Nonnegative mixed states in Weyl–Wigner–Moyal theory

José M. Gracia-Bondı́a and Joseph C. Várilly

Escuela de Matemática, Universidad de Costa Rica, 11501 San José, Costa Rica

Phys. Lett. A 128 (1988), 20–24

Abstract

We classify the gaussian Wigner functions corresponding to mixed states and show that,
unlike the case of pure states, not all nonnegative mixed states are gaussian.

A theorem of Hudson [1], generalized to the 𝑛-dimensional case by Soto and Claverie [2],
establishes that the only nonnegative distribution functions corresponding to pure states in the
Weyl–Wigner–Moyal (WWM) formulation of Quantum Mechanics are the Wigner transforms of
gaussian wavefunctions.

Recently, Littlejohn [3] posed this problem: which gaussian functions in phase space represent
pure quantum states? These are, of course, all of the aforementioned transforms; they are themselves
gaussians in phase space. But Littlejohn solved the question essentially by an elegant calculation
within the autonomous, algebraic formulation of WWM theory based on the twisted product.

In this letter we enlarge on Littlejohn’s work by establishing which gaussians in phase space
correspond to states, pure or mixed. Furthermore, we show that the analogue of the Hudson–Soto–
Claverie–Littlejohn result fails for mixed states: there are non-gaussian nonnegative functions in
phase space representing quantum states.

We start by recalling the notion of twisted product of two functions in phase space:

( 𝑓 × 𝑔) (𝑢) :=
∫
ℝ2𝑛

∫
ℝ2𝑛

𝑓 (𝑣) 𝑔(𝑤) exp[𝑖(𝑢𝑡𝐽𝑣 + 𝑣𝑡𝐽𝑤 + 𝑤𝑡𝐽𝑢)] 𝑑𝑣 𝑑𝑤

where 𝑢𝑡 = (𝑞1, . . . , 𝑞𝑛, 𝑝1, . . . , 𝑝𝑛) ∈ ℝ2𝑛 (𝑡 denotes transpose), 𝑣, 𝑤 ∈ ℝ2𝑛, 𝐽 =

(
0 1𝑛

−1𝑛 0

)
, and

𝑑𝑣 = (2𝜋)−𝑛 𝑑2𝑛𝑣, with 𝑑2𝑛𝑣 being Lebesgue measure. We have chosen units so that ℏ = 2. This is
a continuous bilinear operation on A := 𝐿2(ℝ2𝑛). Let I := { 𝑓 ×𝑔 : 𝑓 , 𝑔 ∈ A }. A and I correspond
respectively to the ideals of Hilbert–Schmidt and trace-class operators on 𝐿2(ℝ𝑛), via the Weyl
correspondence rule. We do not use that correspondence directly here; but we can norm I so as
to be isomorphic to the space of trace-class operators with trace norm. Let B be the dual normed
space of I. We consider B as the space of quantum-mechanical observables; it is an algebra under
the twisted product (isomorphic to the algebra of bounded operators on 𝐿2(ℝ𝑛).)

A state is an element of I which is positive in the algebraic sense, i.e.,

⟨𝜌, 𝑓 ∗ × 𝑓 ⟩ :=
∫
ℝ2𝑛

𝜌(𝑢) ( 𝑓 ∗ × 𝑓 ) (𝑢) 𝑑𝑢 ⩾ 0 for all 𝑓 ∈ B.

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Then 𝜌 = 𝜔∗ ×𝜔 for some 𝜔 ∈ A. Also, one may show that 𝜌 is the Wigner transform of a “density
matrix” 𝑑:

𝜌(𝑞, 𝑝) = (2𝜋)−𝑛
∫
ℝ𝑛

𝑑 (𝑞 + 𝑞′, 𝑞 − 𝑞′)𝑒𝑖𝑝𝑞′ 𝑑𝑛𝑞′.

Pure states are the Wigner transforms of density matrices of the form 𝜓∗(𝑞′)𝜓(𝑞) and are
identified by the condition 𝜌 × 𝜌 = 𝜌. Mixed states are (generalized) convex combinations of pure
states; our states are thus Wigner functions seen from a different viewpoint.

We remark the important equality:

⟨ 𝑓 , 𝑔⟩ :=
∫
ℝ2𝑛

𝑓 (𝑢) 𝑔(𝑢) 𝑑𝑢 =

∫
ℝ2𝑛

( 𝑓 × 𝑔) (𝑢) 𝑑𝑢,

true whenever both sides of the equation make sense. We normalize the states by the condition that
2−𝑛

∫
𝜌2(𝑢) 𝑑𝑢 = 1.

After these preliminaries, we recall Littlejohn’s theorem.

Theorem 1 (Littlejohn). The normalized gaussian

𝜌(𝑢) := 2𝑛 (det 𝐹)1/4 exp(−1
2𝑢

𝑡𝐹𝑢) (1)

is a pure state if and only if the matrix 𝐹 (necessarily positive definite) is moreover symplectic.

