Unitary matrix

In linear algebra, a complex square matrix $U$ is unitary if its conjugate transpose $U *$ is also its inverse, that is, if

U^{*}U=UU^{*}=UU^{-1}=I,

where $I$ is the identity matrix.

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

U^{\dagger }U=UU^{\dagger }=I.

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

Properties

For any unitary matrix $U$ of finite size, the following hold:

Given two complex vectors $x$ and $y$ , multiplication by $U$ preserves their inner product; that is, $⟨ U x, U y ⟩ = ⟨ x, y ⟩$ .
$U$ is normal ( $U^{*}U=UU^{*}$ ).
$U$ 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 $U=VDV^{*},$ where $V$ is unitary, and $D$ is diagonal and unitary.
$\left|\det(U)\right|=1$ .
Its eigenspaces are orthogonal.
$U$ can be written as $U = e iH$ , where $e$ indicates the matrix exponential, $i$ is the imaginary unit, and $H$ is a Hermitian matrix.

For any nonnegative integer $n$ , the set of all $n \times n$ unitary matrices with matrix multiplication forms a group, called the unitary group $U(n)$ .

Any square matrix with unit Euclidean norm is the average of two unitary matrices.[1]

Equivalent conditions

If U is a square, complex matrix, then the following conditions are equivalent:[2]

$U$ is unitary.
$U^{*}$ is unitary.
$U$ is invertible with $U^{-1}=U^{*}$ .
The columns of $U$ form an orthonormal basis of $\mathbb {C} ^{n}$ with respect to the usual inner product. In other words, $U^{*}U=I$ .
The rows of $U$ form an orthonormal basis of $\mathbb {C} ^{n}$ with respect to the usual inner product. In other words, $UU^{*}=I$ .
$U$ is an isometry with respect to the usual norm. That is, $\|Ux\|_{2}=\|x\|_{2}$ for all $x\in \mathbb {C} ^{n}$ , where ${\textstyle \|x\|_{2}={\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}}$ .
$U$ is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of $U$ ) with eigenvalues lying on the unit circle.

Elementary constructions

2 × 2 unitary matrix

The general expression of a 2 × 2 unitary matrix is

U={\begin{bmatrix}a&b\\-e^{i\varphi }b^{*}&e^{i\varphi }a^{*}\\\end{bmatrix}},\qquad \left|a\right|^{2}+\left|b\right|^{2}=1,

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 determinant of such a matrix is

\det(U)=e^{i\varphi }.

The sub-group of those elements $U$ with $\det(U)=1$ is called the special unitary group SU(2).

The matrix $U$ can also be written in this alternative form:

U=e^{i\varphi /2}{\begin{bmatrix}e^{i\varphi _{1}}\cos \theta &e^{i\varphi _{2}}\sin \theta \\-e^{-i\varphi _{2}}\sin \theta &e^{-i\varphi _{1}}\cos \theta \\\end{bmatrix}},

which, by introducing φ₁ = ψ + Δ and φ₂ = ψ − Δ, takes the following factorization:

U=e^{i\varphi /2}{\begin{bmatrix}e^{i\psi }&0\\0&e^{-i\psi }\end{bmatrix}}{\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}e^{i\Delta }&0\\0&e^{-i\Delta }\end{bmatrix}}.

This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle $θ$ .

Another factorization is[3]

U={\begin{bmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \\\end{bmatrix}}{\begin{bmatrix}e^{i\xi }&0\\0&e^{i\zeta }\end{bmatrix}}{\begin{bmatrix}\cos \beta &\sin \beta \\-\sin \beta &\cos \beta \\\end{bmatrix}}.

Many other factorizations of a unitary matrix in basic matrices are possible.[4][5][6][7]

References

Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices". Linear and Multilinear Algebra. 50 (4): 321–326. doi:10.1080/03081080290025507.
Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis. Cambridge University Press. doi:10.1017/9781139020411. ISBN 9781139020411.
Führ, Hartmut; Rzeszotnik, Ziemowit (2018). "A note on factoring unitary matrices". Linear Algebra and Its Applications. 547: 32–44. doi:10.1016/j.laa.2018.02.017. ISSN 0024-3795.
Williams, Colin P. (2011), Williams, Colin P. (ed.), "Quantum Gates", Explorations in Quantum Computing, Texts in Computer Science, London: Springer, p. 82, doi:10.1007/978-1-84628-887-6_2, ISBN 978-1-84628-887-6, retrieved 2021-05-14
Nielsen, Michael A.; Chuang, Isaac (2010). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. p. 20. ISBN 978-1-10700-217-3. OCLC 43641333.
Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald (1995-11-01). "Elementary gates for quantum computation". Physical Review A. American Physical Society (APS). 52 (5): 3457–3467. arXiv:quant-ph/9503016. doi:10.1103/physreva.52.3457. ISSN 1050-2947., page 8
Marvian, Iman (2022-01-10). "Restrictions on realizable unitary operations imposed by symmetry and locality". Nature Physics: 1–7. arXiv:2003.05524. doi:10.1038/s41567-021-01464-0. ISSN 1745-2481.See also:"Forbidden by symmetry"

External links

Weisstein, Eric W. "Unitary Matrix". MathWorld. Todd Rowland.
Ivanova, O. A. (2001) [1994], "Unitary matrix", Encyclopedia of Mathematics, EMS Press
"Show that the eigenvalues of a unitary matrix have modulus 1". Stack Exchange. March 28, 2016.

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[1] Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices". Linear and Multilinear Algebra. 50 (4): 321–326. doi:10.1080/03081080290025507.

[2] Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis. Cambridge University Press. doi:10.1017/9781139020411. ISBN 9781139020411.

[3] Führ, Hartmut; Rzeszotnik, Ziemowit (2018). "A note on factoring unitary matrices". Linear Algebra and Its Applications. 547: 32–44. doi:10.1016/j.laa.2018.02.017. ISSN 0024-3795.

[4] Williams, Colin P. (2011), Williams, Colin P. (ed.), "Quantum Gates", Explorations in Quantum Computing, Texts in Computer Science, London: Springer, p. 82, doi:10.1007/978-1-84628-887-6_2, ISBN 978-1-84628-887-6, retrieved 2021-05-14

[5] Nielsen, Michael A.; Chuang, Isaac (2010). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. p. 20. ISBN 978-1-10700-217-3. OCLC 43641333.

[Barenco-6] Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald (1995-11-01). "Elementary gates for quantum computation". Physical Review A. American Physical Society (APS). 52 (5): 3457–3467. arXiv:quant-ph/9503016. doi:10.1103/physreva.52.3457. ISSN 1050-2947., page 8

[7] Marvian, Iman (2022-01-10). "Restrictions on realizable unitary operations imposed by symmetry and locality". Nature Physics: 1–7. arXiv:2003.05524. doi:10.1038/s41567-021-01464-0. ISSN 1745-2481.See also:"Forbidden by symmetry"

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