Unitary matrix

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

where I is the identity matrix.

In physics, especially in quantum mechanics, the Hermitian adjoint of a matrix is denoted by a dagger (†) and the equation above becomes

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, Ux, Uy⟩ = ⟨x, y.
  • U is normal ().
  • 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.
  • .
  • Its eigenspaces are orthogonal.
  • U can be written as U = eiH, 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 × 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]

  1. is unitary.
  2. is unitary.
  3. is invertible with .
  4. The columns of form an orthonormal basis of with respect to the usual inner product. In other words, .
  5. The rows of form an orthonormal basis of with respect to the usual inner product. In other words, .
  6. is an isometry with respect to the usual norm. That is, for all , where .
  7. is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of ) with eigenvalues lying on the unit circle.

Elementary constructions

2 × 2 unitary matrix

The 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 determinant of such a matrix is

The sub-group of those elements with is called the special unitary group SU(2).

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

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

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

Another factorization is[3]

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

See also

References

  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.
  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
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