THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
arXiv:1611.02838v1 [math.FA] 9 Nov 2016
3 ˇ ˇ ZALAR4 IGOR KLEP1 , SCOTT MCCULLOUGH2 , KLEMEN SIVIC , AND ALJAZ
Abstract. A linear map Φ between real symmetric matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations In ⊗ Φ are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps which are not completely positive is given.
1. Introduction For n ∈ N, let Mn be the vector space of real n×n matrices equipped with transposition as the involution ∗ and let Sn = {A ∈ Mn : A∗ = A} be its subspace of real symmetric matrices. A symmetric matrix A ∈ Sn is positive semidefinite if and only if all of its eigenvalues are nonnegative; equivalently, v ∗ Av ≥ 0 for all v ∈ Rn . We write A 0. For n, m ∈ N, a linear map Φ : Mn → Mm is ∗-linear if Φ(A∗ ) = Φ(A)∗ for every A ∈ Mn . A ∗-linear map Φ : Mn → Mm (resp. Φ : Sn → Sm ) is positive if Φ(A) 0 for every A 0. For k ∈ N, a ∗-linear map Φ : Mn → Mm (resp. Φ : Sn → Sm ) induces a ∗-linear map Φ(k) : Mk ⊗ Mn = Mkn → Mk ⊗ Mm = Mkm , (resp. Φ(k) : Mk ⊗ Sn → Mk ⊗ Sm ,
M ⊗ A 7→ M ⊗ Φ(A)
M ⊗ A 7→ M ⊗ Φ(A)),
where ⊗ stands for the Kronecker tensor product of matrices. A ∗-linear map Φ is kpositive if Φ(k) is positive. If Φ is k-positive for every k ∈ N, then Φ is completely positive (cp). Obviously, every cp map is positive, and the transpose map M2 → M2 is positive but not 2-positive and thus not cp. Positive maps occur frequently in matrix theory [Hog14, Wor76] and functional analysis (e.g., positive linear functionals). Cp maps are ubiquitous in quantum physics (where they are called quantum channels or operations) [NC10], and operator algebra [Pau02]. Both types of maps are also important topics in random matrix theory and free probability [VDN92]. In quantum information theory cp maps are used to describe the quantum mechanical generalization of a noisy channel. The Stinespring representation theorem Date: November 10, 2016. 2010 Mathematics Subject Classification. 13J30, 46L07, 52A40 (Primary); 47L25, 81P45, 90C22 (Secondary). Key words and phrases. positive map, completely positive map, positive polynomial, sum of squares, convex cone. 1 Supported by the Marsden Fund Council of the Royal Society of New Zealand. Partially supported by the Slovenian Research Agency grants P1-0222 and L1-6722. 2 Research supported by the NSF grant DMS-1361501. 3 Partially supported by the Slovenian Research Agency grants P1-0222 and L1-6722. 4 Supported by the Slovenian Research Agency. 1
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
2
[Pau02, Theorem 4.1] implies that under a cp map each state is mapped into a separable state, so the quantum state of each particle in the channel can be described independently. The states that are not separable are called entangled. In the entangled case the quantum states of the particles are correlated and cannot be described independently; the quantum state must be described for the entire system. Positive maps that are not cp preserve positivity on separable states, but they may fail to preserve it on the entangled ones. Consequently, such maps are of great importance for detecting entanglement of a system. ˙ for a small sample of the vast We refer to [AS06, ASY14, HSR03, P-GWPR06, SWZ11] quantum information theory literature on entanglement breaking maps; see also [JKPR11, Stø08, PTT11]. Verifying whether a linear map is positive is computationally intractable; numerical algorithms, based on Lasserre’s [Las09] polynomial sum of squares relaxations for detecting positivity are given in [NZ+]. Recently Collins, Hayden, Nechita [CHN+] studied entanglement breaking maps from a free probability viewpoint [VDN92] using von Neumann algebras. Among other results they present random techniques for constructing k-positive maps which are not k + 1positive in large dimensions [CHN+, Theorem 4.2]. The gap between positive and cp maps was also investigated by Arveson [Arv09a, Arv09b], and Aubrun, Szarek, Werner, ˙ ˙ Ye, Zyczkowski [SWZ08, ASY14]. Arveson used operator algebra to establish: Theorem 1.1 (Arveson [Arv09a]). Let n, m ≥ 2. Then the probability p that a positive map ϕ : Mn (C) → Mm (C) is cp satisfies 0 < p < 1. ˙ Szarek, Werner and Zyczkowski use classical convexity and geometry of Banach spaces to improve upon Arveson’s results by providing quantitative bounds on the probability p ˙ Theorem 5]. (in the case where n = m) and establish its asymptotic behavior, see [SWZ08, In this paper we investigate the gap between positive and completely positive maps by translating the problem into the language of real algebraic geometry [BCR98]. 1.1. Main results and reader’s guide. The contribution of this paper is threefold. First, we will study nonnegative biquadratic biforms which are not sums of squares by estimating volumes of appropriate cones of positive polynomials. The study of positive polynomials is one of the pillars of real algebraic geometry, starting with Artin’s solution of Hilbert’s 17th problem, cf. [Mar08, Lau09, Rez95, Put93, Sce09, Scw03, KS10, Pow11, Cim12, Oza13]. To estimate the ratio between compact base sections of the cones of sums of squares biforms and nonnegative biquadratic biforms we shall employ powerful techniques, based on harmonic analysis and classical convexity, developed by Blekherman [Ble06] and Barvinok-Blekherman [BB05]. Let R[x, y] be the vector space of real polynomials in the variables x := (x1 , . . . , xn ) and y := (y1 , . . . , ym ). Let R[x, y]k1 ,k2 be the subspace of biforms of bidegree (k1 , k2 ), i.e., polynomials from R[x, y] which are homogeneous of degree k1 in x and of degree k2 in y. Let (n,m)
Pos(2k1 ,2k2 ) = {f ∈ R[x, y]2k1 ,2k2 : f (x, y) ≥ 0 for all (x, y) ∈ Rn × Rm } , ) ( X (n,m) Sos(2k1 ,2k2 ) = f ∈ R[x, y]2k1 ,2k2 : f = fi2 for some fi ∈ R[x, y]k1 ,k2 , i
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
3
be the cone of nonnegative biforms and the cone of sums of squares biforms; respectively. In all but a few stray cases the cone of sums of squares biforms is strictly contained in the cone of nonnegative biforms. (n,m)
(n,m)
Theorem 1.2 (Choi, Lam, Reznick [CLR80, Theorem 8.4]). Pos(2k1 ,2k2 ) = Sos(2k1 ,2k2 ) if and only if n = 2 and k2 = 1 or m = 2 and k1 = 1. (n,m)
(n,m)
We shall estimate the gap between the cones Pos(2k1 ,2k2 ) and Sos(2k1 ,2k2 ) by comparing volumes of suitably chosen compact sections of these cones. Let T := S n−1 × S m−1 and consider the product measure σ = σ1 × σ2 on T , where S n−1 ⊆ Rn , S m−1 ⊆ Rm are the unit spheres and σ1 , σ2 are the normalized Lebesgue measures on S n−1 and S m−1 , respectively. The Lp norm of a biform f ∈ R[x, y]2k1 ,2k2 on T is given by Z Z Z p p p |f | dσ = kf kp = |f (x, y)| dσ2 (y) dσ1 (x). x∈S n−1
T
y∈S m−1
(n,m)
Let H(2k1 ,2k2 ) be the hyperplane of biforms from R[x, y]2k1 ,2k2 of average 1 on T , i.e., Z (n,m) H(2k1 ,2k2 ) = f ∈ R[x, y]2k1 ,2k2 : f dσ = 1 . T
(n,m)
Let Pos(2k1 ,2k2 )
′
(n,m)
and Sos(2k1 ,2k1 )
′
(n,m)
(n,m)
be the sections of the cones Pos(2k1 ,2k2 ) and Sos(2k1 ,2k2 ) ,
′ \ (n,m) (n,m) (n,m) Pos(2k1 ,2k2 ) = Pos(2k1 ,2k2 ) H(2k1 ,2k2 ) , ′ \ (n,m) (n,m) (n,m) Sos(2k1 ,2k2 ) = Sos(2k1 ,2k2 ) H(2k1 ,2k2 ) .
′ ′ (n,m) (n,m) Thus Pos(2k1 ,2k2 ) and Sos(2k1 ,2k2 ) are convex and compact full-dimensional sets in the (n,m)
finite dimensional hyperplane H(2k1 ,2k2 ) . For technical reasons we translate these sections P P 2 k2 by subtracting the polynomial ( ni=1 x2i )k1 ( m j=1 yj ) , i.e., ( ) n m ′ X X (n,m) (n,m) g Pos = f ∈ R[x, y]2k1 ,2k2 : f + ( x2 )k1 ( y 2)k2 ∈ Pos , (2k1 ,2k1 )
f (n,m) Sos (2k1 ,2k2 )
i
=
(
f ∈ R[x, y]2k1 ,2k2 : f + (
j
i=1
j=1
n X
m X
x2i )k1 (
i=1
j=1
yj2)k2 ∈
(2k1 ,2k2 )
(n,m) Sos(2k1 ,2k2 )
′
)
.
