R.L. Hudson: "When is the Wigner quasi-probability density nonnegative ?", Rep.Math.Phys. 6(1974) 249--252 Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
The Classical Limit of Quantum Theory - Werner (1995) (Correct)
....as (W h ) 2=h) d Gamma ff h ( Pi) Delta ; 4:14) where ( Pi ) x) Gammax) is the parity operator [Gro] Here we have chosen the normalization such that formally, or with suitable regularization, 2) Gammad R dx dp (W h ) x; p) 1. Of course, W h is rarely positive [Hud,BW] and in general not even integrable. Ignoring such technical quibbles, however, as most of the literature on Wigner functions does, we get a simplified formulation of the classical limit, and also an interesting class of convergent states. The modified definition of the classical limit is ....
R.L. Hudson: "When is the Wigner quasi-probability density nonnegative ?", Rep.Math.Phys. 6(1974) 249--252
Sparse Representations with Chirplets via Maximum.. - O'Neill, Flandrin, Karl (Correct)
....to satisfy the generalized uncertainty principle [14] with equality oe 2 t oe 2 Gamma cov 2 t 1 4 where oe t is the duration, oe is the bandwidth, and cov t is the time frequency covariance. 2 ffl Chirplets are the only signals for which the Wigner distribution is non negative [15]. ffl Calculations involving chirplets can often be expressed in closed form. The paper proceeds as follows. In Section II we develop the maximum likelihood estimator (MLE) for a model of one chirplet in noise ( 1) with q = 1) and in Section III we derive the Cram er Rao lower bound (CRLB) for ....
R.L. Hudson. When is the Wigner quasi-probability density non negative? Rep. Math. Phys., 6:249--252, 1974.
Mixed States with Positive Wigner Functions - Bröcker, Werner (1994) (Correct)
....a joint probability density for position and momentum. There is no contradiction, however, because the Wigner function is usually not a proper probability distribution after all: it can take negative values. Perhaps the sharpest formulation of this fact is in a beautiful Theorem by R. Hudson [3], proved in 1974, showing that the Wigner distribution function of a pure state (for a single degree of freedom) is everywhere positive if and only if the state is coherent, i.e. if its wave function is a complex Gaussian. This result was subsequently generalized to multi dimensional quantum ....
R. Hudson:"When is the Wigner quasi-probability density non-negative", Rep.Math.Phys. 6(1974) 249-252
On Separability, Positivity and Minimum Uncertainty in.. - Flandrin (1998) (1 citation) (Correct)
.... phase terms the only signals for which the Wigner Ville distribution W x (t; f) j Z 1 Gamma1 x i t 2 j x i t Gamma 2 j e Gammai2 f d = Z 1 Gamma1 X i f 2 j X i f Gamma 2 j e i2 t d (5) is everywhere non negative (Hudson s theorem [16], cf. Proposition 1 below) 3. Separability. Their Wigner Ville distribution (5) is separable, namely: W g (t; f) jCj 2 r 2 ff e Gamma2fft 2 e Gamma 2 2 ff f 2 : 6) These three properties are important ones for attaching to Gaussian signals a specific status and for offering a ....
....with extensions beyond the Wigner Ville case. The purpose of this paper is not to review all of them and only those results which are connected in some way with the two other issues of separability and minimum uncertainty will be considered here. 2. 1 Positivity Proposition 1 (Hudson s theorem [16]) The Wigner Ville distribution is positive for signals of the type g ff;fi;fl (t) j e Gamma(fft 2 fit fl) 7) with (ff; fi; fl) 2 C 3 and Refffg 0, and only for them. Proof. The fact that (generalized) Gaussian signals of the form (7) have a positive WignerVille distribution follows ....
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R.L. Hudson, "When is the Wigner quasi-probability density non-negative?," Rep. Math. Phys., 6 (2), pp. 249--252, 1974.
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