Complex matrix whose conjugate transpose equals its inverse
For matrices with orthogonality over the real number field, see
orthogonal matrix. For the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1, see
unitarity.
In physics, especially in quantum mechanics, the conjugate transpose is referred to as the
Hermitian adjoint of a matrix and is denoted by a
dagger (†), so the equation above is written
A complex matrix U is special unitary if it is unitary and its
matrix determinant equals 1.
U is
diagonalizable; that is, U is
unitarily similar to a diagonal matrix, as a consequence of the
spectral theorem. Thus, U has a decomposition of the form where V is unitary, and D is diagonal and unitary.
. That is, will be on the unit circle of the complex plane.
One general expression of a 2 × 2 unitary matrix is
which depends on 4 real parameters (the phase of a, the phase of b, the relative magnitude between a and b, and the angle φ). The form is configured so the
determinant of such a matrix is
^Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices". Linear and Multilinear Algebra. 50 (4): 321–326.
doi:
10.1080/03081080290025507.
S2CID120125694.
^Williams, Colin P. (2011). "Quantum gates". In Williams, Colin P. (ed.). Explorations in Quantum Computing. Texts in Computer Science. London, UK: Springer. p. 82.
doi:
10.1007/978-1-84628-887-6_2.
ISBN978-1-84628-887-6.
^Jarlskog, Cecilia (2006). "Recursive parameterisation and invariant phases of unitary matrices".
arXiv:math-ph/0510034.
^Alhambra, Álvaro M. (10 January 2022).
"Forbidden by symmetry". News & Views. Nature Physics. 18 (3): 235–236.
doi:
10.1038/s41567-021-01483-x.
ISSN1745-2481.
S2CID256745894. The physics of large systems is often understood as the outcome of the local operations among its components. Now, it is shown that this picture may be incomplete in quantum systems whose interactions are constrained by symmetries.