Quantum Brascamp-Lieb Dualities

Brascamp-Lieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theory -- including entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of `geometric' Brascamp-Lieb inequalities.


I. INTRODUCTION
The classical Brascamp-Lieb (BL) problem asks, given a finite sequence of surjective linear maps L k : R m → R m k and q k ∈ R + for k ∈ [n], for the optimal constant C ∈ R such that [7,10,14,47] holds for all non-negative functions f k : R m k → R + , k ∈ [n], where • p denotes the p-norm.
Many classical integral inequalities fall into this framework, such as the Hölder inequality, Young's inequality, and the Loomis-Whitney inequality.A celebrated theorem by Lieb asserts that the optimal constant in Eq. ( 1) can be computed by optimizing over centred Gaussians f k alone [47].Remarkably, Eq. ( 1) has a dual, entropic formulation in terms of the differential entropy H(g) := − g(x) log g(x) dx.Namely, Eq. ( 1) holds for all f 1 , . . ., f n as above if, and only if, for all probability densities g on R m with finite differential entropy, we have [19] Here, g k denotes the marginal probability density on R m k corresponding to L k , i.e., the push-forward of g along L k defined by R m φ(L k x)g(x)dx = R m k φ(y)g k (y)dy for all bounded, continuous functions φ on R m k .The duality between Eq. (1) and Eq. ( 2) readily generalizes to arbitrary measure spaces and measurable maps [19].
Of particular interest is the so-called geometric case where each L k is a surjective partial isometry and n k=1 q k L † k L k = 1 R m [2][3][4][5][6][7].In this case, Eq. ( 1) and Eq. ( 2) hold with C = 0.This setup includes the Hölder and Loomis-Whitney inequalities.Equivalently, we are given n subspaces V k ⊆ R m (the supports of the L k ) such that n k=1 q k Π k = 1 R m , where Π k denotes arXiv:1909.02383v3[quant-ph] 20 Feb 2023 the orthogonal projection onto V k .In this case we can think of the marginal densities g k as functions on V k , namely In particular, if V k is a coordinate subspace of R m then g V k is nothing but the usual marginal probability density of the corresponding random variables, justifying our terminology.As a concrete example, let V 1 , V 2 be the two coordinate subspaces of R 2 and q 1 = q 2 = 1; then Eq. ( 2) amounts to the sub-additivity property of the differential entropy, which is dual to the trivial estimate In contrast, already for three equiangular lines in R 2 (a 'Mercedes star' configuration) and q 1 = q 2 = q 3 = 2 3 , neither inequality is immediate.
Recently, the BL duality has been extended on the entropic side to not only include entropy inequalities as in Eq. ( 2) but also relative entropy inequalities in terms of the Kullback-Leibler divergence [52].The dual analytic form then again corresponds to generalized Young inequalities as in Eq. ( 1) but now for weighted p-norms.Interestingly, this extended BL duality covers many fundamental entropic statements from information theory and more.This includes, e.g., hypercontractivity inequalities, strong data processing inequalities, and transportation-cost inequalities [53].
Here, we raise the question how aforementioned BL dualities can be extended in the noncommutative setting.Our main motivation comes from quantum information theory, where quantum entropy inequalities are pivotal and dual formulations often promise new insights.BL dualities for non-commutative integration have previously been studied by Carlen and Lieb [20].Amongst other contributions, they gave BL dualities similar to Eq. (1) -Eq.( 2) leading to generalized sub-additivity inequalities for quantum entropy.
In this paper, we extend the classical duality results of [52,53] to the quantum settingthereby generalizing Carlen and Lieb's BL duality to the quantum relative entropy and general quantum channel evolutions.In particular, we derive in Section II a fully quantum BL duality for quantum relative entropy and discuss its properties.In Section III we then discuss a plethora of examples from quantum information theory that are covered by our quantum BL duality.As novel inequalities, we give quantum versions of the geometric Brascamp-Lieb inequalities discussed above, whose entropic form can be interpreted as an uncertainty relation for certain Gaussian quantum operations (Section III B).
Note added: Since the first version of our manuscript, our geometric quantum Brascamp-Lieb inequalities from Section III B have been extended to the conditional case [50] and to more general Gaussian quantum operations [29].We briefly mention these extensions in Section III B.