(n,m)
Let M := M(2k1 ,2k2 ) be the hyperplane of biforms from R[x, y]2k1 ,2k2 with average 0 on T , Z M = f ∈ R[x, y]2k1 ,2k2 : f dσ = 0 . T
Notice that g (n,m) ⊆ M and Sos f (n,m) ⊆ M. Pos (2k1 ,2k1 ) (2k1 ,2k1 )
Let DM be the dimension of M and let SM , BM be the unit sphere and the unit ball in M equipped with the L2 norm, respectively. Then M is isomorphic to RDM as normed spaces. Let ψ : RDM → M be an isomorphism and ψ∗ µ the pushforward of the Lebesgue measure µ on RDM to M, i.e., ψ∗ µ(E) := µ(ψ −1 (E)) for every Borel measurable set E ⊆ M.
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
4
Lemma 1.3. The measure of a Borel set E ⊆ M does not depend on the choice of the isomorphism ψ, i.e., if ψ1 : RDM → M and ψ2 : RDM → M are isomorphisms of normed spaces, then (ψ1 )∗ µ(E) = (ψ2 )∗ µ(E). Proof. We have (ψ2 )∗ µ(E) = µ(ψ2−1 (E)) = µ((ψ2−1 ◦ ψ1 ◦ ψ1−1 )(E)) = µ((ψ2−1 ◦ ψ1 )(ψ1−1 (E))) = µ(ψ1−1 (E)) = (ψ1 )∗ µ(E),
where the first equality in the second line holds since ψ2−1 ◦ ψ1 is a linear isometry and µ is a Lebesgue measure. The bounds for the volume of the section of nonnegative biforms are as follows. Theorem 1.4. For n, m ∈ N we have D1 (n,m) 21 M g 2 2 Vol Pos(2k1 ,2k2 ) 2k 2k 1 2 ≤ 4 min , , c2k1 ,2k2 ≤ Vol BM 2k12 + n 2k22 + m
where
c2k1 ,2k2 =
(
1
2−4 max(n, m)− 2 ,
if k1 = k2 = 1
exp(−3) (2⌈max(n, m) ln(2 max(k1 , k2 ) + 1)⌉)
− 21
,
otherwise.
Next we give bounds for the volume of the section of sums of squares biforms. Theorem 1.5. For integers n, m ≥ 3 we have D1 (n,m) M f Vol Sos(2k1 ,2k2 ) ≤ e2k1 ,2k2 , d2k1 ,2k2 ≤ Vol BM
where
d2k1 ,2k2 =
e2k1 ,2k2 =
1 2−8 · 6− 2 ·
(
√
nm+n+m , (n+4)(m+4) 3 k2 !) 2
(k1 ! √ √ 2 6·42k1 +2k2 · (2k1 )! (2k2 )!
√ 1 , 210 6 · √nm+n+m q √ 2 6 · 42k1 +2k2 · (2kk11)!!
if k1 = k2 = 1 k2 k1 n 2 m 2 n m k ( 2 +2k1 ) 1 ( 2 +2k2 )k2
,
otherwise if k1 = k2 = 1
(2k2 )! k2 !
k − 21
·n
k − 22
m
,
otherwise
Combining the previous two theorems we obtain: Corollary 1.6. For integers n, m ≥ 3 we have D1 (n,m) M f Vol Sos(2k1 ,2k2 ) ≤ g2k1 ,2k2 , f2k1 ,2k2 ≤ (n,m) g Vol Pos(2k1 ,2k2 )
where
f2k1 ,2k2 =
g2k1 ,2k2 =
√
√3 2
211 7
3 , min(n,m) nk1 mk2 2+max
C2k1 ,2k2 · √ ( 14 √ 2 6
min(n,m)+1
if k1 = k2 = 1
12
n m , k2 k2 1 2 n m k 1 ( 2 +2k1 ) ( 2 +2k2 )k2
,
otherwise,
,
D2k1 ,2k2 · (nk1 −1 mk2 −1 min(n, m))
if k1 = k2 = 1 − 21
,
otherwise,
,
.
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
5
and 3
C2k1 .2k2 D2k1 ,2k2
(k ! k !) 2 p1 2 = √ , 3 · 42(k1 +k2 +1) · (2k1 )! (2k2 )! · min(k1 , k2) s √ (2k1 )! (2k2 )! ⌈ln(2 max(k1 , k2 ) + 1)⌉. 3 · e3 · 42k1 +2k2 +1 · = k1 ! k2 !
Proof. For the case k1 = k2 = 1 see Theorem 3.2. For the other constants f2k1 ,2k2 , g2k1 ,2k2 , C2k1 ,2k2 , D2k1 ,2k2 combine Theorems 1.4 and 1.5 together with the estimate ⌈Nt⌉ ≤ N⌈t⌉, used for N = max(n, m) and t = ln(2 max(k1 , k2 ) + 1). Section 2 is devoted to Theorems 1.4 and 1.5. The proof of Theorem 1.4 is given in Subsection 2.1, while for the proof of Theorem 1.5 we need some preliminary results about the apolar inner product on the vector space of biforms introduced and studied in Subsection 2.2. Finally, Theorem 1.5 is established in Subsection 2.3. The second contribution of the paper is the estimate on the gap between the cones of positive and completely positive maps. By converting the problem into the language of real algebraic geometry, the following estimate will follow by Theorems 1.4 and 1.5. Corollary 1.7. For integers n, m ≥ 3 the probability pn,m that a positive linear map Φ : Sn → Sm is completely positive, is bounded by !DM !DM √ √ 3 3 214 6 p < pn,m < p , 211 72 min(n, m) min(n, m) + 1 m+1 where DM = n+1 − 1. In particular, if min(n, m) ≥ 229 · 3, then 2 2 lim
max(n,m)→∞
pn,m = 0.
Here, the probability pn,m is defined as the ratio between the volumes of the sections f (n,m) and Pos g (n,m) in M. Sos (2,2)
(2,2)
Remark 1.8. ˙ ˙ provide similar bounds in the case of (1) Szarek, Werner, and Zyczkowski in [SWZ08] complex matrix algebras with n = m. However, their normalization is different from ours. We normalize using tr(Φ(In )) = nm (see Proposition 3.4), whereas in ˙ the compact cross-section is obtained by fixing tr(Φ(In )) = n. [SWZ08] 1 (2) We note that the normalized probability pn,m DM (as in Corollary 1.6) does not go to 0 if min(m, n) is bounded and max(m, n) → ∞.
Section 3 converts the positive–cp gap problem into the language of real algebraic geometry [BCR98]. To each linear map Φ : Sn → Sm we associate the biquadratic biform pΦ ∈ R[x, y], pΦ = y∗ Φ(xx∗ )y. Then Φ is positive if and only if pΦ is nonnegative on Rn+m , and Φ is cp if and only if pΦ is a sum of squares of polynomials, see Proposition 3.1 below. Therefore positive maps which are not cp correspond exactly to nonnegative biquadratic biforms which are not sums of squares biforms. We note that a different connection between (completely) positive maps and real algebraic geometry was introduced and investigated in [HKM13, HKM+]. The third contribution of the paper is the construction (from random input data) of positive maps Φ : Sn → Sm (n, m ≥ 3) that are not completely positive, see Section 4.
6
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
Again, by Proposition 3.1, it suffices to construct nonnegative biquadratic biforms which are not sums of squares biforms. This is done in Algorithm 4.1 by specializing the [BSV16] algorithm to our context. Algorithm 4.1 depends on semidefinite programming [WSV00], so produces a floating point output. We discuss implementation and rationalization, i.e., producing exact output, in Subsection 4.3. 1.1.1. Comparison with the original work of Blekherman. In [Ble06] Blekherman established estimates on the volumes of compact sections of the cones of nonnegative forms and sums of squares forms. If the degree is fixed and the number of variables goes to infinity the ratio between the volumes goes to 0. We restrict ourselves to special subcones of these cones, i.e., the cones of nonnegative biforms and sums of squares biforms. It is not clear how to directly apply the estimates from [Ble06] to this special case. In fact, [BR+] gives the example of symmetric nonnegative forms vs sums of squares, where the ratio between the corresponding volumes behaves differently, i.e., does not tend to 0. Regarding biforms as tensor products of forms we establish estimates for biforms following the techniques of [Ble06]. In [BSV16] there is an explicit construction of nonnegative quadratic forms on special projective varieties which are not sums of squares forms. We specialize their construction to the context of biquadratic biforms to produce nonnegative biforms which are not sums of squares biforms. Recently, Erg¨ ur posted the preprint [Erg+] on arXiv. There he extends some of Blekherman’s volume estimates to biforms; like our results in Section 2 his results readily generalize to multiforms. While there is certain overlap with our results, we explicitly compute all constants appearing in the estimates. Furthermore, some of our estimates are strictly better than the ones of [Erg+]; cf. Theorem 2.1 and [Erg+, Section 3]. Acknowledgments. The authors thank Greg Blekherman for many inspiring discussions and for bringing the preprint [Erg+] to their attention. We would also like to thank Benoit Collins for helpful suggestions. 2. Blekherman type estimates for biforms In this section we extend the estimates on the volumes of compact sections of the cones of nonnegative forms and sums of squares forms established in [Ble06] to biforms. Our proofs borrow heavily from [Ble06] and to a lesser extent from [BB05]. For clarity and completeness of exposition we give proofs with all the details, even if some of the reasoning repeats arguments from [Ble06]. At various places we will regard the vector space R[x, y]2k1 ,2k2 of biforms of bidegree (2k1, 2k2 ) as a module over the product SO(n) × SO(m) of special orthogonal groups with the action given by rotating the coordinates, i.e., for (A, B) ∈ SO(n) × SO(m) and f ∈ R[x, y]2k1 ,2k2 we define (2.1)
(A, B) · f (x, y) = f (A−1 x, B −1 y).
