Brascamp-Lieb Inequalities

for Non-Commutative Integration

Eric A. Carlen^{1} and Elliott H. Lieb^{2}

Received: August 6, 2008 Revised: September 22, 2008

Communicated by Heinz Siedentop

Abstract. We formulate a non-commutative analog of the Brascamp-Lieb inequality, and prove it in several concrete settings.

2000 Mathematics Subject Classification: 47C15, 15A45, 26D15 Keywords and Phrases: inequalities, traces, non-commutative integration

1 Introduction

1.1 Young’s inequality in the context of ordinary Lebesgue integration

In this paper, we shall extend the class of generalized Young’s inequalities known as Brascamp-Lieb inequalities (B-L inequalities) to an operator algebra setting entailing non-commutative integration. The original Young’s inequality [40] states that for non negative measurable functions f_{1}, f_{2} and f_{3} on ℝ, and 1 ≤ p_{1}, p_{2}, p_{3} ≤ ∞, with 1/p_{1} + 1/p_{2} + 1/p_{3} = 2,

∫_{ℝ^{2}} f_{1}(x)f_{2}(x − y)f_{3}(y)dxdy

≤ (∫_{ℝ} f_{1}^{p_{1}}(t)dt)^{1/p_{1}} (∫_{ℝ} f_{2}^{p_{2}}(t)dt)^{1/p_{2}} (∫_{ℝ} f_{3}^{p_{3}}(t)dt)^{1/p_{3}} .  (1.1)

1Work partially supported by U.S. National Science Foundation grant DMS 06-00037.

2Work partially supported by U.S. National Science Foundation grant PHY 06-52854.

\begin{matrix}c \\ ◯ \end{matrix} 2008 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Documenta Mathematica 13 (2008) 553–584

Thus, it provides an estimate of the integral of a product of functions in terms of a product of L^{p} norms of these functions. The crucial difference with a Hölder type inequality is that the integrals on the right are integrals over only ℝ, while the integral on the left is an integral over ℝ^{2}, and none of the three factors in the product on the left – f(x), g(x − y) or h(y) – are integrable (to any power) on ℝ^{2}. To frame the inequality in terms that are more amenable to the generalizations considered here, define the maps ϕ_{j} : ℝ^{2} → ℝ, j = 1, 2, 3, by

ϕ_{1}(x, y) = x  ϕ_{2}(x, y) = x − y  and  ϕ_{3}(x, y) = y .

Then (1.1) can be rewritten as

∫_{ℝ^{2}} (∏_{j=1}^{3} f_{j} ∘ ϕ_{j}) d^{2}x ≤ ∏_{j=1}^{3} (∫_{ℝ} f_{j}^{p_{j}}(t)dt)^{1/p_{j}} .  (1.2)

There is now no particular reason to limit ourselves to products of only three functions, or to integrals over ℝ^{2} and ℝ, or even any Euclidean space for that matter:

1.1 DEFINITION. Given measure spaces (Ω, S, μ) and (M_{j}, M_{j}, ν_{j}), j = 1, …, N, not necessarily distinct, together with measurable functions ϕ_{j} : Ω → M_{j} and numbers p_{1}, …, p_{N} with 1 ≤ p_{j} ≤ ∞, 1 ≤ j ≤ N, we say that a B-L inequality holds for {ϕ_{1}, …, ϕ_{N}} and {p_{1}, …, p_{N}} in case there is a finite constant C such that

∫_{Ω} ∏_{j=1}^{N} f_{j} ∘ ϕ_{j}dμ ≤ C ∏_{j=1}^{N} ∥f_{j}∥_{L^{p_{j}} (ν_{j})}  (1.3)

holds whenever each f_{j} is non-negative and measurable on M_{j}, j = 1, …, N. There are by now many examples. One of the oldest is the original discrete Young’s inequality. In the current notation, this concerns the case in which Ω = ℤ^{2} equipped with counting measure, N = 3, and each M_{j} is ℤ, equipped with counting measure. Then with

ϕ_{1}(m, n) = m  ϕ_{2}(m, n) = m − n  and  ϕ_{3}(m, n) = n ,

(1.2) holds for any three non-negative functions f_{j} : ℤ → ℝ_{+} under the same conditions on the p_{j} as in the continuous case; i.e., 1/p_{1} + 1/p_{2} + 1/p_{3} = 2. There is a significant difference: In the discrete case, the constant C = 1 is sharp, and there is equality if and only if one of the f_{j} is identically zero, or else f_{1} vanishes except at some m_{0}, f_{3} vanishes except at some n_{0}, and f_{2} vanishes except at m_{0} − n_{0}. This discrete inequality is also due to Young [40], while the statement about cases of equality is proved in [20], where the authors also consider extensions to more than three functions.

In the continuous case, a much wider generalization to more than three functions was made by B-L in [5], where the sharp constant in Young’s inequality – which is strictly less than 1 unless p_{1} = p_{2} = 1 – was obtained, with a proof that the only non-negative functions yielding equality are certain Gaussian functions. (This best constant was also obtained at the same time by Beckner [4], for three functions.) These inequalities generalize from ℝ to ℝ^{n}. The complete generalization to the case in which the M_{j} are all Euclidean spaces, but of different dimension, and the ϕ_{j} are linear transformations from ℝ^{n} to M_{j}, was proved by Lieb [23]. Again, the maximizers are Gaussians. Another proof of this generalized version, together with a reverse form, was obtained by Barthe [1], who also provided a detailed analysis of the cases of equality in the original B-L inequality from [5]. The cases of equality in the higher dimensional generalization from [23] were analyzed in detail in [7, 8]. Examples in which Ω is the sphere S^{N−1} or the permutation group S^{N} were proved in [13, 14], and the above definition of B-L inequalities in arbitrary measure spaces is taken from [10], where a duality between B-L inequalities and subadditivity of entropy inequalities is proved.

1.2 A generalized Young’s inequality in the context of noncommutative integration

In non-commutative integration theory, as developed by Irving Segal [31, 32, 33], the basic framework is a triple (H, 𝔄, λ) where H is a Hilbert space, 𝔄 is a W^{∗} algebra (a von Neumann algebra) of operators on H, and λ is a positive linear functional on the finite rank operators in 𝔄. In Segal’s picture, the algebra 𝔄 corresponds to the algebra of bounded measurable functions, and applying the positive linear functional λ to a positive operator corresponds to taking the integral of a positive function. That is,

A ↦ λ(A)  corresponds to  f ↦ ∫_{M} fdν .

Such a triple (H, 𝔄, λ) is called a non-commutative integration space. Certain natural regularity properties must be imposed on λ if one is to get a wellbehaved non-commutative integration theory, but we shall not go into these here as the examples that we consider are all based on the case in which λ is the trace on operators on H, or some closely related functional, for which discussion of these extra conditions would be a digression. In this operator algebra setting there are natural non-commutative analogs of the usual L^{p} spaces: If A is a finite rank operator in 𝔄, and 1 ≤ q < ∞, define

∥A∥_{q,λ} = (λ(A^{∗}A)^{q/2})^{1/q} .

This defines a norm (under appropriate conditions on λ that are obvious for the trace), and the completion of the space of finite rank operators in 𝔄 under this norm defines a non-commutative L^{p} space. (The completion may contain unbounded operators “affiliated” to 𝔄.) For more on the general theory of non-commutative integration, see the early papers [15, 31, 33, 34] and the more recent work in [16, 19, 21, 25]. To frame an analog of (1.3) in an operator algebra setting, we replace the measure spaces by non-commutative integration spaces:

(Ω, S, μ) ⟶ (H, 𝔄, λ) and (M_{j}, M_{j}, ν_{j}) ⟶ (H_{j}, 𝔄_{j}, λ_{j}) j = 1, …, N .

The right hand side of (1.3) has an obvious generalization to the operator algebra setting in terms of the non-commutative L_{p} norms introduced above. As for the left hand side of (1.3), regard f_{j} ↦ f_{j} ∘ ϕ_{j} as a W^{∗} algebra homomorphism (which, restricted to the W^{∗} algebra L^{∞}(M_{j}), it is), and suppose we are given W^{∗} homomorphisms

ϕ_{j} : 𝔄_{j} → 𝔄 .

