Guillaume Aubrun
email: aubrun@math.univ-lyon1.fr

Cécilia Lancien
email: lancien@math.univ-lyon1.fr

ZONOIDS AND SPARSIFICATION OF QUANTUM MEASUREMENTS

Keywords: 1991 Mathematics Subject Classification. 52A21, 81P15, 81P45 . positive operator-valued measure, zonoid, sparsification

In this paper, we establish a connection between zonoids (a concept from classical convex geometry) and the distinguishability norms associated to quantum measurements or POVMs (Positive Operator-Valued Measures), recently introduced in quantum information theory.

This correspondence allows us to state and prove the POVM version of classical results from the local theory of Banach spaces about the approximation of zonoids by zonotopes. We show that on C d , the uniform POVM (the most symmetric POVM) can be sparsified, i.e. approximated by a discrete POVM having only O(d 2 ) outcomes. We also show that similar (but weaker) approximation results actually hold for any POVM on C d .

By considering an appropriate notion of tensor product for zonoids, we extend our results to the multipartite setting: we show, roughly speaking, that local POVMs may be sparsified locally. In particular, the local uniform POVM on

be approximated by a discrete POVM which is local and has O(d 2 1 × • • • × d 2 k ) outcomes.

Introduction

A classical result by Lyapounov ( [START_REF] Rudin | Functional analysis, McGraw-Hill International Series in pure and applied Mathematics[END_REF], Theorem 5.5) asserts that the range of a non-atomic R n -valued vector measure is closed and convex. Convex sets in R n obtained in this way are called zonoids. Zonoids are equivalently characterized as convex sets which can be approximated by finite sums of segments.

In this paper we consider a special class of vector measures: Positive Operator-Valued Measures (POVMs). In the formalism of quantum mechanics, POVMs represent the most general form of a quantum measurement. Recently, Matthews, Wehner and Winter [START_REF] Matthews | Distinguishability of quantum states under restricted families of measurements with an application to data hiding[END_REF] introduced the distinguishability norm associated to a POVM. This norm has an operational interpretation as the bias of the POVM for the state discrimination problem (a basic task in quantum information theory) and is closely related to the zonoid arising from Lyapounov's theorem.

A well-studied question in high-dimensional convexity is the approximation of zonoids by zonotopes. The series of papers [START_REF] Figiel | The dimension of almost spherical sections of convex bodies[END_REF][START_REF] Schechtman | More on embedding subspaces of Lp in l n r[END_REF][START_REF] Bourgain | Approximation of zonoids by zonotopes[END_REF][START_REF] Talagrand | Embedding subspaces of L 1 into ℓ N 1[END_REF] culminates in the following result: any zonoid in R n can be approximated by the sum of O(n log n) segments. The aforementioned connection between POVMs and zonoids allows us to state and prove approximation results for POVMs, which improve on previously known bounds. Precise statements appear as Theorem 4.3 and 4.4.

Our article is organized as follows. Section 1 introduces POVMs and their associated distinguishability norms. Section 2 connects POVMs with zonoids. Section 3 introduces a notion of tensor product for POVMs, and the corresponding notion for zonoids. Section 4 pushes forward this connection to state the POVM version of approximation results for zonoids, which are proved in Sections 6, 7 and 8. Section 5 provides sparsification results for local POVMs on multipartite systems.

The reader may have a look at Table 1, which summarizes analogies between zonoids and POVMs. Let us recall a few standard concepts from classical convex geometry that we will need throughout our proofs. The support function h K of a convex compact set K ⊂ R n is the function defined for x ∈ R n by h K (x) = sup{ x, y : y ∈ K}. Moreover, for a pair K, L of convex compact sets, the inclusion K ⊂ L is equivalent to the inequality h K h L . The polar of a convex set K ⊂ R n is K • = {x ∈ R n : x, y 1 whenever y ∈ K}. The bipolar theorem (see e.g. [START_REF] Barvinok | A course in convexity[END_REF]) states that (K • ) • is the closed convex hull of K and {0}. A convex body is a convex compact set with non-empty interior. Whenever we apply tools from convex geometry in the (real) space H(C d ) (e.g. polar or support function), we use the Hilbert-Schmidt inner product (A, B) → Tr AB to define the Euclidean structure.

The letters C, c, c 0 , . . . denote numerical constants, independent from any other parameters such as the dimension. The value of these constants may change from occurrence to occurrence. Similarly c(ε) denotes a constant depending only on the parameter ε. We also use the following convention: whenever a formula is given for the dimension of a (sub)space, it is tacitly understood that one should take the integer part.

POVMs and distinguishability norms

In quantum mechanics, the state of a d-dimensional system is described by a positive operator on C d with trace 1. The most general form of a measurement that may be performed on such a quantum system is encompassed by the formalism of Positive Operator-Valued Measures (POVMs). Given a set Ω equipped with a σ-algebra F , a POVM on C d is a map M : F → H + (C d ) which is σ-additive and such that M(Ω) = Id. In this definition the space (Ω, F ) could potentially be infinite, so that the POVMs defined on it would be continuous. However, we often restrict ourselves to the subclass of discrete POVMs, and a main point of this article is to substantiate this "continuous to discrete" transition.

A discrete POVM is a POVM in which the underlying σ-algebra F is required to be finite. In that case there is a finite partition

Ω = A 1 ∪ • • • ∪ A n generating F .
The positive operators M i = M(A i ) are often referred to as the elements of the POVM, and they satisfy the condition

M 1 + • • • + M n = Id.
We usually identify a discrete POVM with the set of its elements by writing M = (M i ) 1 i n . The index set {1, . . . , n} labels the outcomes of the measurement. The integer n is thus the number of outcomes of M and can be seen as a crude way to measure the complexity of M.

What happens when measuring with a POVM M a quantum system in a state ρ ? In the case of a discrete POVM M = (M i ) 1 i n , we know from Born's rule that the outcome i is output with probability Tr(ρM i ). This simple formula can be used to quantify the efficiency of a POVM to perform the task of state discrimination. State discrimination can be described as follows: a quantum system is prepared in an unknown state which is either ρ or σ (both hypotheses being a priori equally likely), and we have to guess the unknown state. After measuring it with the discrete POVM M = (M i ) 1 i n , the optimal strategy, based on the maximum likelihood probability, leads to a probability of wrong guess equal to [START_REF] Holevo | Statistical decision theory for quantum systems[END_REF][START_REF] Helstrom | Quantum detection and estimation theory[END_REF]

P error = 1 2 1 - 1 2 n i=1 |Tr(ρM i ) -Tr(σM i )| .
In this context, the quantity 1 2 n i=1 |Tr(ρM i ) -Tr(σM i )| is therefore called the bias of the POVM M on the state pair (ρ, σ).

Following [START_REF] Matthews | Distinguishability of quantum states under restricted families of measurements with an application to data hiding[END_REF], we introduce a norm on H(C d ), called the distinguishability norm associated to M, and defined for ∆ ∈ H(C d ) by ( 1)

∆ M = n i=1 |Tr(∆M i )| .
It is such that P error = 1 2 1 -1 2 ρσ M , and thus quantifies how powerful the POVM M is in discriminating one state from another with the smallest probability of error.

The terminology "norm" is slightly abusive since one may have

∆ M = 0 for a nonzero ∆ ∈ H(C d ).
The functional • M is however always a semi-norm, and it is easy to check that • M is a norm if and only if the POVM elements (M i ) 1 i n span H(C d ) as a vector space. Such POVMs are called informationally complete in the quantum information literature.

Similarly, the distinguishability norm associated to a general POVM M, defined on a set Ω equipped with a σ-algebra F , is described for

∆ ∈ H(C d ) by (2) ∆ M = Tr(∆M(•)) TV = sup A∈F Tr(∆M(A)) -Tr(∆M(Ω \ A)) = sup M∈M(F )
Tr(∆(2M -Id)).

Here µ TV denotes the total variation of a measure µ. When M is discrete, formulae (1) and ( 2) coincide. Note also that the inequality

K M = 2 conv(M(F )) -Id.
By Lyapounov's theorem, the convex hull operation is not needed when M is non-atomic. For a discrete POVM M = (M i ) 1 i n , equation (3) may be rewritten in the form (4)

K M = conv{±M 1 } + • • • + conv{±M n },
where the addition of convex sets should be understood as the Minkowski sum:

A + B = {a + b : a ∈ A, b ∈ B}.
We are going to show that POVMs can be sparsified, i.e approximated by discrete POVMs with few outcomes. The terminology "approximation" here refers to the associated distinguishability norms: a POVM M is considered to be "close" to a POVM M ′ when their distinguishability norms satisfy inequalities of the form

(1 -ε) • M ′ • M (1 + ε) • M ′ .
This notion of approximation has an operational significance: two POVMs are comparable when both lead to comparable biases when used for any state discrimination task. Let us perhaps stress that point: if one has additional information on the states to be discriminated, it may of course be used to design a POVM specifically efficient for those (one could for instance be interested in the problem of distinguishing pairs of low-rank states [START_REF] Sen | Random measurement bases, quantum state distinction and applications to the hidden subgroup problem[END_REF][START_REF] Ambainis | Quantum t-designs: t-wise independence in the quantum world[END_REF]).

In this paper, we study the distinguishability norms from a functional-analytic point of view. We are mostly interested in the asymptotic regime, when the dimension d of the underlying Hilbert space is large.

POVMs and zonoids

2.1. POVMs as probability measures on states. The original definition of a POVM involves an abstract measure space, and the specification of this measure space is irrelevant when considering the distinguishability norms. The following proposition, which is probably well-known, gives a more concrete look at POVMs as probability measures on the set S(C d ) of states on C d . Proposition 2.1. Let M be a POVM on C d . There is a unique Borel probability measure µ on S(C d ) with barycenter equal to Id/d and such that, for any ∆ ∈ H(C d ),

(5)

∆ M = d S(C d ) |Tr(∆ρ)| dµ(ρ).
Conversely, given a Borel probability measure µ with barycenter equal to Id/d, there is a POVM M such that (5) is satisfied.

