Generalizations of Pauli matrices

This method of generalizing the Pauli matrices refers to a generalization from a single 2-level system (qubit) to multiple such systems. In particular, the generalized Pauli matrices for a group of $N$ qubits is just the set of matrices generated by all possible products of Pauli matrices on any of the qubits.{{cite journal |last1=Brown |first1=Adam R. |last2=Susskind |first2=Leonard |title=Second law of quantum complexity |journal=Physical Review D |date=2018-04-25 |volume=97 |issue=8 |pages=086015 |doi=10.1103/PhysRevD.97.086015|arxiv=1701.01107 |bibcode=2018PhRvD..97h6015B |s2cid=119199949 }}

The vector space of a single qubit is $V\_1\; =\; \backslash mathbb\{C\}^2$ and the vector space of $N$ qubits is $V\_N\; =\; \backslash left(\backslash mathbb\{C\}^2\backslash right)^\{\backslash otimes\; N\}\backslash cong\; \backslash mathbb\{C\}^\{2^N\}$. We use the tensor product notation

:$\backslash sigma\_a^\{(n)\}\; =\; I^\{(1)\}\; \backslash otimes\; \backslash dotsm\; \backslash otimes\; I^\{(n-1)\}\; \backslash otimes\; \backslash sigma\_a\; \backslash otimes\; I^\{(n+1)\}\; \backslash otimes\; \backslash dotsm\; \backslash otimes\; I^\{(N)\},\; \backslash qquad\; a\; =\; 1,\; 2,\; 3$

to refer to the operator on $V\_N$ that acts as a Pauli matrix on the $n$th qubit and the identity on all other qubits. We can also use $a\; =\; 0$ for the identity, i.e., for any $n$ we use $\backslash sigma\_0^\{(n)\}\; =\; \backslash bigotimes\_\{m=1\}^N\; I^\{(m)\}$. Then the multi-qubit Pauli matrices are all matrices of the form

:$\backslash sigma\_\{\backslash ,\backslash vec\; a\}\; :=\; \backslash prod\_\{n=1\}^N\; \backslash sigma\_\{a\_n\}^\{(n)\}\; =\; \backslash sigma\_\{a\_1\}\; \backslash otimes\; \backslash dotsm\; \backslash otimes\; \backslash sigma\_\{a\_N\},\; \backslash qquad\; \backslash vec\{a\}\; =\; (a\_1,\; \backslash ldots,\; a\_N)\; \backslash in\; \backslash \{0,\; 1,\; 2,\; 3\backslash \}^\{\backslash times\; N\}$,

i.e., for $\backslash vec\{a\}$ a vector of integers between 0 and 4. Thus there are $4^N$ such generalized Pauli matrices if we include the identity $I\; =\; \backslash bigotimes\_\{m=1\}^N\; I^\{(m)\}$ and $4^N\; -\; 1$ if we do not.

In quantum computation, it is conventional to denote the Pauli matrices with single upper case letters

:$I\; \backslash equiv\; \backslash sigma\_0,\; \backslash qquad\; X\; \backslash equiv\; \backslash sigma\_1,\; \backslash qquad\; Y\; \backslash equiv\; \backslash sigma\_2,\; \backslash qquad\; Z\; \backslash equiv\; \backslash sigma\_3.$

This allows subscripts on Pauli matrices to indicate the qubit index. For example, in a system with 3 qubits,

:$X\_1\; \backslash equiv\; X\; \backslash otimes\; I\; \backslash otimes\; I,\; \backslash qquad\; Z\_2\; \backslash equiv\; I\; \backslash otimes\; Z\; \backslash otimes\; I.$

Multi-qubit Pauli matrices can be written as products of single-qubit Paulis on disjoint qubits. Alternatively, when it is clear from context, the tensor product symbol $\backslash otimes$ can be omitted, i.e. unsubscripted Pauli matrices written consecutively represents tensor product rather than matrix product. For example:

:$XZI\; \backslash equiv\; X\_1Z\_2\; =\; X\; \backslash otimes\; Z\; \backslash otimes\; I.$

The traditional Pauli matrices are the matrix representation of the $\backslash mathfrak\{su\}(2)$ Lie algebra generators $J\_x$, $J\_y$, and $J\_z$ in the 2-dimensional irreducible representation of SU(2), corresponding to a spin-1/2 particle. These generate the Lie group SU(2).

For a general particle of spin $s=0,1/2,1,3/2,2,\backslash ldots$, one instead utilizes the $2s+1$-dimensional irreducible representation.

{{main article|Spin (physics)#Higher spins}}

This method of generalizing the Pauli matrices refers to a generalization from 2-level systems (Pauli matrices acting on qubits) to 3-level systems (Gell-Mann matrices acting on qutrits) and generic $d$-level systems (generalized Gell-Mann matrices acting on qudits).

