We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson's Theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counter-example. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson's Theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counter-example. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.

Feb 1, 20061 citation

Classicality in discrete Wigner functions

Cecilia Cormick, Ernesto F. Galvão, Daniel Gottesman +2 more

Gibbons et al., [Phys. Rev. A 70, 062101 (2004)] have recently defined discrete Wigner functions $W$ to represent quantum states in a Hilbert space with finite dimension. We show that such a class of Wigner functions $W$ can be defined so that the only pure states having non-negative $W$ for all such functions are stabilizer states, as conjectured by Galv\~ao, [Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving non-negativity of $W$ for all definitions of $W$ in the class form a subgroup of the Clifford group. This means pure states with non-negative $W$ and their associated unitary dynamics are classical in the sense of admitting an efficient classical simulation scheme using the stabilizer formalism.Gibbons et al., [Phys. Rev. A 70, 062101 (2004)] have recently defined discrete Wigner functions $W$ to represent quantum states in a Hilbert space with finite dimension. We show that such a class of Wigner functions $W$ can be defined so that the only pure states having non-negative $W$ for all such functions are stabilizer states, as conjectured by Galv\~ao, [Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving non-negativity of $W$ for all definitions of $W$ in the class form a subgroup of the Clifford group. This means pure states with non-negative $W$ and their associated unitary dynamics are classical in the sense of admitting an efficient classical simulation scheme using the stabilizer formalism.

arXiv:quant-ph/0506222Physical Review AJan 3, 2006129 citations

The Hudson theorem in LCA groups and infinite quantum spin systems

Fabio Nicola, Federico Riccardi

The celebrated Hudson theorem states that the Gaussian functions in $\mathbb{R}^d$ are the only functions whose Wigner distribution is everywhere positive. Motivated by quantum information theory, D. Gross proved an analogous result on the Abelian group $\mathbb{Z}_d^n$ , for $d$ odd - corresponding to a system of $n$ qudits - showing that the Wigner distribution is nonnegative only for the so-called stabilizer states. Extending this result to the thermodynamic limit of finite-dimensional systems naturally leads us to consider general $2$ -regular LCA groups that possess a compact open subgroup, where the issue of the positivity of the Wigner distribution is currently an open problem. We provide a complete solution to this question by showing that if the map $x\mapsto 2x$ is measure-preserving, the functions whose Wigner distribution is nonnegative are exactly the subcharacters of second degree, up to translation and multiplication by a constant. Instead, if the above map is not measure-preserving, the Wigner distribution always takes negative values. We discuss in detail the particular case of infinite sums of discrete groups and infinite products of compact groups, which correspond precisely to infinite quantum spin systems. Further examples include $n$ -adic systems, where $n\geq 2$ is an arbitrary integer (not necessarily a prime), as well as solenoid groups.The celebrated Hudson theorem states that the Gaussian functions in $\mathbb{R}^d$ are the only functions whose Wigner distribution is everywhere positive. Motivated by quantum information theory, D. Gross proved an analogous result on the Abelian group $\mathbb{Z}_d^n$, for $d$ odd - corresponding to a system of $n$ qudits - showing that the Wigner distribution is nonnegative only for the so-called stabilizer states. Extending this result to the thermodynamic limit of finite-dimensional systems naturally leads us to consider general $2$-regular LCA groups that possess a compact open subgroup, where the issue of the positivity of the Wigner distribution is currently an open problem. We provide a complete solution to this question by showing that if the map $x\mapsto 2x$ is measure-preserving, the functions whose Wigner distribution is nonnegative are exactly the subcharacters of second degree, up to translation and multiplication by a constant. Instead, if the above map is not measure-preserving, the Wigner distribution always takes negative values. We discuss in detail the particular case of infinite sums of discrete groups and infinite products of compact groups, which correspond precisely to infinite quantum spin systems. Further examples include $n$-adic systems, where $n\geq 2$ is an arbitrary integer (not necessarily a prime), as well as solenoid groups.

arXiv:2507.13154Jul 17, 20250 citations

Hudson's theorem fails for the SU(1,1) discrete series

Chon-Fai Kam

Hudson's theorem states that a pure state of a bosonic mode has a non-negative Wigner function if and only if it is Gaussian. It underwrites the reading of Wigner negativity as a faithful signature of pure-state non-classicality. We show the statement has no analogue on curved phase space. For the positive discrete series of $SU(1,1)$ , realised on the upper sheet of a two-sheeted hyperboloid, the Wigner-positive pure states form a strictly larger set than the Perelomov coherent orbit. Superpositions of the lowest weight state with the first excited state stay positive up to a mixing angle of $24.93^\circ$ at Bargmann index $k=1$ , and the admissible set has positive volume, with a maximal width that is not attained in the two-state direction. We show analytically that the quadratic form controlling positivity degenerates in the far field onto a single mixing angle. That degeneracy bounds the window by $\arctan(1/\sqrt{2k})$ and leaves the threshold itself fixed at intermediate hyperbolic distance.Hudson's theorem states that a pure state of a bosonic mode has a non-negative Wigner function if and only if it is Gaussian. It underwrites the reading of Wigner negativity as a faithful signature of pure-state non-classicality. We show the statement has no analogue on curved phase space. For the positive discrete series of $SU(1,1)$, realised on the upper sheet of a two-sheeted hyperboloid, the Wigner-positive pure states form a strictly larger set than the Perelomov coherent orbit. Superpositions of the lowest weight state with the first excited state stay positive up to a mixing angle of $24.93^\circ$ at Bargmann index $k=1$, and the admissible set has positive volume, with a maximal width that is not attained in the two-state direction. We show analytically that the quadratic form controlling positivity degenerates in the far field onto a single mixing angle. That degeneracy bounds the window by $\arctan(1/\sqrt{2k})$ and leaves the threshold itself fixed at intermediate hyperbolic distance.