Proof. We use the polar decomposition for vector measures, which follows from applying the Radon-Nikodym theorem to vector measures (see [START_REF] Rudin | Real and complex analysis[END_REF], Theorem 6.12): a vector measure µ defined on a σ-algebra F on Ω and taking values in a normed space (R n , • ) satisfies dµ = hd|µ| for some measurable function h : Ω → R n . Moreover, one has h = 1 |µ|-a.e. Here |µ| denotes the total variation measure of µ.

Let M be a POVM on C d , defined on a σ-algebra F on Ω. We equip H(C d ) with the trace norm, so that we simply have |M| = Tr M and |M|(Ω) = d. The polar decomposition yields a measurable function h

: Ω → H(C d ) such that h 1 = 1 |M|-a.e. Moreover, the fact that M(F ) ⊂ H + (C d ) implies that h ∈ H + (C d ) |M|-a.e.
Let µ be the push forward of 1 d |M| under the map h. We have

Id = M(Ω) = Ω h d|M| = d H(C d ) ρ dµ(ρ).
And since h ∈ S(C d ) a.e., µ is indeed a Borel probability measure on S(C d ), with barycenter equal to Id/d. Finally, for any ∆ ∈ H(C d ),

∆ M = Ω | Tr(∆h)| d|M| = d S(C d ) | Tr(∆ρ)| dµ(ρ).
We postpone the proof of uniqueness to the next subsection (see after Proposition 2.5).

Conversely, given a Borel probability measure µ on S(C d ) with barycenter at Id/d, consider the vector measure M : B → H(C d ), where B is the Borel σ-algebra on S(C d ), defined by

M(A) = d A ρ dµ(ρ).
It is easily checked that M is a POVM and that formula ( 5) is satisfied.

Note that in the case of a discrete POVM M = (M i ) 1 i n , the corresponding probability measure is

µ = 1 d n i=1 (Tr M i ) δ M i Tr M i . Corollary 2.2. Given a POVM M on C d , there is a sequence (M n ) of discrete POVMs such that K Mn converges to K M in Hausdorff distance. Moreover, if µ (resp. µ n ) denotes the probability measure on S(C d ) associated to M (resp. to M n ) as in (5)
, we can guarantee that the support of µ n is contained into the support of µ.

Proof. Let µ be the probability measure associated to M. Given n, let (Q k ) be a finite partition of S(C d ) into sets of diameter at most 1/n with respect to the trace norm. Let ρ k ∈ S(C d ) be the barycenter of the restriction of µ to Q k (only defined when µ(Q k ) > 0). The probability measure

µ n = k µ(Q k )δ ρ k
has the same barycenter as µ, and the associated POVM M n satisfies

h KM (∆) -h KM n (∆) d ∆ ∞ n ,
and therefore K Mn converges to K M . The condition on the supports can be enforced by changing slightly the definition of µ n . For each k we can write

ρ k = λ k,j ρ k,j , where (λ k,j ) is a convex combination and (ρ k,j ) belong to the support of µ restricted to Q k . The measure µ ′ n = k µ(Q k ) j λ k,j δ ρ k,j
satisfies the same properties as µ n , and its support is contained into the support of µ.

POVMs and zonoids.

We connect here POVMs with zonoids, which form an important family of convex bodies (see [START_REF] Bolker | A class of convex bodies[END_REF][START_REF] Schenider | Zonoids and related topics[END_REF][START_REF] Goodey | Zonoids and Generalizations[END_REF] for surveys on zonoids to which we refer for all the material presented here). A zonotope Z ⊂ R n is a closed convex set which can be written as the Minkowski sum of finitely many segments, i.e. such that there exist finite sets of vectors

(u i ) 1 i N and (v i ) 1 i N in R n such that (6) Z = conv{u 1 , v 1 } + • • • + conv{u N , v N }.
A zonoid is a closed convex set which can be approximated by zonotopes (with respect to the Hausdorff distance). Every zonoid has a center of symmetry, and therefore can be translated into a (centrally) symmetric zonoid. Note that for a centrally symmetric zonotope, we can choose v i = -u i in [START_REF] Bolker | A class of convex bodies[END_REF].

Here are equivalent characterizations of zonoids.

Proposition 2.3. Let K ⊂ R n be a symmetric closed convex set. The following are equivalent.

(i) K is a zonoid.

(ii) There is a Borel positive measure ν on the sphere S n-1 which is even (i.e. such that ν(A) = ν(-A) for any Borel set A ⊂ S n-1 ) and such that, for every x ∈ R n ,

(7) h K (x) = S n-1 | x, θ | dν(θ).
(iii) There is a vector measure µ : (Ω, F ) → R n such that K = µ(F ). Moreover, when these conditions are satisfied, the measure ν is unique.

Remark 2.4. Having the measure ν supported on the sphere and be even is only a matter of normalization and a way to enforce uniqueness: if ν is a Borel measure on R n for which linear forms are integrable, there is a symmetric zonoid

K ⊂ R n such that h K (x) = R n | x, y | dν(y).
As an immediate consequence, we characterize which subsets of [-Id, Id] arise as K M for some POVM M.

Proposition 2.5. Let K ⊂ H(C d ) be a symmetric closed convex set. Then the following are equivalent.

(i) K is a zonoid such that K ⊂ [-Id, Id] and ±Id ∈ K.

(ii) There exists a POVM M on C d such that K = K M . Moreover, K is a zonotope only if the POVM M can be chosen to be discrete.

Proof. Let K be a zonoid such that ±Id ∈ K ⊂ [-Id, Id]. From Proposition 2.3, there is a vector measure µ defined on a σ-algebra F on a set Ω, whose range is K. Let A ∈ F such that µ(A) = -Id. The vector measure M defined for B ∈ F by

M(B) = 1 2 (µ(B \ A) -µ(B ∩ A)) = 1 2 (µ(B∆A) + Id)
is a POVM. Indeed, its range, which equals 1 2 (K + Id), lies inside the positive semidefinite cone, and contains Id. We get from (3) that K M = K.

Conversely, for any POVM M, formula (3) implies that ±Id ∈ K ⊂ [-Id, Id]. The fact that K is a zonoid follows, using the general fact that the convex hull of the range of a vector measure is a zonoid (see [START_REF] Bolker | A class of convex bodies[END_REF], Theorem 1.6).

In the case of zonotopes and discrete POVMs, these arguments have more elementary analogues which we do not repeat.

We can now argue about the uniqueness part in Proposition 2.1. This is indeed a consequence of the uniqueness of the measure associated to a zonoid in Proposition 2.3: after rescaling and symmetrization, a measure µ on S(C d ) satisfying [START_REF] Bennett | Schur multipliers[END_REF] naturally induces a measure ν on the Hilbert-Schmidt sphere satisfying [START_REF] Bourgain | Approximation of zonoids by zonotopes[END_REF]

for K = K M .
Another characterization of zonoids involves the Banach space L 1 = L 1 ([0, 1]). A symmetric convex body K is a zonoid if and only if the normed space (R n , h K ) embeds isometrically into L 1 . Therefore, Proposition 2.5 can be restated as a characterization of distinguishability norms on H(C d ).

Local POVMs and tensor products of zonoids

Tensor products for zonoids.

There is a natural notion of tensor product for subspaces of L 1 which appeared in the Banach space literature (see e.g. [START_REF] Figiel | Large subspaces of ℓ n ∞ and estimates of the Gordon-Lewis constant[END_REF]).

Definition 3.1. Let X, Y be two Banach spaces which can be embedded isometrically into L 1 , i.e. such that there exist linear norm-preserving maps i : X → L 1 (µ) and j : Y → L 1 (ν). Then, the 1-tensor product of X and Y is defined as the completion of the algebraic tensor product X ⊗ Y for the norm

k x k ⊗ y k X⊗ 1 Y = k i(x k )(s)j(y k )(t) dµ(s)dν(t).
It can be checked that the norm above is well-defined and does not depend on the particular choice of the embeddings i, j (see e.g. [START_REF] Figiel | Large subspaces of ℓ n ∞ and estimates of the Gordon-Lewis constant[END_REF] or Lemma 2 in [START_REF] Rosental | On tensor products of operators from L p to L q[END_REF]).

In the finite-dimensional case, subspaces of L 1 are connected to zonoids. Therefore, Definition 3.1 leads naturally to a notion of tensor product for (symmetric) zonoids. Definition 3.2. Let K ⊂ R m and L ⊂ R n be two symmetric zonoids, and suppose that ν K and ν L are Borel measures on S m-1 and S n-1 respectively, such that for any x ∈ R m and y ∈ R n ,

h K (x) = S m-1 | x, θ | dν K (θ) and h L (y) = S n-1 | y, φ | dν L (φ).
The zonoid tensor product of K and L is defined as the zonoid K ⊗ Z L ⊂ R n ⊗ R m whose support function satisfies

(8) h K⊗ Z L (z) = S m-1 S n-1 | z, θ ⊗ φ | dν K (θ)dν L (φ) for any z ∈ R m ⊗ R n .
As in Definition 3.1, this construction does not depend on the choice of the measures ν K and ν L . This can be seen directly: given z ∈ R m ⊗ R n and φ ∈ S n-1 , set z(φ) = (Id ⊗ φ|) (z). We have [START_REF] Chafaï | Interactions between compressed sensing, random matrices and high dimensional geometry[END_REF] h

K⊗ Z L (z) = S n-1 h K ( z(φ)) dν L (φ),
and therefore K ⊗ Z L does not depend on ν K . The same argument applies for ν L .

In the case of zonotopes, the zonoid tensor product takes a simpler form :

i conv{±v i } ⊗ Z   j conv{±w j }   = i j conv{±v i ⊗ w j }.
Here is a first simple property of the zonoid tensor product.

