proposition A.612 The dictionary between $A$ and $\Gamma$

Rests on

• depends_on lemma A.601 Several variables, complex symmetric matrix ¶
  • depends_on lemma A.599 One variable, complex coefficient ¶
    • depends_on equation 17.42 eq:ft-gaussian ¶
    • depends_on theorem 7.109 Leibniz integral rule ¶
      • depends_on definition 7.125 Multiple integral ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶
        • depends_on definition 7.39 Darboux sums and the definite integral ¶
      • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
        • depends_on proposition 7.22 Sequential characterization ¶
        • depends_on theorem 7.7 Bolzano–Weierstrass ¶
        • proves proof ch:05-real-analysis@proof-10 ¶
      • depends_on theorem 7.35 Mean value theorem ¶
        • depends_on proposition 7.29 Linearity ¶
        • depends_on theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-19 ¶
      • proves proof ch:05-real-analysis@proof-67 ¶
    • depends_on theorem 8.6 Cauchy–Riemann equations ¶
      • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.97 Partial derivative; gradient ¶
      • depends_on equation 8.4 eq:cpx-derivative ¶
      • proves proof ch:06-complex-analysis@proof-3 ¶
    • proves proof app:A-long-proofs@proof-358 ¶
  • depends_on lemma A.600 The real part of an inverse ¶
    • depends_on theorem 5.40 Rank–nullity ¶
      • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶
        • depends_on definition 5.14 Linear independence ¶
      • depends_on definition 5.39 Kernel, image, nullity, rank ¶
        • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 5.7 Vector subspace ¶
        • proves proof ch:03-linear-algebra-representations@proof-12 ¶
      • depends_on lemma 5.38 Exchange and completion ¶
        • depends_on definition 5.15 Basis ¶ ↺
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
        • depends_on definition 5.14 Linear independence ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-11 ¶
      • proves proof ch:03-linear-algebra-representations@proof-13 ¶
    • depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
      • depends_on proposition 5.28 Gram–Schmidt ¶
        • depends_on definition 5.14 Linear independence ¶ ↺
        • depends_on definition 5.18 Inner product ¶
        • depends_on definition 5.27 Orthonormal basis ¶
        • depends_on equation 5.44 eq:lin-norm-assoc ¶
        • proves proof ch:03-linear-algebra-representations@proof-5 ¶
      • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
        • depends_on definition 5.70 Characteristic polynomial ¶
        • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
        • depends_on proposition 5.49 Injective, surjective, invertible ¶
        • proves proof ch:03-linear-algebra-representations@proof-26 ¶
      • depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
        • depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
        • depends_on definition 5.41 Adjoint ¶
        • depends_on theorem 5.40 Rank–nullity ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-31 ¶
      • proves proof ch:03-linear-algebra-representations@proof-34 ¶
    • proves proof app:A-long-proofs@proof-359 ¶
  • depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
  • proves proof app:A-long-proofs@proof-360 ¶
• depends_on lemma A.600 The real part of an inverse ¶ ↺
• depends_on proposition A.611 The constraint on $\Gamma$ ¶
  • depends_on lemma A.610 Block positivity ¶
    • depends_on theorem 5.40 Rank–nullity ¶ ↺
    • depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
    • proves proof app:A-long-proofs@proof-368 ¶
  • depends_on lemma A.603 Overlaps and the resolution of unity ¶
    • depends_on equation A.918 eq:app-hudson-gaussian-coherent ¶
    • depends_on equation 25.56 eq:pq-wigner-overlap ¶
    • depends_on theorem 25.53 Properties of the Wigner function ¶
      • depends_on equation 25.53 eq:pq-wigner ¶
      • depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
        • depends_on definition 12.2 Hilbert space ¶
        • depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
        • proves proof ch:10-hilbert-spaces@proof-1 ¶
      • depends_on theorem 17.32 Fourier inversion ¶
        • depends_on definition 17.29 Fourier transform; the treatise convention ¶
        • depends_on equation 17.42 eq:ft-gaussian ¶ ↺
        • proves proof ch:15-fourier-integral-transforms@proof-20 ¶
      • proves proof ch:08-poisson-quantum-bridge@proof-20 ¶
    • proves proof app:A-long-proofs@proof-362 ¶
  • depends_on proposition A.609 The exponent is a quadratic polynomial ¶
    • depends_on lemma A.607 A zero-free entire function is an exponential ¶
      • depends_on lemma A.606 $F$ is entire, and is the overlap in disguise ¶
        • depends_on definition A.605 The analytic continuation of the overlaps ¶
        • depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶ ↺
        • depends_on theorem 8.6 Cauchy–Riemann equations ¶ ↺
        • proves proof app:A-long-proofs@proof-364 ¶
      • depends_on theorem 7.109 Leibniz integral rule ¶ ↺
      • depends_on theorem 8.6 Cauchy–Riemann equations ¶ ↺
      • proves proof app:A-long-proofs@proof-365 ¶
    • depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
      • depends_on equation 8.13 eq:cpx-derivative-formula ¶
      • depends_on theorem 8.18 Liouville ¶
        • depends_on proposition 8.10 Fundamental theorem for contours ¶
        • depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
        • proves proof ch:06-complex-analysis@proof-12 ¶
      • depends_on theorem 8.20 Taylor expansion ¶
        • depends_on equation 8.12 eq:cpx-cif ¶
        • depends_on equation 8.13 eq:cpx-derivative-formula ¶ ↺
        • depends_on lemma 8.9 ML estimate ¶
        • depends_on proposition 7.46 Geometric series ¶
        • proves proof ch:06-complex-analysis@proof-14 ¶
      • proves proof app:A-long-proofs@proof-366 ¶
    • depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
      • depends_on equation 25.54 eq:pq-wigner-marginal-x ¶
      • depends_on lemma A.603 Overlaps and the resolution of unity ¶ ↺
      • depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
        • depends_on equation A.910 eq:app-hudson-gaussian-form ¶
        • depends_on equation 25.53 eq:pq-wigner ¶ ↺
        • depends_on lemma A.601 Several variables, complex symmetric matrix ¶ ↺
        • proves proof app:A-long-proofs@proof-361 ¶
      • proves proof app:A-long-proofs@proof-363 ¶
    • proves proof app:A-long-proofs@proof-367 ¶
  • proves proof app:A-long-proofs@proof-369 ¶
• proves proof app:A-long-proofs@proof-370 ¶

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