Then each ϕ_{j}(A_{j}) belongs to 𝔄. However in the non-commutative case, the product of the ϕ_{j}(A_{j}) depends on their order in the product, and need not be self-adjoint even – let alone positive – even if each of the A_{j} are positive. Therefore, let us return to the left side of (1.3) and suppose that each f_{j} is strictly positive. Then defining

h_{j} = ln(f_{j})  so that  f_{j} ∘ ϕ_{j} = e^{h∘ϕ_{j}} ,

we can then rewrite (1.3) as

∫_{Ω} exp (∑_{j=1}^{N} h_{j} ∘ ϕ_{j}) dμ ≤ C ∏_{j=1}^{N} ∥e^{h_{j}}∥_{L^{p_{j}} (ν_{j})} ,  (1.4)

We can now formulate our operator algebra analog of (1.3):

1.2 DEFINITION. Given non-commutative integration spaces (H, 𝔄, λ) and (H_{j}, 𝔄_{j}, λ_{j}), j = 1, …, N, together with W^{∗} algebra homomorphisms ϕ_{j} : 𝔄_{j} → 𝔄, j = 1, …, N, and indices 1 ≤ p_{j} ≤ ∞, j = 1, …, N, a noncommutative B-L inequality holds for {ϕ_{1}, …, ϕ_{N}} and {p_{1}, …, p_{N}} if there is a finite constant C so that

λ (exp [∑_{j=1}^{N} ϕ_{j}(H_{j})]) ≤ C ∏_{j=1}^{N} (λ_{j} exp [p_{j}H_{j}])^{1/p_{j}}  (1.5)

whenever H_{j} is self-adjoint in 𝔄_{j}, j = 1, …, N.

In this paper, we are concerned with determining the indices and the best constant C for which such an inequality holds, and shall focus on two examples: The first concerns operators on tensor products of Hilbert spaces, and the second concerns Clifford algebras.

1.3 A generalized Young’s inequality for tensor products

1.3 EXAMPLE. Let H_{j}, j = 1, …, n be separable Hilbert spaces, and let Let K denote the tensor product

K = H_{1} ⊗ ⋯ ⊗ H_{n} .

Define 𝔄 to be 𝔅(K), the algebra of bounded operators on K, and define λ to be Tr, the trace Tr on K, so that (H, 𝔄, λ) = (K, 𝔅(K), Tr). For any non-empty subset J of {1, …, n}, let K_{J} denote the tensor product

K_{J} = ⨂_{j∈J} H_{j} .

Define 𝔄_{J} to be 𝔅(K_{J}), the algebra of bounded operators on K_{J}, and define λ_{J} to be Tr_{J}, the trace on K_{J}, so that (H_{J}, 𝔄_{J}, λ_{J}) = (K_{J}, 𝔅(K_{J}), Tr_{J}). There are natural homomorphisms ϕ_{J} embedding the 2^{n} − 1 algebras 𝔄_{J} into 𝔄. For instance, if J = {1, 2},

ϕ_{{1,2}}(A_{1} ⊗ A_{2}) = A_{1} ⊗ A_{2} ⊗ I_{H_{3}} ⊗ ⋯ ⊗ I_{H_{N}} ,  (1.6)

and ϕ_{{1,2}} is extended linearly. It is obvious that in case J ∩ K = ∅ and J ∪ K = {1, …, n}, then for all H_{J} ∈ 𝔄_{J} and H_{K} ∈ 𝔄_{K},

Tr (e^{H_{J} +H_{K}} ) = Tr_{J} (e^{H_{J}} ) Tr_{K} (e^{H_{K}} ) ,  (1.7)

but things are more interesting when J ∩ K ≠ ∅ and J and K are both proper subsets of {1, …, n}. If H_{J} and H_{K} do not commute, which is the typical situation for J ∩ K ≠ ∅, one can estimate the left hand side of (1.7) by first applying the Golden–Thompson inequality [17, 36], which says that for selfadjoint operators H_{J} and H_{K},

Tr (e^{H_{J} +H_{K}} ) ≤ Tr (e^{H_{J}} e^{H_{K}} ) .

One might then apply Hölder’s inequality – but if J and K are proper subsets of {1, …, n}, this will yield a finite bound if and only if all of the Hilbert spaces whose indices are not included in both J and K are finite dimensional. Even then, the bound depends on the dimension in an unpleasant way. The non-commutative B-L Inequalities provided by the next theorem do not have this defect.

1.4 THEOREM. Let J_{1}, …, J_{N} be N non-empty subsets of {1, …, n} For each i ∈ {1, …, n}, let p(i) denote the number of the sets J_{1}, …, J_{N} that contain i, and let p denote the minimum of the p(i). Then, for self-adjoint operators H_{j} on K_{J_{j}}, j = 1, …, N,

Tr (exp [∑_{j=1}^{N} ϕ_{J_{j}}(H_{j})]) ≤ ∏_{j=1}^{N} (Tr_{J_{j}} e^{qH_{j}})^{1/q}  (1.8)

Documenta Mathematica 13 (2008) 553–584

for q = p (and hence all 1 ≤ q ≤ p), while for all q > p, it is possible for the left hand side to be infinite, while the right hand side is finite.

Note that in Theorem 1.4, the constant C in Definition (1.2) is 1. The fact that the constant C = 1 is best possible, and that the inequality cannot hold for q > p = min{p(1), …, p(N)} is easy to see by considering the case that each H_{j} has finite dimension d_{j}, and H_{j} = 0 for each j. Then

Tr (exp [∑_{j=1}^{N} ϕ_{J_{j}}(H_{j})]) = ∏_{j=1}^{n} d_{j}

and

∏_{j=1}^{N} (Tr_{J_{j}}e^{qH_{j}})^{1/q} = ∏_{j=1}^{N} ∏_{k∈J_{j}} d_{k}^{1/q} = ∏_{j=1}^{n} d_{j}^{p(j)/q} .

We will prove inequality (1.8) for q = p in Section 3. As an example, consider the case in which n = 6, N = 3 and

J_{1} = {1, 2, 3} J_{2} = {3, 4, 5} and J_{3} = {5, 6, 1} .

Here, p = 1, and hence

Tr (exp [∑_{j=1}^{3} ϕ_{J_{j}}(H_{j})]) ≤ ∏_{j=1}^{3} (Tr_{J_{j}} e^{H_{j}}) .  (1.9)

Theorem 1.4 says that the inequality (1.9) can be extended to larger tensor products, and this obviously has an statistical mechanical interpretation as a bound on the partition function of a collection of interacting spins in terms of a product of partition functions of simple constituent sub-systems. For more background on this, see Thirring’s book [35]. To estimate the left side of (1.9) without using Theorem 1.4, one might use the Golden-Thompson inequality and then Schwarz’s inequality to write

Tr (exp [∑_{j=1}^{3} ϕ_{J_{j}}(H_{j})]) ≤ Tr (e^{ϕ_{1}(H_{1})+ϕ_{3}(H_{3})}e^{ϕ_{2}(H_{2})})

≤ (Tr e^{2[ϕ_{1}(H_{1})+ϕ_{3}(H_{3})]})^{1/2} (Tr e^{2ϕ_{2}(H_{2})})^{1/2} .

While the L^{2} norms are an improvement over the L^{1} norms in (1.9), the traces are now over the entire tensor product space. Thus, for example,

(Tr e^{2ϕ_{2}(H_{2})})^{1/2} = (d_{1}d_{2}d_{6})^{1/2} (Tr_{J_{2}} e^{2H_{2}})^{1/2}

where d_{j} is the dimension of Hilbert space H_{j}. This dimension dependence may be unfavorable if any of the dimensions is large.

1.4 A generalized Young’s inequality in Clifford algebras

Our next example concerns Clifford algebras, which, as Segal emphasized [32], allow one to represent fermion Fock space as an L^{2} space – albeit a noncommutative L^{2} space, but still with many of the advantages of having a Hilbert space represented as a function space, as in the usual Schrödinger representation in quantum mechanics. In the finite dimensional setting, with n degrees of freedom, one starts with n operators Q_{1}, …, Q_{n} on some Hilbert space H that satisfy the canonical anticommutation relations

Q_{i}Q_{j} + Q_{j}Q_{i} = 2δ_{i,j}I .

One can concretely construct such operators acting on H = (ℂ^{2})^{⊗n}, the n–fold tensor product of ℂ^{2} with itself; see [6]. The Clifford algebra ℭ is the operator algebra on H that is generated by Q_{1}, …, Q_{n}. The Clifford algebra ℭ itself is 2^{n} dimensional. In fact, let α = (α_{1}, …, α_{n}) be a fermionic multi-index, which means that each α_{j} is either 0 or 1. Then define

Q^{α} = Q_{1}^{α_{1}}Q_{2}^{α_{2}} ⋯ Q_{n}^{α_{n}} .  (1.10)

it is easy to see that the 2^{n} operators Q^{α} are a basis for the Clifford algebra, so that any operator A in ℭ has a unique expression

A = ∑_{α} x_{α}Q^{α} .