Lemma 3.3. Given symmetric zonoids K, L and linear maps S, T , we have

S(K) ⊗ Z T (L) = (S ⊗ T )(K ⊗ Z L)
Additionally, and crucially for the applications we have in mind, the zonoid tensor product is compatible with inclusions.

Lemma 3.4. Let K, K ′ be two symmetric zonoids in R m with K ⊂ K ′ , and let L, L ′ be two symmetric zonoids in

R n with L ⊂ L ′ . Then K ⊗ Z L ⊂ K ′ ⊗ Z L ′ .
Proof. This is a special case of Lemma 2 in [START_REF] Rosental | On tensor products of operators from L p to L q[END_REF]. Here is a proof in the language of zonoids. We may assume that L = L ′ , the general case following then by arguing that

K ⊗ Z L ⊂ K ′ ⊗ Z L ⊂ K ′ ⊗ Z L ′ .
In terms of support functions, we are thus reduced to showing that the inequality h K h K ′ implies the inequality h K⊗ Z L h K ′ ⊗ Z L , which is an easy consequence of (9). Suppose that (X, • X ) and (Y, • Y ) are Banach spaces with Euclidean norms, i.e. induced by some inner products •, • X and •, • Y . Their Euclidean tensor product X ⊗ 2 Y is defined (after completion) by the norm induced by the inner product on the algebraic tensor product which satisfies

x ⊗ y, x ′ ⊗ y ′ = x, x ′ X y, y ′ Y .
It turns out that, for Euclidean norms, the tensor norms ⊗ 1 and ⊗ 2 are equivalent. Proposition 3.5 (see [START_REF] Rosental | On tensor products of operators from L p to L q[END_REF][START_REF] Bennett | Schur multipliers[END_REF]). If X and Y are two Banach spaces equipped with Euclidean norms, then

2 π • X⊗ 2 Y • X⊗ 1 Y • X⊗ 2 Y .
3.2. Local POVMs. In quantum mechanics, when a system is shared by several parties, the underlying global Hilbert space is the tensor product of the local Hilbert spaces corresponding to each of the subsystems. A physically relevant class of POVMs on such a multipartite system is the one of local POVMs, describing the situation where each party is only able to perform measurements on his own subsystem.

Definition 3.6. For i = 1, 2, let M i denote a POVM on C di , defined on a σ-algebra F i on a set Ω i . The tensor POVM M 1 ⊗ M 2 is the unique map defined on the product σ-algebra F 1 ⊗ F 2 on Ω 1 × Ω 2 , and such that

(M 1 ⊗ M 2 )(A 1 × A 2 ) = M 1 (A 1 ) ⊗ M 2 (A 2 )
for every

A 1 ∈ F 1 , A 2 ∈ F 2 . By construction, M 1 ⊗ M 2 is a POVM on C d1 ⊗ C d2 .
In the discrete case, this definition becomes more transparent: if M = (M i ) 1 i m and N = (N j ) 1 j n are discrete POVMs, then M ⊗ N is also discrete, and

M ⊗ N = (M i ⊗ N j ) 1 i m,1 j n .
POVMs on C d1 ⊗ C d2 which can be decomposed as tensor product of two POVMs are called local POVMs. If we identify the POVMs M 1 and M 2 with measures µ 1 and µ 2 as in Proposition 2.1, then the measure corresponding to M 1 ⊗ M 2 is the image of the product measure µ 1 × µ 2 under the map (ρ, σ) → ρ ⊗ σ. It thus follows that Proposition 3.7. If M and N are two POVMs, then

• M⊗N = • M ⊗ 1 • N and K M⊗N = K M ⊗ Z K N .
These definitions and statements are given here only in the bipartite case for the sake of clarity, but can be extended to the situation where a system is shared between any number k of parties.

Sparsifying POVMs

4.1. The uniform POVM. It has been proved in [START_REF] Matthews | Distinguishability of quantum states under restricted families of measurements with an application to data hiding[END_REF] that, in several senses, the "most efficient" POVM on C d is the "most symmetric" one, i.e. the uniform POVM U d , which corresponds to the uniform measure on the set of pure states in the representation (5) from Proposition 2.5.

The corresponding norm is

(10) ∆ U d = d E | ψ|∆|ψ |,
where ψ is a random Haar-distributed unit vector. An important property is that the norm • U d is equivalent to a "modified" Hilbert-Schmidt norm.

Proposition 4.1 ([15, 20]). For every ∆ ∈ H(C d ), we have

(11) 1 √ 18 ∆ 2(1) ∆ U d ∆ 2(1) ,
where the norm • 2(1) is defined as

(12) ∆ 2(1) = Tr(∆ 2 ) + (Tr∆) 2 .
One can check that ∆ 2(1) equals the L 2 norm of the random variable g|∆|g , where g is a standard Gaussian vector in C d , while the L 1 norm of this random variable is nothing else than ∆ U d . Therefore Proposition 4.1 can be seen as a reverse Hölder inequality, and an interesting problem would be to find the optimal constant in that inequality (the factor √ 18 is presumably far from optimal). This dimension-free lower bound on the distinguishing power of the uniform POVM is of interest in quantum information theory. One could cite as one of its applications the possibility to establish lower-bounds on the dimensionality reduction of quantum states [START_REF] Harrow | Limitations on quantum dimensionality reduction[END_REF]. However, from a computational or algorithmic point of view, this statement involving a continuous POVM is of no practical use. There has been interest therefore in the question of sparsifying U d , i.e. of finding a discrete POVM, with as few outcomes as possible, which would be equivalent to U d in terms of discriminating efficiency. Examples of such constructions arise from the theory of projective 4-designs.

Given an integer t 1, an (exact) t-design is a finitely supported probability measure µ on S C d such that

S C d |ψ ψ| ⊗t dµ(ψ) = S C d |ψ ψ| ⊗t dσ(ψ) = d + t -1 t -1 P Sym t (C d ) .
Here, σ denotes the Haar probability measure on S C d , and P Sym t (C d ) denotes the orthogonal projection onto the symmetric subspace Sym

t (C d ) ⊂ (C d ) ⊗t .
Note that a t-design is also a t ′ -design for any t ′ t. Let µ be a 1-design. The map ψ → |ψ ψ| pushes forward µ into a measure μ on the set of (pure) states, with barycenter equal to Id/d. By Proposition 2.5, this measure corresponds to a POVM, and in the following we identify t-designs with the associated POVMs. For example the uniform POVM U d is a t-design for any t.

This notion can be relaxed: define an ε-approximate t-design to be a finitely supported measure µ on S C d such that

(1 -ε) S C d |ψ ψ| ⊗t dσ(ψ) S C d |ψ ψ| ⊗t dµ(ψ) (1 + ε) S C d |ψ ψ| ⊗t dσ(ψ).
It has been proved in [START_REF] Ambainis | Quantum t-designs: t-wise independence in the quantum world[END_REF] that a 4-design (exact or approximate) supported on N points yields a POVM M with N outcomes such that ( 13)

C -1 • U d • M C • U d
for some constant C. The proof is based on the fourth moment method, which is used to control the first absolute moment of a random variable by its second and fourth moments. Now, what is the minimal cardinality of a 4-design? The support of any exact or ε-approximate (provided ε < 1) 4-design must contain at least dim(Sym 4 (C d )) = d+3 4 = Ω(d 4 ) points. Conversely, an argument based on Carathéodory's theorem shows that there exist exact 4-designs with O(d 8 ) points. Starting from such an exact 4-design, the sparsification procedure from [START_REF] Batson | Twice-Ramanujan sparsifiers[END_REF] gives a deterministic and efficient algorithm which outputs an εapproximate 4-design supported by O(d 4 /ε 2 ) points.

However, this approach has two drawbacks: the constant C from ( 13) cannot be taken close to 1, and the number of outcomes has to be Ω(d 4 ). We are going to remove both inconveniences in our Theorem 4.3. 4.2. Euclidean subspaces. How do these ideas translate into the framework of zonoids? The analogue of U d is the most symmetric zonoid, namely the Euclidean ball B n 2 ⊂ R n . To connect with literature from functional analysis, it is worth emphasizing that approximating B n 2 by a zonotope which is the sum of N segments is equivalent to embedding the space

ℓ n 2 = (R n , • 2 ) into the space ℓ N 1 = (R N , • 1 ). Indeed, assume that x 1 , . . . , x N are points in R n such that, for some constants c, C, cZ ⊂ B n 2 ⊂ CZ, where Z = conv{±x 1 } + • • • + conv{±x N }. Then the map u : R n → R N defined by u(x) = x, x 1 , • • • , x, x N satisfies c u(x) 1
x 2 C u(x) 1 for any x ∈ R n . In this context, the ratio C/c is often called the distortion of the embedding.

An early result by Rudin [START_REF] Rudin | Trigonometric series with gaps[END_REF] shows an explicit embedding of

ℓ n 2 into ℓ O(n 2 ) 1
with distortion √ 3. This is proved by the fourth moment method and can be seen as the analogue of the constructions based on 4-designs. The following theorem (a variation on Dvoretzky's theorem) has been a major improvement on Rudin's result, showing that ℓ N 1 has almost Euclidean sections of proportional dimension.

Theorem 4.2 ([11]

). For every 0 < ε < 1, there exists a subspace E ⊂ R N of dimension n = c(ε)N such that for any x ∈ E,

(14) (1 -ε)M x 2 x 1 (1 + ε)M x 2 ,
where M denotes the average of the 1-norm over the Euclidean unit sphere S N -1 .

