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Physica Scripta

Wigner function for a phase state

Ulrike Herzog, Henryk Paul, Th. Richter

Published Jan 1, 1993• 23 citations
Publisher

Abstract

The Wigner function for a phase state |ϕ⟩ is expanded, as a function of polar coordinates ϱ' and ϕ', in terms of cos [ k (ϕ' − ϕ)] ( k = 0, 1, 2,...). The ϱ'-dependent expansion coefficients are evaluated numerically. The Wigner function thus obtained fits well to an approximate analytic solution valid for ϱ' ≫ 1.

DOI
10.1088/0031-8949/1993/t48/009

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The Wigner function for a phase state |ϕ⟩ is expanded, as a function of polar coordinates ϱ' and ϕ', in terms of cos [ k (ϕ' − ϕ)] ( k = 0, 1, 2,...). The ϱ'-dependent expansion coefficients are evaluated numerically. The Wigner function thus obtained fits well to an approximate analytic solution valid for ϱ' ≫ 1.
We review some of the difficulties previously encountered in defining the phase operator for finite quantum systems. We then propose the Wigner–Weyl quantization of the angle function φ on phase space as the phase operator, and derive a closed expression for its matrix elements with respect to the Hermite functions. We also determine the quantization of e^{iφ} , which turns out to be a weighted shift operator, its spectrum and that of its adjoint. This is done in the framework of quantization of a certain symbol class of phase space distributions, specialized to those which depend on one variable only. After recalling some results for the position and momentum variables, we apply the scheme to functions of radius or angle. We give necessary and sufficient conditions for operators obtained by quantizing functions of the angle to be elements of \mathscr L^+ [\mathscr S(\mathbf R)] and \mathscr L^+ [\mathscr S (\mathbf R), L^2(\mathbf R)] , and a sufficient condition for boundedness. We then consider the associated questions of commutation relations and uncertainties for operators in \mathscr L^+ [\mathscr S(\mathbf R), L^2(\mathbf R)] , which we define as bilinear forms. As must be the case, the commutator between our phase operator and the number operator exhibits noncanonical terms. Not surprisingly, the Poisson bracket of their phase space symbols also exhibits a noncanonical term.
Magic -- the resource enabling quantum computational advantage beyond stabilizer circuits -- has a clean phase-space characterization in odd prime dimensions that qubits notoriously lack. We study C(rho), the l_1 distance from a state's discrete Wigner function to the stabilizer polytope, and determine its exact geometry. For a single qubit with Bloch vector r we prove the closed form C = max(0, ||r||_1 - 1)/2, so the maximum is (sqrt(3)-1)/2, attained at the eight face states. For products of single-qubit states, C(rho_1 x ... x rho_n) <= prod_i (1 + C(rho_i)) - 1, with equality whenever every factor has Bloch sign product r_x r_y r_z <= 0; this includes the exact tensor powers ((1+sqrt(3))/2)^n - 1 and the exact maximum of C over fully separable n-qubit states. Because no discrete Wigner function for qubits is Clifford covariant, any such measure is frame dependent, and we identify which of its features are not. For a single qubit there are exactly two valid frames up to phase-space translation and C is identical on both. Pauli-covariant two-qubit frames number 2048 up to translation, and across them the ratio C(rho_Rx)/C(rho_Ry) takes the values 1/2, 1 and 2. In every one of these frames some stabilizer state has negative Wigner entries, a consequence of the Mermin-Peres square, so the two-qubit bound Gamma >= 1 + C/2 against the robustness of magic Gamma is frame independent. Within the Wootters frame we prove integer values 1, 2, 1 of the tightness ratio kappa := (Gamma-1)/C for three families in the two-qubit repetition codespace, with explicit certificates. For n-qubit GHZ phase states we prove Gamma = |sin phi| + |cos phi| for all n and C >= 2^(floor(n/2)-1) (|sin phi| + |cos phi| - 1), tight for n = 2 and, by exact linear programming, for n = 3, 4, so kappa decays exponentially. Finally, C is not a magic monotone, so it cannot replace Gamma in distillation bounds.
Abstract We consider the Wetterich exact renormalization group (RG) equation. Approximate closed equations are obtained from it, applying certain truncation schemes for the effective average action. These equations are solved either purely numerically or by certain extra truncations for the potential and related quantities, called the functional truncations. Traditionally, the functional truncations consist of truncated expansions in powers of ρ ˜ − ρ ˜ 0 , where ρ ˜ ∝ ϕ 2 , φ is the averaged order parameter, and ρ ˜ 0 corresponds to the minimum of the dimensionless potential u k ( ρ ˜ ) , depending on the infrared cut-off scale k . We propose a new approach of functional truncations, using the expansion u k ( ρ ˜ ) − u k ( 0 ) = ( 1 − s ) − μ u 1 , k s + u 2 , k s 2 + ⋯ , where s = ρ ˜ / ( ρ ˜ 0 + ρ ˜ ) , ρ ˜ 0 is an optimization parameter and µ is the exponent, describing the ρ ˜ → ∞ asymptotic. The newly developed method provides accurate estimates of the critical exponents η , ν and als