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Often, in the theoretical practice, the second moments of a bosonic or canonical quantum system are grouped together in a matrix called "correlation" or "covariance" matrix.
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Often, in the theoretical practice, the second moments of a bosonic or canonical quantum system are grouped together in a matrix called "correlation" or "covariance" matrix.
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It is trivial to say that a global invariant is also a local invariant.
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In fact, the matrix iΩV is Hermitian and, therefore, diagonalizable by a unitary transformation. Then, by taking the modulus of its 2n real eigenvalues, one gets the n symplectic eigenvalues of V.
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In fact, the matrix iΩV is Hermitian and, therefore, diagonalizable by a unitary transformation. Then, by taking the modulus of its 2n real eigenvalues, one gets the n symplectic eigenvalues of V.
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66149119459
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For every M M (n,R) there always exists a pair of proper rotations R1, R2 SO (n) such that M= R1 D R2T with D diagonal and real.
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For every M M (n,R) there always exists a pair of proper rotations R1, R2 SO (n) such that M= R1 D R2T with D diagonal and real.
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52
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66149087395
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For the possibility of symplectically diagonalizing quadratic forms under different positivity conditions, see Appendix 6 of Ref..
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For the possibility of symplectically diagonalizing quadratic forms under different positivity conditions, see Appendix 6 of Ref..
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53
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66149093929
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According to the Sylvester's law of inertia, congruence transformations preserve the signs of the eigenvalues.
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According to the Sylvester's law of inertia, congruence transformations preserve the signs of the eigenvalues.
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Notice that Ref. adopts the commutation relations [x l, x m] =i Ωlm, so that the variance of the vacuum noise is equal to 1/2. In this notation, our Eq. 85 becomes detAdetB+ [(1/4) -detC] 2 - I4 (detA+detB) /4, which is exactly the Eq. (17) of Ref..
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Notice that Ref. adopts the commutation relations [x l, x m] =i Ωlm, so that the variance of the vacuum noise is equal to 1/2. In this notation, our Eq. 85 becomes detAdetB+ [(1/4) -detC] 2 - I4 (detA+detB) /4, which is exactly the Eq. (17) of Ref..
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57
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This theorem is a specialization of the orthogonal block diagonalization which is valid for all the normal matrices [see, e.g., Cambridge University Press, New York
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Matrix Analysis
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