Theorem 4.2 was first proved in [START_REF] Figiel | The dimension of almost spherical sections of convex bodies[END_REF], making a seminal use of measure concentration in the form of Lévy's lemma. The argument shows that a generic subspace E (i.e. picked uniformly at random amongst all c(ε)N -dimensional subspaces of R N ) satisfies the conclusion of the theorem with high probability for c(ε) = O ε 2 | log ε| -1 . This was later improved in [START_REF] Gordon | Some inequalities for Gaussian processes and applications[END_REF] to c(ε) = O ε 2 . 4.3. Sparsification of the uniform POVM. Translated in the language of zonotopes, Theorem 4.2 states that the sum of O(n) randomly chosen segments in R n is close to the Euclidean ball B n 2 . More precisely, for any

0 < ε < 1, the zonotope Z = conv{±x 1 } + • • • + conv{±x N }, with N = c(ε) -1 n and x 1 , . . . , x N randomly chosen points in R n , is ε-close to the Euclidean ball B n 2 , in the sense that (1 -ε)Z ⊂ B n 2 ⊂ (1 + ε)Z.
By analogy, we expect a POVM constructed from O(d 2 ) randomly chosen elements to be close to the uniform POVM. This random construction can be achieved as follows: let (|ψ i ) 1 i n be independent random vectors, uniformly chosen on the unit sphere of C d . Set P i = |ψ i ψ i |, 1 i n, and S = P 1 + • • • + P n . When n d, S is almost surely invertible, and we may consider the random POVM [START_REF] Harrow | Limitations on quantum dimensionality reduction[END_REF] M = (S -1/2 P i S -1/2 ) 1 i n .

Theorem 4.3. Let M be a random POVM on C d with n outcomes, defined as in [START_REF] Harrow | Limitations on quantum dimensionality reduction[END_REF], and let 0 < ε < 1. If n Cε -2 | log ε|d 2 , then with high probability the POVM M satisfies the inequalities

(1 -ε) ∆ U d ∆ M (1 + ε) ∆ U d for every ∆ ∈ H(C d ).
By "with high probability" we mean that the probability that the conclusion fails is less than exp(-c(ε)d) for some constant c(ε). Theorem 4.3 is proved in Section 6, the proof being based on a careful use of ε-nets and deviation inequalities. It does not seem possible to deduce formally Theorem 4.3 from the existing Banach space literature.

Theorem 4.3 shows that the uniform POVM on C d can be ε-approximated (in the sense of closeness of distinguishability norms) by a POVM with n = O(ε -2 | log ε|d 2 ) outcomes. Note that the dependence of n with respect to d is optimal: since a POVM on C d must have at least d 2 outcomes to be informationally complete, one cannot hope for a tighter dimensional dependence. The dependence with respect to ε is less clear: the factor | log ε| can probably be removed but we do not pursue this direction.

Our construction is random and a natural question is whether deterministic constructions yielding comparable properties exist. A lot of effort has been put in derandomizing Theorem 4.2. We refer to [START_REF] Indyk | Almost-Euclidean subspaces of l N 1 via tensor products: a simple approach to randomness reduction[END_REF] for bibliography and mention two of the latest results. Given any 0 < γ < 1, it is shown in [START_REF] Indyk | Almost-Euclidean subspaces of l N 1 via tensor products: a simple approach to randomness reduction[END_REF] how to construct, from cn γ random bits (i.e. an amount of randomness sub-linear in n) a subspace of ℓ N 1 satisfying [START_REF] Harrow | The Church of the Symmetric Subspace[END_REF] with N (γε) -Cγ n. A completely explicit construction appears in [START_REF] Indyk | Uncertainty Principles, Extractors, and Explicit Embeddings of L 2 into L 1[END_REF], with N n2 C(ε)(log log n) 2 = n 1+C(ε)o(n) . It is not obvious how to adapt these constructions to obtain sparsifications of the uniform POVM using few or no randomness. 4.4. Sparsification of any POVM. Theorem 4.2 initiated intensive research in the late 80's [START_REF] Schechtman | More on embedding subspaces of Lp in l n r[END_REF][START_REF] Bourgain | Approximation of zonoids by zonotopes[END_REF][START_REF] Talagrand | Embedding subspaces of L 1 into ℓ N 1[END_REF] on the theme of "approximation of zonoids by zonotopes", trying to extend the result for the Euclidean ball (the most symmetric zonoid) to an arbitrary zonoid. This culminated in Talagrand's proof [START_REF] Talagrand | Embedding subspaces of L 1 into ℓ N 1[END_REF] that for any zonoid Y ⊂ R n and any 0 < ε < 1, there exists a zonotope Z ⊂ R n which is the sum of O(ε -2 n log n) segments and such that

(1 -ε)Y ⊂ Z ⊂ (1 + ε)Y .
A more precise version is stated in Section 8. Whether the log n factor can be removed is still an open problem.

This result easily implies a similar result for POVMs, provided we consider the larger class of sub-POVMs. A discrete sub-POVM with n outcomes is a finite family M = (M i ) 1 i n of n positive operators such that S = n i=1 M i Id. As for POVMs, the norm associated to a sub-POVM M is defined for ∆ ∈ H(C d ) by

∆ M = n i=1 | Tr(∆M i )|.
We prove the following result in Section 8.

< ε < 1, there is a sub-POVM M ′ = (M ′ i ) 1 i n , with n Cε -2 d 2 log(d) such that, for any ∆ ∈ H(C d ), (1 -ε) ∆ M ∆ M ′ ∆ M .
Moreover, we can guarantee that the states M ′ i / Tr(M ′ i ) belong to the support of the measure µ associated to M. We do not know whether Theorem 4.4 still holds if we want M ′ to be a POVM. Given a sub-POVM (M i ) 1 i n , there are at least two natural ways to modify it into a POVM. A solution is to add an extra outcome corresponding to the operator Id -S, and another one is to substitute S -1/2 M i S -1/2 in place of M i , as we proceeded in [START_REF] Harrow | Limitations on quantum dimensionality reduction[END_REF]. However for a general POVM, the error terms arising from this renormalization step may exceed the quantity to be approximated.

Sparsifying local POVMs

Proposition 5.1 below is an immediate corollary of Lemma 3.4 and Proposition 3.7. In words, it shows that, on a multipartite system, a local POVM can be sparsified by tensorizing sparsifications of each of its factors.

Proposition 5.1. Let 0 < ε < 1. Let M 1 , . . . , M k be POVMs and M ′ 1 , . . . , M ′ k be (sub-)POVMs, on C d1 , . . . , C d k respectively, satisfying, for all 1 i k, and for all ∆ ∈ H(C di ), (1 -ε) ∆ Mi ∆ M ′ i (1 + ε) ∆ Mi .
Then, for any

∆ ∈ H(C d1 ⊗ • • • ⊗ C d k ), (1 -ε) k ∆ M1⊗•••⊗M k ∆ M ′ 1 ⊗•••⊗M ′ k (1 + ε) k ∆ M1⊗•••⊗M k .
Let us give a concrete application of Proposition 5.1. We consider k finite-dimensional Hilbert spaces C d1 , . . . , C d k and define the local uniform POVM on the k-partite Hilbert space C d1 ⊗ • • • ⊗ C d k as the tensor product of the k uniform POVMs U d1 , . . . , U d k . We will denote it by LU. The corresponding distinguishability norm can be described, for any

∆ ∈ H(C d1 ⊗ • • • ⊗ C d k ), as ∆ LU = d E | ψ 1 ⊗ • • • ⊗ ψ k |∆|ψ 1 ⊗ • • • ⊗ ψ k | , where d = d 1 × • • • × d k
is the dimension of the global Hilbert space, and where the random unit vectors ψ 1 , . . . , ψ k are independent and Haar-distributed in C d1 , . . . , C d k respectively.

The following multipartite generalization of Proposition 4.1 shows that the norm • LU , in analogy to the norm • U , is equivalent to a "modified" Hilbert-Schmidt norm. Proposition 5.2 ([20]). For every

∆ ∈ H(C d1 ⊗ • • • ⊗ C d k ), we have (16) 1 18 k/2 ∆ 2(k) ∆ LU ∆ 2(k) ,
where the norm • 2(k) is defined as

(17) ∆ 2(k) = I⊂{1,...,k}
Tr Tr I ∆ 2 .

Here Tr I denotes the partial trace over all parties I ⊂ {1, . . . , k}.

Proof of Proposition 5.2. A direct proof appears in [START_REF] Lancien | Distinguishing multi-partite states by local measurements[END_REF], but we find interesting to show that in can be deduced (with a worst constant) from Proposition 4.1. If we denote by •, • H the inner product inducing a Euclidean norm • H , we have

A 1 ⊗ • • • ⊗ A k , B 1 ⊗ • • • ⊗ B k 2(k) = A 1 , B 1 2(1) × • • • × A k , B k 2(1)
which is equivalent to saying that

• 2(k) = • 2(1) ⊗ 2 • • • ⊗ 2 • 2(1)
.

We thus get by Proposition 3.5,

c k-1 0 • 2(k) • 2(1) ⊗ 1 • • • ⊗ 1 • 2(1) • 2(k)
with c 0 = 2/π. Now, we also know by Proposition 3.7 that on

H(C d1 ⊗• • •⊗C d k ), • LU = • U d 1 ⊗ 1 • • •⊗ 1 • U d k ,
and by Proposition 4.1 that c • 2(1)

• U d • 2(1)
for some constant c (c = 1/ √ 18 works). So by Lemma 3.4,

c k • 2(1) ⊗ 1 • • • ⊗ 1 • 2(1) • LU • 2(1) ⊗ 1 • • • ⊗ 1 • 2(1) ,
and therefore

c k-1 0 c k • 2(k) • LU • 2(k) .
Remarkably, local dimensions do not appear in equation ( 16). This striking fact that local POVMs can have asymptotically non-vanishing distinguishing power can be used to construct an algorithm that solves the Weak Membership Problem for separability in quasi-polynomial time (see [START_REF] Brandão | Faithful Squashed Entanglement[END_REF] for a description in the bipartite case). Hence the importance of being able to sparsify the local uniform POVM by a POVM for which the locality property is preserved and which has a number of outcomes that optimally scales as the square of the global dimension. We state the corresponding multipartite version of Theorem 4.3, which is straightforwardly obtained by combining the unipartite version with Proposition 5.1.