(n,m) (n,m) g (n,m) f (n,m) Note that the cones Pos(2k1 ,2k2 ) and Sos(2k1 ,2k2 ) , and the sections Pos (2k1 ,2k2 ) and Sos(2k1 ,2k2 ) are invariant under this action.
2.1. Nonnegative biforms. In this subsection we establish bounds for the volume of the section of nonnegative biforms. The main result is the following.
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
7
Theorem 2.1. For n, m ∈ N we have: D1 (n,m) 21 M g 2 2 Vol Pos(2k1 ,2k2 ) 2k 2k 2 1 , c2k1 ,2k2 ≤ ≤ 4 min , Vol BM 2k12 + n 2k22 + m
where
c2k1 ,2k2 =
(
1 24
1
max(n, m)− 2 ,
if k1 = k2 = 1
exp(−3) (2⌈max(m, n) ln(2 max(k1 , k2 ) + 1)⌉)
− 12
,
otherwise.
The proof of Theorem 2.1 occupies the next two subsections. It is inspired by Blekherman’s proof of [Ble06, Theorem 4.1]. Let V be a real vector space. Recall that for a convex body K in V with origin in its interior the gauge GK is defined by GK : V → R,
GK (p) = inf {λ > 0 : p ∈ λ · K} . (n,m)
g 2.1.1. Proof of the lower bound in Theorem 2.1. We denote K = Pos (2k1 ,2k2 ) . Note that K is a convex body in M with origin in its interior and the boundary of K consists of biforms with minimum −1 on T . Indeed, it is easy to see that K consists exactly of biforms from M with minimum at least −1 on T and that every biform with minimum −1 on T belongs to its boundary. However, if f ∈ K satisfies mf := min(x,y)∈T f (x, y) > −1, then the ball B(f, mf + 1) := {g ∈ R[x, y]2k1 ,2k2 : kf − gk∞ < mf + 1}
also belongs to K and hence f belongs to the interior of K. Therefore the gauge GK : M → R of K in M is given by GK (f ) = | min f (v)| for f ∈ M. v∈T
Let µ e be the Haar probability measure on SM . By [Pis89, p. 91], 1 Z D1 M Vol K DM −DM = . GK de µ Vol BM SM
By H¨older’s inequality we have Z
SM
and so
D1
M
−DM GK de µ
Vol K Vol BM
D1
M
≥
≥ Z
Z
SM
SM
G−1 µ, K de
G−1 µ. K de
By Jensen’s inequality (applied to the convex function y = x1 on R>0 ), Z −1 Z −1 GK de µ≥ GK de µ . SM
SM
Since kf k∞ = maxv∈T |f (v)| ≥ | minv∈T f (v)|, it follows that Z −1 1 Vol K DM ≥ kf k∞ de µ . Vol BM SM
The proof the lower bound in Theorem 2.1 now reduces to proving the following claim.
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
8
Claim 1:
Z
SM
kf k∞ de µ≤
1 c2k1 ,2k2
.
To prove this claim we will use [Bar02, Corollary 2]. Write G = SO(n) × SO(m) and consider the tensor product (Rn )⊗2k1 ⊗ (Rm )⊗2k2 . Let e1 ∈ Rn , f1 ∈ Rm be standard unit vectors and let w be the tensor w := (e1 )⊗2k1 ⊗ (f1 )⊗2k2 ∈ (Rn )⊗2k1 ⊗ (Rm )⊗2k2 . We also define v := w − q,
where q =
Z
(g, h)w d(g, h),
(g,h)∈G
and we integrate w.r.t. the Haar measure on G. Similarly as in [BB05, Example 1.2] we proceed as follows: (1) We identify the vector space of biforms from R[x, y]2k1 ,2k2 with the vector space V1 of the restrictions of linear functionals ℓ : (Rn )⊗2k1 ⊗ (Rm )⊗2k2 → R to the orbit {(g, h)w : (g, h) ∈ G} . (2) We identify the vector space of biforms from M with the vector space V2 of the restrictions of linear functionals ℓ : (Rn )⊗2k1 ⊗ (Rm )⊗2k2 → R to B = conv((g, h)v : (g, h) ∈ G). (3) We introduce an inner product on V2 by defining Z hℓ1 , ℓ2 i := ℓ1 ((g, h)v) · ℓ2 ((g, h)v) d(g, h). G
This also induces the dual inner product on V2∗ ∼ = V2 which we also denote by h·, ·i.
By [Bar02, Corollary 2],
1
kf k∞ ≤ (Dk ) 2k · kf k2k ,
where Dk = dim span{(g, h)w ⊗k : (g, h) ∈ G}. Clearly,
1k Dk = dim span{ge⊗2k : g ∈ SO(n)} · dim span{hf1⊗2k2 k : h ∈ SO(m)} 1 2k1 k + n − 1 2k2 k + m − 1 , = 2k2 k 2k1 k
where the second equality follows as in [Bar02, p. 404]. 2 2 max(k1 , k2 )k + max(n, m) − 1 2k1 k + n − 1 2k2 k + m − 1 . ≤ (2.2) Dk = 2 max(k1 , k2 )k 2k2 k 2k1 k We now distinguish two cases. Case 1: k1 = k2 = 1. If max(n, m) is odd, we let k0 = 21 (max(n, m) − 1). Otherwise take k0 = to get k1 1 4k0 0 2k0 Dk 0 ≤ ≤ 24 , 2k0 where the last inequality follows by a simple induction.
1 2
max(n, m)
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
Case 2: k1 > 1 or k2 > 1. Claim 2: For k0 ≥ ⌈max(m, n) ln(2 max(k1 , k2 ) + 1)⌉, 1
Dk2k0 0 ≤ exp(3). We define the function 1
H(x) = −x ln(x) − (1 − x) ln(1 − x) for x ∈ (0, 1).
For λ ∈ 0, 2 we have the estimate i n n X X λ n n i n n−i n (1 − λ) λ (1 − λ) = 1 = (λ + (1 − λ)) = i i 1−λ i=0 i=0 λn ⌊λn⌋ ⌊λn⌋ X n X n λ n (1 − λ) > exp(−nH(λ)), = i i 1 − λ i=0 i=0
λ ≤ 1 for λ ∈ (0, 21 ] in the inequality. It follows that where we used that 1−λ a b a ≤ exp aH for a, b ∈ N and b ≤ ⌊ ⌋. b a 2 a Since ab = a−b and H ab = H a−b , we conclude a b a ≤ exp aH for a, b ∈ N and b ≤ a. (2.3) b a
Writing C1 = max(k1 , k2 ), C2 = max(m, n) and using (2.2), (2.3) we get k1 1 2C1 k 2k exp (2C1 k + C2 − 1) H Dk ≤ 2C1 k + C2 − 1 C2 − 1 C2 − 1 2C1 k = exp 2C1 ln 1 + + ln 1 + 2C1 k k C2 − 1 C2 − 1 2C1 k C2 − 1 · exp ln 1 + = exp 2C1 ln 1 + 2C1 k k C2 − 1 C2 − 1 C2 − 1 2C1 k ≤ exp · exp , ln 1 + k k C2 − 1
where we used ln(1 + x) ≤ x for x > −1 in the second inequality. Let as assume that k0 ≥ ⌈C2 ln(2C1 + 1)⌉. Then
C2 − 1 exp < exp(1), k0 since k0 ≥ C2 . To prove Claim 2 it remains to establish C2 − 1 2C1 k0 (2.4) exp ≤ exp(2). ln 1 + k0 C2 − 1
Notice that (2.4) holds if and only if 2C1 k0 2k0 ln 1 + ≤ . C2 − 1 C2 − 1
9
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
10
Now
2C1 k0 (2C1 + 1)k0 ln 1 + ≤ ln . C2 − 1 |{z} C2 − 1 k0 ≥C2
Thus it suffices to prove that
ln or equivalently
(2C1 + 1)k0 C2 − 1
2k0 , C2 − 1
≤
k0 ≤ 2k0 . C2 − 1 Using ln(x) ≤ x − 1 < x for x > 0 we estimate the left hand side from above by (C2 − 1) ln (2C1 + 1) + ln
(C2 − 1) ln (2C1 + 1) + k0 and since (C2 − 1) ln (2C1 + 1) + k0 ≤ 2k0
if and only if
(C2 − 1) ln (2C1 + 1) ≤ k0 ,
(2.4) holds. Hence Claim 2 follows.