The condition on 𝐹 may be rephrased as follows: 𝐹 = 𝑆𝑡𝑆 for some symplectic matrix 𝑆. In
fact, if 𝐹 = exp(𝐽𝐵) with 𝐵𝑡 = 𝐵, one can take 𝑆 = 𝑆𝑡 = exp( 1

2𝐽𝐵).
Thus we assert the next result.

Theorem 2. A normalized gaussian of the form (1) represents a quantum state if and only if there
is a symplectic matrix 𝑆 so that

𝐹 = 𝑆𝑡 diag[_1, . . . , _𝑛, _1, . . . , _𝑛] 𝑆

where 0 < _𝑖 ⩽ 1 for 𝑖 = 1, . . . , 𝑛. Except for trivial translations this gives all states which may be
represented by gaussians in phase space.

We break the proof into several steps.

1. Any positive definite real quadratic form is equivalent by symplectic conjugation to one given
by the matrix diag[_1, . . . , _𝑛, _1, . . . , _𝑛] with _𝑖 > 0 for 𝑖 = 1, . . . , 𝑛.
This is a relatively old result in Classical Mechanics [4, 5]. For instance, when 𝑛 = 1, if

𝐹 =

(
𝑎 𝑏

𝑏 𝑐

)
has 𝑎 > 0, 𝑎𝑐 − 𝑏2 > 0, and if 𝑑 =

√
𝑎𝑐 − 𝑏2, then

𝐹 = 𝑆𝑡
(
𝑑 0
0 𝑑

)
𝑆 with 𝑆 =

(√︁
𝑎/𝑑 𝑏/

√
𝑎𝑑

0
√︁
𝑑/𝑎

)
.

2. If Ξ is a “unitary” element of B, i.e., Ξ∗ × Ξ = Ξ × Ξ∗ = 1, then the inner automorphism
𝜌 ↦→ Ξ∗ × 𝜌 × Ξ transforms states into states, preserving their pure or mixed character.

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3. Linear canonical changes of coordinates in phase space are realized by automorphisms of this
form; namely, for any symplectic matrix 𝑆 there is a unitary Ξ𝑆 ∈ B such that

(Ξ∗
𝑆 × 𝑓 × Ξ𝑆) (𝑢) = 𝑓 (𝑆𝑢)

for any 𝑓 ∈ B and in particular for a state.
Take Ξ𝑆 (𝑢) := 2𝑛 det(1 + 𝑆)−1/2 exp[ 𝑖2𝑢

𝑡𝐽 (1 + 𝑆)−1(1 − 𝑆)𝑢]. We shall omit the calculation
of Ξ𝑆, referring instead to [6, 7]. We remark that the functions Ξ𝑆 realize the metaplectic
representation of Shale, Segal and Weil [8–10] in the framework of twisted product theory.
(The formula should be modified when det(1+𝑆) = 0, but this is immaterial for our purposes.)
Thus we need only consider matrices of the form diag[_1, . . . , _𝑛, _1, . . . , _𝑛]. In what follows
we use the orthonormal basis of functions on phase space given in [11] as follows. For 𝑛 = 1,
𝑘 ⩾ 𝑙 (nonnegative integers) and 𝑢 = (𝑞, 𝑝):

𝑓𝑘𝑙 (𝑢) := 2(−1)𝑙
√︂

𝑙!
𝑘!

(𝑞 − 𝑖𝑝)𝑘−𝑙𝐿𝑘−𝑙
𝑙 (𝑞2 + 𝑝2) 𝑒−(𝑞2+𝑝2)/2,

where the 𝐿𝑘−𝑙
𝑙

are Laguerre polynomials; 𝑓𝑘𝑙 := 𝑓 ∗
𝑙𝑘

for 𝑘 < 𝑙. For 𝑛 > 1, write 𝑢𝑖 := (𝑞𝑖, 𝑝𝑖)
and 𝑘 = (𝑘1, . . . , 𝑘𝑛), 𝑙 = (𝑙1, . . . , 𝑙𝑛); then 𝑓𝑘𝑙 (𝑢) := 𝑓𝑘1𝑙1 (𝑢1) · · · 𝑓𝑘𝑛𝑙𝑛 (𝑢𝑛).
The 𝑓𝑘𝑙 satisfy the important property: 𝑓𝑘𝑙 × 𝑓𝑟𝑠 = 𝛿𝑙𝑟 𝑓𝑘𝑠 (Kronecker delta). The “diagonal”
elements 𝑓𝑘𝑘 correspond to pure states of the 𝑛-dimensional harmonic oscillator.

4. A function of the form (1) with 𝐹 = diag[_1, . . . , _𝑛, _1, . . . , _𝑛] and some _𝑖 > 1 is not a
state.
It evidently suffices to consider the case 𝑛 = 1. The first excited state of the harmonic oscillator
is 𝑓11(𝐻) = 2(2𝐻 − 1)𝑒−𝐻 where 𝐻 = 1

2𝑢
2 = 1

2 (𝑞
2 + 𝑝2). Recalling that 𝑓11 × 𝑓11 = 𝑓11, we

get

⟨𝜌, 𝑓 ∗11 × 𝑓11⟩ = (some positive constant)
∫ ∞

0
(2𝐻 − 1) 𝑒−(_+1)𝐻 𝑑𝐻

= (some positive constant)
1 − _

(1 + _)2 < 0.

