Unitary matrix

In linear algebra, an invertible complex square matrix $U$ is unitary if its matrix inverse $U -1$ equals its conjugate transpose $U *$ , that is, if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U^* U = UU^* = 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U^\dagger U = UU^\dagger = I.}

A complex matrix $U$ is special unitary if it is unitary and its matrix determinant equals $1$ .

For real numbers, the 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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U = VDV^*,} where $V$ is unitary, and $D$ is diagonal and unitary.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \left|\det(U)\right| = 1} . That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \det(U)} will be on the unit circle of the complex plane.
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 $U(n)$ .

Every 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]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U} is unitary.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U^*} is unitary.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U} is invertible with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U^{-1} = U^*} .
The columns of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U} form an orthonormal basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \Complex^n} with respect to the usual inner product. In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U^*U = I} .
The rows of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U} form an orthonormal basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \Complex^n} with respect to the usual inner product. In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle UU^* = I} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U} is an isometry with respect to the usual norm. That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \|Ux\|_2 = \|x\|_2} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle x \in \Complex^n} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\textstyle \|x\|_2 = \sqrt{\sum_{i=1}^n |x_i|^2}} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U} is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U} ) with eigenvalues lying on the unit circle.

Elementary constructions

2 × 2 unitary matrix

One general expression of a 2 × 2 unitary matrix is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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 form is configured so the determinant of such a matrix is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \det(U) = e^{i \varphi} ~. }

The sub-group of those elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \ U\ } with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \ \det(U) = 1\ } is called the special unitary group SU(2).

Among several alternative forms, the matrix $U$ can be written in this form: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \ U = e^{i\varphi / 2} \begin{bmatrix} e^{i\alpha} \cos \theta & e^{i\beta} \sin \theta \\ -e^{-i\beta} \sin \theta & e^{-i\alpha} \cos \theta \\ \end{bmatrix}\ ,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \ e^{i\alpha} \cos \theta = a\ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \ e^{i\beta} \sin \theta = b\ ,} above, and the angles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \ \varphi, \alpha, \beta, \theta\ } can take any values.

By introducing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \ \alpha = \psi + \delta\ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \ \beta = \psi - \delta\ ,} has the following factorization:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U = \begin{bmatrix} \cos \rho & -\sin \rho \\ \sin \rho & \;\cos \rho \\ \end{bmatrix} \begin{bmatrix} e^{i\xi} & 0 \\ 0 & e^{i\zeta} \end{bmatrix} \begin{bmatrix} \;\cos \sigma & \sin \sigma \\ -\sin \sigma & \cos \sigma \\ \end{bmatrix} ~. }

Many other factorizations of a unitary matrix in basic matrices are possible.^[4]^[5]^[6]^[7]^[8]^[9]

References

^ Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices". Linear and Multilinear Algebra. 50 (4): 321–326. doi:10.1080/03081080290025507. S2CID 120125694.
^ Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis. Cambridge University Press. doi:10.1017/CBO9781139020411. 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. S2CID 125455174.
^ 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. ISBN 978-1-84628-887-6.
^ Nielsen, M.A.; Chuang, Isaac (2010). Quantum Computation and Quantum Information. Cambridge, UK: 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; et al. (1 November 1995). "Elementary gates for quantum computation". Physical Review A. 52 (5). American Physical Society (APS): 3457–3467, esp.p. 3465. arXiv:quant-ph/9503016. doi:10.1103/physreva.52.3457. ISSN 1050-2947. PMID 9912645. S2CID 8764584.
^ Marvian, Iman (10 January 2022). "Restrictions on realizable unitary operations imposed by symmetry and locality". Nature Physics. 18 (3): 283–289. arXiv:2003.05524. doi:10.1038/s41567-021-01464-0. ISSN 1745-2481. S2CID 245840243.
^ 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. ISSN 1745-2481. S2CID 256745894. 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.

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.

[1] Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices". Linear and Multilinear Algebra. 50 (4): 321–326. doi:10.1080/03081080290025507. S2CID 120125694.

[2] Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis. Cambridge University Press. doi:10.1017/CBO9781139020411. 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. S2CID 125455174.

[4] 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. ISBN 978-1-84628-887-6.

[5] Nielsen, M.A.; Chuang, Isaac (2010). Quantum Computation and Quantum Information. Cambridge, UK: 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; et al. (1 November 1995). "Elementary gates for quantum computation". Physical Review A. 52 (5). American Physical Society (APS): 3457–3467, esp.p. 3465. arXiv:quant-ph/9503016. doi:10.1103/physreva.52.3457. ISSN 1050-2947. PMID 9912645. S2CID 8764584.

[7] Marvian, Iman (10 January 2022). "Restrictions on realizable unitary operations imposed by symmetry and locality". Nature Physics. 18 (3): 283–289. arXiv:2003.05524. doi:10.1038/s41567-021-01464-0. ISSN 1745-2481. S2CID 245840243.

[8] Jarlskog, Cecilia (2006). "Recursive parameterisation and invariant phases of unitary matrices". arXiv:math-ph/0510034.

[9] Alhambra, Álvaro M. (10 January 2022). "Forbidden by symmetry". News & Views. Nature Physics. 18 (3): 235–236. doi:10.1038/s41567-021-01483-x. ISSN 1745-2481. S2CID 256745894. 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.

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