Theorem 5.3. Let 0 < ε < 1. For all 1 i k, let M i be a random POVM on C di with n i Cε -2 | log ε|d 2 i outcomes, defined as in [START_REF] Harrow | Limitations on quantum dimensionality reduction[END_REF]. Then, with high probability, the local POVM

M 1 ⊗ • • • ⊗ M k on C d1 ⊗ • • • ⊗ C d k is such that, for any ∆ ∈ H(C d1 ⊗ • • • ⊗ C d k ), (1 -ε) k ∆ LU ∆ M1⊗•••⊗M k (1 + ε) k ∆ LU .
Let us rephrase the content of Theorem 5.3: the local uniform POVM on C d1 ⊗ • • •⊗ C d k can be kε-approximated (in terms of distinguishability norms) by a POVM which is also local and has a total number of outcomes n

= O(C k ε -2k | log ε| k d 2 ), where d = d 1 × • • • × d k .
Note that the dimensional dependence of n is optimal. On the contrary, the dependence of n on ε deteriorates as k grows. The high-dimensional situation our result applies to is thus really the one of a "small" number of "large" subsystems (i.e. k fixed and d 1 , . . . , d k → +∞), and not of a "large" number of "small" subsystems.

Proof of Theorem 4.3

In this section we prove Theorem 4.3. Let n ∈ N and (|ψ i ) 1 i n be independent random unit vectors, uniformly distributed on the unit sphere of C d . Our main technical estimates are a couple of probabilistic inequalities. Proposition 6.1 is an immediate consequence of Theorem 1 in [START_REF] Aubrun | On almost randomizing channels with a short Kraus decomposition[END_REF]. Proposition 6.2 is a consequence of Bernstein inequalities. However, its proof requires some careful estimates which we postpone to Section 7. Proposition 6.1. If (|ψ i ) 1 i n are independent random vectors, uniformly distributed on the unit sphere of C d , then for every 0 < η < 1

P (1 -η) Id d 1 n n i=1 |ψ i ψ i | (1 + η) Id d 1 -C d exp(-cnη 2 ).
Proposition 6.2. Let ∆ ∈ H(C d ), and (|ψ i ) 1 i n be independent random vectors, uniformly distributed on the unit sphere of C d . For 1 i n, consider the random variables

X i = d| ψ i |∆|ψ i | and Y i = X i -E X i = X i -∆ U d .
Then, for any t > 0,

P 1 n n i=1 Y i t ∆ U d 2 exp(-c ′ 0 n min(t, t 2 )).
We now show how to derive Theorem 4.3 from the estimates in Propositions 6.1 and 6.2. For each 1 i n, set P i = |ψ i ψ i |, and introduce the (random) norm defined for any ∆ ∈ H(C d ) as

|||∆||| = d n n i=1 | Tr(∆P i )|.
We will now prove that |||•||| is, with probability close to 1, a good approximation to • U d . First, using Proposition 6.2, we obtain that for any 0 < ε < 1 and any ∆ ∈ H(C d )

(18) P ((1 -ε) ∆ U d |||∆||| (1 + ε) ∆ U d ) 1 -2 exp(-c ′ 0 nε 2 ).
We next use a net argument. Fix 0 < ε < 1/3 and a ε-net N inside the unit ball for the norm • U d , with respect to the distance induced by • U d . A standard volumetric argument (see [START_REF] Pisier | The Volume of Convex Bodies and Banach Spaces Geometry[END_REF], Lemma 4.10) shows that we may assume card(

N ) (1 + 2/ε) d 2 (3/ε) d 2 . Introduce the quantities A := sup{|||∆||| : ∆ U d 1}, A ′ := sup{|||∆||| : ∆ ∈ N }. Given ∆ such that ∆ U d 1, there is ∆ 0 ∈ N with ∆ -∆ 0 U d ε.
By the triangle inequality, we have

|||∆||| A ′ + |||∆ -∆ 0 ||| A ′ + εA. Taking supremum over ∆ yields A A ′ + εA i.e. A A ′ 1-ε . If we introduce B := inf{|||∆||| : ∆ U d = 1} and B ′ := inf{|||∆||| : ∆ ∈ N }, a similar argument shows that B B ′ -εA, so that in fact B B ′ -εA ′ 1-ε . We therefore have the implications (19) 1 -ε B ′ A ′ 1 + ε =⇒ 1 -ε - ε(1 + ε) 1 -ε B A 1 + ε 1 -ε =⇒ 1 -3ε B A 1 + 3ε.
By the union bound, we get from (18) that P(1ε B ′ A ′ 1 + ε) 1 -2 card(N ) exp(-c ′ 0 nε 2 ). Combined with [START_REF] Indyk | Almost-Euclidean subspaces of l N 1 via tensor products: a simple approach to randomness reduction[END_REF], and using homogeneity of norms, this yields ( 20)

P (1 -3ε) • U d ||| • ||| (1 + 3ε) • U d 1 -2 3 ε d 2 exp(-c ′ 0 nε 2 ).
This probability estimate is non-trivial, and can be made close to 1, provided n d 2 ε -2 | log ε|.

Whenever n d, the vectors (|ψ i ) 1 i n generically span C d , and therefore the operator S = P 1 + • • • + P n is invertible. We may then define P i = S -1/2 P i S -1/2 so that M = (

P i ) 1 i n is a POVM. The norm associated to M is, for any ∆ ∈ H(C d ), ∆ M = n i=1 | Tr(∆ P i )|.
We now argue that the norms ||| • ||| and • M are similar enough (modulo normalization), because the modified operators P i are close enough to the initial ones P i . This is achieved by showing that T :=

P( T -Id ∞ η) P( T -2 -Id ∞ η) C d exp(-c ′ nε 2 /d).
This upper bound is much smaller than

1 provided n C 1 ε -2 d 2 . Also, note that the event T -Id ∞ η implies that ∆ -T ∆T M ∆ -T ∆T 1 ∆ 1 Id -T ∞ (1 + T ∞ ) 2η ∆ 1 = 2ε ∆ U d .
Using the cyclic property of the trace, we check that T ∆T M = |||∆|||. Now, choose n larger than both C 0 ε -2 | log ε|d 2 and C 1 ε -2 d 2 . With high probability, the events from equations ( 20) and ( 21) both hold. We then obtain for every ∆ ∈ H(C d ),

∆ M T ∆T M + ∆ -T ∆T M |||∆||| + 2ε ∆ U d (1 + 5ε) ∆ U d
and similarly ∆ M (1 -5ε) ∆ U d . This is precisely the result from Theorem 4.3 with 5ε instead of ε, which of course can be absorbed by renaming the constants appropriately.

7. Proof of Proposition 6.2

The proof is a direct application of a large deviation inequality for sums of independent sub-exponential (or ψ 1 ) random variables. Recall that the ψ 1 -norm of a random variable X (which quantifies the exponential decay of the tail) may be defined via the growth of even moments

X ψ1 := sup q∈N 1 2q E |X| 2q 1/2q .
This definition is more practical than the standard definition through the Orlicz function x → exp(x) -1, and leads to an equivalent norm (see [START_REF] Chafaï | Interactions between compressed sensing, random matrices and high dimensional geometry[END_REF], Corollary 1.1.6). The large deviation inequality for a sum of independent ψ 1 random variables is known as Bernstein's inequality.

Theorem 7.1 (Bernstein's inequality, see [START_REF] Chafaï | Interactions between compressed sensing, random matrices and high dimensional geometry[END_REF], Theorem 1.2.5.). Let X 1 , . . . , X n be n independent ψ 1 random variables with mean zero. Setting M = max Proof of Lemma 7.2. For each integer q, we compute E [Tr(∆P )] 2q = E Tr ∆ ⊗2q P ⊗2q = Tr ∆ ⊗2q E P ⊗2q .

We use the fact (see e.g. [START_REF] Harrow | The Church of the Symmetric Subspace[END_REF]) that Here is a version of Talagrand's theorem which is suitable for our purposes. Theorem 8.1 ([31]). Let Z ⊂ R n be a symmetric zonotope, with Z = i∈I conv{±u i } for a finite family of vectors (u i ) i∈I . Then for every ε > 0 there exists a subset J ⊂ I with card J Cn log n/ε 2 , and positive numbers (λ i ) i∈J such that the zonotope

Z ′ = i∈J conv{±λ i u i } satisfies Z ′ ⊂ Z ⊂ (1 + ε)Z ′ .
Theorem 4.4 is a very simple consequence of Theorem 8.1. Let M be a POVM to be sparsified. Using Corollary 2.2, we may assume that M = (M i ) i∈I is discrete. Applying Theorem 8.1 to the zonotope K M = i∈I conv{±M i } (which lives in a d 2 -dimensional space), we obtain a zonotope Z ′ = i∈J conv{±λ i M i } with card J Cd 2 log d/ε 2 such that Z ′ ⊂ K M ⊂ (1+ε)Z ′ . It remains to show that M ′ = (λ i M i ) i∈J is a sub-POVM. We know that h Z ′ h KM . Therefore, given a unit vector x ∈ C d , the inequality h Z ′ (∆) h KM (∆) applied with ∆ = |x x| shows that

Notation.

  We denote by H(C d ) the space of Hermitian operators on C d , and by H + (C d ) the subset of positive operators. We denote by • 1 the trace class norm, by • ∞ the operator norm and by • 2 the Hilbert-Schmidt norm. Notation [-Id, Id] stands for the set of self-adjoint operators A such that -Id A Id. In other words [-Id, Id] is the self-adjoint part of the unit ball for • ∞ . We denote by S(C d ) the set of states on C d (a state is a positive operator with trace 1).

Corollary 2 . 6 .

 26 Let • be a norm on H(C d ). The following are equivalent (1) There is POVM M on C d such that • = • M . (2) The normed space (H(C d ), • ) is isometric to a subspace of L 1 , and the following inequality is satisfied for any ∆ ∈ H(C d ) | Tr ∆| ∆ Tr |∆|.