Notation.Let A and B be separable Hilbert spaces.We denote the set of bounded operators on A by L(A), the set of trace-class operators on A by T(A), the set of Hermitian operators on A by Herm(A), the set of positive operators on A by P (A), and the set of positive semi-definite operators on A by P (A).A density operator or quantum state is a positive semi-definite trace-class operator with unit trace; we denote the set of density operators on A by S(A).The set of trace-preserving and positive maps from T(A) to T(B) is denoted by TPP(A, B) and the set of trace-preserving and completely positive maps from T(A) to T(B) is denoted by TPCP(A, B).For E ∈ TPP(A, B) the adjoint map E † , which is a unital and positive map from L(B) to L(A), is defined by tr E(X) † Y = tr X † E † (Y ) for all X ∈ T(A) and Y ∈ L(B).When it is clear from the context, we sometimes leave out identity operators, i.e., we may write The where ω τ denotes that the support of ω is contained in the support of τ .The von Neumann entropy can be expressed as a relative entropy, H(ρ) = −D(ρ 1), where 1 denotes the identity operator.For ρ AB ∈ S(A ⊗ B) with H(A) ρ < ∞, the conditional entropy of A given B is defined as [45] where the notation H(A) ρ := H(ρ A ) refers to the entropy of the reduced density operator ρ A = tr B (ρ) on A. For A and B finite-dimensional we can also write Throughout this manuscript the default is that Hilbert spaces are finite-dimensional unless explicitly stated otherwise (such as in Section III B).

II. BRASCAMP-LIEB DUALITY FOR QUANTUM RELATIVE ENTROPIES
In this section, we describe our main result (Theorem II.1) and discuss some of its mathematical properties.

A. Main result
The following result establishes a version of the Brascamp-Lieb dualities of [19,52,53] for quantum relative entropies.
and C ∈ R.Then, the following two statements are equivalent: tr exp log σ+ where L p := (tr|L| p ) 1 p is the Schatten p-norm for p ∈ [1, ∞] and an anti-norm for p ∈ (0, 1]. 2 Moreover, Eq. (5) holds for all ω k ∈ P (B k ) if and only if it holds for all ω k ∈ S(B k ) with full support. 1The case when ρA does not have full support is covered by the convention 0 log 0 = 0. Unless specified otherwise, we choose to leave the basis of the logarithm function log(•) unspecified and write exp(•) for its inverse function. 2An anti-norm is a non-negative function on P (A) that is homogeneous ( αω = α ω for α > 0) and super-additive ( ω + ω ≥ ω + ω ) for ω, ω ∈ P (A) [13].NB: • 1 is both a norm and an anti-norm.
We refer to Eq. ( 4) as a quantum Brascamp-Lieb inequality in entropic form, and to Eq. ( 5) as a quantum Brascamp-Lieb inequality in analytic form.The latter can be understood as a quantum version of a Young-type inequality.The two formulations in Eq. ( 4) and Eq. ( 5) encompass a large class of concrete inequalities, as we will see in Section III below; we are also often interested in identifying the smallest constant C ∈ R such that either inequality holds.
To this end, both directions of Theorem II.1 are of interest: 1. To prove quantum entropy inequalities, Theorem II.1 allows us to alternatively work with matrix exponential inequalities in the analytic form.That this approach can give crucial insights was already discovered in the original proof of strong sub-additivity of the von Neumann entropy [49], which relied on Lieb's triple matrix inequality for the exponential function (see also [31,60] for more recent works).We discuss similar examples in Section III C.
2. In the commutative setting, we know that for deriving Young-type inequalities it can be beneficial to work in the entropic form [19,21].As the quantum relative entropy has natural properties mirroring its classical counterpart, this translates to the non-commutative setting.We discuss corresponding examples in Section III A and Section III B.
The proof of Theorem II.1 relies on the following formula for the Legendre transform of the quantum relative entropy and its dual.
These variational formulas are powerful on their own for proving quantum entropy inequalities, as, e.g., the first term in Eq. ( 6) only depends on ρ (but not on σ) and the second term only on σ (but not on ρ).We refer to [60] for a more detailed discussion.We mention that Carlen-Lieb use the variational characterization of the von Neumann entropy to derive Brascamp-Lieb dualities and [20, bottom of page 564] commented that their proof strategy extends to the relative entropy via Petz's variational expression for the relative entropy (Fact II.2), which is what is done here.