The linear functional τ on ℭ is defined by

τ (∑_{α} x_{α}Q^{α}) = x_{(0,...,0)} .  (1.11)

That is, τ acting on A picks out the coefficient of the identity in A = ∑_{α} x_{α}Q^{α}. It turns out that when the Clifford algebra is constructed in the way described here, as an algebra operators on the 2^{n} dimensional space H = (ℂ^{2})^{⊗n}, τ is nothing other than the normalized trace:

τ(A) = \frac{1}{2^{n}} Tr_{H}(A) .

Hence τ is a positive linear functional, and ((ℂ^{2})^{⊗n}, ℭ, τ) is a non-commutative integration space in the sense of Segal. Clifford algebras have infinitely many subalgebras that are also Clifford algebras of lower dimension. This is in contrast to the setting described in Example 1.3, in which the only natural subalgebras are the 2^{n} −1 subalgebras corresponding to the 2^{n} − 1 non-empty subsets of the index set {1, …, n}.

To describe these subalgebras, let J be the canonical injection of ℝ^{n} into ℭ, which is given by

J ((x_{1}, …, x_{n})) = ∑_{j=1}^{n} x_{j}Q_{j} .  (1.12)

If x and y are any two vectors in ℝ^{n}, it is easy to see from the canonical anticommutation relations that

(J (x))(J (y)) = 2(x · y)I .

Hence if V is any m dimensional subspace of ℝ^{n}, and {u_{1}, …, u_{m}} is any orthonormal basis for V, the m operators

J (u_{1}), …, J (u_{m})

again satisfy the canonical anticommutation relations, and generate a subalgebra of ℭ that is denoted by ℭ(V), and referred to as the Clifford algebra over V. In the same vein, it is convenient to refer to ℭ itself as the Clifford algebra over ℝ^{n}. Obviously, ℭ(V) is naturally isomorphic to ℭ(ℝ^{m}), and for A ∈ ℭ(V) one may compute τ(A) using either the normalized trace τ inherited from ℭ, or the normalized trace τ_{V} induced by the identification of ℭ(V) with ℭ(ℝ^{m}). As Segal emphasized, ((ℂ^{2})^{⊗n}, ℭ, τ) is, in many ways, a non-commutative analog of the Gaussian measure space (ℝ^{n}, γ(x)dx) where

γ(x) = \frac{1}{(2π)^{n/2}} e^{−|x|^{2}/2} .  (1.13)

In fact, just as orthogonality implies independence in (ℝ^{n}, γ(x)dx), if V and W are two orthogonal subspaces of ℝ^{n}, and if A ∈ ℭ(V) and B ∈ ℭ(W), then

τ(AB) = τ(A)τ(B) .

The results we prove here reinforce this analogy. We are now ready to introduce our next example:

1.5 EXAMPLE. For some n > 1, let 𝔄 be the Clifford algebra over ℝ^{n} with its usual inner product, and let 𝔄 be equipped with its unique tracial state τ, which is simply the normalized trace. For each j = 1, …, N, let V_{j} be a subspace of ℝ^{n}, and let 𝔄_{j} be ℭ(V_{j}), the Clifford algebra over V_{j} with the inner product that V_{j} inherits from ℝ^{n}. Let 𝔄_{j} be equipped with its unique tracial state τ_{j}. The natural embedding of V_{j} into ℝ^{n} induces a homomorphism of 𝔄_{j} into 𝔄, and we define this to be ϕ_{j}. In this setting, we shall prove

1.6 THEOREM. Let V_{1}, …, V_{N} be N subspaces of ℝ^{n}, and let 𝔄_{j} be the Clifford algebra over V_{j} with the inner product that V_{j} inherits from ℝ^{n}, and let 𝔄_{j} be equipped with its unique tracial state τ_{j}. Let ϕ_{j} be the natural homomorphism of 𝔄_{j} into 𝔄 induced by the natural embedding of V_{j} into ℝ^{n}. Then

τ (exp [∑_{j=1}^{N} ϕ_{j}(H_{j})]) ≤ ∏_{j=1}^{N} (τ_{j} e^{p_{j}H_{j}})^{1/p_{j}}  (1.14)

for all self-adjoint operators H_{j} ∈ 𝔄_{j} if and only if

∑_{j=1}^{N} \frac{1}{p_{j}} P_{j} ≤ I_{ℝ^{n}} .  (1.15)

where P_{j} is the orthogonal projection onto V_{j} in ℝ^{n}.

In the special case in which dim(V_{j}) = 1 for each j, (1.14) reduces to an interesting inequality for the hyperbolic cosine. Indeed, let u_{j} be one of the two unit vectors in V_{j}. Then, with u_{j} ⊗ u_{j} denoting the orthogonal projection onto the span of u_{j}, (1.15) reduces to

∑_{j=1}^{N} \frac{1}{p_{j}} u_{j} ⊗ u_{j} ≤ I_{ℝ^{n}} .  (1.16)

The greater simplification, however, is that in this case, the space of self-adjoint operators in each 𝔄_{j} is just two dimensional, and, with J denoting the canonical injection defined in (1.12),

H_{j} = a_{j}I + b_{j}J (u_{j})

for some real numbers a_{j} and b_{j}. Then

∑_{j=1}^{N} H_{j} = (∑_{j=1}^{N} a_{j}) I + J (∑_{j=1}^{N} b_{j}u_{j}) .

This operator has exactly two eigenvalues,

(∑_{j=1}^{N} a_{j}) ± |∑_{j=1}^{N} b_{j}u_{j}|

with equal multiplicities. Likewise, p_{j}H_{j} has exactly two eigenvalues p_{j}a_{j} ±p_{j}b_{j} with equal multiplicities. Hence, in this case, (1.14) reduces to

cosh (|∑_{j=1}^{N} b_{j}u_{j}|) ≤ ∏_{j=1}^{N} (cosh(p_{j}b_{j}))^{1/p_{j}} for all (b_{1}, …, b_{N}) ∈ ℝ^{N} ,

(1.17) which, according to Theorem 1.6, must hold whenever (1.16) is satisfied. (The a_{j}’s make the same contribution to both sides, and may be cancelled away.) Taking the logarithm of both sides, this can be rewritten as

ln cosh (|∑_{j=1}^{N} b_{j}u_{j}|) ≤ ∑_{j=1}^{N} \frac{1}{p_{j}} ln cosh(p_{j}b_{j}) for all (b_{1}, …, b_{N}) ∈ ℝ^{N} ,

(1.18)

and this inequality must hold whenever the unit vectors {u_{1}, …, u_{N}} and the positive numbers {p_{1}, …, p_{N}} satisfy (1.16). Later on, we shall give an elementary proof of this inequality, and hence an elementary proof of Theorem 1.6 when each V_{j} is one dimensional. Our proof of the other cases is less than elementary, and even our elementary proof of (1.18) is less than direct.

2 Subadditivty of Entropy and Generalized Young’s Inequalities

In the examples we have introduced in the previous section, the positive linear functionals λ under consideration are either traces or normalized traces. Throughout this section, we assume that our non-commutative integration spaces (H, 𝔄, λ) are based on tracial positive linear functionals λ. That is, we require that for all A, B ∈ 𝔄,

λ(AB) = λ(BA) .

In such a non-commutative integration space (H, 𝔄, λ), a probability density is a non-negative element ρ of 𝔄 such that λ(ρ) = 1. Indeed, the tracial property of λ ensures that

λ(ρA) = λ(Aρ) = λ(ρ^{1/2}Aρ^{1/2})

so that A ↦ λ(ρA) is a positive linear functional that equals 1 on the identity. Now suppose we have N non-commutative integration spaces (H_{j}, 𝔄_{j}, λ_{j}) and W^{∗} homomorphisms ϕ_{j} : 𝔄_{j} → 𝔄. Then these homomorphisms induce maps from the space of probability densities on 𝔄 to the spaces of probability densities on the 𝔄_{j}: For any probability density ρ on (𝔄, λ), let ρ_{j} be the probability density on (𝔄_{j}, λ_{j}) defined by

λ_{j}(ρ_{j}A) = λ(ρϕ_{j}(A))

for all A ∈ 𝔄_{j}. For example, in the setting of Example 1.3, ρ_{J_{j}} is just the partial trace of ρ over ⨂_{k∈J_{j}^{c}} H_{k}, leaving an operator on ⨂_{k∈J_{j}} H_{k}. In the Clifford algebra setting of Example 1.5, ρ_{j} is simply the orthogonal projection of ρ in L^{2}(ℭ, τ) onto ℭ(V_{j}), which is also known as the conditional expectation [38] of ρ given ℭ(V_{j}). In this section, we are concerned with the relations between the entropies of ρ and the ρ_{1}, …, ρ_{N}. The entropy of a probability density ρ, S(ρ), is defined by

S(ρ) = −λ(ρ ln ρ) .

