Quantum information processing and composite quantum fields

Ramgoolam, Sanjaye; Sedlák, Michal

doi:10.1007/jhep01(2019)170

Cited by 5 publications

(2 citation statements)

References 39 publications

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“…There is a 1-parameter algebra K(n) relevant to 3-index tensor systems, which is closely related to Kronecker coefficients. The algebras A(m, n) were recently used to derive identities involving contents of Young diagrams, which have applications in quantum information processing tasks [55]: the present paper is another link between permutation algebras and information theoretic perspectives directly motivated, in the present case, by information theoretic questions in AdS/CFT. Another connection between the 2-matrix system and small black holes in AdS/CFT is proposed in [56].…”

Section: Discussionmentioning

confidence: 93%

BPS states, conserved charges and centres of symmetric group algebras

Kemp

Ramgoolam

2020

J. High Energ. Phys.

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In N = 4 SYM with U(N) gauge symmetry, the multiplicity of half-BPS states with fixed dimension can be labelled by Young diagrams and can be distinguished using conserved charges corresponding to Casimirs of U(N). The information theoretic study of LLM geometries and superstars in the dual AdS 5 × S 5 background has raised a number of questions about the distinguishability of Young diagrams when a finite set of Casimirs are known. Using Schur-Weyl duality relations between unitary groups and symmetric groups, these questions translate into structural questions about the centres of symmetric group algebras. We obtain algebraic and computational results about these structural properties and related Shannon entropies, and generate associated number sequences. A characterization of Young diagrams in terms of content distribution functions relates these number sequences to diophantine equations. These content distribution functions can be visualized as connected, segmented, open strings in content space.

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Section: Discussionmentioning

confidence: 93%

BPS states, conserved charges and centres of symmetric group algebras

Kemp

Ramgoolam

2020

J. High Energ. Phys.

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“…With all the results on matrix correlators, it may come as no surprise that a similar approach extends to tensor models and yields applications even beyond the realm of theoretical physics. In quantum information processing [31] and linguistics [32] [33] one also finds that matrix models and symmetric groups gather a renewed attention.…”

Section: Introductionmentioning

confidence: 99%

On the counting tensor model observables as U(N) and O(N) classical invariants

Geloun¹

2020

Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019)

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Real or complex tensor model observables, the backbone of the tensor theory space, are classical (unitary, orthogonal, symplectic) Lie group invariants. These observables represent as colored graphs, and that representation gives an handle to study their combinatorial, topological and algebraic properties. We give here an overview of the symmetric group-theoretic formulation of the enumeration of unitary and orthogonal invariant observables which turns out to bear a rich structure. From their counting formulae, one finds a correspondence with topological field theory on 2-cellular complexes that brings other interpretations of the same countings. Furthermore, tensor model observables span an algebra that turns out to be semi-simple. Dealing with complex tensors, we discuss the representation theoretic base of the algebra making explicit its Wedderburn-Artin decomposition. The real case is more subtle as a base of its Wedderburn-Artin decomposition is yet unknown.

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A note on some new hook-content identities

Sedlák

Bisio

2020

J Algebr Comb

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Based on the work of A. Vershik[1], we introduce two new combinatorial identities. We show how these identities can be used to prove a new hook-content identity. The main motivation for deriving this identity was a particular optimization problem in the field of quantum information processing.for the representation of GL(d) × S n , where V λ is either zero or a polynomial irreducible representation of GL(d), S λ is an irreducible representation of S n and λ runs over the partitions of n and is conveniently represented by Young diagram. Both the symmetric group and GL(d) (and especially its compact subgroups U (d) and SU (d)) are of paramount importance in theoretical physics, especially in quantum mechanics. For example, S n is a fundamental symmetry of systems of identical particles and unitary groups represent the set of reversible (finite-dimensional) transformations. Therefore, it is not surprising that physics community keeps a steady interest in the representation theory of these groups and in the Shur-Weyl duality, from the early work of Weyl [3] up to the most recent applications in quantum computing and quantum information processing. For example, Equation (1) denotes a subsystem decomposition (induced by the symmetry of the system-environment interaction) in which one can identify error free subsystems [4,5], or the relevant subsystems for quantum estimation [6]. These are just a couple of examples of a much wider variety of applications (see e.g. Ref.[7] for a review). In the light of this discussion it is clear that the dimensions of the irreducible spaces V λ and S λ , are, more often than not, a crucial piece of information. The value of dim(S λ ) and dim(V λ ) are given by the hook length formula [8] and the hook-content formula [9] respectively.

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Quantum information processing and composite quantum fields

Cited by 5 publications

References 39 publications

BPS states, conserved charges and centres of symmetric group algebras

BPS states, conserved charges and centres of symmetric group algebras

On the counting tensor model observables as U(N) and O(N) classical invariants

A note on some new hook-content identities

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