Proof of Theorem II.1.We first show that Eq. (4) implies Eq. (5).Let H k := log ω k and define H ∈ Herm(A) and ρ ∈ S(A) by respectively.Then, where we used Eq. ( 7) in both the second and the last step and Eq. ( 4) in the penultimate step.By substituting H k = log ω k and taking the exponential on both sides we obtain Eq. ( 5).We now show that, conversely, Eq. ( 5) implies Eq. ( 4).Let ω = exp(H), with H defined as in Eq. ( 8) in terms of H k = log(ω k ) for ω k ∈ P σ k (B k ) that we will choose later.Then, using Eq. ( 6), where the last inequality uses Eq. ( 5) and the final step follows from Eq. ( 6) provided we choose ω 1/q k k as the maximizer for the variational expression of D E k (ρ) σ k .
Remark II.3.As the variational characterizations from Fact II.2 hold in the general W *algebra setting [56], the BL duality in Theorem II.1 extends to separable Hilbert spaces.
Remark II.4.The BL duality in Theorem II.1 can be extended to σ ∈ P (A) and σ Then, the BL duality still holds but for the alternative conditions ρ ∈ S(A) with ρ σ in Eq. ( 4) and ω k ∈ P (B k ) with ω k σ k in Eq. ( 5).
To see this, note that the variational formula in Eq. ( 6) still holds for σ ∈ P (A) as long as ρ σ with the supremum taken over ω ∈ P (A) with ω σ.Similarly, Eq. ( 7) still holds for H ∈ Herm(A) for H σ with the supremum taken over ω ∈ S(A) with ω σ.The proof of Theorem II.1 then also goes through in the more general form.
In many important applications, we are interested in using Theorem II.1 either in the situation that In the latter case, Theorem II.1 specializes to the following equivalence between von Neumann entropy inequalities and Young-type inequalities: and C ∈ R.Then, the following two statements are equivalent: tr exp Carlen and Lieb previously proved a variant of Corollary II.5 in the W * -algebra setting assuming that the maps E † k are W * -homomorphisms and that q k ∈ [0, 1] [20, Theorem 2.2].One interesting special case is when the E k are partial trace maps.The entropic form Eq. ( 9) then corresponds to generalized sub-additivity inequalities for the von Neumann entropy (cf.Section III A).

B. Weighted anti-norms
In the commutative setting, the right-hand side of Eq. ( 5) can conveniently be understood as a product of σ k -weighted norms or anti-norms of the operators ω k [52,53].It is natural to ask whether such an interpretation also holds quantumly.To this end, given p ∈ (0, 1] and σ ∈ P (A), define |||ω||| σ,p := tr exp(log ω p + log σ) for all ω ∈ P (A).The following proposition, which follows readily from [41], shows that |||•||| σ,p is an anti-norm provided that p ≤ 1.For p > 1, it is easy to find σ ∈ P (A) such that the functional |||•||| σ,p is neither a norm nor an anti-norm.
Proof.Clearly, |||•||| σ,p is homogeneous.Since moreover p ∈ (0, 1], [41, Lemma D.1] asserts that its concavity on the set of positive matrices is equivalent to the concavity of its p-th power, i.e., where H = log σ.A well-known result of Lieb [46] states that Eq. ( 11) is indeed concave for any Hermitian matrix H. Thus, |||•||| σ,p is concave.As a consequence of homogeneity and concavity, we obtain that Thus, the quantum Brascamp-Lieb inequality in its analytic form Eq. ( 5) can be written as where, assuming that all q k ≥ 1, the right-hand side can be interpreted in terms of anti-norms, pleasantly generalizing Eq. (10).
We record the following elementary property.
In the commutative case, the BL set satisfies a tensorization property [53, Section V.B], and we can ask if a similar property holds in the non-commutative case as well.Namely, do we have that for q, C (i) ∈ BL E (i) , σ (i) , σ (i) with i ∈ {1, 2} and 1 , . . ., E (1)  n ⊗ E (2)   n as well as σ := σ (1) As we will see in several examples (Section III), tensorization does in general not hold due to the potential presence of entanglement.Indeed, the problem of deciding in which case Eq. ( 13) holds can be understood as a general information-theoretic additivity problem, which contains the (non-)additivity for the minimum output entropy as a special case (cf.Eq. ( 39) in Section III D).

III. APPLICATIONS OF QUANTUM BRASCAMP-LIEB DUALITY
The purpose of this section is to present examples from quantum information theory where the duality from Theorem II.1 is applicable.The majority of examples concern entropy inequalities that are of interest from an operational viewpoint.Theorem II.1 then shows that all entropy inequalities of suitable structure have a dual formulation as an analytic inequality, and vice versa.Depending on the scenario, one form may be easier to prove than the other, and we find that these reformulations often give additional insight.