To prove Claim 1 it remains to estimate the average L2k0 norm, i.e., 2k1 Z Z Z 0 2k0 (2.5) A= kf k2k0 de µ= f dσ de µ. SM
Notice that (2.6)
Z
SM
Z
T
f
2k0
dσ
SM
2k1
0
de µ=
Z
SV2
T
Z
G
hc, (g, h)vi
2k0
d(g, h)
2k1
0
dc,
where SV2 is the unit sphere in V2 endowed with the Haar probability measure dc. Combining (2.5), (2.6) we obtain s 2k1 Z Z 0 2k0 hv, vi p A= hc, (g, h)vi2k0 d(g, h) dc ≤ = 2k0 , DM SV2 G
where we used [BB05, Lemma 3.5] for the inequality and [BB05, Remark p. 62] for the last equality. This proves Claim 1 and establishes the lower bound in Theorem 2.1.
2.1.2. Proof of the upper bound in Theorem 2.1. Before proving the upper bound in Theorem 2.1 we introduce the gradient inner products needed in the proof. Let f ∈ R[x, y]2k1 ,2k2 be a biform. For every fixed y ∈ Rm we define a form f y ∈ R[x]2k1 by f y (x) := f (x, y).
Recall [Ble06, p. 367] that the gradient inner product on R[x]2k1 is defined by Z 1 h∇h1 , ∇h2 i dσ1 for h1 , h2 ∈ R[x]2k1 , hh1 , h2 igr = 2 4k1 S n−1
where
∂h ∂h ∇h = ( ,..., ) for h ∈ R[x]2k1 ∂x1 ∂xn
and
h∇h1 , ∇h2 i =
n X ∂h1 ∂h2 i=1
∂xi ∂xi
.
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
11
We define the x-gradient inner product on R[x, y]2k1 ,2k2 by Z hf, gigrx = hf y , g y igr dσ2 (y). S m−1
Note that positive definiteness follows by noticing that if f ∈ R[x, y]2k1 ,2k2 is a nonzero biform, then there exists (x, y) ∈ S n−1 × S m−1 such that f (x, y) 6= 0. By continuity it follows that f y0 is nonzero for every y0 in some neighbourhood of y. Thus hf y0 , f y0 igr > 0 for every y0 in some neighbourhood of y. Hence hf, f igrx > 0. Let kf kgrx be the x-gradient norm of f and let Bgrx be the unit ball in the x-gradient norm. (n,m)
g◦ be the polar dual of the section Pos g Proof of the upper bound in Theorem 2.1. Let Pos (2k1 ,2k2 ) in M, n o g◦ = f ∈ M : hf, gi ≤ 1 for all g ∈ Pos g (n,m) Pos . (2k1 ,2k2 )
g (n,m) Pos (k1 ,k2 )
we get that By the Blaschke-Santal´o inequality [MP90] applied to g◦ ≤ (Vol BM )2 . g (n,m) (2.7) Vol Pos Vol Pos (2k1 ,2k2 )
(Note that for the validity of (2.7) we used the fact that the origin is the Santal´o point g (n,m) of Pos (2k1 ,2k2 ) . This follows by observing that the origin is the unique point in the convex g (n,m) g (n,m) body Pos (2k ,2k ) fixed by the action of SO(n)×SO(m) and that Pos(2k ,2k ) is also invariant 1
2
1
2
under the action of SO(n) × SO(m).) Hence it suffices to prove that ! 1 21 2 g◦ DM 2k1 + n 2k22 + m 1 Vol Pos max , . ≥ (2.8) Vol BM 4 2k12 2k22
Let B∞ be the unit ball of the supremum norm in M. We notice that \ g (n,m) g (n,m) , B∞ = Pos −Pos (2k1 ,2k2 ) (2k1 ,2k2 )
and by taking polar duals we get
◦ g◦ , −Pos g◦ } ⊆ Pos g◦ B∞ = ConvexHull{Pos
M
g◦ ), (−Pos
L where stands for the Minkowski addition. By a theorem of Rogers and Shephard [RS57, Theorem 1], it follows that 2DM ◦ g◦ . Vol Pos Vol B∞ ≤ DM M Since by induction 2D ≤ 4DM , we see DM ! 1 g◦ DM 1 Vol Pos (2.9) ≥ . ◦ Vol B∞ 4 Using (2.8) and (2.9) the proof of the upper bound in Theorem 2.1 reduces to establishing 12 D1 2 ◦ M Vol B∞ 2k1 + n 2k22 + m . (2.10) ≥ max , Vol BM 2k12 2k22 D1 1 2 ◦ M 2k1 + n 2 Vol B∞ ≥ Claim: . Vol BM 2k12
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
12
We estimate (2.11)
kf k∞ = max m−1 y∈S
y
max |f (x)|
x∈S n−1
≥ max kf y (x)kgr ≥ kf kgrx , m−1 y∈S
where the first inequality follows by [Kel28, Theorem IV], kh∇f y , ∇f y ik∞ ≤ 4k12 kf y k2∞ . Using (2.11) we get the inclusion (2.12)
B∞ ⊆ Bgrx
◦ ◦ and hence Bgr ⊆ B∞ , x
◦ ◦ where B∞ and Bgr are the polar duals of B∞ and Bgrx , respectively. Since Bgrx is an x ellipsoid (the x-gradient norm is induced from an inner product), we deduce ◦ Vol Bgr = x
(Vol BM )2 Vol Bgrx
and hence by (2.12), (Vol BM )2 . Vol Bgrx Therefore the proof of the Claim reduces to showing that ◦ Vol B∞ ≥
hf, f igrx ≥ We estimate
Z
2k12 + n hf, f i . 2k12
hf y , f y igr dσ2 (y) Z 2k12 + n 2k12 + n y y ≥ hf , f i dσ (y) = hf, f i , 2 2k12 2k12 S m−1
hf, f igrx =
S m−1
where the inequality follows by [Ble06, (4.3.1)]. This proves the Claim. By interchanging the roles of x and y in the Claim we also obtain the inequality 2 1 D1 ◦ M Vol B∞ 2k2 + m 2 , ≥ Vol BM 2k22 which proves (2.10) and concludes the proof of the upper bound in Theorem 2.1. 2.2. The apolar inner product on R[x, y]2k1 ,2k2 . Before tackling the bounds for the volume of the section of sum of squares biforms we have to extend some of the results on the apolar inner product given in [Ble06, §5]. For technical reasons we identify R[x, y]2k1 ,2k2 with R[x]2k1 ⊗ R[y]2k2 in the natural way. Recall from [Rez92, p. 11] that for a form X r= cα xi11 · · · xinn ∈ R[x]2k1 , α=(i1 ,...,in )
the associated differential operator Drx is defined by X ∂ i1 ∂ in Drx = c α i1 · · · in . ∂xn ∂x1 α=(i ,...,i ) 1
n
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
13
The operator Drx induces the inner product on R[x]2k1 , called the x-apolar inner product, defined by hr, sidx = Drx (s) for s ∈ R[x]2k1 . P Note that positive definiteness follows from Drx (r) = α c2α · i1 ! · · · in !. Analogously we define the differential operator Dty for a form t ∈ R[y]2k2 and the y-apolar inner product h·, ·idy on R[y]2k2 . To every form f ∈ R[x]2k1 ⊗ R[y]2k2 , X f = fℓ1 (x) ⊗ fℓ2 (y) ℓ
X
=
ℓ
X
α=(i1 ,...,in )
i1 in c(ℓ) α x1 · · · xn ⊗
X
(ℓ)
β=(j1 ,...,jm )
we associate the differential operator Df by X Df = Dfxℓ1 ⊗ Dfyℓ2 ℓ
=
X ℓ
X
c(ℓ) α
α=(i1 ,...,in )
X ∂ ∂ ⊗ i1 · · · ∂xinn ∂x1 β=(j ,...,j
jm dβ y1j1 · · · ym ,
in
i1
1
(ℓ)
dβ m)
j1
jm
∂ ∂ , j1 · · · jm ∂y1 ∂ym
and the corresponding inner product, called the apolar inner product, by hf, gid = Df (g) for g ∈ R[x]2k1 ⊗ R[y]2k2 .