Evidently, the entropy functional is concave on the set of probability densities.

2.1 DEFINITION. Given tracial non-commutative integration spaces (H, 𝔄, λ) and (H_{j}, 𝔄_{j}, λ_{j}), j = 1, …, N, together with W^{∗} algebra homomorphisms ϕ_{j} : 𝔄_{j} → 𝔄, j = 1, …, N, and numbers 1 ≤ p_{j} ≤ ∞, j = 1, …, N, a generalized subadditivity of entropy inequality holds if there is a finite constant C so that

∑_{j=1}^{N} \frac{1}{p_{j}} S(ρ_{j}) ≥ S(ρ) − ln C  (2.1)

for all probability densities ρ in 𝔄.

It turns out that for tracial non-commutative integration spaces, generalized subadditivity of entropy inequalities and B-L inequalities are dual to one another, just as they are in the commutative case [10], so that if one holds, so does the other, with the same values of p_{1}, …, p_{N} and C. The following is, in fact, a direct non-commutative analog of the main theorem of [10].

2.2 THEOREM. Let (H, 𝔄, λ) and (H_{j}𝔄_{j}, λ_{j}), j = 1, …, N, be tracial noncommutative integration spaces. Let ϕ_{j} : 𝔄_{j} → 𝔄, j = 1, …, N be W^{∗} algebra homomorphisms. Then for any numbers 1 ≤ p_{j} ≤ ∞, j = 1, …, N, and any finite constant C, the generalized subadditivity of entropy inequality (2.1) is true for all probability densities ρ on 𝔄 if and only if the non-commutative B-L inequality (1.5) is true for all self-adjoint H_{j} ∈ 𝔄_{j}, j = 1, …, N, with the same p_{1}, …, p_{N} and the same C.

As a consequence of Theorem 2.2, one strategy for proving a non-commutative B-L inequality is to prove the corresponding generalized subadditivity of entropy inequality. We shall see in our examples that this is an effective strategy; indeed, this is how we prove Theorems 1.4 and 1.6. In the current tracial context, the proof of Theorem 2.2 is a direct adaptation of the proof of the corresponding result in the context of Lebesgue integration given in [10]. It turns on a well–known formula for the Legendre transform of the entropy. For completeness, we give this formula in Lemma 2.3 below. Except for the S(A) = −∞ case in (2.2) below, it is the same as [26, Prop. 1.10]. The Legendre transform is usually defined for functions on a dual pair of real linear spaces. Therefore, it is convenient to extend the definition of S to all of 𝔄_{h,1}, the (real) subspace of self-adjoint elements A of 𝔄 with λ(|A|) < ∞. as follows:

S(A) = \begin{cases}−λ(A ln A) & if A ≥ 0 and λ(A) = 1, \\ −∞ & otherwise. \end{cases}  (2.2)

2.3 LEMMA. Let 𝔄 be 𝔅(H), the algebra of bounded operators on a separable Hilbert space H. Let λ denote either the trace Tr on H, or, if H is finite dimensional, the normalized trace τ. Then for all A ∈ 𝔄_{h,1},

−S(A) = sup_{H self adjoint, semibounded above} {λ(AH) − ln (λ (e^{H}))} .  (2.3)

The supremum is an attained maximum if and only if A is a strictly positive probability density, in which case it is attained at H if and only if H = ln A+cI for some c ∈ ℝ. Consequently, for all self adjoint H with H semibounded above,

ln (λ (e^{H})) = sup_{A∈𝔄_{h,1}} {λ(AH) + S(A)} .  (2.4)

The supremum is a maximum at all points of the domain of ln (λ (e^{H})), in which case it is attained only at the single point A = e^{H}/(λ(e^{H})).

Proof: We consider first the case that λ = Tr, and H has finite dimension d. To see that the supremum is ∞ unless 0 ≤ A ≤ I, let c be any real number, and let u be any unit vector. Then let H be c times the orthogonal projection onto u. For this choice of H,

λ(AH) − ln (λ (e^{H})) = c⟨u, Au⟩ − ln(e^{c} + (d − 1)) .

If ⟨u, Au⟩ < 0, this tends to ∞ as c tends to −∞. If ⟨u, Au⟩ > 1, this tends to ∞ as c tends to ∞. Hence we need only consider 0 ≤ A ≤ I. Next, taking H = cI, c ∈ ℝ,

λ(AH) − ln (λ (e^{H})) = cλ(A) − c − ln(d) .

Unless λ(A) = 1, this tends to ∞ as c tends to ∞. Hence we need only consider the case that A is a density matrix ρ. Let ρ be any density matrix on H and let H be any self-adjoint operator such that Tr(e^{H}) < ∞, and define the density matrix σ by

σ = \frac{e^{H}}{Tr(e^{H})} .

Then, by the positivity of the relative entropy,

Tr(ρ ln ρ − ρ ln σ) ≥ 0 ,

with equality if and only if σ = ρ. But by the definition of σ, this reduces to

Tr(ρ ln ρ) ≥ Tr(ρH) − ln (Tr (e^{H})) ,

with equality if and only if H = ln ρ. From here, the rest is very simple, including the treatment of the normalized trace.

Petz [27] has shown how to extend Lemma 2.3 to a much more general context, and his result can be used to extend the validity Theorem 2.2 beyond the tracial case. However, since the examples in which we prove the generalized subadditivity inequality here are tracial, we shall not go into this.

Proof of Theorem 2.2: Suppose, first, that the non-commutative B-L inequality (1.5) holds. Then, for any probability density ρ in 𝔄, and any selfadjoint H_{j} ∈ 𝔄_{j}, j = 1, …, N, apply (2.3) with A = ρ and H = ∑_{j=1}^{N} ϕ_{j}(H_{j}) to obtain

−S(ρ) ≥ λ (ρ [∑_{j=1}^{N} ϕ_{j}(H_{j})]) − ln [λ (exp [∑_{j=1}^{N} ϕ_{j}(H_{j})])]

= ∑_{j=1}^{N} λ_{j}(ρ_{j}H_{j}) − ln [λ (exp [∑_{j=1}^{N} ϕ_{j}(H_{j})])]

≥ ∑_{j=1}^{N} λ_{j}(ρ_{j}H_{j}) − ln [C ∏_{j=1}^{N} λ_{j} (e^{p_{j}H_{j}})^{1/p_{j}} ]

= ∑_{j=1}^{N} \frac{1}{p_{j}} [λ_{j}(ρ_{j}[p_{j}H_{j}]) − ln (λ_{j} (e^{[p_{j}H_{j}]}))] − ln C .

(2.5)

The first inequality here is (2.3), and the second is the non-commutative B-L inequality (1.5).

Now choosing p_{j}H_{j} to maximize λ_{j}(ρ_{j}[p_{j}H_{j}]) − ln (λ_{j} (e^{[p_{j}H_{j}]})), we get from (2.3) once more that

λ_{j}(ρ_{j}[p_{j}H_{j}]) − ln (λ_{j} (e^{[p_{j}H_{j}]})) = −S(ρ_{j}) = λ_{j}(ρ_{j} ln ρ_{j}) .

Thus, we have proved (2.1) with the same p_{1}, …, p_{N} and C that we had in (1.5).

Next, suppose that (2.1) is true. We shall show that in this case the noncommutative B-L inequality (1.5) holds with the same p_{1}, …, p_{N} and C. To do this, let the self-sadjoint operators H_{1}, …, H_{N} be given, and define

ρ = [λ (exp [∑_{j=1}^{N} ϕ_{j}(H_{j})])]^{−1} exp [∑_{j=1}^{N} ϕ_{j}(H_{j})] .

Then by Lemma 2.3,

ln [λ (exp [∑_{j=1}^{N} ϕ_{j}(H_{j})])] = λ (ρ [∑_{j=1}^{N} ϕ_{j}(H_{j})]) + S(ρ)

= ∑_{j=1}^{N} λ_{j} [ρ_{j}H_{j}] + S(ρ)

≤ ∑_{j=1}^{N} \frac{1}{p_{j}} [λ_{j} [ρ_{j}(p_{j}H_{j})] + S(ρ_{j})] + ln C

≤ ∑_{j=1}^{N} \frac{1}{p_{j}} ln [λ_{j} (exp(p_{j}H_{j}))] + ln C

(2.6)

The first inequality is the generalized subadditivity of entropy inequality (2.1), and the second is (2.4). Exponentiating both sides of (2.6), we obtain (1.5) with the same p_{1}, …, p_{N} and C that we had in (2.1).