A. Generalized (strong) sub-additivity
In this section, we discuss entropy inequalities that generalize the sub-additivity and strong sub-additivity properties of the von Neumann entropy.Recall that the latter states that H(AB) + H(BC) ≥ H(ABC) + H(B) for ρ ABC ∈ S(A ⊗ B ⊗ C) [49].
We first state the following result from [20, Theorem 1.4 & Theorem 3.1], which gives generalized sub-additivity relations and their dual analytic form.Here, the second argument in the relative entropy is always equal to the identity.Throughout this section, all quantum channels are given by partial trace channels.
Corollary III.1 (Quantum Shearer and Loomis-Whitney inequalities, [20]).Let S 1 , . . ., S n be non-empty subsets of [m] such that every s ∈ [m] belongs to at least p of those subsets.Then, the following inequalities hold and are equivalent: where S denotes the complement of a subset S of [m].
Inequalities in the form of Eq. ( 14) have been termed quantum Shearer's inequalities and their analytic counterparts as in Eq. ( 15) are known as quantum Loomis-Whitney inequalities.
Interestingly, and as explained in [20, Section 1.3], the latter cannot directly be deduced from standard matrix trace inequalities such as Golden-Thompson combined with Cauchy-Schwarz.That Eq. ( 14) and Eq. ( 15) are equivalent follows from Corollary II.5 by choosing C = 0, q k = 1 p , and The following result provides a conditional version of the quantum Shearer inequality with side information.
Proposition III.2 (Conditional quantum Shearer inequality).Let S 1 , . . ., S n be non-empty subsets of [m] such that every s ∈ [m] belongs to exactly p of those subsets.Then, For n = 2, S 1 = {1}, S 2 = {2}, p = 1, Eq. ( 16) reduces to which is equivalent to the strong sub-additivity of von Neumann entropy. 3ote that, in contrast to Corollary III.1, in the conditional case it is not enough to assume that every s ∈ [m] belongs to at least p of the subsets.This is clear from the following proof.For a concrete counterexample, note that for n = 2, S 1 = S 2 = {1}, S 3 = {2}, p = 1, Eq. ( 16) is violated for, e.g., a maximally entangled state between A 1 and B.
Proof of Corollary III.1 and Proposition III.2.We adapt the argument of [20] to the conditional case.If S and T are two subsets of [m] then strong sub-additivity implies that This means that we obtain a stronger version of Eq. ( 16) if we replace any two subsets S k , S l by S k ∪ S l , S k ∩ S l .Moreover, each such replacement preserves the number of times that any s ∈ [m] is contained in the subsets S 1 , . . ., S n .We can successively apply replacement steps until we arrive at the situation where S k ⊆ S l or S l ⊆ S k for any two subsets.Without loss of generality, this means that it suffices to prove Corollary III.1 and Proposition III.2 in the case that is contained in at least p of the subsets.The corresponding inequality Eq. ( 16) can thus be simplified to Remark III.3.Corollary III.1 and Proposition III.2 also hold for separable Hilbert spaces, as the variational characterizations from Fact II.2 hold in the general W * -algebra setting [56].

B. Brascamp-Lieb inequalities for Gaussian quantum operations
In this section, we present quantum versions of the classical Brascamp-Lieb inequalities as in Eq. (1) and Eq. ( 2), where probability distributions on R m are replaced by quantum states on L 2 (R m ), the Hilbert space of square-integrable wave functions on R m .We focus on the geometric case discussed in the introduction.The marginal distribution with respect to a subspace X ⊆ R m has the following natural quantum counterpart.Define a TPCP map E X as the composition of the unitary L 2 (R m ) ∼ = L 2 (X) ⊗ L 2 (X ⊥ ) with the partial trace over the second tensor factor.Given a density operator ρ on L 2 (R m ), we can think of as the reduced density operator corresponding to X.This is the natural quantum version of the marginal probability density in Eq. (3) of the introduction.Indeed, if we identify ρ with its kernel in L 2 (R m × R m ), and likewise for ρ k , then we have the completely analogous formula This definition is very similar in spirit to the quantum addition operation in the quantum entropy power inequality of [43] (see also [28,44]) and in fact contains the latter as a special case.In linear optical terms, ρ X can be interpreted as the reduced state of dim X many output modes obtained by subjecting ρ to a network of beamsplitters with arbitrary transmissivities.
The following result establishes quantum versions of the Brascamp-Lieb dualities as in Eq. ( 1) and Eq. ( 2) for the geometric case.
