Symplectic matrix

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In mathematics, a symplectic matrix is a 2 n × 2 n {\displaystyle 2n\times 2n} {\displaystyle 2n\times 2n} matrix M {\displaystyle M} {\displaystyle M} with real entries that satisfies the condition

M T Ω M = Ω , {\displaystyle M^{\text{T}}\Omega M=\Omega ,} {\displaystyle M^{\text{T}}\Omega M=\Omega ,} 1

where M T {\displaystyle M^{\text{T}}} {\displaystyle M^{\text{T}}} denotes the transpose of M {\displaystyle M} {\displaystyle M} and Ω {\displaystyle \Omega } {\displaystyle \Omega } is a fixed 2 n × 2 n {\displaystyle 2n\times 2n} {\displaystyle 2n\times 2n} nonsingular, skew-symmetric matrix. This definition can be extended to 2 n × 2 n {\displaystyle 2n\times 2n} {\displaystyle 2n\times 2n} matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.

Typically Ω {\displaystyle \Omega } {\displaystyle \Omega } is chosen to be the block matrix Ω = [ 0 I n − I n 0 ] , {\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},} {\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},} where I n {\displaystyle I_{n}} {\displaystyle I_{n}} is the n × n {\displaystyle n\times n} {\displaystyle n\times n} identity matrix. The matrix Ω {\displaystyle \Omega } {\displaystyle \Omega } has determinant + 1 {\displaystyle +1} {\displaystyle +1} and its inverse is Ω − 1 = Ω T = − Ω {\displaystyle \Omega ^{-1}=\Omega ^{\text{T}}=-\Omega } {\displaystyle \Omega ^{-1}=\Omega ^{\text{T}}=-\Omega }.

Properties

Generators for symplectic matrices

Every symplectic matrix has determinant + 1 {\displaystyle +1} {\displaystyle +1}, and the 2 n × 2 n {\displaystyle 2n\times 2n} {\displaystyle 2n\times 2n} symplectic matrices with real entries form a subgroup of the general linear group G L ( 2 n ; R ) {\displaystyle \mathrm {GL} (2n;\mathbb {R} )} {\displaystyle \mathrm {GL} (2n;\mathbb {R} )} 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 n ( 2 n + 1 ) {\displaystyle n(2n+1)} {\displaystyle n(2n+1)}, and is denoted S p ( 2 n ; R ) {\displaystyle \mathrm {Sp} (2n;\mathbb {R} )} {\displaystyle \mathrm {Sp} (2n;\mathbb {R} )}. 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 D ( n ) = { ( A 0 0 ( A T ) − 1 ) : A ∈ GL ( n ; R ) } N ( n ) = { ( I n B 0 I n ) : B ∈ Sym ( n ; R ) } {\displaystyle {\begin{aligned}D(n)=&\left\{{\begin{pmatrix}A&0\\0&(A^{T})^{-1}\end{pmatrix}}:A\in {\text{GL}}(n;\mathbb {R} )\right\}\\N(n)=&\left\{{\begin{pmatrix}I_{n}&B\\0&I_{n}\end{pmatrix}}:B\in {\text{Sym}}(n;\mathbb {R} )\right\}\end{aligned}}} {\displaystyle {\begin{aligned}D(n)=&\left\{{\begin{pmatrix}A&0\\0&(A^{T})^{-1}\end{pmatrix}}:A\in {\text{GL}}(n;\mathbb {R} )\right\}\\N(n)=&\left\{{\begin{pmatrix}I_{n}&B\\0&I_{n}\end{pmatrix}}:B\in {\text{Sym}}(n;\mathbb {R} )\right\}\end{aligned}}} where Sym ( n ; R ) {\displaystyle {\text{Sym}}(n;\mathbb {R} )} {\displaystyle {\text{Sym}}(n;\mathbb {R} )} is the set of n × n {\displaystyle n\times n} {\displaystyle n\times n} symmetric matrices. Then, S p ( 2 n ; R ) {\displaystyle \mathrm {Sp} (2n;\mathbb {R} )} {\displaystyle \mathrm {Sp} (2n;\mathbb {R} )} is generated by the set[1]p. 2 { Ω } ∪ D ( n ) ∪ N ( n ) {\displaystyle \{\Omega \}\cup D(n)\cup N(n)} {\displaystyle \{\Omega \}\cup D(n)\cup N(n)} of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in D ( n ) {\displaystyle D(n)} {\displaystyle D(n)} and N ( n ) {\displaystyle N(n)} {\displaystyle N(n)} together, along with some power of Ω {\displaystyle \Omega } {\displaystyle \Omega }.

