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| In [[mathematics]] and [[physics]], in particular [[quantum information]], the term '''generalized Pauli matrices''' refers to families of matrices which generalize the (linear algebraic) properties of the [[Pauli matrices]]. Here, a few classes of such matrices are summarized.
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| == Generalized Gell-Mann matrices (Hermitian)==
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| === Construction ===
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| Let {{math|''E''<sub>''jk''</sub>}} be the matrix with 1 in the {{math|''jk''}}-th entry and 0 elsewhere. Consider the space of ''d''×''d'' complex matrices, {{math|ℂ<sup>''d''×''d''</sup>}}, for a fixed ''d''.
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| Define the following matrices,
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| * For {{math|''k'' < ''j''}}, {{math| ''f''<sub>''k,j''</sub><sup>''d''</sup> {{=}} ''E<sub>kj</sub>''+''E<sub>jk</sub>''}} .
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| * For {{math|''k'' > ''j''}}, {{math| ''f''<sub>''k,j''</sub><sup>''d''</sup> {{=}} − ''i'' (''E<sub>jk</sub>'' − ''E<sub>kj</sub>'')}} .
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| * Let {{math|''h''<sub>1</sub><sup>''d''</sup> {{=}} ''I''<sub>''d''</sub>}}, the identity matrix.
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| * For {{math|1 < ''k'' < ''d''}}, <math>h_k ^d = h ^{d-1} _k \oplus 0 ~.</math>
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| * For {{math|''k'' {{=}} ''d''}}, <math>~~~h_d ^d = \sqrt{\tfrac{2}{d(d-1)}} \left ( h_1 ^{d-1} \oplus (1-d)\right )~.</math>
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| The collection of matrices defined above are called the ''generalized Gell-Mann matrices'', in dimension {{mvar|d}}.
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| The symbol ⊕ (utilized in the [[Cartan subalgebra]] above) means [[Matrix_addition#Direct_sum|matrix direct sum]].
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| The generalized Gell-Mann matrices are [[Hermitian matrix|Hermitian]] and [[traceless]] by construction, just like the Pauli matrices. One can also check that they are orthogonal in the [[Hilbert-Schmidt operator|Hilbert-Schmidt]] [[inner product]] on {{math|ℂ<sup>''d''×''d''</sup>}}. By dimension count, one sees that they span the vector space of {{math|''d'' × ''d''}} complex matrices, <math>\mathfrak{gl}</math>(''d'',ℂ).
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| In dimensions ''d''=2 and 3, the above construction recovers the Pauli and [[Gell-Mann matrices]], respectively.
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| == A non-Hermitian generalization of Pauli matrices ==
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| The Pauli matrices <math>\sigma _1</math> and <math>\sigma _3</math> satisfy the following:
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| :<math>
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| \sigma _1 ^2 = \sigma _3 ^2 = I, \; \sigma _1 \sigma _3 = - \sigma _3 \sigma _1 = e^{\pi i} \sigma _3 \sigma_1.
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| </math>
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| The so-called [[Hadamard matrix|Walsh-Hadamard conjugation matrix]] is
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| :<math>
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| W = \tfrac{1}{\sqrt{2}}
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| \begin{bmatrix}
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| 1 & 1 \\ 1 & -1
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| \end{bmatrix}.
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| </math>
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| Like the Pauli matrices, ''W'' is both [[Hermitian matrix|Hermitian]] and [[Unitary matrix|unitary]]. <math>\sigma _1, \; \sigma _3</math> and ''W'' satisfy the relation
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| :<math>\; \sigma _1 = W \sigma _3 W^* .</math>
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| The goal now is to extend the above to higher dimensions, ''d'', a problem solved by [[James Joseph Sylvester|J. J. Sylvester]] (1882).
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| === Construction: The clock and shift matrices===
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| Fix the dimension {{mvar|d}} as before. Let {{math|''ω'' {{=}} exp(2''πi''/''d'')}}, a root of unity. Since {{math|''ω''<sup>''d''</sup> {{=}} 1}} and {{math|''ω'' ≠ 1}}, the sum of all roots annuls:
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| :<math>1 + \omega + \cdots + \omega ^{d-1} = 0 .</math>
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| Integer indices may then be cyclically identified mod {{mvar|d}}.
