjugation by elements of S n × S m . This is an example of what was called a permutation centralizer algebra (PCA) in [4,5], and which has many applications in the study of gauge invariant operators with relevance to AdS/CFT. It is denoted A(n, m). For m = 1 we have A(n, 1). The element n a=1 (a, n + 1), which we will use here, is an element of A(n, 1). It is very interesting that PCAs are also finding a use in quantum information processing (QIP). We discuss this further in Section 5. The Consider the projector P r in the group algebra of S n , denoted C(S n ) We choose an embedding C(S n ) → C(S n+1 ), where S n acts on {1, 2, · · · , n} and S n+1 acts on {1, · · · , n, n + 1}, and construct an element in C(S n+1 ) (a, n + 1) (2.14) We then consider the trace in W This is a sum of permutations in S n ⊂ S n+1 . Doing the multiplication of P r with n a=1 (a, n + 1) and taking the trace, we get See Appendix eq. A.10 for the derivation. Let us consider another way to compute the same trace. We observe that n a=1 (a, n + 1) = T is the sum of all permutations in S n which consist of a single swop. Using Eqs. (A.11) and (A.16) we get We made use of the fact This is a special case k = 1, n 1 = n, n 2 = 1 of an identity A.27 derived in Appendix A. Now we have Dividing by Dim N r on both sides we obtain The first term in the above equation is zero due to Eq. (2.11). Thus, we get which is the desired identity.

A generalization with S m × S n
Consider S m × S n → S m+n and the vector space W = V ⊗m+n N . For Young diagrams r, s with m, n boxes respectively, consider which gives the decomposition into irreducible representations of the U(N) which acts diagonally on V U r ⊗ V U s . First consider the dimension on both sides of the equation The expression f R frfs has a large N expansion 1 + O(1/N). Comparing the two equations, we conclude that all the 1/N corrections in lead to identities. For example, There will also be higher order equations: at each order in 1/N the equation involves Littlewood-Richardson coefficients and Young diagram contents, hence just data pertaining to the symmetric groups. All the equations arise from the large N expansion of Now consider the trace It follows that On the other hand, using (A.11), (A.12) and (A.27), we derive Comparing (3.10) and (3.11), we have Consider the large N expansion.
Fraction f R frfs is the only term in the summand of the RHS in (3.12) which contains N dependence. Considering the order N term of the RHS, we get zero using the first identity in (3.6). Considering the constant term, we get Equivalently, Now it is easy to see that this is a generalization of (2.22).

Multi-partite generalization
Consider S n 1 × . . . × S n k → S n 1 +...+n k and tensor space W = V ⊗n 1 +...+n k N . For Young diagrams r 1 , r 2 , · · · , r k with n 1 , n 2 , · · · , n k boxes, we have representations Considering the decomposition of the tensor product under the diagonal action of U(N), we have The multiplicities g(r 1 , · · · , r k ; R) can be expressed in terms of Littlewood-Richardson coefficients. For example By considering the dimensions on the two sides of (4.1), we have Using (2.6) the above equation can be rewritten as Now we switch to considering the induction of representations of symmetric groups associated with the above Young diagrams. Using induction of V Comparing equations (4.4), (4.5) we have Again we consider a large N expansion of Validity of Eq. (4.6) for all N ≥ k j=1 n j leads to identities for every power of 1/N. For example, the 1/N terms give It is useful to be explicit about the embedding of S n 1 × · · · S n k in S n 1 +···n k . Let S n 1 be the group of permutations of [n 1 ] = {1, 2, · · · , n 1 }. Let S n 2 be the group of permutations of [n 2 ] = {n 1 + 1, · · · , n 1 + n 2 }. And S n i for 1 ≤ i ≤ k be the group of permutations of We also let S n 1 +n 2 +···n 1 +n 2 +···n k be the group of permutations of {1, 2, · · · , n 1 + n 2 + · · · + n k }. Let us evaluate the trace in two ways. We observe that Direct calculation (analogous to Eq.(3.11)) gives On the other hand, using (A.11), (A.12) and (A.27), we have Comparing (4.12) and (4.13), we have Consider the large N expansion.