Theorem 4 . 4 .

 44 Given any POVM M on C d and any 0

For

  ∆ ∈ H(C d ), consider the random variablesX i = d| Tr(∆P i )| with P i = |ψ i ψ i |, and Y i = X i -E X i = d| Tr(∆P i )| -∆ U d .The random variables Y i are independent and have mean zero. The key lemma is a bound on their ψ 1 norm. Lemma 7.2. Let ∆ ∈ H(C d ) and consider the random variable X := d| Tr(∆P )|, where P = |ψ ψ|, with ψ uniformly distributed on the unit sphere ofC d . Then X ψ1 ∆ 2(1) and X -E X ψ1 3 ∆ 2(1) 3 √ 18 ∆ U d .Therefore, we may apply Bernstein's inequality with M = σ 3 √ 18 ∆ U d , yielding Proposition 6.2.

E 8 .

 8 P ⊗2q = (2q)! (d + 2q -1) × • • • × d P Sym 2q (C d ) = 1 (d + 2q -1) × • • • × d π∈S2q U (π),where P Sym 2q (C d ) denotes the orthogonal projection onto the symmetric subspace Sym 2q (C d ) ⊂ (C d ) ⊗2q , and for each permutation π ∈ S 2q , U (π) denotes the associated permutation unitary on (C d ) ⊗2q . This yieldsE [Tr(∆P )] 2q = 1 (d + 2q -1) × • • • × d π∈S2q Tr ∆ ⊗2q U (π) .If ℓ 1 , . . . , ℓ k denote the lengths of the cycles appearing in the cycle decomposition of a permutation π ∈ S 2q , we haveℓ 1 + • • • + ℓ k = 2qandTr ∆ ⊗2q U (π) = k i=1Tr(∆ ℓi ).Now, for any integerℓ 2, we have | Tr(∆ ℓ )| [Tr(∆ 2 )] ℓ/2 ∆ ℓ 2(1) . The inequality | Tr(∆ ℓ )| ∆ ℓ 2(1) is also (trivially) true for ℓ = 1. Therefore Tr ∆ ⊗2q U (π) ∆ 2q 2(1) . It follows that E [Tr(∆P )] 2q (2q)! d 2q ∆ 2q 2(1) 2q ∆ 2(1) d 2q ,so that E X 2q 1/2q 2q ∆ 2(1) , and thus X ψ1 ∆ 2[START_REF] Ambainis | Quantum t-designs: t-wise independence in the quantum world[END_REF] . The last part of the Lemma follows from the triangle inequality, since E X ψ1 = | E X| 2 X ψ1 , and from the equivalence[START_REF] Figiel | The dimension of almost spherical sections of convex bodies[END_REF] between the norms • U d and • 2[START_REF] Ambainis | Quantum t-designs: t-wise independence in the quantum world[END_REF] . Proof ofTheorem 4.4

  i∈J λ i | x|M i |x | |x x| M |x x| 1 = 1,and therefore i∈J λ i M i Id, as required. Since the inclusionsZ ′ ⊂ K M ⊂ (1 + ε)Z ′ are equivalent to the inequalities • M ′ • M (1 + ε) • M ′ , Theorem 4.4 follows.

Acknowledgements

We thank Andreas Winter for having first raised the general question of finding POVMs with few outcomes but good discriminating power. We also thank Marius Junge for suggesting the possible connection between POVMs and zonoids, and for pointing out to us relevant literature.

This research was supported by the ANR project OSQPI ANR-11-BS01-0008.


Guillaume Aubrun
email: aubrun@math.univ-lyon1.fr

AND Fedor Sukochev
email: f.sukochev@unsw.edu.au

Dmitriy Zanin
email: d.zanin@unsw.edu.au

CATALYSIS IN THE TRACE CLASS AND WEAK TRACE CLASS IDEALS

Given operators A, B in some ideal I in the algebra LpHq of all bounded operators on a separable Hilbert space H, can we give conditions guaranteeing the existence of a trace-class operator C such that B b C is submajorized (in the sense of Hardy-Littlewood) by A b C ? In the case when I " L 1 , a necessary and almost sufficient condition is that the inequalities TrpB p q ď TrpA p q hold for every p P r1, 8s. We show that the analogous statement fails for I " L 1,8 by connecting it with the study of Dixmier traces.

Introduction

Let H be an infinite-dimensional separable Hilbert space, LpHq be the algebra of all bounded operators on H and C 0 " C 0 pHq the set of compact operators.

Given A P C 0 , we denote by µpAq :" tµpk, Aqu kě0 the sequence of singular values of the operator A (that is, eigenvalues of the operator |A|) arranged in the decreasing order and taken with multiplicities (if any). We say that B P C 0 is submajorized by A P C 0 in the sense of Hardy-Littlewood (written B ăă A) if for every integer n If A, B P C 0 are such that B ăă A, then B b C ăă A b C for every C P C 0 . 1 The converse does not hold, even in the finite-dimensional setting: if A, B, C are such that µpAq " p0.5, 0.25, 0.25, 0, ¨¨¨q, µpBq " p0.4, 0.4, 0.1, 0.1, 0, ¨¨¨q and µpCq " p0.6, 0.4, 0, ¨¨¨q, one checks easily that B b C ăă A b C while B is not submajorized by A. This example appears in [START_REF] Plenio | Entanglement-assisted local manipulation of pure quantum states[END_REF] and is related to the phenomenon of catalysis in quantum information theory (the operator C being called a catalyst). This corresponds to the situation where the transformation of some quantum state (in that case, B) into another quantum state (in that case, A) is only possible in the presence of an extra quantum state (in that case, C) although the latter is not consumed in the process. It is argued in [START_REF] Plenio | Entanglement-assisted local manipulation of pure quantum states[END_REF] that this phenomenon can be used to improve the efficiency of entanglement concentration procedures.

The research of GA was supported by the ANR projects OSQPI (ANR-11-BS01-0008) and StoQ (ANR-14-CE25-0003). The research of FS and DZ has been supported by the ARC projects DP140100906 and DP 120103263.

1 Suppose first that C ě 0 has finite rank. That is, C " ř n´1 k"0 µpk, Cqp k , where p k , 0 ď k ă n, are pairwise orthogonal rank one projections.

Set A k " A b µpk, Cqp k and B k " B b µpk, Cqp k . It is immediate that B k ăă A k for 0 ď k ă n. It follows from Lemma 2.3 in [4] that ř n´1 k"0 B k ăă ř n´1 k"0 A k or, equivalently, B bC ăă AbC.
For an arbitrary C, the assertion follows by approximation.

In the following we restrict ourselves to A, B being positive elements in Ş pą1 L p (L p denoting the Schatten-von Neumann ideal) and compare the following statements.

(i) There exists a nonzero

C P L 1 such that B b C ăă A b C.
(ii) For every p ą 1, we have TrpB p q ď TrpA p q.

One checks that (i) implies (ii). This follows from the monotonicity of A Þ Ñ TrpA p q with respect to submajorization and from the formula TrpS b T q " TrpSq ¨TrpT q, S, T P L 1 .

There is some hope to reverse the implication (i) ñ (ii) if we allow closure of the set tB :

DC P L 1 such that B b C ăă A b Cu
with respect to some topology (for the finite-dimensional case, see [START_REF] Aubrun | Nechita I. Catalytic majorization and lp norms[END_REF][START_REF] Klimesh | Inequalities that completely characterize the Catalytic Majorization Relation[END_REF][START_REF] Turgut | Catalytic transformations for bipartite pure states[END_REF]).

To explain why some closure is needed, we give an example of a pair A, B of positive operators satisfying (ii) but not (i). Consider positive operators with µpAq ‰ µpBq and such that TrpB p q ď TrpA p q for p P p1, 8q, while TrpB p0 q " TrpA p0 q for some p 0 P p1, 8q (such an example exists among finite rank operators). Note that the norm in L p0 is strictly monotone with respect to submajorization (see Proposition 2.1 in [START_REF] Chilin | Characterisations of Kadec-Klee properties in symmetric spaces of measurable functions[END_REF]). That is, if K P L p0 pHq and if L ăă K, then either µpLq " µpKq or }L} p0 ă }K} p0 . Suppose that (i) holds, i.e. that B b C ăă A b C for some nonzero C P L 1 (that is, no closure is taken). We then have TrppB b Cq p0 q " TrppA b Cq p0 q and, by strict monotonicity, µpk, B b Cq " µpk, A b Cq for all k ě 0. Now, taking into account that the sequences µpB b Cq and µpA b Cq coincide with decreasing rearrangements of sequences µpBq b µpCq and µpAq b µpCq respectively, we infer that µpAq " µpBq.

As we shall see, the choice of the topology plays a crucial role. Prior to stating the precise question, we recall a few definitions and relevant facts.

There is a remarkable correspondence between sequence spaces and two-sided ideals in LpHq due to J.W. Calkin, [START_REF] Calkin | Two-sided ideals and congruences in the ring of bounded operators in Hilbert space[END_REF]. Recall, that a linear subspace J in LpHq is a two-sided ideal if X P J and Y P LpHq imply Y X, XY P J . Every non-trivial ideal necessarily consists of compact operators. A Calkin space J is a subspace of c 0 (the space of all vanishing sequences) such that x P J and µpyq ď µpxq imply y P J, where µpxq is the decreasing rearrangement of the sequence |x|. The Calkin correspondence may be explained as follows. If J is a Calkin space then associate to it the subset J in LpHq J :" tX P C 0 : µpXq P Ju.

Conversely, if J is a two-sided ideal, then associate to it the sequence space J :" tx P c 0 : µpxq " µpXq for some X P J u.

For the proof of the following theorem we refer to Calkin's original paper, [START_REF] Calkin | Two-sided ideals and congruences in the ring of bounded operators in Hilbert space[END_REF], and to B. Simon's book, [START_REF] Simon | Trace ideals and their applications[END_REF]Theorem 2.5].