Example 2.2. For f = f1 ⊗ f2 ∈ R[x]2k1 ⊗ R[y]2k2 and g = g1 ⊗ g2 ∈ R[x]2k1 ⊗ R[y]2k2 , we have Df (g) = Df1 (g1 )Df2 (g2 ). Note that this inner product is invariant under the action of SO(n) × SO(m). For a point v = (v1 , . . . , vn ) ∈ S n−1 , we denote by v 2k1 the form v 2k1 := (v1 x1 + · · · + vn xn )2k1 ∈ R[x]2k1 .
We define a linear operator Tv : R[x]2k1 → R[x]2k1 by Z (2.13) Tv (r) = r(v)v 2k1 dσ1 (v). S n−1
Analogously for a point u = (u1 , . . . , um ) ∈ S m−1 we denote by u2k2 and T u the form and the linear operator on R[y]2k2 given by Z m X 2k2 2k2 u u =( uj yj ) ∈ R[y]2k2 and T (t) = t(u)u2k2 dσ2 (u). S m−1
j=1
Finally let
T : R[x]2k1 ⊗ R[y]2k2 → R[x]2k1 ⊗ R[y]2k2
be the linear operator defined by T
X ℓ
fℓ1 ⊗ fℓ2
!
=
X ℓ
Tv (fℓ1 ) ⊗ T u (fℓ2 ) .
Some properties of the operator T we will need are collected in the following lemma.
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
14
Lemma 2.3. The following statements hold: (1) The operator T relates the two inner products by the following identity, hT f, gid = (2k1 )!(2k2 )! hf, gi . (2) The operator T is bijective. o nP P 2 k2 , i.e., (3) The operator T has eigenspace span ( ni=1 x2i )k1 ⊗ ( m y ) j=1 j ! ! n m n m X X X X x2i )k1 ⊗ ( yj2)k2 , T ( x2i )k1 ⊗ ( yj2)k2 = c · ( i=1
where
i=1
j=1
j=1
Γ(k1 + 21 )Γ( n2 ) Γ(k2 + 21 )Γ( m2 ) √ c= √ . πΓ(k1 + n2 ) πΓ(k2 + m2 )
Proof. By bilinearity it suffices to prove Lemma 2.3 ((1)) only for elementary tensors f = f1 ⊗ f2 , g = g1 ⊗ g2 ∈ R[x]2k1 ⊗ R[y]2k2 . Since hT f, gid = hTv f1 , g1 idx hT u f2 , g2idy , hf, gi = hf1 , g1 i hf2 , g2 i ,
Lemma 2.3 ((1)) follows by [Ble06, Lemma 5.1]. Since T maps from the finite-dimensional vector space into itself, to prove Lemma 2.3 ((2)), it suffices to prove that the kernel of T is trivial. Let us assume that T f = 0 for some f ∈ R[x]2k1 ⊗ R[y]2k2 . By Lemma 2.3 ((1)) it follows that hf, f i = 0. Hence f = 0 and the kernel of T is trivial. Finally, Lemma 2.3 ((3)) follows by ! ! ! m n n m X X X X yj2)k2 x2i )k1 ⊗ T u ( T ( x2i )k1 ⊗ ( yj2 )k2 = Tv ( i=1
=
j=1
n Γ(k1 + 12 )Γ( n2 ) X 2 k1 √ ( x) πΓ(k1 + n2 ) i=1 i
!
j=1
i=1
⊗
m Γ(k2 + 21 )Γ( m2 ) X 2 k2 √ ( y ) πΓ(k2 + m2 ) j=1 j
!
,
where the second equality follows by [Ble06, p. 371] used for Tv and T u .
P Let L be a full-dimensional cone in R[x]2k1 ⊗R[y]2k2 such that the polynomial ( ni=1 x2i )k1 ⊗ R P 2 k2 ( m is an interior point of L and T f dσ > 0 for all non-zero f ∈ L. Let Le be the j=1 yj ) subset of M defined by ( ) n m X X Le = f ∈ M : f + ( x2 )k1 ⊗ ( y 2 )k 2 ∈ L . i
i=1
j
j=1
Let L∗ be the dual cone of L w.r.t. the L2 inner product and L∗d the dual cone of L in the apolar inner product, L∗ = {f ∈ R[x]2k1 ⊗ R[y]2k2 : hf, gi ≥ 0 for all g ∈ L} ,
L∗d = {f ∈ R[x]2k1 ⊗ R[y]2k2 : hf, gid ≥ 0 for all g ∈ L} . P P 2 k2 belongs to the interiors of L∗ Proposition 2.4. The biform ( ni=1 x2i )k1 ⊗ ( m j=1 yj ) ∗ and Ld .
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
Pn
15
Pm
Proof. The biform rxk1 ⊗ sky2 := ( i=1 x2i )k1 ⊗ ( j=1 yj2 )k2 is in the interior of L∗ (resp. L∗d )
if and only if rxk1 ⊗ sky2 , g > 0 (resp. rxk1 ⊗ sky2 , g d > 0) is true for all g ∈ L. Since Z Z
k1 k2 k1 k2 rx ⊗ sy , g = (rx ⊗ sy ) · gdσ = gdσ, T
T
(resp.
k1 rx ⊗ sky2 , g d =
−1 k1
k1 1 1 T rx ⊗ sky2 , g = rx ⊗ sky2 , g (2k1 )!(2k2 )! (2k1 )!(2k2 )!c Z 1 = gdσ, (2k1 )!(2k2 )!c T
where c is defined as in Lemma 2.3 ((3)), and the first equality follows by Lemma 2.3 ((1)), ((2)), while the second one by Lemma 2.3 ((3))), this is true by definition of L. f∗ and L f∗ be defined by Let L d ) ( n m X X f∗ = L f ∈ M: f + ( x2i )k1 ⊗ ( yj2)k2 ∈ L∗ , f∗ = L d
(
f ∈ M: f + (
i=1
j=1
n X
m X
i=1
x2i )k1 ⊗ (
j=1
yj2)k2 ∈ L∗d
)
.
The following is an analog of [Ble06, Lemma 5.2]. Lemma 2.5. Let L be a full dimensional cone in R[x]2k1R⊗R[y]2k2 such that the polynomial P P 2 k2 is the interior point of L and T f dσ > 0 for all non-zero f ∈ L. ( ni=1 x2i )k1 ⊗( m j=1 yj ) f∗ and L f∗ : Then we have the following relationship between the volumes of L d
f∗ Vol L d f∗ Vol L
k1 ! k2 ! ≤ n m k 1 ( 2 + 2k1 ) ( 2 + 2k2 )k2
where α =1−
2k1 − 1 n + 2k1 − 1
2
−
! D1
≤
+
M
2k2 − 1 m + 2k2 − 1
2
k1 ! k2 ! n m k 1 ( 2 + k 1 ) ( 2 + k 2 )k 2
α
,
2k2 2k1 n + 2k1 − 2 m + 2k2 − 2
2
.
Proof. From Lemma 2.3 ((1)) it follows that for all f, g ∈ R[x]2k1 ⊗ R[y]2k2 , hf, gi ≥ 0 if and only if hT f, gid ≥ 0. Therefore, T maps L∗ to L∗d . By Lemma 2.3 it follows that for all f, g ∈ R[x]2k1 ⊗ R[y]2k2 , hf, gid ≥ 0 if and only if hT −1 f, gi ≥ 0, where T −1 is the inverse of T . Therefore, T maps L∗ onto L∗d , (2.14) Let ∆x =
Pn
∂2 i=1 ∂x2i
(resp. ∆y =
Pm
T (L∗ ) = L∗d .
∂2 j=1 ∂yj2 )
be the Laplace differential operator on R[x]
(resp. R[y]). Then R[x]2k1 (resp. R[y]2k2 ) splits into irreducible SO(n)-modules (resp. SO(m)-modules) [Vil68, Chapter IX §2], 1 R[x]2k1 = ⊕ki=0 rxi Hn,2k1 −2i ,
where rx =
n X i=1
x2i
2 (resp. R[y]2k2 = ⊕kj=0 sjy Hm,2k2 −2j ),
and Hn,2i = {r ∈ R[x]2i : ∆x r = 0}
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
16
Pm 2 (resp. sy = j=1 yj and Hm,2j = {s ∈ R[y]2j : ∆y s = 0}). Then the SO(n) × SO(m)module R[x]2k1 ⊗ R[y]2k2 splits into submodules as follows: 1 2 R[x]2k1 ⊗ R[y]2k2 = ⊕ki=0 ⊕kj=0 (rxi Hn,2k1−2i ⊗ sjy Hm,2k2 −2j ).