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| Now define, with Sylvester, the
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| '''shift matrix'''<ref>Sylvester, J. J., (1882), ''Johns Hopkins University Circulars'' '''I''': 241-242; ibid '''II''' (1883) 46;
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| ibid '''III''' (1884) 7-9. Summarized in ''The Collected Mathematics Papers of James Joseph Sylvester'' (Cambridge University Press, 1909) v '''III''' .
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| [http://quod.lib.umich.edu/u/umhistmath/aas8085.0003.001/664?rgn=full+text;view=pdf;q1=nonions online] and [http://quod.lib.umich.edu/u/umhistmath/AAS8085.0004.001/165?cite1=Sylvester;cite1restrict=author;rgn=full+text;view=pdf further].
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| </ref>
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| :<math>
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| \Sigma _1 =
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| \begin{bmatrix}
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| 0 & 0 & 0 & \cdots &0 & 1\\
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| 1 & 0 & 0 & \cdots & 0 & 0\\
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| 0 & 1 & 0 & \cdots & 0 & 0\\
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| 0 & 0 & 1 & \cdots & 0 & 0 \\
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| \vdots & \vdots & \vdots & \ddots &\vdots &\vdots \\
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| 0 & 0 &0 & \cdots & 1 & 0\\
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| \end{bmatrix}
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| </math>
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| and the '''clock matrix''',
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| :<math>
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| \Sigma _3 =
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| \begin{bmatrix}
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| 1 & 0 & 0 & \cdots & 0\\
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| 0 & \omega & 0 & \cdots & 0\\
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| 0 & 0 &\omega ^2 & \cdots & 0\\
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| \vdots & \vdots & \vdots & \ddots & \vdots\\
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| 0 & 0 & 0 & \cdots & \omega ^{d-1}
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| \end{bmatrix}.
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| </math>
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| These matrices generalize ''σ''<sub>1</sub> and ''σ''<sub>3</sub>, respectively.
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| Note that the unitarity and tracelessness of the two Pauli matrices is preserved, but not Hermiticity in dimensions higher than two. Since Pauli matrices describe [[Quaternions]], Sylvester dubbed the higher-dimensional analogs "nonions", "sedenions", etc.
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| These two matrices are also the cornerstone of '''quantum mechanical dynamics in finite-dimensional vector spaces'''<ref>[[Hermann Weyl|Weyl, H.]], "Quantenmechanik und Gruppentheorie", ''Zeitschrift für Physik'', '''46''' (1927) pp. 1–46, {{doi|10.1007/BF02055756}}.</ref><ref>Weyl, H., ''The Theory of Groups and Quantum Mechanics'' (Dover, New York, 1931)</ref><ref>{{cite doi|10.1007/BF00715110|noedit}}</ref> as formulated by [[Hermann Weyl]], and find routine applications in numerous areas of mathematical physics.<ref>For a serviceable review,
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| see Vourdas A. (2004), "Quantum systems with finite Hilbert space", ''Rep. Prog. Phys.'' '''67''' 267. doi: 10.1088/0034-4885/67/3/R03.</ref> The clock matrix amounts to the exponential of position in a "clock" of ''d'' hours, and the shift matrix is just the translation operator in that cyclic vector space, so the exponential of the momentum. They are (finite-dimensional) representations of the corresponding elements of the [[Heisenberg group]] on a ''d''-dimensional Hilbert space.
