In mathematics, a symplectic matrix is a matrix with real entries that satisfies the condition

(1)

where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix. This definition can be extended to matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.

Typically is chosen to be the block matrix where is the identity matrix. The matrix has determinant and its inverse is .

Properties

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Generators for symplectic matrices

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Every symplectic matrix has determinant  , and the   symplectic matrices with real entries form a subgroup of the general linear group   under matrix multiplication since being symplectic is a property stable under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension  , and is denoted  . The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space.

This symplectic group has a distinguished set of generators, which can be used to find all possible symplectic matrices. This includes the following sets   where   is the set of   symmetric matrices. Then,   is generated by the set[1]p. 2   of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in   and   together, along with some power of  .

Inverse matrix

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Every symplectic matrix is invertible with the inverse matrix given by   Furthermore, the product of two symplectic matrices is, again, a symplectic matrix. This gives the set of all symplectic matrices the structure of a group. There exists a natural manifold structure on this group which makes it into a (real or complex) Lie group called the symplectic group.

Determinantal properties

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It follows easily from the definition that the determinant of any symplectic matrix is ±1. Actually, it turns out that the determinant is always +1 for any field. One way to see this is through the use of the Pfaffian and the identity   Since   and   we have that  .

When the underlying field is real or complex, one can also show this by factoring the inequality  .[2]

Block form of symplectic matrices

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Suppose Ω is given in the standard form and let   be a   block matrix given by  

where   are   matrices. The condition for   to be symplectic is equivalent to the two following equivalent conditions[3]

  symmetric, and  

  symmetric, and  

The second condition comes from the fact that if   is symplectic, then   is also symplectic. When   these conditions reduce to the single condition  . Thus a   matrix is symplectic iff it has unit determinant.

Inverse matrix of block matrix

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With   in standard form, the inverse of   is given by   The group has dimension  . This can be seen by noting that   is anti-symmetric. Since the space of anti-symmetric matrices has dimension   the identity   imposes   constraints on the   coefficients of   and leaves   with   independent coefficients.

Symplectic transformations

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In the abstract formulation of linear algebra, matrices are replaced with linear transformations of finite-dimensional vector spaces. The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space. Briefly, a symplectic vector space   is a  -dimensional vector space   equipped with a nondegenerate, skew-symmetric bilinear form   called the symplectic form.

A symplectic transformation is then a linear transformation   which preserves  , i.e.

 

Fixing a basis for  ,   can be written as a matrix   and   as a matrix  . The condition that   be a symplectic transformation is precisely the condition that M be a symplectic matrix:

 

Under a change of basis, represented by a matrix A, we have

 
 

One can always bring   to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of A.

The matrix Ω

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Symplectic matrices are defined relative to a fixed nonsingular, skew-symmetric matrix  . As explained in the previous section,   can be thought of as the coordinate representation of a nondegenerate skew-symmetric bilinear form. It is a basic result in linear algebra that any two such matrices differ from each other by a change of basis.

The most common alternative to the standard   given above is the block diagonal form

 

This choice differs from the previous one by a permutation of basis vectors.

Sometimes the notation   is used instead of   for the skew-symmetric matrix. This is a particularly unfortunate choice as it leads to confusion with the notion of a complex structure, which often has the same coordinate expression as   but represents a very different structure. A complex structure   is the coordinate representation of a linear transformation that squares to  , whereas   is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which   is not skew-symmetric or   does not square to  .

Given a hermitian structure on a vector space,   and   are related via

 

where   is the metric. That   and   usually have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that the metric g is usually the identity matrix.

Diagonalization and decomposition

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  • For any positive definite symmetric real symplectic matrix S there exists U in   such that
 
where the diagonal elements of D are the eigenvalues of S.[4]
  for   and  
  • Any real symplectic matrix can be decomposed as a product of three matrices:
 
(2)

such that O and O' are both symplectic and orthogonal and D is positive-definite and diagonal.[5] This decomposition is closely related to the singular value decomposition of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition.