By Lemma 2.3 ((3)), where
T rxk1 ⊗ sky2 = c · rxk1 ⊗ sky2 ,
Γ(k1 + 21 )Γ( n2 ) Γ(k2 + 21 )Γ( m2 ) √ . c= √ πΓ(k1 + n2 ) πΓ(k2 + m2 )
Since 1c T commutes with the action of SO(n) × SO(m) and fixes rxk1 ⊗ sky2 , it also fixes the orthogonal complement of rxk1 ⊗ sky2 , which is the hyperplane of all biforms with integral f∗ to L f∗ . Applying [Ble04, 0 on T . Using this and (2.14), we conclude that 1c T maps L d Lemma 7.4] componentwise for 1c T we have that ! ! k1 X k2 X X X 1 T fℓ1 ⊗ fℓ2 = cjk ℓx2j (fℓ1 ) ⊗ ℓy2k (fℓ2 ) c j=0 k=0 ℓ ℓ ! k k 1 X 2 X X ℓx2j (fℓ1 ) ⊗ ℓy2k (fℓ2 ) , cjk = j=0 k=0
where cjk
ℓ
k1 ! Γ(k1 + n2 ) k2 ! Γ(k2 + m2 ) = (k1 − j)! Γ(k1 + j + n2 ) (k2 − k)! Γ(k2 + k +
m ) 2
and ℓx2j (fℓ1 ) (resp. ℓy2k (fℓ2 )) denotes the orthogonal projection of fℓ1 to rxk1 −j Hn,2j (resp. fℓ2 to sky2 −k Hm,2k ). Note that ck1 k2 is the smallest among the coefficients cjk and the lower bound on the change in volume is ! 1 f∗ DM k1 ! Γ(k1 + n2 ) k2 ! Γ(k2 + m2 ) Vol L d ≥ = ck 1 k 2 . f∗ Γ(2k1 + n2 ) Γ(2k2 + m2 ) Vol L Estimate
k1 ! Γ(k1 + n2 ) k2 ! Γ(k2 + m2 ) k1 ! k2 ! ≥ n . n m m k Γ(2k1 + 2 ) Γ(2k2 + 2 ) ( 2 + 2k1 ) 1 ( 2 + 2k2 )k2
This proves the lower bound in Lemma 2.5. To prove the upper bound in Lemma 2.5 observe that the largest coefficient of contraction occurs in the submodule Hn,2k1 ⊗ Hm,2k2 which has dimension DH = (dim R[x]2k1 − dim R[x]2k1 −2 )(dim R[y]2k2 − dim R[y]2k2 −2 ) n + 2k − 1 n + 2k − 3 m + 2k − 1 m + 2k − 3 2 2 1 1 . − · − = 2k2 − 2 2k2 2k1 − 2 2k1
The dimension DM of the ambient space M is n + 2k1 − 1 m + 2k2 − 1 − 1. DM = 2k2 2k1 We have
n + 2k1 − 1 m + 2k2 − 1 · C, DH = 2k2 2k1
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
where C=
2k1 − 1 2k1 1− n + 2k1 − 2 n + 2k1 − 1
1−
17
2k2 − 1 2k2 . m + 2k2 − 2 m + 2k2 − 1
Thus DH = DM · C + C |{z} < DM · C + 1. C 1 or k2 > 1, then
DH 1 2k2 2k1 r
(2.31)
v u u 2DU < t2 DM
Using (2.31) together with the estimates n+k1 −1 (2k1)! −k1 k1 (2.32) < n n+2k1 −1 k1 ! −1 2k 1
n+k1 −1 k1 n+2k1 −1 − 2k1
(resp.
1
m+k2 −1 k2 m+2k2 −1 − 2k2
m+k2 −1 k2 m+2k2 −1 − 2k2
1
.
1
0 ), Z −1 Z −1 GBsq de µ≥ µ GBsq de . SM
SM
Therefore using (2.33) we have
(2.34)
Vol Bsq Vol BM
D1
M
≥ e−1 2k1 ,2k2 .
′ ◦ (n,m) Let (Sos′ ) be the polar dual of Sos′ := Sos(2k1 ,2k2 ) . By definition, ◦
(Sos′ ) = {f ∈ M : hf, gi ≤ 1 for all g ∈ Sos′ } . ◦T ◦ Claim: Bsq = (Sos′ ) − (Sos′ ) .
First we prove the inclusion (⊆) in the Claim. Let us take f ∈ Bsq and show that ◦T ◦ ◦ f ∈ (Sos′ ) − (Sos′ ) . By definition of (Sos′ ) we have to prove that
(2.35)
|hf, hi| ≤ 1 for all h ∈ Sos′ .
By assumption f ∈ Bsq we have 2 f, g ≤ 1 for all g ∈ SR[x,y] (2.36) . k1 ,k2
2 P P ◦ Notice that every h = ki=1 h2i ∈ (Sos′ ) can be written as h = ki=1 λi √hλi i , where λi = D E R 2 Pk hi √ √hi f, that , it follows by (2.36) ∈ S λ = 1. Since h dσ and ≤ 1. i R[x,y] k1 ,k2 i=1 T i λi λi Hence |hf, hi| ≤ 1. This proves (2.35).
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
24
The inclusion (⊇) in the Claim is trivial (since the definition (2.36) is a special case of the definition (2.35)). Let Sos∗ be the dual cone of Sos in the L2 inner product, Sos∗ = {f ∈ R[x, y]2k1 ,2k2 : hf, gi ≥ 0 for all g ∈ Sos}, g∗ be the set and let Sos ( g∗ = Sos f ∈ R[x, y]2k (
=
(
=
: f +(
1 ,2k2
X
x2i )k1 (
i
f ∈ M : hf + ( f ∈ M : hf + (
X
x2i )k1 (
i
X
j
X j
x2i )k1 (
i
X
X j
yj2 )k2 ∈ Sos∗
\
(n,m) H(2k1 ,2k2 )
yj2 )k2 , gi ≥ 0 for all g ∈ Sos
)
yj2 )k2 , gi ≥ 0 for all g ∈ Sos′
)
)
= {f ∈ M : hf, gi ≥ −1 for all g ∈ Sos′ }
= {f ∈ M : h−f, gi ≤ 1 for all g ∈ Sos′ } ◦
= − (Sos′ ) ,
(n,m)
where the second equality follows by definitions of Sos∗ and H(2k1 ,2k2 ) , the third by homoDP E P geneity of the inner product and the forth by ( i x2i )k1 ( j yj2 )k2 , g = 1 for g ∈ Sos′ . g∗ = − (Sos′ )◦ together with (2.34) and the Claim we get Using Sos g∗ Vol Sos Vol BM
(2.37)
Since ( (n,m)
P
i
x2i )k1 (
Sos(2k1 ,2k2 ) we have (2.38) where g∗ = Sos d
(
P R
! D1
M
≥ e−1 2k1 ,2k2 .
(n,m)
j
T
yj2 )k2 is in the interior of Sos(2k1 ,2k2 ) and for all non-zero f ∈
f dσ > 0, it follows by Lemma 2.5 that g∗ Vol Sos d g Vol Sos∗
! D1
M
≥
f ∈ R[x, y]2k1 ,2k2 : f + (
( n2
X
k1 ! k2 ! , m + 2k1 )k1 ( 2 + 2k2 )k2
x2i )k1 (
i
X j
yj2 )k2 ∈ Sos∗d
\
(n,m) H(2k1 ,2k2 )
)
,
(n,m)
and Sos∗d ⊂ R[x, y]2k1 ,2k2 is the dual cone in the apolar inner product of Sos(2k1 ,2k2 ) . Combining (2.37) and (2.38) we obtain g∗ Vol Sos d Vol BM
! D1
M
≥ e−1 2k1 ,2k2
( n2
k1 ! k2 ! . m k + 2k1 ) 1 ( 2 + 2k2 )k2
(n,m) g∗ ⊆ Sos^ By Lemma 2.6, Sos d (2k1 ,2k2 ) and the lower bound of Theorem 2.7 is proved.
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
25
3. Positive maps and biforms In this section we connect linear maps on symmetric matrices with biforms. This will translate the question of comparing the size of the cone of completely positive maps with the size of the cone of positive maps to the question of comparing the size of the cone of sums of squares biforms with the size of the cone of positive biforms. We denote by L(Sn , Sm ) the vector space of all linear maps from Sn to Sm . There is a linear bijection Γ between linear maps L(Sn , Sm ) and biforms R[x, y]2,2 of bidegree (2, 2) given by (3.1)
Γ : L(Sn , Sm ) → R[x, y]2,2 ,
Φ 7→ pΦ (x, y) := y∗ Φ(xx∗ )y.
Thus Γ translates between properties of linear maps from L(Sn , Sm ) and the corresponding properties of biforms from R[x, y]2,2 . Positivity (resp. complete positivity) of a map Φ corresponds to nonnegativity (resp. being a sum of squares) of the polynomial pΦ : Proposition 3.1. Let Φ : Sn → Sm be a linear map. Then (1) Φ is positive iff pΦ is nonnegative; (2) Φ is completely positive iff pΦ is a sum of squares.