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| The following relations echo those of the Pauli matrices:
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| :<math>\Sigma _ 1 ^d = \Sigma _ 3 ^d = I</math>
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| and the braiding relation,
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| :<math>\; \Sigma_3 \Sigma _1 = \omega \Sigma_1 \Sigma _3 = e^{2 \pi i / d} \Sigma_1 \Sigma _3 ,</math>
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| the [[Stone–von_Neumann_theorem#Uniqueness_of_representation|Weyl formulation of the CCR]], or
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| :<math>\; \Sigma_3 \Sigma _1 \Sigma _3^{d-1} \Sigma_1 ^{d-1} = \omega ~.</math>
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| On the other hand, to generalize the Walsh-Hadamard matrix ''W'', note
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| :<math>
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| W = \tfrac{1}{\sqrt{2}}
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| \begin{bmatrix}
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| 1 & 1 \\ 1 & \omega ^{2 -1}
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| \end{bmatrix}
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| =
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| \tfrac{1}{\sqrt{2}}
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| \begin{bmatrix}
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| 1 & 1 \\ 1 & \omega ^{d -1}
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| \end{bmatrix}.
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| </math>
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| Define, again with Sylvester, the following analog matrix,<ref>J.J. Sylvester, J. J. (1867) . ''Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers.'' [[Philosophical Magazine]], 34:461–475. [http://www.tandfonline.com/doi/pdf/10.1080/14786446708639914 online]</ref> still denoted by ''W'' in a slight abuse of notation,
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| :<math>
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| W =
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| \frac{1}{\sqrt{d}}
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| \begin{bmatrix}
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| 1 & 1 & 1 & \cdots & 1\\
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| 1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)^2}\\
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| 1 & \omega^{d-2} & \omega^{2(d-2)} & \cdots & \omega^{(d-1)(d-2)}\\
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| \vdots & \vdots & \vdots & \ddots & \vdots \\
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| 1 &\omega &\omega ^2 & \cdots & \omega^{d-1}
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| \end{bmatrix}~.
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| </math>
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| It is evident that ''W'' is no longer Hermitian, but is still unitary. Direct calculation yields
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| :<math>\; \Sigma_1 = W \Sigma_3 W^* ~,</math>
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| which is the desired analog result. Thus, {{mvar|W}} , a [[Vandermonde matrix]], arrays the eigenvectors of {{math|Σ<sub>1</sub>}}, which has the same eigenvalues as {{math|Σ<sub>3</sub>}}.
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| When ''d'' = 2<sup>k</sup>, ''W'' * is precisely the matrix of the [[discrete Fourier transform#The unitary DFT|discrete Fourier transform]],
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| converting position coordinates to momentum coordinates and vice-versa.
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| The family of ''d'' <sup>2</sup> unitary (but non-Hermitian) independent matrices
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| {{Equation box 1
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| |indent =::
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| |equation = <math>
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| (\Sigma_1)^k (\Sigma_3)^j =\sum_{m=0}^{d-1} |m+k\rangle \omega^{jm} \langle m| ,
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| </math>|cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |bgcolor=#F9FFF7}}
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| provides Sylvester's well-known basis for <math>\mathfrak{gl}</math>(''d'',ℂ), known as "nonions" <math>\mathfrak{gl}</math>(3,ℂ), "sedenions" <math>\mathfrak{gl}</math>(4,ℂ), etc...<ref>{{cite doi|10.1063/1.528006|noedit}}</ref>
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| This basis can be systematically connected to the above Hermitian basis.<ref>{{cite doi|10.1063/1.528788|noedit}}</ref> (For instance, the powers of {{math|Σ<sub>3</sub>}}, the [[Cartan subalgebra]],
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| map to linear combinations of the {{math|''h''<sub>''k''</sub><sup>''d''</sup>}}s.) It can further be used to identify <math>\mathfrak{gl}</math>(''d'',ℂ) , as {{math|''d'' → ∞}}, with the algebra of [[Poisson brackets]].
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| == See also ==
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| * [[Hermitian matrix]]
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| * [[Bloch sphere]]
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| * [[Discrete Fourier transform]]
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| * [[Generalized Clifford algebra]]
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| * [[Circulant matrix]]
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| * [[Shift operator]]
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| == Notes ==
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| {{Reflist}}
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| {{DEFAULTSORT:Generalizations Of Pauli Matrices}}
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| [[Category:Linear algebra]]
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