proposition A.604 Non-negativity forbids a vanishing overlap

open in the book · appendices/A-long-proofs.tex:28956 · p. 3082

Rests on

depends_on equation 25.54 eq:pq-wigner-marginal-x ¶
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 definition 5.18 Inner product ¶
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
    - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-4 ¶
    - 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.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 ¶
  - 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 theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - 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 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 5.39 Kernel, image, nullity, rank ¶
      - depends_on lemma 5.38 Exchange and completion ¶
      - 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 theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
      - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
      - 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 ¶
- proves proof app:A-long-proofs@proof-361 ¶
proves proof app:A-long-proofs@proof-363 ¶

Supports

depends_on proposition A.609 The exponent is a quadratic polynomial ¶
- depends_on proposition A.611 The constraint on $\Gamma$ ¶
  - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:pq-wigner-marginal-x	declared	appendices/A-long-proofs.tex:28961
`depends_on`	→	Overlaps and the resolution of unity	declared	appendices/A-long-proofs.tex:28961
`depends_on`	→	The Wigner function of a Gaussian is a positive Gaussian	declared	appendices/A-long-proofs.tex:28961
`depends_on`	←	The exponent is a quadratic polynomial	declared	appendices/A-long-proofs.tex:29168
`proves`	←	app:A-long-proofs@proof-363	declared	appendices/A-long-proofs.tex:28965