In Eq. (4.14) the only term in the summand of the RHS, which contains N dependence is the fraction f R fr 1 ...fr k . Considering the order N term of the RHS, we get zero using the first identity in (4.8). Considering the constant term, we get This can be equivalently rewritten as Wee see that this is a generalization of (3.15).

Permutation centralizer algebras, Composite gauge invariant operators and AdS/CFT
The identities above have been derived by calculating the trace in tensor spaces of some elements in the group algebra of S n 1 +n 2 +···+n k , which are invariant under conjugation of by permutations in S n 1 × S n 2 × · · · S n k . Let us specialise to the case k = 2. The subspace of C(S m+n ) which is invariant under conjugation by S m × S n forms an algebra which has been studied in detail in [4,5]. The motivation came from the role these played in the construction of bases of gauge invariant operators which diagonalise an inner product coming from free quantum field theory [6,7,8,9,10]. Key insights into the construction of these bases came from the physics of strings attached to branes in the context of the AdS/CFT correspondence [11,12,13]. Consider quantum fields X, Y which are N × N matrices transforming in the adjoint of a U(N) gauge symmetry.
For large N, the space of gauge invariant operators is in 1-1 correspondence with the elements of A(m, n). One way to count the dimension of this space is to count the traces, which amounts to counting cyclic words built from two letters. As explained in the references above (and reviewed in [14]) the dimension of the space of gauge invariant operators is also given in terms of Littlewood-Richardson coefficients g(R 1 , R 2 , R 3 ) which are multiplicities for the U(N) representation associated with Young diagram R 3 (having m + n boxes) to appear in the tensor product of R 1 ⊗ R 2 , where R 1 and R 2 have m and n boxes.

Dim(A(m, n))
The finite N counting is given simply by restricting R 3 to have no more than N rows. This follows by application of Schur-Weyl duality. The reason these permutation equivalences arise in constructing gauge invariants is that if we consider a general operator The bosonic symmetry leads to an equivalence for all γ ∈ S m × S n . Fourier transformation on A(m, n) using representation theory of symmetric groups leads to a Young diagram basis Q R 3 R 1 ,R 2 ;ν 1 ,ν 2 , with 1 ≤ ν 1 , ν 2 ≤ g(R 1 , R 2 , R 3 ). Representation theoretic formulae for the Fourier coefficients giving the transformation from trace basis to the Young diagram basis are given in the papers above. Structural questions about A(m, n), notably regarding minimal sets of generators for maximal commuting subalgebras, are related to the question of how many charges (generalized Casimirs) are needed to specify a state in the 2-matrix system [15,4]. This can be considered to be a measure of complexity of this state space.
A(m, n) is an example of a permutation centralizer algebra. A(n 1 , · · · , n k ) is analogously defined, and is relevant to gauge invariant operators made from k flavours of matrix quantum fields. The elements we have used to get the identities above are in fact special elements which are central in A(n 1 , · · · , n k ). The central subspace is spanned by products of elements from the centre of C(S i n i ) with elements from the centre of i C(S n i ). (these properties of the centre of PCAs are explained in [4] and the special role of the centre in terms of the complexity of correlator computations is discussed). The traces of central elements can thus be obtained using character formulae for symmetric groups [16].
In fact any central element of A(n 1 , · · · , n k ) will lead to an identity of the kind we discussed in the earlier sections.
We may make a few remarks about the analogies which are emerging between quantum information processing and gauge invariant composite quantum fields through the shared feature of permutation centralizer algebras and Schur-Weyl duality. In QIP, multiple uses of a unitary operation occur in the computational tasks like oracle based algorithms, estimation problems or arbitrary protocols, which should perform equally well for all states or channels 3 . In QFT, these multiple uses of a unitary U arise through the action on a polynomial composite quantum field.