Let $E\_\{jk\}$ be the matrix with 1 in the {{math|jk}}-th entry and 0 elsewhere. Consider the space of $d\backslash times\; d$ complex matrices, $\backslash Complex^\{d\backslash times\; d\}$, for a fixed $d$.

Define the following matrices,

: $f\_\{k,j\}^\{\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; d\}\; =$

\begin{cases}E_{kj} + E_{jk} & {\text{for }}k < j,\\

-i(E_{jk} - E_{kj})&{\text{for }} k > j.\end{cases}

and

: $h\_\{k\}^\{\backslash ,\backslash ,\backslash ,\; d\}\; =$

\begin{cases}I_d & {\text{for }} k = 1,\\

h_{k}^{\,\,\, d-1} \oplus 0 &{\text{for }} 1 < k < d, \\

\sqrt{\tfrac{2}{d(d - 1)}} \left( h_1^{d-1} \oplus (1 - d)\right) = \sqrt{\tfrac{2}{d(d - 1)}} \left( I_{d-1} \oplus (1 - d)\right) &{\text{for }} k = d

\end{cases}

The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension $d$.{{Cite journal | doi = 10.1016/S0375-9601(03)00941-1| title = The Bloch vector for N-level systems| journal = Physics Letters A| volume = 314| issue = 5–6| pages = 339–349| year = 2003| last1 = Kimura | first1 = G. |arxiv = quant-ph/0301152 |bibcode = 2003PhLA..314..339K | s2cid = 119063531}}{{Cite journal

| doi = 10.1088/1751-8113/41/23/235303

| issn = 1751-8121

| last = Bertlmann | first = Reinhold A.

| author2 = Philipp Krammer

| title = Bloch vectors for qudits

| journal = Journal of Physics A: Mathematical and Theoretical

| volume = 41 | issue = 23 | pages = 235303

| date = 2008-06-13

| bibcode = 2008JPhA...41w5303B

| arxiv = 0806.1174 | s2cid = 118603188

}} The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum.

The generalized Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli matrices. One can also check that they are orthogonal in the Hilbert–Schmidt inner product on $\backslash Complex^\{d\backslash times\; d\}$. By dimension count, one sees that they span the vector space of $d\backslash times\; d$ complex matrices, $\backslash mathfrak\{gl\}(d,\backslash Complex)$. They then provide a Lie-algebra-generator basis acting on the fundamental representation of $\backslash mathfrak\{su\}(d)$.

In dimensions $d$ = 2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.

A particularly notable generalization of the Pauli matrices was constructed by James Joseph Sylvester in 1882.Sylvester, J. J., (1882), Johns Hopkins University Circulars I: 241-242; ibid II (1883) 46;

ibid III (1884) 7–9. Summarized in The Collected Mathematics Papers of James Joseph Sylvester (Cambridge University Press, 1909) v III .

[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].

These are known as "Weyl–Heisenberg matrices" as well as "generalized Pauli matrices".{{Cite journal|last=Appleby|first=D. M.|date=May 2005|title=Symmetric informationally complete–positive operator valued measures and the extended Clifford group|url=http://aip.scitation.org/doi/10.1063/1.1896384|journal=Journal of Mathematical Physics|language=en|volume=46|issue=5|pages=052107|arxiv=quant-ph/0412001|bibcode=2005JMP....46e2107A|doi=10.1063/1.1896384|issn=0022-2488}}{{Cite journal|last1=Howard|first1=Mark|last2=Vala|first2=Jiri|date=2012-08-15|title=Qudit versions of the qubit π / 8 gate|url=https://link.aps.org/doi/10.1103/PhysRevA.86.022316|journal=Physical Review A|language=en|volume=86|issue=2|pages=022316|arxiv=1206.1598|bibcode=2012PhRvA..86b2316H|doi=10.1103/PhysRevA.86.022316|s2cid=56324846|issn=1050-2947}}

The Pauli matrices $\backslash sigma\; \_1$ and $\backslash sigma\; \_3$ satisfy the following:

:$$

\sigma_1^2 = \sigma_3^2 = I, \quad \sigma_1 \sigma_3 = - \sigma_3 \sigma_1 = e^{\pi i} \sigma_3 \sigma_1.

The so-called Walsh–Hadamard conjugation matrix is

:$$

W = \frac{1}{\sqrt{2}}

\begin{bmatrix}

1 & 1 \\ 1 & -1

\end{bmatrix}.