3 Proof of the generalized subadditivity of entropy inequality for tensor products of Hilbert spaces

The crucial tool that we use in this section is the strong subadditivity of entropy [24], which we now recall in a formulation that is suited to our purposes. Suppose, as in Example 1.3, that we are given n separable Hilbert spaces H_{1}, …, H_{n}. As before, let K denote their tensor product, and for any nonempty subset J of {1, …, n}, let K_{J} denote ⨂_{j∈J} H_{j}. For a density matrix ρ on K, and any non-empty subset J of {1, …, n}, define ρ_{J} = Tr_{J^{c}}ρ to be the density matrix on K_{J} induced by the natural injection of 𝔅(K_{J}) into 𝔅(K). As noted above, ρ_{J} is nothing other than the partial trace of ρ over the complementary product of Hilbert spaces, ⨂_{j∉J} H_{j}. The strong subadditivity of entropy is expressed by the inequality stating that for all nonempty J, K ⊂ {1, …, n},

S(ρ_{J}) + S(ρ_{K}) ≥ S(ρ_{J∪K}) + S(ρ_{J∩K}) .  (3.1)

In case J ∩ K = ∅, it reduce to the ordinary subadditivity of entropy, which is the elementary inequality

S(ρ_{J}) + S(ρ_{K}) ≥ S(ρ_{J∪K})  for J ∩ K = ∅ .  (3.2)

Combining these, we have

S(ρ_{{1,2}}) + S(ρ_{{2,3}}) + S(ρ_{{3,1}}) ≥ S(ρ_{{1,2,3}}) + S(ρ_{{2}}) + S(ρ_{{1,3}})
≥ 2S(ρ_{{1,2,3}}) ,

(3.3) where the first inequality is the strong subadditivity (3.1) and the second is the ordinary subadditivity (3.2). Thus, for n = 3 and J_{1} = {1, 2}, J_{2} = {2, 3} and J_{3} = {3, 1}, we obtain

\frac{1}{2} ∑_{j=1}^{N} S(ρ_{J_{j}}) ≥ S(ρ) .

In this example, each index i ∈ {1, 1, 3} belonged to exactly two of the set J_{1}, J_{2} and J_{3}, and this is the source of the factor of 1/2 in the inequality. The same procedure leads to the following result:

3.1 THEOREM. Let J_{1}, …, J_{N} be N non-empty subsets of {1, …, n} For each i ∈ {1, …, n}, let p(i) denote the number of the sets J_{1}, …, J_{N} that contain i, and let p denote the minimum of the p(i). Then

\frac{1}{p} ∑_{j=1}^{N} S(ρ_{J_{j}}) ≥ S(ρ)  (3.4)

for all density matrices ρ on K = H_{1} ⊗ ⋯ ⊗ H_{n}.

Proof: Simply use strong subadditivty to combine overlapping sets to produce as many “complete” sets as possible, as in the example above. Clearly, there can be no more than p of these. If p(i) > p for some indices i, there will be “left over” partial sets. The entropy is always non-negative, and therefore, discarding the corresponding entropies gives us ∑_{j=1}^{N} S(ρ_{J_{j}}) ≥ pS(ρ), and hence the inequality. It is now a very simple matter to prove Theorem 1.4: Proof of Theorem 1.4: By the remarks made after the statement of the theorem, all that remains to be proved is the inequality (1.8) for q = p. However, this follows directly from Theorem 2.2 and Theorem 3.4.

4 On the generalized Young’s inequality with a Gaussian reference measure

Before turning to the proof of our non-commutative B-L inequality in Clifford algebras, we discuss the commutative case in which the reference measures is Gaussian. We do this here for two reasons: First, as noted, a Clifford algebra ℭ with its normalized trace τ is a non-commutative analog of a Gaussian measure space. This analogy is strong enough that we shall be able to pattern our analysis in the Clifford algebra case on an analysis of the Gaussian case. Second, the Gaussian inequality is of interest in itself, and seems not to have been fully studied before. Suppose that V_{1}, …, V_{N} are N non-zero subspaces of ℝ^{n}, and for each j, define ϕ_{j} = P_{j} to be the orthogonal projection of ℝ^{n} onto V_{j}. Equip ℝ^{n} and equip each V_{j} with Lebesgue measure. Then the problem of determining for which sets of indices {p_{1}, …, p_{N}} there exists a finite constant C so that (1.3) holds for all non-negative measurable functions f_{j} on V_{j}, j = 1, …, N is highly non trivial, and has only recently been fully solved [7, 8]. Moreover, determining the value of the best constant C for those choices of {p_{1}, …, p_{N}} is still a challenging finite dimensional variational problem for which there is no general explicit solution. In contrast, suppose we are given a non-degenerate Gaussian measure on ℝ^{n}. It will be convenient to take the covariance matrix of the Gaussian to define the inner product, so that the Gaussian becomes a unit Gaussian. For each positive integer m, define γ_{m}(x) = (2π)^{−m/2}e^{−|x|^{2}/2} on ℝ^{m}. Then equipping ℝ^{n} with the measure γ_{n}(x)dx and equipping each V_{j} with the γ_{d_{j}}(x)dx, d_{j} being the dimension of V_{j}, it turns out that there is a very simple necessary and sufficient condition on the indices {p_{1}, …, p_{N}} for the constant C to be finite, and better yet, the best constant C is always 1 whenever it is finite:

4.1 THEOREM. Let V_{1}, …, V_{N} be N non-zero subspaces of ℝ^{n}, and for each j, and let d_{j} denote the dimension of V_{j}. Define P_{j} to be the orthogonal projection of ℝ^{n} onto V_{j}. Given the numbers p_{j}, 1 ≤ p_{j} < ∞ for j = 1, …, N, there exists a finite constant C such that

∫_{ℝ^{n}} ∏_{j=1}^{N} f_{j} ∘ P_{j}(x)γ_{n}(x)dx ≤ C ∏_{j=1}^{N} (∫_{V_{j}} f_{j}^{p_{j}}(y)γ_{d_{j}}(y)dy)^{1/p_{j}}  (4.1)

holds for all non-negative f_{j} on V_{j}, j = 1, …, N, if and only if

∑_{j=1}^{N} \frac{1}{p_{j}} P_{j} ≤ Id_{ℝ^{n}}  (4.2)

and in this case, C = 1.

We hasten to point out that this theorem is partially known. In the special case that each of the subspaces V_{j} is one dimensional, Barthe and Cordero-Erausquin [2], have proved that it is sufficient to have equality in (4.2); in this one dimensional context, their sufficient condition then is

∑_{j=1}^{N} \frac{1}{p_{j}} u_{j} ⊗ u_{j} = Id_{ℝ^{n}}  (4.3)

with each u_{j} being a unit vector spanning V_{j}. They did this as an intermediate step in a short proof of the Lebesgue measure version of the B-L inequality under the condition (4.3) – the so-called geometric case. Perhaps because their main focus was the Lebesgue measure case, in which (4.3) is not a necessary condition for finiteness of the constant C, they did not address the necessity of this condition in the Gaussian case. Indeed, under the condition that equality holds in (4.2), the inequality (4.1) is equivalent to its Lebesgue measure analog, which is known to hold with the constant C = 1 under this same condition [7, 8]. To see this, define g_{1}, …, g_{N} by

g_{j}(y) = f_{j}(y)(γ_{j}(y))^{1/d_{j}} j = 1, …, N .

As noted in [2] for the one dimensional case, this change of variable allows one to pass back and forth between the Gaussian and Lebesgue measure version of the B-L inequality – under the condition (1.15). It is also perhaps worth remarking that once one has proved the sufficiency of equality in (4.2), the inequality case follows easily: By the spectral theorem, one can write

I − ∑_{j=1}^{N} \frac{1}{p_{j}} P_{j} = ∑_{k=1}^{n} ν_{k}Q_{k}

where the Q_{k} are rank one projections, and each ν_{k} satisfies 0 ≤ ν_{k} ≤ 1. For each k such that ν_{k} > 0, define q_{k} = 1/ν_{k}, and augment the sets of p_{j}’s and P_{j}’s by adding in all such q_{k}’s and Q_{k}’s. The augmented sets now satisfy (4.2) with equality, but the augmentation is “invisible” in the augmented version of (4.1) if we specialize to f_{k} = 1 identically for each index k in the augmentation. Hence, the equality case discussed in [2] really is the heart of the matter. Nonetheless, it is worthwhile to give a proof of Theorem 4.1 here for two reasons: First, it may be surprising that the condition (1.15) is necessary for the inequality to hold with any finite constant at all. Second, the proof we will give of sufficiency of the condition (1.15) serves as a model for the proof of the corresponding theorem in the Clifford algebra case that we consider in the next section. In proving Theorem 4.1, our first step is to pass to the problem of proving a generalized subadditivity inequality. Because the commutative version of Theorem 2.2 has been proved in [10], Theorem 4.2 below on subbadditivity of entropy with respect to a Gaussian reference measure is equivalent to Theorem 4.1. Hence, it suffices to prove one or the other. Before stating and proving the subadditivty theorem, we first recall that for any probability density ρ on (ℝ^{m}, dγ_{m}), the entropy of ρ, is defined by

S(ρ) = − ∫_{ℝ^{m}} ρ(y) ln ρ(y)γ_{m}(y)dy .

