Non-negative Wigner functions in prime dimensions

0.14 MB·Italian

by  David Gross

Non-negative Wigner functions in prime dimensions
D. Gross
Institute for Mathematical Sciences, Imperial College London, London SW7 2BW, UK and
QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK ∗
(Dated: February 1, 2008)
Accordingtoaclassical resultduetoHudson,theWignerfunctionofapure,continuousvariable
quantum state is non-negative if and only if the state is Gaussian. We have proven an analogous
statement for finite-dimensional quantum systems. In this context, the role of Gaussian states is
taken on bystabilizer states. Thegeneral results havebeen published in [D. Gross, J. Math. Phys.
47, 122107 (2006)]. For the case of systems of odd prime dimension, a greatly simplified proof can
be employed which still exhibits the main ideas. The present paper gives a self-contained account
of these methods.
7
0
I. INTRODUCTION In the previous theorem, := {0,...,d−1} denotes
0 d
Z
2 the set of integers modulo d.
The Wigner distribution establishes a correspondence Itturnsoutthat,althoughtheformulationofHudson’s
n
a between quantum mechanical states and real pseudo- resultcarriesnaturallyovertofinitedimensionalsystems,
J probability distributions on phase space. ’Pseudo’ refers the respectiveproofsareradicallydifferent. The original

1 to the fact that, while the Wigner function resembles argument relies crucially on function theory, which is of
3 many of the properties of probability distributions, it course not available in the setting that this paper ad-
can take on negative values. It is therefore of interest dresses.
1
to characterize those quantum states that are classical Recently, Galvao et. al. took a first step into the di-
v
in the sense of giving rise to non-negative phase space rection of classifying the quantum states with positive
4
distributions. Wigner function (see Ref. [3]). To explain the relation-
0
0 For the case of pure states described by vectors in ship of their results to the present paper, we have to
2 H = L2( ), the resolution of this problem was given comment shortly on two different approaches to defin-
0 by HudsoRn in Ref. [1], and later extended to multiple- ing discrete Wigner functions. On the one hand, it
7 particles by Soto and Claverie (Ref. [2]). has long been realized that Wigner’s definition carries
0
overnaturally to discrete odd-dimensional systems (Ref.
/ Theorem 1. (Hudson, Soto, Claverie) Let ψ ∈L2( n)
h [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and Section III). This
R
p be a state vector. The Wigner function of ψ is non- approach is the one used in the present paper. On the
- negative if and only if ψ is a Gaussian state. other hand, Gibbons, Hoffmann, and Wootters listed a
t
n By definition, a vector is Gaussian if and only if it is set of axioms that candidate definitions have to fulfill in
a of the form
order to resemble the properties of the well-known con-
u
tinuous case (Ref. [13]). Let us call functions that fall
q ψ(q)=e2πi(qθq+xq),
: into this class generalized Wigner functions. The char-
v where x ∈ n and θ is a symmetric matrix with entries acterization does not specify a unique solution: for a d-
Xi in [20]. R dimensional Hilbert space, there exist dd−1 distinct gen-
C eralized Wigner functions. The construction of Gibbons
r
a It is our objective to prove that the situation for dis- et. al. has been described only for the case where d is
crete quantum systems is very similar, at least when the the power of a prime.
dimension of the Hilbert space is odd. The following Ifthe dimensionofthe Hilbertspaceisoftheformdn,
Theorem states the main result. a second ambiguity arises. We are free to conceive such
a space either as being associated to a system of n con-
Theorem 2. (Discrete Hudson’s Theorem) Let d be odd
stituents,eachofdimensiond,ortoasingleoneofdimen-
and ψ ∈ L2( n) be a state vector. The Wigner function
Zd sion dn. While the Wigner function is the same for both
of ψ is non-negative if and only if ψ is a stabilizerstate.
cases,thesetofstabilizerstatesisnot(seeRefs. [5,12]).
