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African Journal of Biomedical Research

Schur-Weyl Duality Theorem

Swmaih Almarshi

Published Jan 1, 2025• 0 citations
Publisher

Abstract

This paper introduce the idea of Schur Weyl Duality Theorem which is a mathematical theorem in representation theory. And it has a numerous applications in quantum information theory. Specifically, in classical information theory, the method of types can be used to carry out some tasks such as estimating probability distribution, randomness concentration and data compression.

DOI
10.53555/ajbr.v28i2s.6403

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This paper introduce the idea of Schur Weyl Duality Theorem which is a mathematical theorem in representation theory. And it has a numerous applications in quantum information theory. Specifically, in classical information theory, the method of types can be used to carry out some tasks such as estimating probability distribution, randomness concentration and data compression.
A number of important results in quantum information theory can be connected quite elegantly to the representation theory of the symmetric group through a quantum analogue of the classical information-theoretic "method of types" that arises naturally from the Schur-Weyl duality.We will give a brief introduction to this connection and briefly discuss some of the results that follow from it, such as quantum source compression rates, entanglement concentration rates, quantum entropy inequalities, and the admissisble spectra of partial density matrices from pure, multipartite entangled states.
This thesis investigates quantum cloning and related quantum entanglement problems using core concepts of representation theory, in particular those associated with the symmetric group. The research explores Schur-Weyl duality and its extensions, which allow efficient representation and manipulation of quantum systems, serving as a valuable tool for quantum information theory. A primary application of Schur-Weyl duality is the quantum cloning problem, which is studied for both the 1→21 \to 21→2 and the more general 1→N1 \to N1→N cases, providing new insights into the constraints imposed by the no-cloning theorem. The investigation extends to a more general quantum entanglement problem on a complete graph.
Entanglement distillation from noisy bipartite states and reliable quantum communication over noisy channels are fundamental tasks in quantum information theory. Yet existing coding schemes typically require prior knowledge of the underlying state or channel, or assume independent or weakly correlated quantum information sources. Here, using Schur-Weyl duality, we establish a universal quantum error-correction principle that overcomes these limitations. We show that the irreducible permutation spaces arising from the local Schur-Weyl decomposition of multiple copies of an unknown bipartite mixed state form maximally entangled quantum code spaces whose error structure is governed entirely by representation theory. Correcting these symmetry-resolved errors yields protocols for entanglement distillation and quantum communication that achieve coherent-information rates with exponentially vanishing error and optimal second-order corrections, even in the case of almost i.i.d. resources. Our approach also provides sharp finite-copy guarantees that are worst-case optimal at every finite block length, including in the sample-efficient regime, where the number of copies scales only polynomially with the number of qubits. Together, these results establish Schur-Weyl duality as a general mechanism for universal, robust, and optimal quantum coding, unifying entanglement distillation, quantum error correction, and quantum communication within a common representation-theoretic framework.
This thesis explores the interplay between representation theory and quantum information. Specifically, we focus on mixed Schur–Weyl duality, which considers the action of the unitary group on mixed tensors. This setting naturally arises in quantum information tasks involving unitary-equivariant channels, such as port-based teleportation, quantum majority vote, and universal transposition of unitary operators. A key contribution of this thesis is an explicit derivation of the action of the generators of the partially transposed permutation matrix algebra—the commutant of the mixed unitary action—in the Gelfand–Tsetlin basis. As another key result of this thesis, we develop efficient quantum circuits for the mixed quantum Schur transform, a novel primitive in quantum information. The key ingredient of our construction is new efficient circuits for the dual Clebsch–Gordan transform of the unitary group. A significant application of our findings is the construction of efficient quantum algorithms for port-based teleportation, a variant of quantum teleportation that eliminates the need for corrective operations. Another application is a symmetry reduction of semidefinite optimisation problems with unitary equivariance symmetry. Finally, we study the extendibility of quantum states possessing unitary, mixed unitary, or orthogonal symmetry on the complete graph. We obtain analytically the exact maximum values for projections onto the maximally entangled state and the antisymmetric state for each of the three symmetry classes. This thesis demonstrates the usefulness of mixed Schur–Weyl duality in quantum information and computing. We expect that our tools will help address other problems in other areas of quantum information processing, such as communication, cryptography, and simulation.
Exploiting permutation invariance to reduce the exponential scaling of semidefinite programs in quantum information has emerged as a powerful computational technique. In this work, we develop a systematic framework for using this reduction via Schur-Weyl duality for optimization problems, and establish methods that allow one to work fully inside the permutation invariant subspace while performing operations such as (partially) applying channels and taking (partial) traces, or computing expressions like the quantum relative entropy. We then apply our techniques to the problem of computing efficient lower bounds on the channel fidelity over nnn parallel uses of a quantum channel. The algorithm, which we call symmetric seesaw method, exploits permutation-invariant codes to yield improved lower bounds on the channel fidelity over nnn uses of the depolarizing and amplitude-damping channel in the regime of tens of channel uses, and was used in [arxiv:2604.27042] to demonstrate non-asymptotic superactivation of quantum capacity for n=17n = 17n=17. An implementation of our methods, aimed at being suitable for various quantum information theoretic optimization problems, is also available as an open-source Python package.
Abstract Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t -fold tensor powers U⊗tU^{\otimes t}U⊗t U ⊗ t of all unitaries U∈U(d)U\in U(d)U∈U(d) U ∈ U ( d ) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries . The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group. We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems , showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well). We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana ) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem . We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.