Proof. The implication (⇒) of (1) is trivial. For the implication (⇐) observe that any P positive semidefinite matrix X ∈ Sn can be written as the sum X = ki=1 vi vi∗ where P vi ∈ Rn for each i. Hence y ∗ Φ(X)y = ki=1 y ∗ pΦ (vi vi∗ )y is positive for every y ∈ Rm . To prove the implication (⇒) of (2) first invoke the Arveson’s extension theorem [Pau02, e : Mn → Mm and then the Theorem 7.5] to extend Φ to a completely positive map Φ Stinespring’s representation theorem (see[Pau02, Theorem 4.1] or [Cho75, Theorem 1]) Pℓ Pℓ ∗ ∗ e in the form X 7→ to represent Φ i=1 Vi XVi for some ℓ ∈ N where i=1 Vi Vi is a bounded operator of norm kΦk. (The proofs of the real finite-dimensional versions of [Pau02, Theorem 7.5] and [Pau02, Theorem 4.1] can be found, for example, in [HKM13, §3.1].) Hence pΦ (x, y) =
ℓ X
y∗ Vi∗ xx∗ Vi y =
i=1
∗
ℓ X
qi2 (x, y),
i=1
where qi (x, y) = x Vi y for each i. It remains to prove the implication (⇐) of (2). It suffices to prove that there is an e : Mn → Mm . Since pΦ (x, y) is a sum of extension of Φ to a completely positive map Φ squares, it is of the form !2 ℓ ℓ n m X ℓ X X X X 2 pΦ (x, y) = qi (x, y) = qijk xk yj = (y∗ (qijk )jk x)2 i=1
i=1
=
ℓ X
j=1 k=1
i=1
y∗ (qijk )jk xx∗ (qijk )∗jk y,
i=1
where qi (x, y) =
Pm Pn j=1
k=1 qijk xk yj
∗
∈ R[x, y]. From pΦ (x, y) := y∗ Φ(xx∗ )y it follows that
Φ(xx ) =
ℓ X i=1
(qijk )jk xx∗ (qijk )∗jk .
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
26
e : Mn → Mm defined by Hence the map Φ e Φ(X) =
ℓ X
(qijk )jk X(qijk )∗jk
i=1
for all X ∈ Mn (R)
is a completely positive extension of Φ. Let Pos(Sn , Sm ) and CP(Sn , Sm ) denote the cone of positive maps and the cone of completely positive maps from Sn to Sm , respectively. By Proposition 3.1, comparing (n,m) the cones Pos(Sn , Sm ) and CP(Sn , Sm ) is equivalent to comparing the cones Pos(2,2) and (n,m)
Sos(2,2) . g (n,m) and Sos f (n,m) . In this subsection we obtain 3.1. Comparing the volumes of Pos (2,2) (2,2) f (n,m) and Pos g (n,m) . By Theorem bounds on the ratio between the volumes of the sets Sos (2,2)
(2,2)
1.2 the sets are the same if and only if n ≤ 2 or m ≤ 2. Here is the main result of this subsection. Theorem 3.2. For integers n, m ≥ 3 we have D1 (n,m) √ √ M f 14 Vol Sos(2,2) 2 3 3 6 p
2, [BSV16, Procedure 3.3] is an explicit construction of nonnegative quadratic forms from R[z]/In,m which are not sums of squares forms from random input data. We now present this procedure specialized to our context of biquadratic biforms. 4.1. Algorithm. Algorithm 4.1. Let n > 2, m > 2, d = n+m−2 = dim σn,m (Pn−1 ×Pm−1 ) and e = (n−1)(m−1) = codim σn,m (Pn−1 ×Pm−1 ). To obtain a quadratic form in Pos(VR (In,m )) \ Sos(VR (In,m )) proceed as follows: Step 1 Construction of linear forms h0 , . . . , hd . Step 1.1 Choose e + 1 random points x(i) ∈ Rn and y (i) ∈ Rm and calculate their Kronecker tensor products z (i) = x(i) ⊗ y (i) ∈ Rnm . Step 1.2 Choose d random vectors v1 , . . . vd ∈ Rnm from the kernel of the matrix ∗ z (1) . . . z (e+1) . The corresponding linear forms h1 , . . . , hd are
hj (z) = vj∗ · z ∈ R[z] for j = 1, . . . , d. If the number of points in the intersection ∗ \ ker( v1 . . . vd ) V (In,m )
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
31
n+m−2
is not equal to deg(V (In,m )) = n−1 or if the points in the intersection are not in linearly general position, then repeat Step 1.1. Step 1.3 Choose a random vector v0 from the kernel of the matrix ∗ z (1) . . . z (e) . (Note that we have omitted z (e+1) .) The corresponding linear form h0 is h0 (z) = v0∗ · z ∈ R[z]. If h0 intersects h1 , . . ., hd in more than e points on V (In,m ), then repeat Step 1.3. Let a be the ideal in R[z]/In,m generated by h0 , h1 , . . . , hd . Step 2 Construction of a quadratic form f ∈ (R[z]/In,m ) \ a2 . Step 2.1 Let g1 (z), . . . , g(n)(m) (z) be the generators of the ideal In,m , i.e., the 2 × 2 2 2 minors zij zkl − zil zkj for 1 ≤ i < k ≤ n, 1 ≤ j < l ≤ m. For each i = 1, . . . , e (i) (i) compute a basis {w1 , . . . , wd+1 } ⊆ Rnm of the kernel of the matrix ∇g1 (z (i) )∗ .. . . ∇g(n)(m) (z (i) )∗ 2
2
(Note that this kernel is always (d + 1)-dimensional, since the variety V (In,m ) is d-dimensional (in Pnm−1 ) and smooth.) Step 2.2 Let ei denote the i-th standard basis vector of the corresponding vector space, i.e., the vector with 1 on the i-th component and 0 elsewhere. Choose a 2 2 random vector v ∈ Rn m from the intersection of the kernels of the matrices ∗ (i) (i) for i = 1, . . . , e z (i) ⊗ w1 · · · z (i) ⊗ wd+1 with the kernels of the matrices ∗ for 1 ≤ i < j ≤ nm. ei ⊗ ej − ej ⊗ ei
(The latter condition ensures v is a symmetric tensor in Rnm ⊗ Rnm . Note also that we have omitted the point z (e+1) .) For 1 ≤ i, k ≤ n and 1 ≤ j, l ≤ m denote Eijkl = (ei ⊗ ej ) ⊗ (ek ⊗ el ) + (ek ⊗ el ) ⊗ (ei ⊗ ej ) ∈ Rn
2 m2
.
If v is in [ span {vi ⊗ vj + vj ⊗ vi : 0 ≤ i ≤ j ≤ d} {Eijkl − Eilkj ; 1 ≤ i < k ≤ n, 1 ≤ j < l ≤ m} , then repeat Step 2.2. Otherwise the corresponding quadratic form f f (z) = v ∗ · (z ⊗ z) ∈ R[z]/In,m , does not belong to a2 .
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
32
Step 3 Construction of a quadratic form in R[z]/In,m that is positive but not a sum of squares. P Calculate the greatest δ0 > 0 such that δ0 f + di=0 h2i is nonnegative on VR (In,m ). Then for every 0 < δ < δ0 the quadratic form (δf +
d X
h2i )(z)
i=0
is nonnegative on VR (In,m ) but is not a sum of squares. 4.2. Correctness of Algorithm 4.1. The main ingredient in the proof is the theory of minimal degree varieties as developed in [BSV16]. Since the Segre variety σn,m (Pn−1 × Pm−1 ) is not of minimal degree for n, m ≥ 3 [BSV16, Example 5.6], Sos(VR (In,m )) ( Pos(VR (In,m )). Hence results of [BSV16, Section 3] apply; their Procedure 3.3 adapted to our set-up is Algorithm 4.1. While Step 1 and Step 3 follow immediately from the 2 that “vanishingto the corresponding steps in [BSV16, Procedure 3.3], we note for Step (i) (i) (i) second order at z ” means f (z ) = 0 and ∇f (z ) ∈ span ∇gj (z (i) ) : 1 ≤ j ≤ n2 m2 . Moreover, the former step is redundant, as the relation )(m2 ) (n2X ∇ f − λj gj (z (i) ) = 0 j=1
together with the well-known identity 2q(z) = (∇q(z))∗ z for any quadratic form q immeP diately yields f (z (i) ) = 0, since z (i) ∈ V (In,m ). The quadratic form δf + di=0 h2i is never a sum of squares, since f 6∈ a2 , while it is nonnegative on VR (In,m ) for sufficiently small P δ > 0 by the positive definiteness of the Hessian of di=0 h2i at its real zeros z (1) , . . . , z (e) , see the proof of the correctness of Procedure 3.3 in [BSV16]. We note that the verification in Step 1.2 is computationally difficult, but since all steps in the algorithm are performed with random data, all the generic conditions from [BSV16, Procedure 3.3] are satisfied with probability 1. Hence, Algorithm 4.1 works well with probability 1 without implementing verifications. 4.3. Implementation and rationalization. Step 1 and Step 2 are easily implemented as they only require linear algebra. (The verification in Step 1.2 can be performed using Gr¨obner basis if m, n are small, but is “always” satisfied with random input data.) On the other hand, Step 3 is computationally difficult; testing nonnegativity even of low degree polynomials is NP-hard, cf. [LNQY09]. We thus employ a sum of squares relaxation technique motivated by (the solution to) Hilbert’s 17th problem [BCR98]. Consider the following polynomial optimization problem: find the maximal δ0 such that (4.3)
# σn,m
δ0 f +
d X i=0
h2i
X j,k
xj yk
2 ℓ
is a sum of squares.