Given that ψ(q)6=0 for all q, a vector ψ is a stabilizer
Indeed, the ’single-particle’ stabilizer states turn out to
state if and only if it is of the form
be a proper subset of the ’multiple-particle’ ones. As
ψ(q)=e2dπi(qθq+xq), a striking example, the generalized Bell and GHZ states
norarenotstabilizerstatesonsingled2ord3-dimensional
where x ∈ n and θ is a symmetric matrix with entries systems.
in . Zd InRef.[3]itwasprovedthatastateofasingle-particle
d
Z system of prime-power dimension is a stabilizer state if
and only if all its generalized Wigner functions are non-
negative. Theauthorsaimtoestablishnecessaryrequire-
∗Electronicaddress: [email protected] ments for quantum computational speedup. Indeed, if
2
the Wigner function of a quantum computer is positive thereadertoRefs. [5]forfurtherdetails. Inwhatfollows
at all times, then it operates only with stabilizer states d denotes an odd prime. All integer arithmetic in this
and hence offers no advantage over classical computers, paper is implicitly assumed to be modd. The symbol
by the Gottesman-Knill Theorem (Ref. [15]). 2−1 =(d+1)/2isthemultiplicativeinversemodulod. All
For the case of pure states, our results imply the ones statevectorsareelementsoftheHilbertspaceHspanned
ofRef.[3]andexceedthemintwoways. Firstly,weshow by {|0i,...,|d−1i}. Lastly, ω = e2dπi is a dth root of
thatitsufficestocheckpositivityforasingledefinitionof unity.
the Wigner function, as opposedto dd−1 ones. Secondly, The relations
our statements hold for multiple-particle systems, which
x(q)|ki=|k+qi, z(p)|ki=ωpk|ki
constitute the proper setting for both quantum compu-
tation and the Gottesman-Knill Theorem. On the other
define the shift and boost operators respectively. The
hand, Ref. [3] makes assertions about mixed states and
mostcentralelementinthetheoryaretheWeyloperators
qubit systems, which are not covered by our findings.
(in quantum information also known as the generalized
Our general results have been published in Ref. [5].
Pauli operators) given by
However, the proof is rather involved. Many technicali-
tiesariseduetothefactthatfornon-primed,arithmetic w(p,q)=ω−2−1pqz(p)x(q).
modulo d lacks the desirable properties of finite fields.
Our aim in writing Ref. [5] was to achieve the broad- The characteristic function of an operator ρ is given by
est possible generality in spite of these difficulties. The the expansion coefficients of ρ in terms of the Weyl op-
downside of this approach is that core ideas of the argu- erators
mentareobscuredbytechnicalissues. Thepresentpaper 1
employs a different method of proof, which is available Ξρ(ξ,x)= tr(w(ξ,x)†ρ).
d
only for systems of odd prime dimension. For this spe-
cialcase,themainresultofRef.[5]canbeobtainedusing We define the Wigner function to be the symplectic
onlyafractionofthespace. Itisourhopethatthispaper Fourier transform of the characteristic function:
makes the ideas accessible to a wider audience. 1
W (p,q) = ωpξ−qxΞ (ξ,x).
The next section summarizes further findings con- ρ ρ
d
tained in Ref. [5]. We go on to recall the definition and ξ,Xx∈ d
Z
properties of discrete Wigner functions in Section III.
It is a tedious yet straight-forwardcomputation to show
Section IV is devoted to a complete proof of the easiest
that the Wigner function of a pure state is given by
special case of Theorem 2, that being given by a single
particle on a Hilbert space of prime dimension. W (p,q) := W (p,q)
ψ |ψihψ|
1
= ω−ξpψ(q+2−1ξ)ψ¯(q−2−1ξ).
d
II. FURTHER RESULTS AND IMPLICATIONS ξX∈ d
Z
IfS isa2×2-matrixwithelementsin anddetermi-
d
It is natural to ask how Hudson’s results generalize to nant 1, then there exists a unitary operZation µ(S) (the
mixed states. Certainly, mixtures of Gaussian states are Weil [18] [21] or metaplectic representation of S) such
positive on phase space and Narcowich in Ref. [16] con- that
jectured that all such quantum states are convex combi-
nations of Gaussian ones. Bro¨cker and Werner refuted µ(S)w(p,q)µ(S)† =w(S(p,q)).
the conjecture by giving a counter-example (Ref. [17]).
The Wigner function is covariant in the sense that, if
We show in Ref. [5] that the situation is similar in the
ρ′ =µ(S)ρµ(S)†, then
finite setting.
Further, we show how to lift the ambiguity in the ax- Wρ′(p,q)=Wρ(S(p,q)). (1)
iomatic characterization of Wigner functions by requir-
ingCliffordcovariance,notethataunitaryoperatorpre- Similarly, the Weyl operators induce translations of
serves positivity if and only it is a Clifford operation, the Wigner function. Letting ρ′ = w(p′,q′)ρw(p′,q′)†,
discuss the relation of various ways to introduce Wigner it holds that
functions and stabilizer states in dimensions of the form
d = pn, and give an explicit account on the connection Wρ′(p,q)=Wρ(p+p′,q+q′). (2)
between stabilizer states and Gaussian states. The Clifford group is the set of unitary matrices that
sendWeyloperatorstoWeyloperatorsunderconjugation
[22]. Every Clifford mapping is of the form w(p,q)µ(S)
III. WIGNER FUNCTIONS
and hence preserves positivity of the Wigner function.