quant-pharXiv:2609.07242Sep 8, 20260 citations

Mixed states with positive Wigner functions

T. Bröcker, Reinhard F. Werner

The Wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent, according to a result of Hudson. The characterization of mixed states with a positive Wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters (i.e., functions with given ‘‘Wigner spectrum’’ in the sense of Narcowich). If a state is a convex combination of coherent states, it has the property that the Wigner spectrum contains the unit interval, which in turn implies that the Wigner function is positive. It is shown by explicit examples that the converses of both implications are false. The examples are taken from a low-dimensional section of the state space, in which all Wigner spectra can be computed. In this set counterexamples to a conjecture by Narcowich concerning the Wigner spectrum of products are also found, as well as a state whose Wigner spectrum is a convergent sequence of discrete points.The Wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent, according to a result of Hudson. The characterization of mixed states with a positive Wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters (i.e., functions with given ‘‘Wigner spectrum’’ in the sense of Narcowich). If a state is a convex combination of coherent states, it has the property that the Wigner spectrum contains the unit interval, which in turn implies that the Wigner function is positive. It is shown by explicit examples that the converses of both implications are false. The examples are taken from a low-dimensional section of the state space, in which all Wigner spectra can be computed. In this set counterexamples to a conjecture by Narcowich concerning the Wigner spectrum of products are also found, as well as a state whose Wigner spectrum is a convergent sequence of discrete points.

Journal of Mathematical PhysicsJan 1, 199570 citations

Discrete Wigner functions and quantum computational speedup

Ernesto F. Galvão

Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] have recently defined a class of discrete Wigner functions $W$ to represent quantum states in a finite Hilbert space dimension $d$ . I characterize the set ${C}_{d}$ of states having non-negative $W$ simultaneously in all definitions of $W$ in this class. For $d\ensuremath{\leqslant}5$ I show ${C}_{d}$ is the convex hull of stabilizer states. This supports the conjecture that negativity of $W$ is necessary for exponential speedup in pure-state quantum computation.Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] have recently defined a class of discrete Wigner functions $W$ to represent quantum states in a finite Hilbert space dimension $d$. I characterize the set ${C}_{d}$ of states having non-negative $W$ simultaneously in all definitions of $W$ in this class. For $d\ensuremath{\leqslant}5$ I show ${C}_{d}$ is the convex hull of stabilizer states. This supports the conjecture that negativity of $W$ is necessary for exponential speedup in pure-state quantum computation.

arXiv:quant-ph/0405070Physical Review AApr 1, 2005122 citations

Gibbons et al., [Phys. Rev. A 70, 062101 (2004)] have recently defined discrete Wigner functions

W

to represent quantum states in a Hilbert space with finite dimension. We show that such a class of Wigner functions

W

can be defined so that the only pure states having non-negative

W

for all such functions are stabilizer states, as conjectured by Galv\~ao, [Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving non-negativity of

W

for all definitions of

W

in the class form a subgroup of the Clifford group. This means pure states with non-negative

W

and their associated unitary dynamics are classical in the sense of admitting an efficient classical simulation scheme using the stabilizer formalism.

The celebrated Hudson theorem states that the Gaussian functions in

\mathbb{R}^d

are the only functions whose Wigner distribution is everywhere positive. Motivated by quantum information theory, D. Gross proved an analogous result on the Abelian group

\mathbb{Z}_d^n

, for

d

odd - corresponding to a system of

n

qudits - showing that the Wigner distribution is nonnegative only for the so-called stabilizer states. Extending this result to the thermodynamic limit of finite-dimensional systems naturally leads us to consider general

2

-regular LCA groups that possess a compact open subgroup, where the issue of the positivity of the Wigner distribution is currently an open problem. We provide a complete solution to this question by showing that if the map

x\mapsto 2x

is measure-preserving, the functions whose Wigner distribution is nonnegative are exactly the subcharacters of second degree, up to translation and multiplication by a constant. Instead, if the above map is not measure-preserving, the Wigner distribution always takes negative values. We discuss in detail the particular case of infinite sums of discrete groups and infinite products of compact groups, which correspond precisely to infinite quantum spin systems. Further examples include

n

-adic systems, where

n\geq 2

is an arbitrary integer (not necessarily a prime), as well as solenoid groups.

SU(1,1)

, realised on the upper sheet of a two-sheeted hyperboloid, the Wigner-positive pure states form a strictly larger set than the Perelomov coherent orbit. Superpositions of the lowest weight state with the first excited state stay positive up to a mixing angle of

24.93^\circ

at Bargmann index

k=1

, and the admissible set has positive volume, with a maximal width that is not attained in the two-state direction. We show analytically that the quadratic form controlling positivity degenerates in the far field onto a single mixing angle. That degeneracy bounds the window by

\arctan(1/\sqrt{2k})

and leaves the threshold itself fixed at intermediate hyperbolic distance.

Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] have recently defined a class of discrete Wigner functions

W

to represent quantum states in a finite Hilbert space dimension

d

. I characterize the set

{C}_{d}

of states having non-negative

W

simultaneously in all definitions of

W

in this class. For $d\ensuremath{\leqslant}5$ I show

{C}_{d}

is the convex hull of stabilizer states. This supports the conjecture that negativity of

W

is necessary for exponential speedup in pure-state quantum computation.