Proposition III.4 (Geometric quantum Brascamp-Lieb inequalities).Let X 1 , . . ., X n ⊆ R m be subspaces and let q 1 , . . ., q n ≥ 0 such that n k=1 q k Π k = 1 R m , where Π k denotes the orthogonal projection onto X k .Then, for all ρ ∈ S(L 2 (R m )) with finite second moments, Furthermore, for all ω Note that if X k is spanned by a subset S k ⊆ [m] of the m coordinates of R m , then ρ X k is nothing but the reduced density matrix of subsystems S k , which appears on the right-hand side of the quantum Shearer inequality Eq. ( 14).Thus, Proposition III.4 implies Corollary III.1 in the case that all s ∈ [m] are contained in exactly p of the subsets S k .
To establish Proposition III.4,we will first prove the entropic form Eq. ( 17) using a quantum version of the heat flow approach from [8,21] (cf. the recent works [22][23][24] on entropy inequalities for quantum Markov semigroups).We assume some familiarity with Gaussian quantum systems (see, e.g., [40]) and follow the framework of König and Smith [43], which holds under regularity assumptions on the quantum state, which were subsequently removed by De Palma and Trevisan [28].
Let X ⊆ R m be a subspace and m X its dimension.For all x ∈ X, define position and momentum operators on L 2 (X) by (Q X,x ψ)(y) := (x • y)ψ(y) and P X,x := −i∂ x .Denote by N X t the non-commutative heat flow or heat semigroup [28,43], which is a one-parameter semi-group, meaning N X 0 = 1 and N X s • N X t = N X s+t for s, t ≥ 0. On a suitable domain it is generated by where {e j } m X j=1 is an arbitrary orthonormal basis of X (but we will not directly use this specific form).For every t ≥ 0, N X t is a Gaussian TPCP map, hence fully determined by its action on covariance matrices and mean vectors, 5 which is given by In particular, the heat flow is independent of the choice of orthonormal basis in X.The generalized partial trace maps E X : ρ → ρ X defined above are also Gaussian and act by where µ| X denotes the restriction of µ onto X ⊕ X and likewise for Σ| X .The non-commutative heat flow is compatible with the maps E X , i.e., Indeed, since both channels (and hence their composition) are Gaussian, it suffices to verify that the action commutes on the level of mean vectors and covariance matrices, and the latter is clear from Eq. ( 19) and Eq.(20).See also [28,Lemma 2].Thus, we may unambiguously introduce the notation for the reduced density operator on L 2 (X) at time t.Similarly, we may show that E X is compatible with phase-space translations (cf.[43, Lemma XI.1]).For x ∈ X, define the unitary one-parameter groups X,x (ρ) := e −itP X,x ρ e itP X,x and P (t) X,x (ρ) := e itQ X,x ρ e −itQ X,x .
They are Gaussian, leave the covariance matrices invariant, and send mean vectors µ → µ + t(x T , 0) and µ + t(0, x T ), respectively.By comparing with Eq. ( 20), we find that In the following we shall make use of two crucial properties of the heat flow that will allow us to 'linearize' the proof of the entropy inequality: First, the entropy of ρ (t) X grows logarithmically as t → ∞ and becomes asymptotically independent of the state ρ, as proved in [43,Corollary III.4] and [28,Theorem 5]: In particular, this implies that any valid inequality of the form Eq. ( 17) must satisfy the inequality since this is precisely equivalent to the validity of Eq. ( 17) as t → ∞.For us, this condition follows by taking the trace on both sides of our assumption that n k=1 q k Π k = 1 R m .To state the second property of the heat flow that we will need, we momentarily assume sufficient regularity of the states under consideration, following [43].Then, the Fisher information of a one-parameter family of states σ (s) is defined as . It satisfies the following version of the data processing inequality [43,Theorem IV.4]:For any TPCP map E, For a covariant family of the form σ (s,K) := e isK σ e −isK , the Fisher information can be computed as [43, Lemma IV.5] We can now state the quantum de Bruijn identity [43, Theorem V.1], which computes the derivative of the entropy along the heat flow in terms of the Fisher information: where the total Fisher information J(σ X ) of a state σ X on L 2 (X) is defined by for an arbitrary orthonormal basis {e j } m X j=1 of X.While above we assumed regularity, the Fisher information J(σ X ) can be defined for any state with finite second moments, and the de Bruijn identity (26) generalizes as well [28, Definition 7 & Proposition 1]. 