Like the Pauli matrices, $W$ is both Hermitian and unitary. $\backslash sigma\_1,\; \backslash ;\; \backslash sigma\_3$ and $W$ satisfy the relation

:$\backslash ;\; \backslash sigma\_1\; =\; W\; \backslash sigma\_3\; W^*\; .$

The goal now is to extend the above to higher dimensions, $d$.

{{see also |Generalized Clifford algebra}}

Fix the dimension $d$ as before. Let $\backslash omega\; =\; \backslash exp(2\; \backslash pi\; i\; /\; d)$, a root of unity. Since $\backslash omega^d\; =\; 1$ and $\backslash omega\; \backslash neq\; 1$, the sum of all roots annuls:

:$1\; +\; \backslash omega\; +\; \backslash cdots\; +\; \backslash omega\; ^\{d-1\}\; =\; 0\; .$

Integer indices may then be cyclically identified mod {{mvar|d}}.

Now define, with Sylvester, the shift matrix

:$$

\Sigma _1 =

\begin{bmatrix}

0 & 0 & 0 & \cdots & 0 & 1\\

1 & 0 & 0 & \cdots & 0 & 0\\

0 & 1 & 0 & \cdots & 0 & 0\\

0 & 0 & 1 & \cdots & 0 & 0\\

\vdots & \vdots & \vdots & \ddots &\vdots &\vdots\\

0 & 0 & 0 & \cdots & 1 & 0\\

\end{bmatrix}

and the clock matrix,

:$$

\Sigma _3 =

\begin{bmatrix}

1 & 0 & 0 & \cdots & 0\\

0 & \omega & 0 & \cdots & 0\\

0 & 0 & \omega^2 & \cdots & 0\\

\vdots & \vdots & \vdots & \ddots & \vdots\\

0 & 0 & 0 & \cdots & \omega^{d-1}

\end{bmatrix}.

These matrices generalize $\backslash sigma\_1$'' and $\backslash sigma\_3$, respectively.

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.

These two matrices are also the cornerstone of quantum mechanical dynamics in finite-dimensional vector spacesWeyl, H., "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1–46, {{doi|10.1007/BF02055756}}.Weyl, H., The Theory of Groups and Quantum Mechanics (Dover, New York, 1931){{Cite journal | last1 = Santhanam | first1 = T. S. | last2 = Tekumalla | first2 = A. R. | doi = 10.1007/BF00715110 | title = Quantum mechanics in finite dimensions | journal = Foundations of Physics | volume = 6 | issue = 5 | pages = 583 | year = 1976 |bibcode = 1976FoPh....6..583S | s2cid = 119936801 }} as formulated by Hermann Weyl, and they find routine applications in numerous areas of mathematical physics.For a serviceable review, see Vourdas A. (2004), "Quantum systems with finite Hilbert space", Rep. Prog. Phys. 67 267. {{doi|10.1088/0034-4885/67/3/R03}}. 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 Weyl-Heisenberg group on a $d$-dimensional Hilbert space.

The following relations echo and generalize those of the Pauli matrices:

:$\backslash Sigma\_1^d\; =\; \backslash Sigma\_3^d\; =\; I$

and the braiding relation,

:$\backslash Sigma\_3\; \backslash Sigma\_1\; =\; \backslash omega\; \backslash Sigma\_1\; \backslash Sigma\_3\; =\; e^\{2\backslash pi\; i\; /\; d\}\; \backslash Sigma\_1\; \backslash Sigma\_3\; ,$

the Weyl formulation of the CCR, and can be rewritten as

:$\backslash Sigma\_3\; \backslash Sigma\_1\; \backslash Sigma\_3^\{d-1\}\; \backslash Sigma\_1^\{d-1\}\; =\; \backslash omega\; ~.$

On the other hand, to generalize the Walsh–Hadamard matrix $W$, note

:$$

W = \frac{1}{\sqrt{2}}

\begin{bmatrix}

1 & 1 \\ 1 & \omega^{2-1}

\end{bmatrix}

=

\frac{1}{\sqrt{2}}

\begin{bmatrix}

1 & 1 \\ 1 & \omega^{d-1}

\end{bmatrix}.

Define, again with Sylvester, the following analog matrix,{{Cite journal|doi=10.1080/14786446708639914|title=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|year=1867|last1=Sylvester|first1=J.J.|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=34|issue=232|pages=461–475}} still denoted by $W$ in a slight abuse of notation,

:$$

W =

\frac{1}{\sqrt{d}}

\begin{bmatrix}

1 & 1 & 1 & \cdots & 1\\

1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)^2}\\

1 & \omega^{d-2} & \omega^{2(d-2)} & \cdots & \omega^{(d-1)(d-2)}\\

\vdots & \vdots & \vdots & \ddots & \vdots\\

1 & \omega & \omega ^2 & \cdots & \omega^{d-1}

\end{bmatrix}~.