Note that the relative entropy of ρ(y)γ_{m}(y)dy to γ_{m}(y)dy is −S(ρ); in the convention employed here, the entropy S is concave, and the relative entropy is convex.

4.2 THEOREM. Let V_{1}, …, V_{N} be N non-zero subspaces of ℝ^{n}, and for each j, let d_{j} denote the dimension of V_{j}. Define P_{j} to be the orthogonal projection of ℝ^{n} onto V_{j}. For any probability density ρ on (ℝ^{n}, dγ_{n}), let ρ_{V_{j}} denote the marginal density on (V_{j}, dγ_{d_{j}}). Then, given the numbers p_{j}, 1 ≤ p_{j} < ∞ for j = 1, …, N, there exists a finite constant C such that

∑_{j=1}^{N} \frac{1}{p_{j}} S(ρ_{V_{j}}) ≥ S(ρ) − ln(C)  (4.4)

holds for all probability densities ρ on (ℝ^{n}, dγ_{n}), if and only if

∑_{j=1}^{N} \frac{1}{p_{j}} P_{j} ≤ I  (4.5)

and in this case, ln(C) = 0.

We first prove necessity of the condition (4.5):

4.3 LEMMA. The condition (4.5) in Theorem 4.2 is necessary.

Proof: It suffices to consider densities of the form

ρ(x) = exp(b · x − |b|^{2}/2) ,

for b ∈ ℝ^{n}. Then

ρ_{V_{j}}(x) = exp(P_{j}b · y − |P_{j}b|^{2}/2) ,

and we compute:

S(ρ) = −\frac{|b|^{2}}{2}  and  S(ρ_{V_{j}}) = −\frac{|P_{j}b|^{2}}{2} .

Thus

∑_{j=1}^{N} \frac{1}{p_{j}} S(ρ_{V_{j}}) − S(ρ) = b · [Id_{ℝ^{n}} − ∑_{j=1}^{N} \frac{1}{p_{j}} P_{j}] b ,

and evidently this is bounded below if and only if (4.5) is satisfied.

4.1 Proof of sufficiency

The sufficiency of the condition (4.5) will be proved using an interpolation between an arbitrary density ρ and the uniform density that is provided by the Mehler semigroup. (Indeed, Barthe and Coredero-Erausquin used the Mehler semigroup in their work [2] mentioned above, but used it in a direct proof of the Gaussian B-L inequality inspired by the heat-flow method introduced in [13]. The heat flow approach to prove subadditivity inequalities was developed in [3] and [10].) The Mehler semigroup is the strongly continuous semigroup of positivity preserving contractions on L^{2}(ℝ^{n}, γ_{n}(x)dx) whose generator −N is given by the Dirichlet form

E(f, g) = ∫_{ℝ^{n}} ∇f^{∗}(x) · ∇g(x)γ_{n}(x)dx  (4.6)

through ⟨f, N g⟩_{L^{2}(γ_{n})} = E(f, g), where f^{∗} is the complex conjugate of f. Integrating by parts, one finds

N = −(Δ − x · ∇) ,

The eigenvalues of N are the non-negative integers, and the eigenfunctions are the Hermite polynomials. (In certain physical contexts, the eigenvalues count occupancy of a quantum state and N is called the Boson number operator.) There is a simple explicit formula for the e^{−tN}:

e^{−tN} f(x) = ∫_{ℝ^{n}} f (e^{−t}x + \sqrt{1 − e^{2t}}y) γ_{n}(y)dy ,  (4.7)

which is easily checked. Since evidently N 1 = 0, and e^{−tN} is self-adjoint, it also preserves integrals against γ_{n}(x)dx, and hence, if ρ is any probability density, so is each ρ_{t} := e^{−tN}. As one sees from (4.7),

lim_{t→∞} e^{−tN} ρ(x) = 1 ,  (4.8)

the uniform probability density on (ℝ^{n}, γ_{n}(x)dx), and thus the Mehler semigroup provides us with an interpolation between any probability density ρ and the uniform density 1. This interpolation is well-behaved with respect to the operation of taking marginals: Consider any probability density ρ on (ℝ^{n}, γ_{n}(x)dx), and any m dimensional subspace V of ℝ^{n}. Let ρ_{V} be the marginal density of ρ as in Theorem 4.2. Then of course, we may regard ρ_{V} as a probability density on (ℝ^{n}, γ_{n}(x)dx) that is constant along directions in V^{⊥}. (Simply compose ρ_{V} with P_{V}.) Interpreted this way, so that both ρ and ρ_{V} are functions on ℝ^{N},

(e^{−tN} ρ)_{V} = e^{−tN} (ρ_{V}) .  (4.9)

That is, the process of taking marginals commutes with the action of the Mehler semigroup. The next point to note is that the entropy is monotone increasing along this interpolation: Differentiating, with ρ_{t} = e^{−tN} ρ,

\frac{d}{dt}S(ρ_{t}) = − ∫_{ℝ^{n}} ln(ρ_{t})(Δ−x·∇)ρ_{t}γ_{n}dx = ∫_{ℝ^{n}} ∇ ln ρ_{t}·∇ρ_{t}γ_{n}dx = E(ln ρ_{t}, ρ_{t}) .

For any smooth density ρ, E(ln ρ, ρ) = ∫_{ℝ^{n}} ∇ ln ρ·∇ργ_{n}dx = ∫_{ℝ^{n}} |∇ ln ρ|^{2}ργ_{n}dx, and hence S(ρ_{t}) is strictly increasing for all t. Moreover, since (x, t) ↦ |x|^{2}/t is jointly convex on ℝ^{n} × ℝ_{+}, ρ ↦ E(ln ρ, ρ) has a unique extension as a convex functional on the set of all probability densities on (ℝ^{n}, γ_{n}(x)dx).

4.4 DEFINITION (Entropy Production). The entropy production functional is the convex functional D(ρ) on probability densities on (ℝ^{n}, γ_{n}(x)dx) given by

D(ρ) = ∫_{ℝ^{n}} ln ρ(x)N ρ(x)γ_{n}(x)dx = E(ln ρ, ρ) .  (4.10)

With this definition,

\frac{d}{dt}S(e^{−tN} ρ) = D(e^{−tN} ρ) .

Now because of (4.9), for any subspace V of ℝ^{n},

\frac{d}{dt}S([e^{−tN} ρ]_{V}) = D([e^{−tN} ρ]_{V}) .

Since [e^{−tN} ρ]_{V} is constant along directions orthogonal to V, the derivatives in those directions that figure in D([e^{−tN} ρ]_{V}) are irrelevant; we need only take derivatives along directions in V. This consideration leads to the definitions of the restricted number operator, and the restricted entropy production: Given an m dimensional subspace V of ℝ^{n}, let P_{V} be the orthogonal projection onto V. The restricted number operator N_{V} is the self-adjoint operator on L^{2}(ℝ^{n}, γ_{n}(x)dx) defined through

⟨f, N_{V} g⟩_{L^{2}(γ_{n})} = ∫_{ℝ^{n}} ∇f^{∗}(x) · P_{V} ∇g(x)γ_{n}(x)dx ,  (4.11)

and the restricted entropy production functional D_{V} (ρ) is the convex functional given by

D_{V} (ρ) = ∫_{ℝ^{n}} (N_{V} ln ρ(x)) ρ(x)γ_{n}(x)dx .  (4.12)

With this definition, D(ρ_{V}) = D_{V} (ρ_{V}). However, there is a crucial difference between D_{V} (ρ) and D(ρ_{V}):

4.5 LEMMA. For any smooth probability density ρ on (ℝ^{n}, γ_{n}(x)dx), and any m dimensional subspace V of ℝ^{n}, let ρ_{V} be the corresponding marginal density regarded as a probability density on (ℝ^{n}, γ_{n}(x)dx). Then

D(ρ_{V}) ≤ D_{V} (ρ) .  (4.13)

Proof: Regard ρ_{V} as a function on ℝ^{n} (by composing it with P_{V}). Assume that ρ is smooth and bounded above and below by strictly positive numbers. Notice that since ρ_{V} is constant along directions in V^{⊥},

N ln ρ_{V} = N_{V} ln ρ_{V} .