Inverse matrix

Every symplectic matrix is invertible with the inverse matrix given by M − 1 = Ω − 1 M T Ω . {\displaystyle M^{-1}=\Omega ^{-1}M^{\text{T}}\Omega .} {\displaystyle M^{-1}=\Omega ^{-1}M^{\text{T}}\Omega .} 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

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 Pf ( M T Ω M ) = det ( M ) Pf ( Ω ) . {\displaystyle {\mbox{Pf}}(M^{\text{T}}\Omega M)=\det(M){\mbox{Pf}}(\Omega ).} {\displaystyle {\mbox{Pf}}(M^{\text{T}}\Omega M)=\det(M){\mbox{Pf}}(\Omega ).} Since M T Ω M = Ω {\displaystyle M^{\text{T}}\Omega M=\Omega } {\displaystyle M^{\text{T}}\Omega M=\Omega } and Pf ( Ω ) ≠ 0 {\displaystyle {\mbox{Pf}}(\Omega )\neq 0} {\displaystyle {\mbox{Pf}}(\Omega )\neq 0} we have that det ( M ) = 1 {\displaystyle \det(M)=1} {\displaystyle \det(M)=1}.

When the underlying field is real or complex, one can also show this by factoring the inequality det ( M T M + I ) ≥ 1 {\displaystyle \det(M^{\text{T}}M+I)\geq 1} {\displaystyle \det(M^{\text{T}}M+I)\geq 1}.[2]

Block form of symplectic matrices

Suppose Ω is given in the standard form and let M {\displaystyle M} {\displaystyle M} be a 2 n × 2 n {\displaystyle 2n\times 2n} {\displaystyle 2n\times 2n} block matrix given by M = ( A B C D ) {\displaystyle M={\begin{pmatrix}A&B\\C&D\end{pmatrix}}} {\displaystyle M={\begin{pmatrix}A&B\\C&D\end{pmatrix}}}

where A , B , C , D {\displaystyle A,B,C,D} {\displaystyle A,B,C,D} are n × n {\displaystyle n\times n} {\displaystyle n\times n} matrices. The condition for M {\displaystyle M} {\displaystyle M} to be symplectic is equivalent to the two following equivalent conditions[3]

A T C , B T D {\displaystyle A^{\text{T}}C,B^{\text{T}}D} {\displaystyle A^{\text{T}}C,B^{\text{T}}D} symmetric, and A T D − C T B = I {\displaystyle A^{\text{T}}D-C^{\text{T}}B=I} {\displaystyle A^{\text{T}}D-C^{\text{T}}B=I}

A B T , C D T {\displaystyle AB^{\text{T}},CD^{\text{T}}} {\displaystyle AB^{\text{T}},CD^{\text{T}}} symmetric, and A D T − B C T = I {\displaystyle AD^{\text{T}}-BC^{\text{T}}=I} {\displaystyle AD^{\text{T}}-BC^{\text{T}}=I}

The second condition comes from the fact that if M {\displaystyle M} {\displaystyle M} is symplectic, then M T {\displaystyle M^{T}} {\displaystyle M^{T}} is also symplectic. When n = 1 {\displaystyle n=1} {\displaystyle n=1} these conditions reduce to the single condition det ( M ) = 1 {\displaystyle \det(M)=1} {\displaystyle \det(M)=1}. Thus a 2 × 2 {\displaystyle 2\times 2} {\displaystyle 2\times 2} matrix is symplectic if and only if it has unit determinant.