It is evident that $W$ is no longer Hermitian, but is still unitary. Direct calculation yields

:$\backslash Sigma\_1\; =\; W\; \backslash Sigma\_3\; W^*\; ~,$

which is the desired analog result. Thus, $W$, a Vandermonde matrix, arrays the eigenvectors of $\backslash Sigma\_1$, which has the same eigenvalues as $\backslash Sigma\_3$.

When $d\; =\; 2^k$, $W^*$ is precisely the discrete Fourier transform matrix, converting position coordinates to momentum coordinates and vice versa.

The complete family of $d^2$ unitary (but non-Hermitian) independent matrices $\backslash \{\backslash sigma\_\{k,j\}\backslash \}\_\{k,j=1\}^d$ is defined as follows:

{{Equation box 1

|indent =::

|equation = $$

\sigma_{k,j}:=

\left(\Sigma_1\right)^k \left(\Sigma_3\right)^j = \sum_{m=0}^{d-1} |m+k\rangle \omega^{jm} \langle m|.

| cellpadding= 6

| border

| border colour = #0073CF

| bgcolor=#F9FFF7

}}

This provides Sylvester's well-known trace-orthogonal basis for $\backslash mathfrak\{gl\}(d,\backslash Complex)$, known as "nonions" $\backslash mathfrak\{gl\}(3,\backslash Complex)$, "sedenions" $\backslash mathfrak\{gl\}(4,\backslash Complex)$, etc...{{Cite journal | last1 = Patera | first1 = J. | last2 = Zassenhaus | first2 = H. | doi = 10.1063/1.528006 | title = The Pauli matrices in n dimensions and finest gradings of simple Lie algebras of type An−1 | journal = Journal of Mathematical Physics | volume = 29 | issue = 3 | pages = 665 | year = 1988 |bibcode = 1988JMP....29..665P }}Since all indices are defined cyclically mod {{mvar|d}}, $\backslash mathrm\{tr\}\backslash Sigma\_1^j\; \backslash Sigma\_3^k\; \backslash Sigma\_1^m\; \backslash Sigma\_3^n\; =\; \backslash omega^\{km\}\; d\; ~\backslash delta\_\{j+m,0\}\; \backslash delta\_\{k+n,0\}$.

This basis can be systematically connected to the above Hermitian basis.{{Cite journal | last1 = Fairlie | first1 = D. B. | last2 = Fletcher | first2 = P. | last3 = Zachos | first3 = C. K. | doi = 10.1063/1.528788 | title = Infinite-dimensional algebras and a trigonometric basis for the classical Lie algebras | journal = Journal of Mathematical Physics | volume = 31 | issue = 5 | pages = 1088 | year = 1990 |bibcode = 1990JMP....31.1088F }} (For instance, the powers of $\backslash Sigma\_3$, the Cartan subalgebra,

map to linear combinations of the $h\_\{k\}^\{\backslash ,\backslash ,\backslash ,\; d\}$ matrices.) It can further be used to identify $\backslash mathfrak\{gl\}(d,\backslash Complex)$, as $d\; \backslash to\; \backslash infty$, with the algebra of Poisson brackets.

With respect to the Hilbert–Schmidt inner product on operators, $\backslash langle\; A,\; B\; \backslash rangle\_\backslash text\{HS\}\; =\; \backslash operatorname\{Tr\}(A^*\; B)$, Sylvester's generalized Pauli operators are orthogonal and normalized to $\backslash sqrt\{d\}$:

:$\backslash langle\; \backslash sigma\_\{k,j\},\; \backslash sigma\_\{k\text{'},j\text{'}\}\; \backslash rangle\_\{\backslash text\{HS\}\}\; =\; \backslash delta\_\{k\; k\text{'}\}\backslash delta\_\{j\; j\text{'}\}\; \backslash |\; \backslash sigma\_\{k,j\}\backslash |^2\_\{\backslash text\{HS\}\}\; =\; d\; \backslash delta\_\{k\; k\text{'}\}\backslash delta\_\{j\; j\text{'}\}$.

This can be checked directly from the above definition of $\backslash sigma\_\{k,j\}$.

{{portal|Physics}}

• {{section link|Heisenberg group|Heisenberg group modulo an odd prime p}}
• Hermitian matrix
• Bloch sphere
• Discrete Fourier transform
• Generalized Clifford algebra
• Weyl–Brauer matrices
• Circulant matrix
• Shift operator
• Quantum Fourier transform
• {{section link|3D rotation group|A note on Lie algebras}}

{{Reflist}}

{{DEFAULTSORT:Generalizations Of Pauli Matrices}}

Category:Linear algebra

Category:Mathematical physics