Complex matrices

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If instead M is a 2n × 2n matrix with complex entries, the definition is not standard throughout the literature. Many authors [6] adjust the definition above to

  (3)

where M* denotes the conjugate transpose of M. In this case, the determinant may not be 1, but will have absolute value 1. In the 2×2 case (n=1), M will be the product of a real symplectic matrix and a complex number of absolute value 1.

Other authors [7] retain the definition (1) for complex matrices and call matrices satisfying (3) conjugate symplectic.

Applications

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Transformations described by symplectic matrices play an important role in quantum optics and in continuous-variable quantum information theory. For instance, symplectic matrices can be used to describe Gaussian (Bogoliubov) transformations of a quantum state of light.[8] In turn, the Bloch-Messiah decomposition (2) means that such an arbitrary Gaussian transformation can be represented as a set of two passive linear-optical interferometers (corresponding to orthogonal matrices O and O' ) intermitted by a layer of active non-linear squeezing transformations (given in terms of the matrix D).[9] In fact, one can circumvent the need for such in-line active squeezing transformations if two-mode squeezed vacuum states are available as a prior resource only.[10]

See also

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References

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  1. ^ Habermann, Katharina, 1966- (2006). Introduction to symplectic Dirac operators. Springer. ISBN 978-3-540-33421-7. OCLC 262692314.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
  2. ^ Rim, Donsub (2017). "An elementary proof that symplectic matrices have determinant one". Adv. Dyn. Syst. Appl. 12 (1): 15–20. arXiv:1505.04240. doi:10.37622/ADSA/12.1.2017.15-20. S2CID 119595767.
  3. ^ de Gosson, Maurice. "Introduction to Symplectic Mechanics: Lectures I-II-III" (PDF).
  4. ^ a b de Gosson, Maurice A. (2011). Symplectic Methods in Harmonic Analysis and in Mathematical Physics - Springer. doi:10.1007/978-3-7643-9992-4. ISBN 978-3-7643-9991-7.
  5. ^ Ferraro, Alessandro; Olivares, Stefano; Paris, Matteo G. A. (31 March 2005). "Gaussian states in continuous variable quantum information". Sec. 1.3, p. 4. arXiv:quant-ph/0503237.
  6. ^ Xu, H. G. (July 15, 2003). "An SVD-like matrix decomposition and its applications". Linear Algebra and Its Applications. 368: 1–24. doi:10.1016/S0024-3795(03)00370-7. hdl:1808/374.
  7. ^ Mackey, D. S.; Mackey, N. (2003). On the Determinant of Symplectic Matrices (Numerical Analysis Report 422). Manchester, England: Manchester Centre for Computational Mathematics.
  8. ^ Weedbrook, Christian; Pirandola, Stefano; García-Patrón, Raúl; Cerf, Nicolas J.; Ralph, Timothy C.; Shapiro, Jeffrey H.; Lloyd, Seth (2012). "Gaussian quantum information". Reviews of Modern Physics. 84 (2): 621–669. arXiv:1110.3234. Bibcode:2012RvMP...84..621W. doi:10.1103/RevModPhys.84.621. S2CID 119250535.
  9. ^ Braunstein, Samuel L. (2005). "Squeezing as an irreducible resource". Physical Review A. 71 (5): 055801. arXiv:quant-ph/9904002. Bibcode:2005PhRvA..71e5801B. doi:10.1103/PhysRevA.71.055801. S2CID 16714223.
  10. ^ Chakhmakhchyan, Levon; Cerf, Nicolas (2018). "Simulating arbitrary Gaussian circuits with linear optics". Physical Review A. 98 (6): 062314. arXiv:1803.11534. Bibcode:2018PhRvA..98f2314C. doi:10.1103/PhysRevA.98.062314. S2CID 119227039.