Theorem 1 (Calkin correspondence). The correspondence J Ø J is a bijection between Calkin spaces and two-sided ideals in LpHq.

In the recent papers [START_REF] Kalton | Symmetric norms and spaces of operators[END_REF], [START_REF] Sukochev | Completeness of quasi-normed symmetric operator spaces[END_REF] this correspondence has been specialised to quasinormed symmetrically-normed ideals and quasi-normed symmetric sequence spaces [START_REF] Lindenstrauss | Classical Banach Spaces I[END_REF]. We use the notation } ¨}8 to denote the uniform norm on LpHq.

Definition 2. (i) An ideal E in LpHq is said to be symmetrically (quasi)-normed if it is equipped with a Banach (quasi)-norm } ¨}E such that }XY } E , }Y X} E ď }X} E }Y } 8 , X P E, Y P LpHq.
(ii) A Calkin space E is a symmetric sequence space if it is equipped with a Banach (quasi)-norm } ¨}E such that }y} E ď }x} E for every x P E and y P c 0 such that µpyq ď µpxq.

For convenience of the reader, we recall that a map } ¨} from a linear space X to R is a quasi-norm, if for all x, y P X and scalars α the following properties hold:

(i) }x} ě 0, and }x} " 0 ô x " 0;

(ii) }αx} " |α|}x};

(iii) }x `y} ď Cp}x} `}y}q for some C ě 1.

The couple pX, } ¨}q is a quasi-normed space and the least constant C satisfying the inequality (iii) above is called the modulus of concavity of the quasi-norm } ¨} and denoted by C X . A complete quasi-normed space is called quasi-Banach.

It easily follows from Definition 2 that if pE, } ¨}E q is a quasi-Banach ideal, X P E and Y P LpHq are such that µpY q ď µpXq, then Y P E and }Y } E ď }X} E . In particular, it is easy to see that if E is Calkin space corresponding to E, then setting }x} E :" }X} E (where X P E is such that µpxq " µpXq) we obtain that pE, } ¨}E q is a quasi-Banach symmetric sequence space. The converse implication is much harder and follows from main results of [START_REF] Kalton | Symmetric norms and spaces of operators[END_REF][START_REF] Sukochev | Completeness of quasi-normed symmetric operator spaces[END_REF].

With these preliminaries out of the way, we are now in a position to formulate the main question. Question 3. Let I be a (quasi-)Banach ideal such that I Ă Ş pą1 L p . Let 0 ď A P I. Consider the sets PMpA, Iq " ! 0 ď B P I : TrpB p q ď TrpA p q @p ą 1

) .

CatalpA, Iq "

! 0 ď B P I : D0 ď C P L 1 : C ‰ 0, B b C ăă A b C
) .

Let also CatalpA, Iq denote the closure of CatalpA, Iq with respect to the quasi-norm of I. Is it true that PMpA, Iq " CatalpA, Iq?

Note that PMpA, Iq is a closed subset in I. Indeed, let B n P PMpA, Iq and let B n Ñ B in I as n Ñ 8. Observe that it follows from Definition 2 that I is continuously embedded into LpHq and therefore, it follows from the Closed Graph Theorem that for every fixed p ą 1, the identical embedding I Ă L p is continuous, in particular, there exists a constant cpp, Iq such that }C} p ď cpp, Iq}C} I , C P I. Thus, ˇˇ}Bn}p ´}B} p ˇˇď }B ´Bn } p ď cpp, Iq}B ´Bn } I Ñ 0.

Hence, TrpB p q " lim nÑ8

TrpB p n q ď TrpA p q, p ą 1. We also have that CatalpA,

Iq Ă PMpA, Iq. Indeed, if B b C ăă A b C, then TrpB p q "
TrppB b Cq p q TrpC p q ď TrppA b Cq p q TrpC p q " TrpA p q, p ą 1.

Since PMpA, Iq is closed, it follows that the inclusion CatalpA, Iq Ă PMpA, Iq always holds.

In this paper, we show that the answer to Question 3 is positive when I " L 1 and negative when I " L 1,8 . Recall that L 1,8 is the principal ideal generated by the element A 0 " diagpt1, 1 2 , 1 3 , ¨¨¨uq. Equivalently, L 1,8 " tA P C 0 : sup kě0 pk `1qµpk, Aq ă `8u.

It becomes a quasi-Banach space (see e.g. [START_REF] Kalton | Symmetric norms and spaces of operators[END_REF][START_REF] Sukochev | Completeness of quasi-normed symmetric operator spaces[END_REF]) when equipped with the quasinorm }A} 1,8 " sup kě0 pk `1qµpk, Aq, A P L 1,8 .

Here are our main results. We leave open the question of giving a complete description of the set CatalpA, L 1,8 q.

Theorem 4. For every 0 ď A P L 1 , the sets PMpA, L 1 q and CatalpA, L 1 q coincide. Theorem 5. There exists 0 ď A P L 1,8 such that the set PMpA, L 1,8 q strictly contains the set CatalpA, L 1,8 q.

It is actually simple to deduce Theorem 4 from the finite-dimensional considerations from [START_REF] Aubrun | Nechita I. Catalytic majorization and lp norms[END_REF], as we explain in Section 2. This is in sharp contrast with Theorem 5, whose proof is infinite-dimensional in its nature and uses crucially fine properties of Dixmier traces, which we introduce in Section 3. The heart of the argument behind Theorem 5 appears in Section 4, while we relegate some needed computations to Section 5.

The authors thank the anonymous referees for careful reading of the manuscript and numerous suggestions which have improved the exposition.

The case of L 1

We derive Theorem 4 from the following result which appears in [START_REF] Turgut | Catalytic transformations for bipartite pure states[END_REF] (see also Lemma 2 in [START_REF] Aubrun | Nechita I. Catalytic majorization and lp norms[END_REF]). Lemma 6. Let A, B be positive finite rank operators. Assume that for every 1 ď p ď `8, we have the strict inequality }B} p ă }A} p . Then there exists a nonzero finite rank operator C such that B b C ăă A b C.

Proof of Theorem 4. Let us show the non-trivial inclusion, i.e. that every B P PMpA, L 1 q belongs to CatalpA, L 1 q.

Let p k , k ě 0, be a rank one eigenprojection of the operator A which corresponds to the eigenvalue µpk, Aq. Similarly, let q k , k ě 0, be a rank one eigenprojection of the operator B which corresponds to the eigenvalue µpk, Bq. We have

A " 8 ÿ k"0 µpk, Aqp k , B " 8 ÿ k"0 µpk, Bqq k .
Without loss of generality, µp0, Aq " 1. It follows that p1 ´p1 ´εq p qTrpA p q ě p1 ´p1 ´εq p qµp0, Aq p " 1 ´p1 ´εq p ě ε p .

The latter readily implies

TrpA p q ´εp ě p1 ´εq p TrpA p q, p ě 1, ε P p0, 1q. Now, fix ε P p0, 1q and select n such that

8 ÿ k"n µpk, Aq ă ε, 8 ÿ k"n µpk, Bq ă ε. Set A n " n´1 ÿ k"0 µpk, Aqp k , B n " n´1 ÿ k"0 µpk, Bqq k .

It is clear that

TrpA p n q " TrpA p q ´8 ÿ k"n µpk, Aq p ě TrpA p q ´p 8 ÿ

k"n µpk, Aqq p ą TrpA p q ´εp ě p1 ´εq p TrpA p q, p ě 1.

Therefore, p1 ´εq p TrpB p n q ď p1 ´εq p TrpB p q ď p1 ´εq p TrpA p q ă TrpA p n q, p ě 1. Since both A n and B n are finite rank operators, it follows from Lemma 6 and the first footnote that there exists a finite rank operator C n such that

p1 ´εqB n b C n ăă A n b C n ăă A b C n .
In particular, we have that p1´εqB n P CatalpA, L 1 q. Observing that }B ´Bn } 1 ď 1, we further obtain

}B ´p1 ´εqB n } 1 ď ε}B} 1 `p1 ´εq}B ´Bn } 1 ď εp}B} 1 `1q.
Since ε is arbitrarily small, it follows that B P CatalpA, L 1 q.

Dixmier traces

The crucial ingredient in the proof is the notion of a Dixmier trace on L 1,8 . Let ℓ 8 stand for the Banach space of all bounded sequences x " px n q ně0 equipped with the usual norm }x} 8 :" sup ně0 |x n |. A generalized limit is any positive linear functional on ℓ 8 which equals the ordinary limit on the subspace c of all convergent sequences.

Remark 7. Given a sequence px n q ně0 P ℓ 8 , there is a generalized limit ω such that ωppx n qq " lim sup nÑ8 x n .

Proof. Fix x " px n q P ℓ 8 and let the sequence pn k q kě0 be such that lim kÑ8 x n k " lim sup nÑ8 x n . Consider the set of functionals pϕ n k q kě0 on ℓ 8 defined by ϕ n k py n q :" y n k , y " py n q P ℓ 8 , k ě 0. The set pϕ n k q kě0 belongs to the unit ball B of the Banach dual ℓ 8. The set B is compact in the weak ˚topology σpℓ 8, ℓ 8 q and therefore the set pϕ n k q kě0 possesses a cluster point ω P ℓ 8 in that topology. The fact that ω is a generalized limit on ℓ 8 such that ωppx n qq " lim sup nÑ8 x n follows immediately from the definition of the weak ˚topology.

The Dixmier traces are defined as follows.

Theorem 8. Let ω be a generalized limit. The mapping Tr ω : L 1,8 Ñ R `defined for 0 ď A P L 1,8 by setting

Tr ω pAq :" ω ˜# 1 logpN `2q N ÿ k"0 µpk, Aq + 8
N "0 is additive and, therefore, extends to a positive unitarily invariant linear functional on L 1,8 called a Dixmier trace.