Then, integrating by parts, and using the definition of ρ_{V} and the Schwarz inequality, we obtain:

D(ρ_{V}) = ∫_{ℝ^{n}} [N_{V} ln ρ_{V} (x)] ρ_{V} (x)γ_{n}(x)dx = ∫_{ℝ^{n}} [N_{V} ln ρ_{V} (x)] ρ(x)γ_{n}(x)dx ,

where we have used the definition of ρ_{V} to replace the second ρ_{V} by ρ itself. Then, by the definition of N_{V}, and the Schwarz inequality,

D(ρ_{V}) = ∫_{ℝ^{n}} (∇ ln ρ_{V} (x)) · P_{V} ∇ρ(x)γ_{n}dx
= ∫_{ℝ^{n}} (∇ ln ρ_{V} (x)) · P_{V} (∇ ln ρ(x)) ρ(x)γ_{n}(x)dx

≤ (∫_{ℝ^{n}} |∇ ln ρ_{V} (x)|^{2} ρ(x)γ_{n}(x)dx)^{1/2}(∫_{ℝ^{n}} |P_{V} ∇ ln ρ(x)|^{2} ργ_{n}dx)^{1/2}

(4.14)

In the first factor in the last line, we may replace ρ by ρ_{V} since |∇ ln ρ_{V} (x)|^{2} depends on x only through P_{V} x. Hence this factor is simply \sqrt{D(ρ_{V})}, and the second factor is \sqrt{D_{V} (ρ)}. The proof we have just given is patterned on the proof of an analogous result in the Lebesgue measure case in [10], which in turn is based on similar arguments in [9] and [3]. It is somewhat more complicated to adapt the argument to the Clifford algebra setting, but this is what we shall do in the next section. We are now ready to prove the sufficiency of condition (4.5):

4.6 LEMMA. The condition (4.5) in Theorem 4.2 is sufficient.

Proof: For a probability density ρ on (ℝ^{n}, dγ_{n}) with S(ρ) > −∞, it is easy to see that

lim_{t→∞} S(e^{−tN} ρ) = S(1) = 0

and hence, lim_{t→∞} S(e^{−tN} (ρ_{V_{j}})) = 0 for each j = 1, …, N. Therefore, it suffices to show that

a(t) := [∑_{j=1}^{N} \frac{1}{p_{j}} S(e^{−tN} ρ_{V_{j}}) − S(e^{−tN} ρ)]

is monotone decreasing in t. Differentiating, and using (4.9), and then Lemma 4.5,

\frac{d}{dt}a(t) = [∑_{j=1}^{N} \frac{1}{p_{j}} D((e^{−tN} ρ)_{V_{j}}) − D(e^{−tN} ρ)]

≤ [∑_{j=1}^{N} \frac{1}{p_{j}} D_{V_{j}}(e^{−tN} ρ) − D(e^{−tN} ρ)]

(4.15)

Now note that by (4.12), whenever (4.5) is satisfied,

∑_{j=1}^{N} \frac{1}{p_{j}} D_{V_{j}}(σ) ≤ D(σ)

for any smooth density σ. Hence the derivative of α(t) is negative for all t > 0.

5 Generalized subadditivity of entropy in Clifford algebras

In this section we shall prove

5.1 THEOREM. Let V_{1}, …, V_{N} be N subspaces of ℝ^{n}, and let 𝔄_{j} be the Clifford algebra over V_{j} with the inner product that V_{j} inherits from ℝ^{n}, and let 𝔄_{j} be equipped with its unique tracial state τ_{j}. For any probability density ρ ∈ 𝔄, let ρ_{V_{j}} be the induced probability density in 𝔄_{j}. Let S(ρ) = −τ(ρ ln ρ) and S(ρ_{V_{j}}) = −τ_{j}(ρ_{V_{j}} ln ρ_{V_{j}}) Then

∑_{j=1}^{N} \frac{1}{p_{j}} S(ρ_{V_{j}}) ≥ S(ρ)  (5.1)

for all probability densities ρ ∈ 𝔄 if and only if

∑_{j=1}^{N} \frac{1}{p_{j}} P_{j} ≤ I_{ℝ^{n}} .  (5.2)

where P_{j} is the orthogonal projection onto V_{j} in ℝ^{n}.

Granted this result, we have: Proof of Theorem 1.6: Theorem 2.2 and Theorem 5.1 together prove Theorem 1.6. We shall now prove Theorem 5.1. As before, we begin by proving the necessity of (5.2).

5.2 LEMMA. The condition (5.2) in Theorem 5.1 is necessary.

Proof: For any vector a = (a_{1}, …, a_{n}) ∈ ℝ^{n}, define

ρ_{a} = I + ∑_{j=1}^{n} a_{j}Q_{j} = I + a · Q .

Then ρ_{a} is a probability density if and only if |a| ≤ 1. Indeed, ρ_{a} has only two eigenvalues, 1 ± |a|, with equal multiplicity. Then (ρ_{a})_{V_{j}} = I + (P_{j}a) · Q, and so (ρ_{a})_{V_{j}} has only two eigenvalues, 1 ± |P_{j}a|, with equal multiplicity. Therefore,

S(ρ_{a}) = −ψ(|a|)  and  S((ρ_{a})_{V_{j}}) = −ψ(|P_{j}a|) .  (5.3)

where ψ(x) is the convex function defined by

ψ(x) = \begin{cases}\frac{1}{2} [(1 + x) ln(1 + x) + (1 − x) ln(1 − x)] & if |x| ≤ 1. \\ ∞ & otherwise, \end{cases}  (5.4)

Thus, for (5.1) to hold for each ρ_{a}, |a| ≤ 1, it must be the case that

∑_{j=1}^{N} \frac{1}{p_{j}} ψ(|P_{j}a|) ≤ ψ(|a|)  for all a with |a| ≤ 1 .  (5.5)

Then, since ψ(x) = x^{2} + O(x^{4}), replacing a by ta, 0 < t < 1, we see that (5.2) must hold. Because of (5.3), once we have proved Theorem 5.1, we will have a proof of (5.5). However, it is of interest to prove this inequality directly, and we do that next.

5.3 PROPOSITION. The inequality (5.5) holds whenever (5.2) is satisfied.

Proof: An easy calculation of derivatives shows that

ψ′(x) = arctanh(x)  and  ψ′′(x) = \frac{1}{1 − x^{2}}

for |x| < 1. Now fix any a with |a| < 1. Then, for t > 0, define

η(t) = ψ(e^{−t}|a|) − ∑_{j=1}^{N} \frac{1}{p_{j}} ψ(e^{−t}|P_{j}a|) .

We have to show that η(t) > 0 for all t > 0. Since evidently lim_{t→∞} η(t) = 0, it suffices to show that η′(t) < 0 for all t > 0. Differentiating, we find

η′(t) = −e^{−t}[|a| arctanh(e^{−t}|a|) − ∑_{j=1}^{N} \frac{1}{p_{j}} |P_{j}a| arctanh(e^{−t}|P_{j}a|)] := e^{−t}θ(t).

Hence, it suffices to show that θ(t) ≥ 0 for all t > 0. Once again, since lim_{t→∞} θ(t) = 0, it suffices to show that θ′(t) < 0 for all t > 0. Differentiating, we find

θ′(t) = −e^{−t} [\frac{|a|^{2}}{1 − e^{−2t}|a|^{2}} − ∑_{j=1}^{N} \frac{1}{p_{j}} \frac{|P_{j}a|^{2}}{1 − e^{−2t}|P_{j}a|^{2}} ] .