6  Proof of Proposition III.4.We first prove the entropic Eq. ( 17) by considering ρ (t) := ρ (t) R m for t ≥ 0. As t → ∞, Eq. ( 17) holds up to arbitrarily small error, as explained below Eq. ( 23).To show its validity at t = 0, we would therefore like to argue that for all t ≥ 0. In view of the de Bruijn identity in Eq. ( 26) and Eq. ( 21), it suffices to establish the following super -addivity property of the Fisher information for all states σ on L 2 (R m ) with finite second moment: 6 The key idea is to first define an integral version of the Fisher information [28,Definition 6].In the setting without side information, this is defined for a state σX on L 2 (X) and for t > 0 by ∆(σX )(t) := I(X : V ) σ XV (t) , where σXV (t) denotes the classical-quantum state with V is a multivariate Gaussian random variable with covariance matrix t(IX ⊕ IX ) and X,y for v = (x, y) ∈ X ⊕ X; one also sets ∆X (σX )(0) := 0. This is well-defined for any state σX and satisfies a finitary version of the de Bruijn identity [28,Theorem 1].Moreover, if σX is a state with finite energy then ∆(σX )(t) is continuous, increasing, and concave as a function of t ≥ 0. Hence, for such states one can define the Fisher information J(σX ) as the (right) derivative of ∆(σX )(t) at t = 0, that is, as We first prove this under the regularity assumptions of [43], so that Eq. ( 25) applies.We will abbreviate Q j := Q R m ,e j and P j := P R m ,e j , where {e j } m j=1 is the standard basis of R m .For all x ∈ X k , it holds that where the second step is by the compatibility of phase-space translations and generalized partial trace (22), the third step uses the data-processing inequality for the Fisher information Eq. ( 24), and the fourth step follows from Eq. ( 25).If we apply the same argument to J σ and sum both inequalities over an orthonormal basis {x} of X k , we obtain where we used the shortcut J(σ, A) for any positive semidefinite m × m matrix A, which is linear in A. Thus, our assumption that n k=1 q k Π k = 1 R m (with all q k ≥ 0) implies the desired inequality: This establishes Eq. ( 28) and hence Eq. ( 17) for states that are sufficiently regular.While Eq. ( 25) need not apply in general, the Fisher information J(σ) and the de Bruijn identity (26) have been generalized to arbitrary states with finite second moments [28], as discussed above.The quantity J(σ, A) can be defined in the same manner so that Eqs. ( 29) and ( 30) hold verbatim, see [29, Definition 6, Propositions 6 & 9]. 7he analytic form in Eq. ( 18) then follows from a slight extension of Theorem II.1, or more specifically the special case discussed in Corollary II.5.Namely, we need to incorporate on the entropic side the finite second moment assumption from Eq. (17).By inspection, the variational formulae from Fact II.2 applied to operators with finite second moment still hold for the respective suprema only taken over operators with finite second moment.Hence, following the proof of the BL duality in Theorem II.1, we can still go from the entropic to the analytic form when assuming that the operator in exponential form on the left hand side of the analytic form has finite second moment.
While the preceding discussion restricted to the geometric case, we can also consider the general case of surjective linear map L k : R m → R m k , as in Section I.For this, write L k as the composition of an invertible map M k ∈ GL(m) and the projection onto the first m k coordinates.Define a unitary operator k ) defines a TPCP map that is the natural quantum version of the marginalization g → g k (same notation as in Eq. ( 2)).We leave it for future work to determine under which conditions such quantum Brascamp-Lieb inequalities hold in general.
Note added : In follow-up work, Eq. ( 17) from Proposition III.4 has been extended to the conditional case with side information [50,Theorem 7.3] for Gaussian states, based on [44].Subsequently, the latter assumption was removed by De Palma and Trevisan [29], who further generalized Proposition III.4 and also fully resolved the aforementioned question.

C. Entropic uncertainty relations
In this section, we explain how the duality of Theorem II.1 and Corollary II.5 offers an elegant way to prove entropic uncertainty relations (cf. the related work [57]).In order to compare our uncertainty bounds with the previous literature, we work in the current subsection with the explicit logarithm function relative to base two.