Inverse matrix of block matrix

With Ω {\displaystyle \Omega } {\displaystyle \Omega } in standard form, the inverse of M {\displaystyle M} {\displaystyle M} is given by M − 1 = Ω − 1 M T Ω = ( D T − B T − C T A T ) . {\displaystyle M^{-1}=\Omega ^{-1}M^{\text{T}}\Omega ={\begin{pmatrix}D^{\text{T}}&-B^{\text{T}}\\-C^{\text{T}}&A^{\text{T}}\end{pmatrix}}.} {\displaystyle M^{-1}=\Omega ^{-1}M^{\text{T}}\Omega ={\begin{pmatrix}D^{\text{T}}&-B^{\text{T}}\\-C^{\text{T}}&A^{\text{T}}\end{pmatrix}}.} The group has dimension n ( 2 n + 1 ) {\displaystyle n(2n+1)} {\displaystyle n(2n+1)}. This can be seen by noting that ( M T Ω M ) T = − M T Ω M {\displaystyle (M^{\text{T}}\Omega M)^{\text{T}}=-M^{\text{T}}\Omega M} {\displaystyle (M^{\text{T}}\Omega M)^{\text{T}}=-M^{\text{T}}\Omega M} is anti-symmetric. Since the space of anti-symmetric matrices has dimension ( 2 n 2 ) , {\textstyle {\binom {2n}{2}},} {\textstyle {\binom {2n}{2}},} the identity M T Ω M = Ω {\displaystyle M^{\text{T}}\Omega M=\Omega } {\displaystyle M^{\text{T}}\Omega M=\Omega } imposes ( 2 n 2 ) {\textstyle 2n \choose 2} {\textstyle 2n \choose 2} constraints on the ( 2 n ) 2 {\displaystyle (2n)^{2}} {\displaystyle (2n)^{2}} coefficients of M {\displaystyle M} {\displaystyle M} and leaves M {\displaystyle M} {\displaystyle M} with n ( 2 n + 1 ) {\displaystyle n(2n+1)} {\displaystyle n(2n+1)} independent coefficients.

Symplectic transformations

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 ( V , ω ) {\displaystyle (V,\omega )} {\displaystyle (V,\omega )} is a 2 n {\displaystyle 2n} {\displaystyle 2n}-dimensional vector space V {\displaystyle V} {\displaystyle V} equipped with a nondegenerate, skew-symmetric bilinear form ω {\displaystyle \omega } {\displaystyle \omega } called the symplectic form.

A symplectic transformation is then a linear transformation L : V → V {\displaystyle L:V\to V} {\displaystyle L:V\to V} which preserves ω {\displaystyle \omega } {\displaystyle \omega }, i.e. ω ( L u , L v ) = ω ( u , v ) . {\displaystyle \omega (Lu,Lv)=\omega (u,v).} {\displaystyle \omega (Lu,Lv)=\omega (u,v).} Fixing a basis for V {\displaystyle V} {\displaystyle V}, ω {\displaystyle \omega } {\displaystyle \omega } can be written as a matrix Ω {\displaystyle \Omega } {\displaystyle \Omega } and L {\displaystyle L} {\displaystyle L} as a matrix M {\displaystyle M} {\displaystyle M}. The condition that L {\displaystyle L} {\displaystyle L} be a symplectic transformation is precisely the condition that M be a symplectic matrix: M T Ω M = Ω . {\displaystyle M^{\text{T}}\Omega M=\Omega .} {\displaystyle M^{\text{T}}\Omega M=\Omega .}

Under a change of basis, represented by a matrix A, we have Ω ↦ A T Ω A {\displaystyle \Omega \mapsto A^{\text{T}}\Omega A} {\displaystyle \Omega \mapsto A^{\text{T}}\Omega A} M ↦ A − 1 M A . {\displaystyle M\mapsto A^{-1}MA.} {\displaystyle M\mapsto A^{-1}MA.} One can always bring Ω {\displaystyle \Omega } {\displaystyle \Omega } 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

Symplectic matrices are defined relative to a fixed nonsingular, skew-symmetric matrix Ω {\displaystyle \Omega } {\displaystyle \Omega }. As explained in the previous section, Ω {\displaystyle \Omega } {\displaystyle \Omega } 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 Ω {\displaystyle \Omega } {\displaystyle \Omega } given above is the block diagonal form Ω = [ 0 1 − 1 0 0 ⋱ 0 0 1 − 1 0 ] . {\displaystyle \Omega ={\begin{bmatrix}{\begin{matrix}0&1\\-1&0\end{matrix}}&&0\\&\ddots &\\0&&{\begin{matrix}0&1\\-1&0\end{matrix}}\end{bmatrix}}.} {\displaystyle \Omega ={\begin{bmatrix}{\begin{matrix}0&1\\-1&0\end{matrix}}&&0\\&\ddots &\\0&&{\begin{matrix}0&1\\-1&0\end{matrix}}\end{bmatrix}}.} This choice differs from the previous one by a permutation of basis vectors.

