The Schur-Weyl states belong to a special class of states with a symmetry described by two Young and Weyl tableaux. Representation of physical systems in Hilbert space spanned on these states enables to extract quantum information hidden in nonlocal degrees of freedom. Such property can be very useful in a broad range of problems in Quantum Computations, especially in quantum algorithms constructions, therefore it is very important to know exact form of these states. Moreover, they allow to reduce significantly the size of eigenproblem, or in general, diminishing the representation matrix of any physical quantities, represented in the symmetric or unitary group algebra. Here we present a new method of Schur-Weyl states construction in a spin chain system representation. Our approach is based on the fundamental shift operators out of which one can build Clebsch-Gordan coefficients for the unitary group U(n) and then derive appropriate Schur-Weyl state probability amplitudes.The Schur-Weyl states belong to a special class of states with a symmetry described by two Young and Weyl tableaux. Representation of physical systems in Hilbert space spanned on these states enables to extract quantum information hidden in nonlocal degrees of freedom. Such property can be very useful in a broad range of problems in Quantum Computations, especially in quantum algorithms constructions, therefore it is very important to know exact form of these states. Moreover, they allow to reduce significantly the size of eigenproblem, or in general, diminishing the representation matrix of any physical quantities, represented in the symmetric or unitary group algebra. Here we present a new method of Schur-Weyl states construction in a spin chain system representation. Our approach is based on the fundamental shift operators out of which one can build Clebsch-Gordan coefficients for the unitary group U(n) and then derive appropriate Schur-Weyl state probability amplitudes.

arXiv:2004.14799Apr 30, 20200 citations

Entanglement of one-dimensional spin chains

Tiefeng Xu, Yu Zhou, Yaxun Zhou +1 more

Physics Letters AJun 1, 20024 citations

Single-spin entanglement

G. B. Furman, V. Meerovich, Vladimir Sokolovsky

arXiv:1707.02407Quantum Information ProcessingJul 18, 20176 citations

A Multigraph Approach for Performing the Quantum Schur Transform

Edward Pearce-Crump

We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of $(\mathbb{C}^d)^{\otimes n}$ and the Schur-Weyl basis of $(\mathbb{C}^d)^{\otimes n}$ . We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a major improvement on Louck's formula for calculating the transition amplitudes between two standard Weyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph for the case $d=2$ , merely by looking at the entries in the two tableaux. The key theoretical component that underpins our results is the discovery of a branching rule for the Schur-Weyl states, which we call the Schur-Weyl branching rule. This branching rule allows us to perform the change of basis transformation described above in a straightforward manner for any $n$ and $d$ .We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of $(\mathbb{C}^d)^{\otimes n}$ and the Schur-Weyl basis of $(\mathbb{C}^d)^{\otimes n}$. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a major improvement on Louck's formula for calculating the transition amplitudes between two standard Weyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph for the case $d=2$, merely by looking at the entries in the two tableaux. The key theoretical component that underpins our results is the discovery of a branching rule for the Schur-Weyl states, which we call the Schur-Weyl branching rule. This branching rule allows us to perform the change of basis transformation described above in a straightforward manner for any $n$ and $d$.

arXiv:2204.10694Apr 22, 20220 citations

Entanglement distillation using Schur-Weyl decomposition for three qubits

Mejia, Jose

The aim of this work is to examine the exponential rates at which entanglement distillation occur in three-qubits systems. The approach we will follow to elucidate the entanglement concentration is based on the Schur-Weyl decomposition and the Keyl-Werner theorem. In order to clearly state the main results of this work we will have to introduce the notion of SLOCC equivalence and the covariant algebra which will allow us to classify states in different \emph{entanglement classes}. Our main results comprehend the asymptotic rates for the probability of being in an invariant subspace in the Wedderburn decomposition of the state with effective Kronecker coefficient $g_{αβγ}=1$ . We will also present a combinatorial formula for the effective Kronecker coefficient for two-row Young diagrams. With this work we hope to shed some light in the way a multipartite system is entangled and at which rate can we convey entanglement for further applications.The aim of this work is to examine the exponential rates at which entanglement distillation occur in three-qubits systems. The approach we will follow to elucidate the entanglement concentration is based on the Schur-Weyl decomposition and the Keyl-Werner theorem. In order to clearly state the main results of this work we will have to introduce the notion of SLOCC equivalence and the covariant algebra which will allow us to classify states in different \emph{entanglement classes}. Our main results comprehend the asymptotic rates for the probability of being in an invariant subspace in the Wedderburn decomposition of the state with effective Kronecker coefficient $g_{αβγ}=1$. We will also present a combinatorial formula for the effective Kronecker coefficient for two-row Young diagrams. With this work we hope to shed some light in the way a multipartite system is entangled and at which rate can we convey entanglement for further applications.

arXiv:1610.09552Oct 29, 20160 citations

Universal quantum coding

Jacopo Rizzo, Filippo Girardi, Ludovico Lami +2 more

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.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.

quant-pharXiv:2609.38038Sep 30, 20260 citations

(\mathbb{C}^d)^{\otimes n}

and the Schur-Weyl basis of

(\mathbb{C}^d)^{\otimes n}

. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a major improvement on Louck's formula for calculating the transition amplitudes between two standard Weyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph for the case

d=2

, merely by looking at the entries in the two tableaux. The key theoretical component that underpins our results is the discovery of a branching rule for the Schur-Weyl states, which we call the Schur-Weyl branching rule. This branching rule allows us to perform the change of basis transformation described above in a straightforward manner for any

n

and

d

g_{αβγ}=1

. We will also present a combinatorial formula for the effective Kronecker coefficient for two-row Young diagrams. With this work we hope to shed some light in the way a multipartite system is entangled and at which rate can we convey entanglement for further applications.