Content deleted Content added
Changed most equations to TeX. Parts to comply with MOS:MATH, parts for consistency/aesthetic reasons |
|||
| (5 intermediate revisions by 4 users not shown) | |||
Line 1:
{{Short description|Families of matrices in mathematics, physics, and quantum information}}
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 matrix|Pauli matrices]]. Here, a few classes of such matrices are summarized.
Line 9:
:<math>\sigma_a^{(n)} = I^{(1)} \otimes \dotsm \otimes I^{(n-1)} \otimes \sigma_a \otimes I^{(n+1)} \otimes \dotsm \otimes I^{(N)}, \qquad a = 1, 2, 3</math>
to refer to the operator on <math>V_N</math> that acts as a Pauli matrix on the <math>n</math>th qubit and the identity on all other qubits. We can also use <math>a = 0</math> for the identity, i.e., for any <math>n</math> we use <math display="inline">\sigma_0^{(n)} = \bigotimes_{m=1}^N I^{(m)}</math>. Then the multi-qubit Pauli matrices are all matrices of the form
:<math>\sigma_{\,\vec a} := \
i.e., for <math>\vec{a}</math> a vector of integers between 0 and 4. Thus there are <math>4^N</math> such generalized Pauli matrices if we include the identity <math display="inline">I = \bigotimes_{m=1}^N I^{(m)}</math> and <math>4^N - 1</math> if we do not.
=== Notations ===
In quantum computation, it is conventional to denote the Pauli matrices with single upper case letters
:<math>I \equiv \sigma_0, \qquad X \equiv \sigma_1, \qquad Y \equiv \sigma_2, \qquad Z \equiv \sigma_3.</math>
This allows subscripts on Pauli matrices to indicate the qubit index. For example, in a system with 3 qubits,
:<math>X_1 \equiv X \otimes I \otimes I, \qquad Z_2 \equiv I \otimes Z \otimes I.</math>
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 <math>\otimes</math> can be omitted, i.e. unsubscripted Pauli matrices written consecutively represents tensor product rather than matrix product. For example:
:<math>XZI \equiv X_1Z_2 = X \otimes Z \otimes I.</math>
== Higher spin matrices (Hermitian) ==
The traditional Pauli matrices are the matrix representation of the <math>\mathfrak{su}(2)</math> Lie algebra generators <math>J_x</math>, <math>J_y</math>, and <math>J_z</math> in the 2-dimensional [[irreducible representation|irreducible]] [[Representation theory of SU(2)|representation of SU(2)]], corresponding to a [[spin-1/2]] particle. These generate the Lie group [[Special unitary group#The group SU(2)|SU(2)]].
For a general particle of spin <math>s=0,1/2,1,3/2,2,\ldots</math>, one instead utilizes the <math>2s+1</math>-dimensional irreducible representation.
Line 41 ⟶ 49:
</math>
The collection of matrices defined above without the identity matrix are called the ''generalized Gell-Mann matrices'', in dimension <math>d</math>.<ref>{{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}}
| doi = 10.1088/1751-8113/41/23/235303
| issn = 1751-8121
| |||