Note that the positivity of generalized limits implies that [START_REF] Aubrun | Nechita I. Catalytic majorization and lp norms[END_REF] |Tr ω pAq| ď }A} 1,8

for every Dixmier trace Tr ω and A P L 1,8 .

Let us comment on how additivity is proved in Theorem 8. This is usually achieved under the extra assumption that ω is scale invariant (see Theorem 1.3.1 in [START_REF] Lord | Singular traces. Theory and applications[END_REF]), i.e. that ω ˝σk " ω for all positive integer k, where σ k : ℓ 8 Ñ ℓ 8 is defined as σ k px 1 , x 2 , . . . , x n , . . . q " px 1 , . Under this extra assumption the map Tr ω is actually additive on the larger ideal M 1,8 (we refer to [START_REF] Lord | Singular traces. Theory and applications[END_REF]Example 1.2.9] for the definition of the latter ideal and to [START_REF] Lord | Singular traces. Theory and applications[END_REF]Section 6.8 ] for historical background). In the form presented here, Theorem 8 follows from Theorem 17 in [START_REF] Sedaev | Dixmier measurability in Marcinkiewicz spaces and applications[END_REF]. For the reader's convenience we reproduce the argument here.

Proof of Theorem 8. Given A P L 1,8 , consider the sequence px N pAqq 8 N "0 defined by

x N pAq " 1 logpN `2q N ÿ j"0 µpj, Aq.
It is not hard to check that for all all positive integer k, [START_REF] Chilin | Characterisations of Kadec-Klee properties in symmetric spaces of measurable functions[END_REF] lim

N Ñ8
x N pAq ´xkN pAq " 0.

Let E Ă ℓ 8 be the subspace E " span tσ k ptx N pAquq : k ě 1, A P L 1,8 u .

It follows from (2) that the equation ω ˝σk pxq " ωpxq is satisfied for x P E. By a version of the Hahn-Banach theorem (see [START_REF] Edwards | Functional Analysis. Theory and applications[END_REF], Theorem 3.3.1), the linear functional ω |E can be extended to a generalized limit ω 1 : ℓ 8 Ñ R which is scale-invariant. The usual argument ( [START_REF] Lord | Singular traces. Theory and applications[END_REF], Theorem 1.3.1) implies that Tr ω 1 (which coincides with Tr ω on L 1,8 ) is additive on L 1,8 .

We also need a version of Fubini's theorem for Dixmier traces.

Theorem 9. For every A P L 1,8 and for every C P L 1 , we have A b C P L 1,8 and

(3)

}A b C} 1,8 ď }A} 1,8 }C} 1 .
Moreover, for every Dixmier trace Tr ω on L 1,8 , we have Proof. We may assume }A} 1,8 " 1. Recall that A 0 " diagpt1, 1 2 , 1 3 , ¨¨¨uq. We have for all k ě 0,

µpk, A b Cq ď µpk, A 0 b Cq ď 1 k `1 k ÿ j"0 µpj, Cq ď }C} 1 k `1 .
where the second inequality follows from Proposition 3.14 in [START_REF] Dykema | Commutator structure of operator ideals[END_REF]. This proves (3).

Observe that both sides of (4) depend linearly on A and C (thanks to Theorem 8). Thus, we can assume without loss of generality that A, C ě 0. When C is a rank one projection, (4) follows from Theorem 8 since in that case µpk, A b Cq " µpk, Aq for all k ě 0. Again appealing to linearity of Dixmier traces, we infer the result for the finite rank operator C and when A P L 1,8 is arbitrary. Consider now a general C P L 1 and let pC n q be a sequence of finite rank operators such that }C ´Cn } 1 Ñ 0. We have Tr ω pAqTrpC n q " Tr ω pAqTrpCq.

As a corollary, we obtain that Dixmier traces give necessary conditions for catalysis.

Corollary 10. Let 0 ď A P L 1,8 and 0 ď B P CatalpA, L 1,8 q. Then for every Dixmier trace Tr ω , one has [START_REF] Dykema | Commutator structure of operator ideals[END_REF] Tr ω pBq ď Tr ω pAq.

Proof. We know from (1) that Dixmier traces are continuous on L 1,8 , and therefore we may assume that B P CatalpA, L 1,8 q. By definition of the latter set (see Question Here is the main technical result used in the proof of Theorem 5. In the lemma below, we tacitly identify a sequence in the space ℓ 8 with the corresponding diagonal operator. For I Ă N, we note by χ I the sequence defined by χ I pnq " 1 if n P I and χ I pnq " 0 otherwise. Lemma 11. Let I be the subset of N defined as I " ď ně0 r2 2n , 2 2n`1 q.

Consider the operator2 B " à mPI 2 ´mχ r0,2 m q .

Then B P L 1,8 . Moreover, TrppαA 0 q 1`s q " lim sÑ0`s ζpsqα 1`s " α, it follows from (6) that there exists δ ą 0 such that the inequality [START_REF] Plenio | Entanglement-assisted local manipulation of pure quantum states[END_REF] TrpB 1`s q ď TrppαA 0 q 1`s q

( 4 )

 4 Tr ω pA b Cq " Tr ω pAqTrpCq.

  |Tr ω pA b C n q ´Tr ω pA b Cq| ď }A b pC ´Cn q} 1,8 ď }A} 1,8 }C ´Cn } 1 and this quantity tends to 0 as n goes to infinity. Consequently, Tr ω pA b Cq " lim nÑ8 Tr ω pA b C n q " lim nÑ8

4 .

 4 3), there exists a nonzero positive C in L 1 with the property that B b C ăă A b C. Combining the definition of Hardy-Littlewood submajorization ăă and the definition of a Dixmier trace Tr ω (see Theorem 8) we infer that the inequality Tr ω pB b Cq ď Tr ω pA b Cq holds for every Dixmier trace Tr ω . Inequality (5) now follows from (4) and from the fact that TrpCq ą 0. The case of L 1,8 : the main argument

5 9 log 2 ă α ă 2 3 log 2 ., 1 3 ,

 523 Let us postpone the proof of Lemma 11 and show how it implies the result stated in Theorem 5. Consider B as in Lemma 11 and fix a number α such that Recall that A 0 " diagpt1,1 2 ¨¨¨uq. Since lim sÑ0`s

  . . , x 1 loooomoooon

		, x 2 , . . . , x 2 loooomoooon , . . . , x n , . . . , x n loooomoooon , . . . q.
	k times	k times	k times

In the subsequent formulas, the symbol ' stands for the direct sum of operators.

holds whenever 0 ă s ď δ. Define the operator A " αA 0 ' }B} 1`δ p where p is a rank one projection. We claim that B P PMpA, L 1,8 q: indeed, for s ą δ we may write TrpB 1`s q " Tr ´pB 1`δ q 1`s 1`δ ¯ď `TrpB 1`δ q ˘1`s 1`δ " }B} 1`s 1`δ ď TrpA 1`s q while for 0 ă s ď δ the inequality TrpB 1`s q ď TrpA 1`s q follows immediately from [START_REF] Plenio | Entanglement-assisted local manipulation of pure quantum states[END_REF].

We now assume by contradiction that B belongs to the set CatalpA, L 1,8 q. We know from Corollary 10 that Tr ω pBq ď Tr ω pAq for every Dixmier trace Tr ω . Observing that any such trace vanishes on finite rank operators, we see that the value Tr ω pAq coincides with Tr ω pαA 0 q and hence is equal to α for for every Dixmier trace Tr ω (see the definition given in Theorem 8). On the other hand, we may choose a generalized limit ω such that Tr ω pBq " lim sup

and obtain from (6) that 2 3 log 2 ď α, a contradiction. We note that the Dixmier trace considered in the proof does not behave in a monotone way with respect to trace of powers: we have TrpB p q ď TrpA p q for every p ą 1, but Tr ω pBq ą Tr ω pAq.

Proof of Lemma 11

Let I and B be as defined in Lemma 11,and denote by E B the spectral measure of B. First, note that for every integer m,

Hence, for every positive integer n, writing 2 m ď n ă 2 m`1 , we infer

Recall also (see e.g. [11, Chapter 2, Section 2.3]) that µpk, Bq, k ě 0 can be computed via the formula µpk, Bq " infts ě 0 : TrpE |B| pps, 8qq ď ku Hence, it follows from (9) that µpk, Bq ď 2 k`1 for every k ě 0 and, in particular, B P L 1,8 . We now prove the right inequality in [START_REF] Edwards | Functional Analysis. Theory and applications[END_REF]. For a given n, let N " TrpE B p2

´22n`1 , 8qq. We know from (8) that N ď 2 2 2n`1 . Therefore,

Hence, for N as above, we have

as needed.

We now focus on the left inequality in [START_REF] Edwards | Functional Analysis. Theory and applications[END_REF] and use the following summation formula, whose proof we postpone. For a given sequence px n q P ℓ 8 and for a given s ą 0, we have that (10)

x l 2 ´ls .

Note that TrpB 1`s q " ř mPI 2 ´ms " ř mě0 χ I pmq2 ´ms (here, χ I p0q " 0). Applying [START_REF] Lindenstrauss | Classical Banach Spaces I[END_REF] It remains to show that C ď 5{18 (we actually show C " 5{18). To that end, we think of χ I as an element of L 8 p0, 8q and define z P L 8 p0, 8q by setting z " χ Ť nPZ r2 2n ,2 2n`1 q . Observe that χ I ď z. Therefore,

Since zp4tq " zptq for every t ą 0, applying Fubini's theorem we have

However,

3t ´8 5t 2 , 2 ď t ď 4 Hence, the latter supremum is, in fact, a maximum which is attained at t " 12 5 and equal to 5 18 .

Proof of [START_REF] Lindenstrauss | Classical Banach Spaces I[END_REF]. Write x l q2 ´ks " p1 ´2´s q ´1 ÿ kělě0

x l 2 ´ks " p1 ´2´s q x l 2 ´ls .