We thus have a first simple interesting analogy, somewhat simplified from the above setup, which seems to hold promise of wider implications :

A unitary quantum channel is analogous to a unitary gauge transformation of an elementary quantum field
In the QIP problem, multiple uses of channels occur within multi step quantum protocols (i.e. within networks of quantum channels). In the composite operator problem of QFT, multiple uses occur in different copies of the elementary quantum field occuring within a composite. We thus have a second simplified analogy to think about.

A multi step quantum protocol is analogous to a composite local operator
The simplicity of these analogies seems to suggest there should be wider applications. For example, for the multi-partite generalization in Section 4 we may ask, is there an appropriate optimization task in quantum information theory involving multiple quantum devices interacting with each other in some way, which employs the multi-partitite identities (4.17) -generalizing the use of (2.22) in perfect probabilistic storage and retrieval [1]?
As noted earlier in this section, structural questions about PCAs have been used to characterize the complexity of quantum states in multi-matrix systems, which have a Young diagram basis as well as a trace basis. The Schur-Weyl duality transformation form tensor product basis to the Young diagram basis for V ⊗n N has been studied from a quantum information perspective [22]. The question of efficient quantum circuits having polynomial number of gates has been addressed. Similar questions can be studied for the transformation from trace basis to Young diagram basis for multi-matrix systems. The definition of complexity of quantum circuits requires a choice of a basic gate set. A reasonable choice in the context of AdS/CFT would be to consider the quantum dilatation operator at one loop and higher loops (see [23] for the 1-loop dilatation operator and [24,25] for applications to brane physics of the action of the one-loop dilatation operator on the Young diagram basis). A challenge would be to identify an AdS/CFT dual for such a notion of circuit complexity involving the quantum dilatation operator in the 2-matrix system.
As we have seen, permutation centralizer algebras, with their traces illuminating aspects of perfect probabilistic storing/retrieving and their structure constants having information about correlators of relevance to AdS/CFT, provide an intriguing mathematical connection between quantum information and AdS/CFT. An interesting question is whether there is a physical interpretation of this mathematical connection between QIP and AdS/CFT. In this connection, it is worth noting that studies of quantum state spaces in AdS/CFT from information theoretic perspectives have been undertaken [26,27,28], primarily in the context of state spaces associated with invariants of a single matrix and the related free fermion system. More broadly on this theme the work of [29] has motivated a rich exploration of connections between AdS/CFT and quantum information. For example it has led to the idea of space-time emerging from entanglement [30] with implications for AdS/CFT holography [31] and black hole physics [32].
the KEK theory group for hospitality during the completion of this project. We are grateful for useful discussions to David Berenstein, Robert de Mello Koch, Costis Papageorgakis, Rodolfo Russo, Masaki Shigemori. MS acknowledges the support by the QuantERA project HIPHOP (project ID 731473), projects QETWORK (APVV-14-0878), MAXAP (VEGA 2/0173/17), GRUPIK (MUNI/G/1211/2017) and GAČR No. GA16-22211S. MS is grateful to A. Bisio and M. Ziman for fruitful discussions and collaborative work, which led to formulation of the identity (1.1), which is re-derived and generalized in this manuscript. We are grateful to the organizers of the Quantum Physics and Logic (QPL2017) conference where this interdisciplinary collaboration was initiated.
A some facts about U (N ), S n and the tensor product V ⊗n N This section is brief review of some key facts about the representation theory of symmetric groups, Unitary groups and their relations following from Schur-Weyl duality. More details are in mathematical physics references such as [33] or mathematics texts such as [3]. We will start with a useful piece of notation. We will use r ⊢ n to denote a partition r of n. Partitions of n correspond to Young diagrams with n boxes, which have row lengths r 1 ≥ r 2 ≥ · · · , with n = r 1 + r 2 + · · · . Young diagrams with n boxes correspond to irreducible representations of S n . Letting V N be the fundamental representation of U(N), the tensor product V ⊗n N is a representation of the diagonal U(N) acting as as well as the symmetric group of all permutations of n objects (S n ). These two actions commute with each other, which leads to Schur-Weyl duality This gives the decomposition of V ⊗n N into irreducible reps of U(N) × S n as a direct sum labelled by Young diagrams.