Example III.5 (Maassen-Uffink).For ρ A ∈ S(A) the Maassen-Uffink entropic uncertainty relation [54] for two arbitrary basis measurements, asserts in its strengthened form [11] that The constant c(X, Z) is tight in the sense that there exist quantum states that achieve equality for certain measurement maps.Eq. ( 10) of Corollary II.5 for n = 2, q 1 = q 2 = 1, E 1 = M X , and E 2 = M Z then immediately gives the equivalent analytic form In other words, in order to prove Eq. ( 31) it suffices to show Eq. ( 32).Now, since the logarithm is operator concave and M † X is a unital map, the operator Jensen inequality [36] implies ) .Together with the monotonicity of X → tr exp(X) [18, Theorem 2.10] and the Golden-Thompson inequality8 [34,61], this establishes the analytic form of Eq. ( 32) Thus, the entropic Maassen-Uffink relation Eq. ( 31) follows from our Corollary II.5.
We note that the approach of proving entropic uncertainty relations via the Golden-Thompson inequality was pioneered by Frank & Lieb [31] and is conceptually different from the original proofs that are either based on complex interpolation theory for Schatten pnorms [54] or the monotonicity of quantum relative entropy [26].We refer to [25] for a review on entropic uncertainty relations.As a possible extension one could choose non-trivial prefactors q k = 1 and study the optimal uncertainty bounds in that setting as well (as done in [57] without the H(A) term).Another natural extension is to general quantum channels instead of measurements (as detailed in [12,32]).The constant c(X, Z) from Eq. ( 31) is multiplicative for tensor product measurements.However, we might ask more generally if for given measurements the optimal lower bound in Eq. ( 31) becomes multiplicative for tensor product measurements.This amounts to an instance of the tensorization question from Eq. ( 13) and we refer to [32,57] for a discussion.
An advantage of our BL analysis is that it suggests tight generalizations to multiple measurements by means of the multivariate extension of the Golden-Thompson inequality [60].A basic example is as follows.
Example III.6 (Six-state [27]).For ρ A ∈ S(A) with dim(A) = 2 and measurement maps Moreover, this relation is tight in the sense that there exist quantum states that achieve equality.Note that applying the Maassen-Uffink relation Eq. (31) for any two of of the three Pauli measurements only yields the weaker bound The equivalent analytic form of Eq. ( 33) is given by Corollary II.5 as The same steps as in the proof of the Maassen-Uffink relation, together with Lieb's triple matrix inequality [46] then yield the upper bound9 In the penultimate step we used that z|ω|z |z z| where |z 0 = (1, 0) T , |z 1 = (0, 1) T . As Together with x|z 0 z 0 |y = 1 2 and x|z 1 z 1 |y = ± i 2 for all x ∈ {x 0 , x 1 }, y ∈ {y 0 , y 1 } we find the upper bound This then concludes the proof of the six-state entropic uncertainty relation Eq. (33).

D. Minimum output entropy
The Brascamp-Lieb duality from Theorem II.1 and Corollary II.5 is also applied usefully to general quantum channels.Recall that the minimum output entropy of a map E ∈ TPCP(A, B) is defined by The computation of minimum output entropy is in general NP-complete [9].Nevertheless, it is a fundamental information measure [58] that has been used, e.g., to prove super-additivity of the Holevo information [38].Corollary II.5 for n = 2, q 1 = q 2 = 1, E 1 = I, and E 2 = E gives the following result.
Corollary III.7 (Minimum output entropy).For E ∈ TPCP(A, B) and C ∈ R, the following two statements are equivalent: tr exp(log Moreover, we have It is unclear if the form Eq. ( 38) could give new insights on the tensorization question of when the minimal output entropy of tensor product channels becomes additive.That is, for which E, F ∈ TPCP(A, B) do we have We note that probabilistic counterexamples are known [38], which shows that the tensorization question Eq. ( 13) is in general answered in the negative.
Proof of Corollary III.7.We give two proofs of Eq. ( 38), one based on the variational characterization of the relative entropy from Eq. ( 6), and the other based on the dual formulation from Eq. (37).Using the former approach, we see that where the final step follows from the variational formula of the largest eigenvalue.Alternatively we can verify Eq. (38) in the analytic picture.To see this, we note that using the equivalence between Eq. (36) and Eq.(37) as well as the monotonicity of the logarithm, log tr exp log ω 1 + E † (log ω 2 ) .
Next, note that, for any Hermitian H, the Golden-Thompson inequality gives max where the second step uses again the variational formula for the largest eigenvalue.This inequality is in fact an equality, since the upper-bound is attained if we choose ω 1 to be a projector onto an eigenvector of H with largest eigenvalue (any such ω 1 commutes with H).If we use this to evaluate Eq. ( 40), then we obtain the desired result.
