Expression of a Tensor Commutation Matrix in Terms of the Generalized Gell-Mann Matrices

Rakotonirina, Christian

doi:10.1155/2007/20672

Cited by 9 publications

(5 citation statements)

References 3 publications

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“…in which U is just an element of a fundamental representation of U(d) on C d , whereas O p is an image of a representation of a permutation group of N elements by orthogonal matrices on (C d ) ⊗N known as tensor permutation operators (see [19][20][21]). Note that π U(d) acts collectively (at the same manner) on all N subsystems, whereas π SN acts globally (permutes subsystems); nevertheless, both representations are reducible.…”

Section: N Independent Finite-dimensional Quantum Systemsmentioning

confidence: 99%

“…The upper bounds on dimensions D J L and D J V can be easily evaluated to D J L ⩽ 2J + 1 and D J V ⩽ |λ CM | N , where |λ CM | denotes the number of different possible values of helicity for each of the particles whereas N denotes total number of particles . However in specific situations both dimensions can be strictly lower due to vanishing of respective Poincare-Clebsch-Gordan coefficients in (21). In order to introduce an invariant encoding it is advisable to separate momentum and angular momentum parts of the state in full analogy with the case of equal momenta encoding (35): T L ρ s,J,P = ˆL dΛf (Λ) N 2 Λ,p |s; ΛP ΛP; s|…”

Section: Non-equal Momentum Encodingsmentioning

confidence: 99%

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Relativistically invariant encoding of quantum information revisited

Schlichtholz,

Markiewicz

2024

New J. Phys.

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In this work, we provide a detailed analysis of the issue of encoding of quantum information which is invariant with respect to arbitrary Lorentz transformations. We significantly extend already known results and provide compliments where necessary. In particular, we introduce novel schemes for invariant encoding which utilize so-called pair-wise helicity -- a physical parameter characterizing pairs of electric-magnetic charges. We also introduce new schemes for ordinary massive and massless particles based on states with fixed total momentum, in contrast to all protocols already proposed, which assumed equal momenta of all the particles involved in the encoding scheme. Moreover, we provide a systematic discussion of already existing protocols and show directly that they are invariant with respect to Lorentz transformations drawn according to any distribution, a fact which was not manifestly shown in previous works.

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Section: N Independent Finite-dimensional Quantum Systemsmentioning

confidence: 99%

Section: Non-equal Momentum Encodingsmentioning

confidence: 99%

Relativistically invariant encoding of quantum information revisited

Schlichtholz,

Markiewicz

2024

New J. Phys.

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“…For further considerations it is necessary to describe the above action by a matrix transformation. Therefore we introduce the following matrix representation of the symmetric group on the complex vector space (C d ) ⊗t known as tensor permutation operators [35,36].…”

Section: Schur-weyl Duality and Duality Of Averaging Over Collective ...mentioning

confidence: 99%

“…Due to lemma 3 the operator T U (ρ) commutes with both operators K ⊗t and A ⊗t n , therefore it can be factored out of the integral in the formula (35):…”

Section: Exact Form Of the Twirling Map For Slocc Operationsmentioning

confidence: 99%

Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur–Weyl duality

Markiewicz,

Przewocki

2023

J. Phys. A: Math. Theor.

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It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging. The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs. In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces.

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“…The d 2 − d witnesses (4) can detect all the coherent quantum states ρ ∈ D(C d ). These optimal coherence witnesses correspond to specific traceless generators of the standard special unitary group SU (d) [30,31]. Together with the operator…”

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confidence: 99%

Tomographic Witnessing and Holographic Quantifying of Coherence

Wang,

Zhou,

et al. 2021

Preprint

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The detection and quantification of quantum coherence play significant roles in quantum information processing. We present an efficient way of tomographic witnessing for both theoretical and experimental detection of coherence. We prove that a coherence witness is optimal if and only if all of its diagonal elements are zero. Naturally, we obtain a bona fide homographic measure of coherence given by the sum of the absolute values of the real and the imaginary parts of the non-diagonal entries of a density matrix, together with its interesting relations with other coherence measures like l1 norm coherence and robust of coherence.

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Expression of a Tensor Commutation Matrix in Terms of the Generalized Gell-Mann Matrices

Abstract: We have expressed the tensor commutation matrix n ⊗ n as linear combination of the tensor products of the generalized Gell-Mann matrices. The tensor commutation matrices 3 ⊗ 2 and 2 ⊗ 3 have been expressed in terms of the classical Gell-Mann matrices and the Pauli matrices.

Cited by 9 publications

References 3 publications

Relativistically invariant encoding of quantum information revisited

Relativistically invariant encoding of quantum information revisited

Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur–Weyl duality

Tomographic Witnessing and Holographic Quantifying of Coherence

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