Sometimes the notation J {\displaystyle J} {\displaystyle J} is used instead of Ω {\displaystyle \Omega } {\displaystyle \Omega } 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 Ω {\displaystyle \Omega } {\displaystyle \Omega } but represents a very different structure. A complex structure J {\displaystyle J} {\displaystyle J} is the coordinate representation of a linear transformation that squares to − I n {\displaystyle -I_{n}} {\displaystyle -I_{n}}, whereas Ω {\displaystyle \Omega } {\displaystyle \Omega } is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which J {\displaystyle J} {\displaystyle J} is not skew-symmetric or Ω {\displaystyle \Omega } {\displaystyle \Omega } does not square to − I n {\displaystyle -I_{n}} {\displaystyle -I_{n}}.

Given a hermitian structure on a vector space, J {\displaystyle J} {\displaystyle J} and Ω {\displaystyle \Omega } {\displaystyle \Omega } are related via Ω a b = − g a c J c b {\displaystyle \Omega _{ab}=-g_{ac}{J^{c}}_{b}} {\displaystyle \Omega _{ab}=-g_{ac}{J^{c}}_{b}} where g a c {\displaystyle g_{ac}} {\displaystyle g_{ac}} is the metric. That J {\displaystyle J} {\displaystyle J} and Ω {\displaystyle \Omega } {\displaystyle \Omega } 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

  • For any positive definite symmetric 2 n × 2 n {\displaystyle 2n\times 2n} {\displaystyle 2n\times 2n} real symplectic matrix S {\displaystyle S} {\displaystyle S}, there is a symplectic unitary U {\displaystyle U} {\displaystyle U}, U ∈ U ( 2 n , R ) ∩ Sp ⁡ ( 2 n , R ) = O ( 2 n ) ∩ Sp ⁡ ( 2 n , R ) , {\displaystyle U\in \mathrm {U} (2n,\mathbb {R} )\cap \operatorname {Sp} (2n,\mathbb {R} )=\mathrm {O} (2n)\cap \operatorname {Sp} (2n,\mathbb {R} ),} {\displaystyle U\in \mathrm {U} (2n,\mathbb {R} )\cap \operatorname {Sp} (2n,\mathbb {R} )=\mathrm {O} (2n)\cap \operatorname {Sp} (2n,\mathbb {R} ),}such that S = U T D U for D = diag ⁡ ( λ 1 , … , λ n , λ 1 − 1 , … , λ n − 1 ) , {\displaystyle S=U^{\text{T}}DU\quad {\text{for}}\quad D=\operatorname {diag} (\lambda _{1},\ldots ,\lambda _{n},\lambda _{1}^{-1},\ldots ,\lambda _{n}^{-1}),} {\displaystyle S=U^{\text{T}}DU\quad {\text{for}}\quad D=\operatorname {diag} (\lambda _{1},\ldots ,\lambda _{n},\lambda _{1}^{-1},\ldots ,\lambda _{n}^{-1}),}where the diagonal elements of D {\displaystyle D} {\displaystyle D} are the eigenvalues of S {\displaystyle S} {\displaystyle S}.[4][5]
  • Any real symplectic matrix S has a polar decomposition of the form:[4] S = U R , {\displaystyle S=UR,} {\displaystyle S=UR,}where U ∈ Sp ⁡ ( 2 n , R ) ∩ U ⁡ ( 2 n , R ) , {\displaystyle U\in \operatorname {Sp} (2n,\mathbb {R} )\cap \operatorname {U} (2n,\mathbb {R} ),} {\displaystyle U\in \operatorname {Sp} (2n,\mathbb {R} )\cap \operatorname {U} (2n,\mathbb {R} ),} and R ∈ Sp ⁡ ( 2 n , R ) ∩ Sym + ⁡ ( 2 n , R ) . {\displaystyle R\in \operatorname {Sp} (2n,\mathbb {R} )\cap \operatorname {Sym} _{+}(2n,\mathbb {R} ).} {\displaystyle R\in \operatorname {Sp} (2n,\mathbb {R} )\cap \operatorname {Sym} _{+}(2n,\mathbb {R} ).}
  • Any real symplectic matrix can be decomposed as a product of three matrices: S = O ( D 0 0 D − 1 ) O ′ , {\displaystyle S=O{\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}O',} {\displaystyle S=O{\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}O',}where O {\displaystyle O} {\displaystyle O} and O ′ {\displaystyle O'} {\displaystyle O'} are both symplectic and orthogonal, and D {\displaystyle D} {\displaystyle D} is positive-definite and diagonal.[6] This decomposition is closely related to the singular value decomposition of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition.
  • The set of orthogonal symplectic matrices forms a (maximal) compact subgroup of the symplectic group.[7] This set is isomorphic to the set of unitary matrices of dimension n {\displaystyle n} {\displaystyle n}, U ( 2 n , R ) ∩ Sp ⁡ ( 2 n , R ) = O ( 2 n ) ∩ Sp ⁡ ( 2 n , R ) ≅ U ( n , C ) {\displaystyle \mathrm {U} (2n,\mathbb {R} )\cap \operatorname {Sp} (2n,\mathbb {R} )=\mathrm {O} (2n)\cap \operatorname {Sp} (2n,\mathbb {R} )\cong \mathrm {U} (n,\mathbb {C} )} {\displaystyle \mathrm {U} (2n,\mathbb {R} )\cap \operatorname {Sp} (2n,\mathbb {R} )=\mathrm {O} (2n)\cap \operatorname {Sp} (2n,\mathbb {R} )\cong \mathrm {U} (n,\mathbb {C} )}. Every symplectic orthogonal matrix can be written as
    ( ℜ ( V ) − ℑ ( V ) ℑ ( V ) ℜ ( V ) ) = [ 1 2 ( I n i I n I n − i I n ) ] † ( V 0 0 V ∗ ) [ 1 2 ( I n i I n I n − i I n ) ] , {\displaystyle {\begin{pmatrix}\Re (V)&-\Im (V)\\\Im (V)&\Re (V)\end{pmatrix}}=\left[{\frac {1}{\sqrt {2}}}{\begin{pmatrix}I_{n}&iI_{n}\\I_{n}&-iI_{n}\end{pmatrix}}\right]^{\dagger }{\begin{pmatrix}V&0\\0&V^{*}\end{pmatrix}}\left[{\frac {1}{\sqrt {2}}}{\begin{pmatrix}I_{n}&iI_{n}\\I_{n}&-iI_{n}\end{pmatrix}}\right],} {\displaystyle {\begin{pmatrix}\Re (V)&-\Im (V)\\\Im (V)&\Re (V)\end{pmatrix}}=\left[{\frac {1}{\sqrt {2}}}{\begin{pmatrix}I_{n}&iI_{n}\\I_{n}&-iI_{n}\end{pmatrix}}\right]^{\dagger }{\begin{pmatrix}V&0\\0&V^{*}\end{pmatrix}}\left[{\frac {1}{\sqrt {2}}}{\begin{pmatrix}I_{n}&iI_{n}\\I_{n}&-iI_{n}\end{pmatrix}}\right],} 2