Example III.8 (Qubit depolarizing channel).The minimal output entropy of the qubit depolarizing channel is given by H min (E p ) = h p/2 with h(x) := −x log x − (1 − x) log(1 − x) is the binary entropy function.In the entropic picture, this follows as the concavity of the entropy ensures that the optimizer in Eq. ( 35) can always be taken to be a pure state; the unitary covariance property of the depolarizing channel then implies that we only need to evaluate the output entropy for a single arbitrary pure state.In the analytic picture, we can use Eq.(38) to see that where the second step follows from unitary covariance and the final step uses that t = 1 − p/2 is the optimizer.

E. Data-processing inequality
The examples given so far employed Corollary II.5, but in this section we give an example that demonstrates Theorem II.1 in its full strength (with σ, σ k = 1).The data-processing inequality (DPI) for the quantum relative entropy is a cornerstone in quantum information theory [51,55,62].It states that, for ρ ∈ S(A) and σ ∈ P (A), the quantum relative entropy cannot increase when applying a channel E ∈ TPP(A, B) to both arguments, i.e.,
Corollary III.9 (DPI duality).For σ ∈ P (A) and E ∈ TPP(A, B) the following inequalities hold and are equivalent: As a simple example for tr σ ≤ tr ρ = 1, one can immediately see that D(ρ σ) ≥ 0 by considering the trace map E(•) = tr(•).Namely, data processing for the trace map takes the trivial analytic form tr log ω ≤ 0 for quantum states ω ∈ S(A).
Given that the DPI is quite powerful, we suspect that Eq. ( 42) may be of interest too.We note that Eq. ( 42) does not immediately follow from existing results and thus seems novel.For example, employing the operator concavity of the logarithm, the operator Jensen inequality, and the Golden-Thompson inequality we get This immediately implies Hansen's multivariate Golden-Thompson inequality [35, Inequality (1)], but is in general still weaker than Eq. ( 42) as the Golden-Thompson inequality applied to the right-hand side of Eq. ( 42) likewise gives tr exp log ω + log E(σ) ≤ tr ωE(σ) .

F. Strong data-processing inequalities
It is a natural to study potential strengthenings of the DPI inequality and a priori it is possible to seek for additive or multiplicative improvements.Additive strengthenings of the DPI have recently generated interest in quantum information theory [30,42,59,60].Here, we consider multiplicative improvements of the DPI, which have been called strong data-processing inequalities in the literature.To this end, define the contraction coefficient of E ∈ TPCP(A, B) at σ ∈ S(A) as The data-processing inequality then ensures that η(σ, E) ≤ 1, and we say that E satisfies a strong data-processing inequality at σ if η(σ, E) < 1. Theorem II.1 for n = 1, C = 0, σ 1 = E(σ), and q 1 = η(σ, E) −1 implies the following equivalence.
In the analytic form of Eq. ( 49), the statement of Eq. ( 51) is equivalent to the claim that η = (1 − p) G. Super-additivity of relative entropy Another type of strengthening of the DPI is as follows.The quantum relative entropy is super-additive for product states in the second argument.That is, for ρ AB , σ AB ∈ S(A ⊗ B) we have Recently, it was shown [17] that Eq. ( 58) indeed holds for Applying Theorem II.1 for n = 2, σ 1 = σ A , σ 2 = σ B , C = 0, E 1 = tr B , E 2 = tr A , q 1 = α, and q 2 = β gives the following BL duality.

IV. CONCLUSION
Our fully quantum Brascamp-Lieb dualities raise a plethora of possible extensions to study.Taking inspiration from the commutative case [53], this could include, e.g., Gaussian optimality questions, hypercontractivity inequalities, transportation cost inequalities, strong converses in Shannon theory, entropy power inequalities [1], or algorithmic and complexity-theoretic questions [15,16,33].For some of these applications it seems that an extension of Barthe's reverse Brascamp-Lieb duality [7] to the non-commutative setting would be useful.
von Neumann entropy of a density operator ρ ∈ S(A) is defined as 1 H(ρ) := − tr ρ log ρ and can be infinite (only) if A is infinite-dimensional.The quantum relative entropy of ω ∈ S(A) with respect to τ ∈ P (A) is given by D(ω τ ) := tr ω(log ω − log τ ) if ω τ and as +∞ otherwise , as in Corollary III.1, this inequality holds since the von Neumann entropy is never negative.And if each s ∈ [m] belongs to exactly p of the subsets, as in Proposition III.2, then S p+1 = • • • = S n = ∅, so the inequality holds trivially.