5. For 𝑛 = 1, 𝛽 > 0, the function 𝑓𝛽 (𝑞, 𝑝) := 2
√︁

tanh(𝛽/2) exp
(
−1

2 (𝑞
2 + 𝑝2) tanh(𝛽/2)

)
is a

(mixed) state.
In fact, it is the Gibbs state of the 1-dimensional oscillator at inverse temperature 𝛽, since
𝑓𝛽 =

√︁
2 sinh 𝛽

∑∞
𝑘=0 𝑒

−(𝑘+ 1
2 )𝛽 𝑓𝑘𝑘 .

6. The function 𝑓 (𝑢) := 2𝑛
√
_1 · · · _𝑛 exp(−1

2_1(𝑞2
1 + 𝑝2

1) + · · · + _𝑛 (𝑞2
𝑛 + 𝑝2

𝑛)) is a (mixed) state
whenever 0 < _1 < 1, . . . , 0 < _𝑛 < 1.
Indeed, with 𝛽𝑖 = 2 arctanh_𝑖, 𝑓 is a tensor product of 𝑛 functions of the previous type:
𝑓 (𝑢) =

∏𝑛
𝑖=1 𝑓𝛽𝑖 (𝑞𝑖, 𝑝𝑖). Thus 𝑓 =

∑
𝑘 𝑎𝑘 𝑓𝑘𝑘 (summed over all 𝑛-tuples of nonnegative

integers); the 𝑓𝑘𝑘 are (pure) states: and the coefficients 𝑎𝑘 =
∏𝑛

𝑖=1
√︁

2 sinh 𝛽𝑖 exp(−(𝑘𝑖 + 1
2 )𝛽𝑖)

are positive and sum to 1.
Theorem 2 follows. □

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Remark. The dimension of the manifold of pure gaussian states is 𝑛2 + 𝑛; the dimension of the
manifold of gaussian states is 𝑛2 + 2𝑛.

To prove that the collection of nonnegative states found does not include every nonnegative
state, an example will suffice. We obtain it from a couple of very interesting results.

Theorem 3 (Bernard et al [12]). The convolution of a quantum state with another quantum state is
nonnegative.

For the proof, we remit to [12]. The proof given there is for pure states, but the result extends
trivially to the general case.

Theorem 4. The convolution of a (suitably normalized) nonnegative function with a quantum state
is a quantum state.

Proof. Our argument is a modified version of that in [13], recast in our algebraic framework based
on twisted products. The point is that translations may be considered as automorphisms. If 𝛿𝑣 is
the Dirac measure on phase space concentrated at 𝑣 (note that 𝛿𝑣 ∈ B) then we can show

(𝛿𝑣 × 𝑓 × 𝛿𝑣) (𝑢) = 𝑓 (2𝑣 − 𝑢).

Let 𝑓 be nonnegative and 𝑔 a state. Then for ℎ ∈ B we have

⟨ 𝑓 ∗ 𝑔, ℎ∗ × ℎ⟩ =
∬

𝑓 (𝑢) 𝑔(−𝑢 − 𝑣) (ℎ∗ × ℎ) (−𝑣) 𝑑𝑢 𝑑𝑣

=

∬
𝑓 (𝑢) (𝛿−𝑢/2 × 𝑔 × 𝛿−𝑢/2) (𝑣) (𝛿0 × ℎ∗ × ℎ × 𝛿0) (𝑣) 𝑑𝑢 𝑑𝑣

=

∫
𝑓 (𝑢) 𝑑𝑢

∫
𝑔(𝑣) (𝑘∗𝑢 × 𝑘𝑢) (𝑣) 𝑑𝑣 ⩾ 0 (with 𝑘𝑢 := ℎ × 𝛿0 × 𝛿−𝑢/2). □

In contrast to step 2 of Theorem 2, here pure states do not remain pure. The last two results
point to a rather mysterious duality between nonnegative functions and quantum states.

Taken together, Theorems 2, 3 and 4 amount to a powerful machine for producing nonnegative
mixed states in WWM theory. We now have, for instance, with 𝑛 = 1:

( 𝑓𝛽 ∗ 𝑓11) (𝑢) = 2𝑒−3𝛽/2√︁2 sinh 𝛽(1 + 𝑢2 sinh2 1
2 𝛽) exp(−1

2𝑢
2𝑒−𝛽/2 sinh 1

2 𝛽),

which must represent a quantum state and it is not gaussian.

A number of recent articles in this Journal [12–16] have discussed quantum-mechanical non-
negative distributions in phase space. We hope this paper helps to throw some light on these
questions.

We gratefully acknowledge support from the Vicerrectorı́a de Investigación of the Universidad
de Costa Rica.

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