Finally, stabilizer states are the images of the com-
This section provides a very superficial introduction putational basis states under the action of the Clifford
todiscreteWigner functions. We allowourselvestorefer group.
3
IV. MAIN THEOREM – SINGLE PARTICLES Proof. Fix a q ∈ . As W is non-negative, so is the
d ψ
IN PRIME DIMENSIONS Fourier transformZof K (q,x) with respect to x. Boch-
ψ
ner’sTheoremimplies thatAx =K(x−y,q) is positive
y
Define the self correlation function semi-definite (psd) which in turn implies that all princi-
pal sub-matrices are psd. In particular the determinant
Kψ(q,x)=ψ(q+2−1x)ψ¯(q−2−1x) of the 2×2 principal sub-matrix
and note that the Wigner function obeys
K (q,0) K (q,2x)
ψ ψ
W(p,q)= 1 ω−pxK (q,x). (cid:18)Kψ(q,−2x) Kψ(q,0) (cid:19)
ψ
dXx = |ψ(q)|2 ψ(q+x)ψ¯(q−x)
(cid:18)ψ¯(q+x)ψ(q−x) |ψ(q)|2 (cid:19)
Recall that the Fourier transform fˆ of a function f :
Zd → C is defined to be fˆ(x) = 1/d qω−qxf(q). must be non-negative. But this means
Therefore, for a fixed q0, W(p,q0) is the FPourier trans-
form of K(q ,x). Hence W is non-negativeif and only if |ψ(q)|4−|ψ¯(q+x)ψ(q−x)|2 ≥0,
0
the d functions K(q ,·) have non-negativeFouriertrans-
0
forms. which proves the theorem.
In harmonic analysis, the set of functions with non-
negative Fourier transforms is characterized by a well- We will call the set of points where a state-vector is
knowntheoremdue to Bochner. We state anelementary non-zero its support.
versionofBochner’sTheorem,alongwith a variationfor
subsequent use. Lemma 5. (Support Lemma) Let ψ be a state vector
with positive Wigner function.
Theorem 3. (Variations of Bochner’s Theorem) Con-
If ψ is supported on two points, then it has maximal
sider a function f : → . It holds that
d
Z C support.
1. The Fourier transform of f is non-negative if and
only if the matrix Proof. Denote by S = suppψ the support of ψ. S has
the property to contain the midpoint of any two of its
Axq =f(x−q) elements. Indeed, if a,b∈S, then setting q =2−1(a+b)
andx=2−1(a−b)intheModulusInequalityshowsthat
is positive semi-definite.
2. The Fourier transform of f has constant modulus |ψ(2−1(a+b))|≥|ψ(a)||ψ(b)|>0,
(i.e. |fˆ(x)| = const) if and only if f is orthogonal
to its translations: hence 2−1(a+b)∈S.
Assume there exist two points a,b ∈ S. Requiring
hf,xˆ(q)fi= f¯(x)f(x−q)=0,
a = 0 is no loss of generality, for else we substitute ψ
Xx by ψ′ = w(0,−a)ψ. By Eq. (2), ψ′ has positive Wigner
for all non-zero q ∈ . function if and only if ψ has.
d
Z We claim that
Proof. The matrix A is circulant. It is well-known that
circulantmatricesarenormal(hencediagonalizable)with 2−lβb∈S (3)
eigenvaluesgivenbytheFouriertransformofthefirstrow
(uptoapositivenormalizationconstant). Thefirstclaim for all l and β ≤ 2l. The proof is by induction on l.
is now immediate. Suppose Eq. (3) holds for some l. If β ≤ 2l+1 is even,
By the same argument, A is proportionalto a unitary then 2−l−1βb=2−l(β/2)b∈S. Else,
matrix if and only if |fˆ(q)| is constant. But a matrix is
unitary if and only if its rows form an ortho-normal set β−1 β+1
2−l−1βb=2−1 2−l b+2−l b ∈S,
of vectors.
2 2
(cid:0) (cid:1)
The next three lemmas harvest some consequences of
which proves the claim.
Bochner’sTheoremto gaininformationonthe pointwise
Now,byFermat’sLittleTheorem2d−1 =1 mod dand
modulus |ψ(q)| of the vector.
hence, setting l = d − 1 in Eq. (3), we conclude that
Lemma 4. (ModulusInequality) Let ψ be a state vector βb∈S forallβ ≤d−1. Buteverypointin d isofthat
Z
with positive Wigner function. form.
