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Physica Scripta

Semiclassical properties of Wigner functions

Asher Peres

Published Dec 1, 1986• 3 citations
Publisher

Abstract

The following conjecture is shown to be plausible: Consider a sequence of quantum theories with different values of ℏ. In each one, let be a non-degenerate self-adjoint operator, having a "classical limit" as ℏ → 0. Let z' be an eigenvalue of in the vicinity of a fixed real number z (independent of ℏ) and let W ( p, q ; z' ) be the corresponding Wigner function. Then, if we subject W ( p, q ; z' ) to a coarse graining over a scale ≫ ℏ, the result, in the limit ℏ → 0, is proportional to the classical distribution δ[ Z ( p, q ) - z ].

DOI
10.1088/0031-8949/34/6b/003

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The following conjecture is shown to be plausible: Consider a sequence of quantum theories with different values of ℏ. In each one, let be a non-degenerate self-adjoint operator, having a "classical limit" as ℏ → 0. Let z' be an eigenvalue of in the vicinity of a fixed real number z (independent of ℏ) and let W ( p, q ; z' ) be the corresponding Wigner function. Then, if we subject W ( p, q ; z' ) to a coarse graining over a scale ≫ ℏ, the result, in the limit ℏ → 0, is proportional to the classical distribution δ[ Z ( p, q ) - z ].
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Let F : ℬ(ℝ)→ℱ(ℋ) be a commutative positive operator-valued measure from the Borel σ-algebra of the reals to the space of bounded positive operators on the Hilbert space ℋ. It is well known that there exist a self-adjoint operator A and a Markov kernel μ(·)(λ) such that F(Δ) = μΔ(A), where Δ ∈ ℬ(ℝ). We prove that, for any Δ ∈ ℬ(ℝ), if F(Δ) is a discrete operator, then A is a linear combination of the moments of F. This result allows us to characterize the quantum observable represented by A by means of the Naimark dilation of F. The result is connected with the following properties of the sequences of linear functionals. Consider an infinite sequence of linear functionals {Ti}i ∈ ℕ, such that Tif = ∫ f(t) dμt(i), corresponding to an infinite sequence of probability measures {μ(·)(i)}i ∈ ℕ on the Borel σ-algebra ℬ([0, 1]) such that μ(·)(i) ≠ μ(·)(j), where i, j ∈ ℕ and i≠j. There exists a real bounded one-to-one continuous function f such that T i f = ∫ f ( t ) d μ t ( i ) ≠ ∫ f ( t ) d μ t ( j ) = T j f , i , j ∊ ℕ , i ≠ j .
Quantum uncertainty limits how strongly Wigner functions can concentrate in phase space. It is commonly conjectured that the vacuum Wigner function continuously majorizes that of any Wigner-positive state. However, by constructing Wigner-positive states whose localized quantum coherence lowers their Wigner entropy below the vacuum value, we show that this conjecture is incorrect. We restore vacuum majorization at finite resolution by integrating single-mode Wigner functions over concentric radial energy shells, obtaining probability vectors majorized by that of the vacuum for all Wigner-positive states and for Wigner-negative states with non-negative shell weights. Anchoring these shells to a fixed oscillator frame breaks affine symplectic invariance, rendering this relation strict for every non-vacuum Gaussian pure state. For Wigner-negative states, the persistence of negative shell weights defines a radial negativity scale with Airy-edge semiclassical asymptotics for highly excited Fock states. Operationally, we combine the majorization relation with partial transposition to obtain a two-mode entanglement criterion directly evaluable from radially binned joint homodyne outcomes without state reconstruction.