    with V ∈ U ( n , C ) {\displaystyle V\in \mathrm {U} (n,\mathbb {C} )} {\displaystyle V\in \mathrm {U} (n,\mathbb {C} )}.

    This equation implies that every symplectic orthogonal matrix has determinant equal to +1 and thus that this is true for all symplectic matrices as its polar decomposition is itself given in terms symplectic matrices.

Complex matrices

If instead M is a 2n × 2n matrix with complex entries, the definition is not standard throughout the literature. Many authors [8] adjust the definition above to

M ∗ Ω M = Ω . {\displaystyle M^{*}\Omega M=\Omega \,.} {\displaystyle M^{*}\Omega M=\Omega \,.} 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 [9] retain the definition (1) for complex matrices and call matrices satisfying (3) conjugate symplectic.

Applications

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.[10] 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).[11] 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.[12]

See also

References

  1. Folland, G. B. (1989). Harmonic analysis in phase space. Princeton University Press. pp. 173 f. ISBN 0-691-08527-7.
  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. 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. Houde, Martin; McCutcheon, Will; Quesada, Nicolás (13 March 2024). "Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson". Canadian Journal of Physics. 102 (10). Sec. V, p. 5. arXiv:2403.04596. Bibcode:2024CaJPh.102..497H. doi:10.1139/cjp-2024-0070.
  6. 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.
  7. Serafini, Alessio (2023). Quantum Continuous Variables. doi:10.1201/9781003250975. ISBN 978-1-003-25097-5.
  8. 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.
  9. Mackey, D. S.; Mackey, N. (2003). On the Determinant of Symplectic Matrices (Numerical Analysis Report 422). Manchester, England: Manchester Centre for Computational Mathematics.
  10. 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.
  11. 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.
  12. 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.