It holds that
Lemma 6. (Constant Modulus) Let ψ be a state vector
|ψ(q)|2 ≥|ψ(q−x)||ψ(q+x)|
with positive Wigner function and maximal support.
for all q,x∈ . Then |ψ(q)|=const.
d
Z
4
Proof. Pick two points x,q ∈ and suppose |ψ(q)| > must hence exist at least two points a,b in the support
d
Z
|ψ(x)|. of W (note that we are now considering the support of
Lettingz =x−q,theassumptionreads|ψ(q)|>|ψ(q+ Wignerfunctions andnolongerthe supportofstatevec-
z)|. Lemma 4 centered at q+z gives tors). Makingoncemoreuseofthefactthattranslations
are implemented by Clifford operations, assume a = 0.
|ψ(q+z)|2 ≥ |ψ(q)||ψ(q+2z)| Thereexistsaunit-determinantmatrixS thatsendsbto
> |ψ(q+z)||ψ(q+2z)|, a vector of the form Sb = (0,q0)T. But then there are
twopoints in the supportofW (p,q ), contradicting
µ(S)ψ 0
therefore |ψ(q+z)| > |ψ(q+2z)|. By inducting on this our earlier derivation.
scheme, we arrive at
|ψ(q)|>|ψ(q+z)|>|ψ(q+2z)|>···
V. SUMMARY
and hence |ψ(q)| > |ψ(q+dz)| = |ψ(q)|, which is a con-
tradiction.
Thus |ψ(q)| ≤ |ψ(x)|. Swapping the roles of x and q We haveproveda ’classicalityresult’for discrete Wig-
proves that equality must hold. ner functions: those state vectors which give rise to a
classical probability distribution in phase space belong
Theorem 7. (Main Theorem – Special Case) Let d be
to the set of stabilizer states. These, in turn, allow for
prime and ψ ∈ L2( ) be a state vector with positive
Zd anefficient classicaldescription. Comparingthe proofof
Wigner function. Then ψ is a stabilizer state.
the special case treated here to the involved argument
Proof. By the Support Lemma, ψ is either a position employed in Ref. [5], it becomes apparent how much the
eigenstate or else it has maximal support. In the former geometrical properties of integer residues modulo prime
case, ψ is manifestly a stabilizer state, so we need only numbers simplify the structure.
treat the latter. Let U be a Clifford operation. Since U
preserves positivity, the Support Lemma applies to Uψ.
Suppose U is such that suppUψ contains just a single
point. Then Uψ belongs to the computational basis and VI. ACKNOWLEDGMENTS
hence, by definition, ψ is a stabilizer state.
Therefore,wearelefttotreatthosestatevectorswhose
The author is grateful for support and advice pro-
imageunderanyCliffordoperationhasmaximalsupport.
vided by Jens Eisert during all stages of this project.
The proof is concluded by showing that such states do
Comments by and discussions with K. Audenaert S.
not exist.
Chaturvedi,H.Kampermann,M.Kleinmann,A.Klimov,
For assume there is such a vector ψ. As ψ has point-
M. Ruzzi, and C.K. Zachos are kindly acknowledged.
wiseconstantmodulus,sodoesK . EmployingTheorem
ψ
3, we find that, for every fixed q , W(p,q ) is orthogo- This work has benefited from funding provided by the
0 0
nalto itsowntranslations. ButsinceW isnon-negative, EuropeanResearchCouncils(EURYI grantofJ.Eisert),
it follows that W(p,q ) can be non-zero on at most one theEuropeanCommission(IntegratedProjectQAP),the
0
point. A Wigner function that is concentrated at a sin- EPSRC (Interdisciplinary Research Collaboration IRC-
gle point can not represent a physical state [23]. There QIP), and the DFG.
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R
[11] S. Chaturvedi E. Ercolessi, G. Marmo, G. Morandi, N. θ has positive semi-definite imaginary part.
5
[21] Thereisaconfusingsimilarity ofnames: theWeilrepre- representation theory of SO(n).
sentation (after Andr´eWeil) acts on theWeyl operators [23] Such a Wigner function corresponds to a Hermitian op-
(after Hermann Weyl). erator with both positive and negative eigenvalues (see
[22] Notethatthe“Cliffordgroup”whichappearsinthecon- Ref. [5]). One can think of this fact as an incarnation of
text of quantum information theory [19] has no connec- the uncertaintyprinciple.
tion to the group by the same name used e.g. in the