A useful formula for the dimension of unitary group U(N) irreps in terms of characters of S n is where C σ is the number of cycles in the permutation σ. This follows from Schur-Weyl duality (A.2). To project to a fixed Young diagram, we can use a projector element in the group If we apply this to the states in V ⊗n N and take a trace, we need to calculate ( usual summation convention, so the i indices are summed from 1 to N). To understand the last line, it is instructive to do some examples at n = 2. If σ = (1)(2), the trace is If σ = (12), the trace is We need to understand some multiplications in the group algebra of S n+1 . The group algebra consists of formal sums of group elements with complex coefficients. What happens when a generic group element σ in the S n subgroup is multiplied with (a, n + 1) for a ∈ {1, · · · , n} ? Example at n = 3, with σ = (1, 2, 3) (a, n + 1) This implies that (a, n + 1)) = 1 n! n a=1 σ∈Sn Central elements ( such as T (Sn) 2 ) multiplying a projector give normalized characters times the projector.
To see this, note that both LHS and RHS are central elements in the group algebra of S n , as a result they are determined by their irreducible characters, and we can easily verify that the two sides have the same irreducible characters. The normalized character is known [16] to be the sum of contents An implication of Schur-Weyl duality is that the Littlewood-Richardson coefficients g(r 1 , r 2 ; R) which give the multiplicities of U(N) tensor product decompositions also have an interpretation purely in terms of symmetric groups. They are the reduction multiplicities for the decomposition of the irrep V Sn R in terms of the subgroup S n 1 × S n 2 . We may express this as where V R r 1 ,r 2 is the multiplicity space, of dimension g(r 1 , r 2 , R). Considering the trace in V Sn R tr(P r 1 • P r 2 ) (A. 15) we arrive at g(r 1 , r 2 , R) = 1 n 1 !n 2 ! σ 1 ∈Sn 1 σ 2 ∈Sn 2 χ r 1 (σ 1 )χ r 2 (σ 2 )χ R (σ 1 • σ 2 ) (A. 16) More generally V U r 1 ⊗ V U r 2 ⊗ · · · ⊗ V U r k = ⊕ R g(r 1 , r 2 , · · · r k ; R)V U R (A.17) and g(r 1 , r 2 , · · · r k ; R) = . Consider the trace where R is a Young diagram with n boxes, r i are Young diagrams with n i boxes, P R and P r i are the corresponding projectors.Using Schur-Weyl duality When the projector P R acts on W , we project to a single factor V U R ⊗V Sn R . We can decompose V Sn R in terms of S n 1 × S n 2 × · · · × S n k . The multiplicities are the Littlewood-Richardson coefficients.
V U R ⊗ V Sn R = V U R ⊗ r 1 ,r 2 ,··· ,r k g(r 1 , r 2 , · · · , r k ; R) V Sn 1 It follows that tr W P R P r 1 d r 1 ⊗ P r 2 d r 2 ⊗ · · · ⊗ P r k d r k = (Dim N R)g(r 1 , r 2 , · · · , r k ; R) (A. 22) This is an important identity we use in the paper. To make the above proof more explicit, we can expand the projectors in terms of characters.
For any permutation τ ∈ S n , Schur-Weyl duality implies that Hence tr W P R P r 1 d r 1 ⊗ P r 2 d r 2 ⊗ · · · ⊗ P r k d r k = σ 1 ∈Sn 1 · · · σ k ∈Sn k σ∈Sn Using the character orthogonality relation which holds for any ρ ∈ S n , we have = g(r 1 , r 2 , · · · , r k ; R)Dim N R (A. 27)