For a given ℓ ∈ N the condition (4.3) can be converted to a linear matrix inequality using Gram matrices of polynomials. Thus maximizing δ0 subject to this constraint is a standard semidefinite programming problem (SDP) [WSV00]. We start by solving (4.3) for ℓ = 1 using one of the standard solvers. If the obtained maximum is δ0 = 0, then we
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
33
increase ℓ and solve another SDP. We repeat this until we obtain a maximum δ0 > 0. In fact, in our numerical experiments this always happened with ℓ = 1 already. Any δ0 > 0 for which (4.3) holds gives an example of a positive biquadratic biform which is not a sum of squares. Together with Proposition 3.1 this yields instances of positive but not completely positive maps. 4.3.1. Rationalization. Step 1 and Step 2 can be performed over Q, leading to rational forms hj , f . But in Step 3 of the algorithm we are using SDPs, so the output δ0 will be floating point. Pick a positive rational δ < δ0 . We now explain how tools from polynomial optimization ([PP08, CKP15]) can be used to provide an exact, symbolic P certificate of positivity for the produced form δf + di=0 h2i by computing a positive P 2 ℓ P # semidefinite rational Gram matrix G for σn,m δ0 f + di=0 h2i x y . That is, j k j,k P # letting p = σn,m δ0 f + di=0 h2i , (4.4)
p
X j,k
xj yk
2 ℓ
= W ∗ GW
where W = W (x, y) is the bihomogeneous vector (xI y J )|I|=|J|=ℓ+1. Since p(x(i) , y (i) ) = 0 for i ≤ e, each positive semidefinite G satisfying (4.4) will have at least an e-dimensional nullspace. Let P be a change of basis matrix containing the vectors W (x(i) , y (i) ), i ≤ e, as the first e columns and a (rational) basis for the orthogonal complement as its remaining columns. With respect to this decomposition, write ˇ 11 G ˇ 12 G ∗ P GP = ˇ ∗ ˇ . G G22 12
ˇ 11 and G ˇ 12 to be equal to 0. Solve these linear equations and By construction, we want G ˇ 22 to produce G. ˇ Then run a SDP to solve G ˇ 0. Use the trivial objective use them in G function, since under a strict feasibility assumption the interior point methods (which all state-of-the-art SDP solvers use) yield solutions in the relative interior of the optimal face, leading to solutions of maximal rank [LSZ98]. If the output of the SDP is a full ˇ simply use a rationalization that is fine enough to yield a positive rank floating point G, semidefinite matrix (cf. [PP08]). 4.4. Example. In this subsection we give an explicit example of a positive map that is not completely positive build off Algorithm 4.1. Let pΦ (x, y) = 104x21 y12 + 283x21 y22 + 18x21 y32 − 310x21 y1 y2 + 18x21 y1 y3 + 4x21 y2 y3 + 310x1 x2 y12
− 18x1 x3 y12 − 16x1 x2 y22 + 52x1 x3 y22 + 4x1 x2 y32 − 26x1 x3 y32 − 610x1 x2 y1 y2 − 44x1 x3 y1 y2
+ 36x1 x2 y1 y3 − 200x1 x3 y1 y3 − 44x1 x2 y2 y3 + 322x1 x3 y2 y3 + 285x22 y12 + 16x23 y12 + 4x2 x3 y12
+63x22 y22 +9x23 y22 +20x2 x3 y22 +7x22 y32 +125x23 y32 −20x2 x3 y32 +16x22 y1 y2 +4x23 y1 y2 −60x2 x3 y1 y2 + 52x22 y1 y3 + 26x23 y1 y3 − 330x2 x3 y1 y3 − 20x22 y2 y3 + 20x23 y2 y3 − 100x2 x3 y2 y3 .
The corresponding linear map Φ : S3 → S3 is as follows: 104 −155 9 285 8 26 Φ(E11 ) = −155 283 2 , Φ(E22 ) = 8 63 −10 , 9 2 18 26 −10 7
ˇ I. KLEP, S. MCCULLOUGH, K. SIVIC, AND A. ZALAR
34
16 2 13 Φ(E33 ) = 2 9 10 , 13 10 125
310 −305 18 Φ(E12 + E21 ) = −305 −16 −22 , 18 −22 4
−18 −22 −100 Φ(E13 + E31 ) = −22 52 161 , −100 161 −26
4 −30 −165 Φ(E23 + E32 ) = −30 20 −50 . −165 −50 −20
We claim that pΦ is nonnegative but not a sum of squares. Equivalently, Φ is positive but not cp. We will establish this by explaining how this example was produced using Algorithm 4.1. Start with the points 1 1 −1 1 1 −1 x(1) y (1) x(2) y (2) 1 −1 1 −1 1 1 (3) (3) 1 1 −1 y = −1 1 1 , x (4) (4) x 1 1 1 y 1 1 1 0 2 2 −3 3 −2 x(5) y (5)
where each x(i) , y (i) ∈ R3 . Find some random linear forms hj from Step 1, e.g., using v0∗ −2 2 −1 −2 0 0 1 0 2 v ∗ 0 2 3 −2 3 0 −3 0 −3 1 ∗ = v −3 7 0 −7 −3 1 0 −1 6 2 . ∗ v3 9 −14 0 14 −3 2 0 −2 −6 v4∗ 0 6 0 −6 −6 0 0 0 6
Finally, a random quadratic form (in z) f satisfying the conditions described in Step 2 is # σ3,3 (f ) = 5x21 y12 −3x21 y22 +4x21 y32 −4x21 y1 y2 +7x21 y1 y3 −2x21 y2 y3 +4x1 x2 y12 −7x1 x3 y12 +x1 x2 y22
+5x1 x3 y22 +2x1 x2 y32 −2x1 x3 y32 +2x1 x2 y1 y2 −3x1 x3 y1 y2 +7x1 x2 y1 y3 −14x1 x3 y1 y3 −10x1 x2 y2 y3
+ x1 x3 y2 y3 − 2x22 y12 + 3x23 y12 − 2x2 x3 y12 + 2x23 y22 + x2 x3 y22 + x22 y32 + 2x23 y32 − 4x2 x3 y32 − x22 y1 y2 + 2x23 y1 y2 + 5x22 y1 y3 + 2x23 y1 y3 − 5x2 x3 y1 y3 − x22 y2 y3 + 4x23 y2 y3 .
P # P4 Next run the SDP maximizing δ0 subject to “σ3,3 ( i=0 h2i + δ0 f ) j,k (xj yk )2 is a sum of squares”. The optimal objective value is δ0 ≈ 3.41628. Choosing δ = 2, let p=
# σ3,3
4 X i=0
h2i + 2f .
Then p = pΦ . As explained above, p is not a sum of squares, whence Φ is not completely positive. Alternately, a SDP can be used to compute an explicit example of a linear functional positive on sum of squares and negative on p. Finally, we used the rationalization procedure described in Subsection 4.3.1 above to prove p is nonnegative (with ℓ = 1). We provide a Mathematica notebook1 where the interested reader can verify the calculations. 1see
https://www.math.auckland.ac.nz/~igorklep/ or the arXiv source of this manuscript
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Igor Klep, Department of Mathematics, The University of Auckland, New Zealand E-mail address: [email protected] Scott McCullough, Department of Mathematics, University of Florida, Gainesville E-mail address: [email protected] Faculty of Mathematics and Physics, University of Ljubljana, Slovenia E-mail address: [email protected] Aljaˇ z Zalar, Institute of Mathematics, Physics, and Mechanics, Ljubljana, Slovenia E-mail address: [email protected]
THERE ARE MANY MORE POSITIVE MAPS THAN COMPLETELY POSITIVE MAPS
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Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Main results and reader’s guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. Comparison with the original work of Blekherman . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Blekherman type estimates for biforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Nonnegative biforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Proof of the lower bound in Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Proof of the upper bound in Theorem 2.1. . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The apolar inner product on R[x, y]2k1 ,2k2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Sums of squares biforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Proof of the upper bound in Theorem 2.7. . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Proof of the lower bound in Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . 3. Positive maps and biforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g (n,m) and Sos f (n,m) . . . . . . . . . . . . . . . . . . . . . . . 3.1. Comparing the volumes of Pos (2,2)
(2,2)
3.2. Comparing the volumes of cones of positive and completely positive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Constructing positive maps that are not completely positive . . . . . . . . . . . . . . . 4.1. Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Correctness of Algorithm 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Implementation and rationalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Rationalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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