Multiplying through by e^{−t}, and absorbing a factor of e^{−t} into a, it suffices to show that

\frac{|a|^{2}}{1 − |a|^{2}} ≥ ∑_{j=1}^{N} \frac{1}{p_{j}} \frac{|P_{j}a|^{2}}{1 − |P_{j}a|^{2}}  (5.6)

for all |a| ≤ 1. However, since |a| ≥ |P_{j}a|,

\frac{|P_{j}a|^{2}}{1 − |a|^{2}} ≥ \frac{|P_{j}a|^{2}}{1 − |P_{j}a|^{2}} ,

Documenta Mathematica 13 (2008) 553–584

and thus (5.6) follows from (5.2). We are now in a position to give an elementary proof of Theorem 1.6 in the special case that each V_{j} is one dimensional. As explained in Example 1.5, it suffices in this case to prove the following:

5.4 PROPOSITION. Suppose {u_{1}, …, u_{N}} is any set of N unit vectors in ℝ^{n}, and {p_{1}, …, p_{N}} is any set of N positive numbers such that

∑_{j=1}^{N} c_{j}u_{j} ⊗ u_{j} = I_{ℝ^{n}} .  (5.7)

Then for any b = (b_{1}, …, b_{N}) inℝ^{N},

ln cosh (|∑_{j=1}^{N} b_{j}u_{j}|) ≤ ∑_{j=1}^{N} \frac{1}{p_{j}} ln cosh(p_{j}b_{j}) .  (5.8)

Proof: Let ψ^{∗} denote the function ψ^{∗}(x) = ln cosh(x), x ∈ ℝ. The notation is meant to indicate the well known fact, easily checked, that ψ^{∗} is the Legendre transform of the function ψ defined in (5.4). Now, given a set of N orthogonal projections {P_{1}, …, P_{N}} satisfying (5.2), we may make any choice of a unit vector u_{j} from the range of P_{j}, and then the N unit vectors {u_{1}, …, u_{N}} will satisfy (5.7). Conversely, given any set of N unit vectors {u_{1}, …, u_{N}} that satisfy (5.7), we may take P_{j} = u_{j} ⊗ u_{j}, and then (5.2) is satisfied. Hence, we suppose we are given a set of N orthogonal projections {P_{1}, …, P_{N}} satisfying (5.2), and for each j, u_{j} is a unit vector in the range of P_{j}. Then for any b ∈ ℝ^{n},

ψ^{∗} (|∑_{j=1}^{N} b_{j}u_{j}|) = sup_{a∈ℝ^{n}} {a · ∑_{j=1}^{N} b_{j}u_{j} − ψ(|a|)}

= sup_{|a|≤1} {∑_{j=1}^{N} P_{j}a · b_{j}u_{j} − ψ(|a|)}

≤ sup_{|a|≤1} \{∑_{j=1}^{N} P_{j}a · b_{j}u_{j} − ∑_{j=1}^{N} \frac{1}{p_{j}} ψ(|P_{j}a|)\}

≤ sup_{|a|≤1} \{∑_{j=1}^{N} |P_{j}a||b_{j}| − ∑_{j=1}^{N} \frac{1}{p_{j}} ψ(|P_{j}a|)\}

= sup_{|a|≤1} \{∑_{j=1}^{N} \frac{1}{p_{j}} [|P_{j}a|p_{j}|b_{j}| − ψ(|P_{j}a|)]\} ,

(5.9) where the first inequality is from (5.5), and the second is from Schwarz. Then, by the definition of the Legendre transform, for any a,

ψ^{∗}(p_{j}b_{j}) ≥ |P_{j}a|(p_{j}|b_{j}|) − ψ(|P_{j}a|) ,

we obtain

ψ^{∗} (|∑_{j=1}^{N} b_{j}u_{j}|) ≤ ∑_{j=1}^{N} \frac{1}{p_{j}} ψ^{∗}(p_{j}b_{j}) ,

which is (5.8). We now prove Theorem 5.1 in full generality. This gives another proof of the last two propositions, but by less elementary means. The proof will follow the basic pattern of the proof of Theorem 4.2 and use the Clifford algebra analog of the Mehler semigroup. This is the so-called Clifford–Mehler semigroup, about which we now recall a few relevant facts.

5.1 About the Clifford–Mehler semigroup

There is also a differential calculus in the Clifford algebra. Let Q_{1}, …, Q_{n} be any set of n generators for the Clifford algebra ℭ over ℝ^{n}. For A ∈ ℭ, define

∇_{i}(A) = \frac{1}{2} [Q_{i}A − Γ(A)Q_{i}] ,

where Γ is the grading operator on ℭ: That is, using the notation in (1.10),

Γ(Q^{α}) = (−1)^{|α|}Q^{α} .

One computes that ∇_{i}(Q^{α}) = 0 if α(i) = 0, and otherwise, ∇_{i}(Q^{α}) is what one gets by anti-commuting the factor of Q_{i} through to the left, and then removing it. In this sense it is like a differentiation operator, and what is more, it is a skew derivation on ℭ, which means that for all and A and B in ℭ, ∇_{j}(AB) = ∇_{j}(A)B + Γ(A)∇_{j}(B). The Clifford algebra analog of the Gaussian energy integral (4.6) is given by

E(A, B) = τ (∑_{j=1}^{n} ∇_{j}A^{∗}∇_{j}B) ,  (5.10)

for all A, B ∈ ℭ. This is the Clifford Dirichlet form studied by Gross [18]. Then, the fermionic number operator, also denoted N, is defined by

E(A, B) = τ(A^{∗}N (B)) .

It is easy to see that the spectrum of N consists of the non negative integers {0, 1, …, n} and that

N Q^{α} = |α|Q^{α} .  (5.11)

The Clifford Mehler semigroup is then given by e^{−tN}. It is clear from this definition, (1.11) and (5.11) that for any A ∈ ℭ, lim_{t→∞} e^{−tN} (A) = τ(A)I. Thus for any probability density ρ in ℭ,

t ↦ ρ_{t} = e^{−tN} (ρ)

provides an interpolation between ρ and I, and each ρ_{t} is a probability density. This corresponds exactly to the Mehler semigroup interpolation that was used to prove Theorem 4.2, and we shall use it here in the same way, though some additional complications shall arise. The operator N does not depend on the choice of the set of generators Q_{1}, …, Q_{n}. Indeed, if {u_{1}, …, u_{n}} is any orthonormal basis of ℝ^{n}, and we define Q̃_{j} = u_{j} · Q for j = 1, …, n, then the Clifford Dirichlet form that one obtains using this basis to define the derivatives is the same as the original. In particular, given an m dimensional subspace V of ℝ^{n}, we may choose {u_{1}, …, u_{n}} so that {u_{1}, …, u_{m}} is an orthonormal basis for V, and then the first m generators will be a set of generators for ℭ_{V}. We then define the reduced Clifford Dirichlet form E_{V} by

E_{V} (A, B) = τ ( ∑_{i,j=1}^{m} ∇_{i}A^{∗}[P_{V}]_{i,j}∇_{j}B) ,  (5.12)

where [P_{V}]_{i,j} is the i, jth entry of the n × n matrix for P_{V}. The restricted number operator N_{V} is then the self-adjoint operator on L^{2}(ℭ) given by τ(A^{∗}N_{V} (B)) = E_{V} (A, B). Now, for any probability density ρ in ℭ let ρ_{V} be the corresponding marginal density regarded as an operator in ℭ by identifying it with ϕ_{V} (ρ_{V}), where ϕ_{V} is the canonical embedding of ℭ(V) into ℭ(ℝ^{n}). Then it is an easy consequence of the definitions that

(e^{−tN} ρ)_{V} = e^{−tN} (ρ_{V}) = e^{−tN_{V}} (ρ_{V}) .  (5.13)

Also, under the condition (5.2), it is easy to see that

∑_{j=1}^{N} \frac{1}{p_{j}} N_{V_{j}} ≤ N .  (5.14)

Finally, we introduce entropy production D(ρ): With ρ_{t} := e^{−tN} ρ, we differentiate and find

\frac{d}{dt}S (ρ_{t}) = τ (ln(ρ_{t})N (ρ_{t})) = E(ln(ρ_{t}), ρ_{t}) .

5.5 DEFINITION (Entropy Production). The entropy production functional at a probability density ρ is the functional defined by

D(ρ) = τ (ln(ρ)N (ρ)) = E(ln(ρ), ρ) .  (5.15)

Given an m dimensional subspace V of ℝ^{n}, the restricted entropy production functional at a probability density ρ is the functional defined by

D_{V} (ρ) = τ (ln(ρ)N_{V} (ρ)) = E_{V} (ln(ρ), ρ) .  (5.16)

The following lemma is the basis of our proof of the sufficiency of (5.2). In the course of proving it, we shall see that both D(ρ) and D_{V} (ρ) are convex functionals, which is somewhat less obvious than in the Gaussian case.

5.6 LEMMA. For any any probability density ρ in ℭ(ℝ^{n}), and any m dimensional subspace V of ℝ^{n}, let ρ_{V} be the corresponding marginal probability density regarded as a probability density in ℭ(ℝ^{n}). Then

D(ρ_{V}) ≤ D_{V} (ρ) .

Proof: We choose an orthonormal basis {u_{1}, …, u_{n}} for ℝ^{n} such that {u_{1}, …, u_{m}} is an orthonormal basis for V. Without loss of generality, we may suppose that {u_{1}, …, u_{n}} is the standard basis so that {Q_{1}, …, Q_{m}} is a set of generators for ℭ(V). Then,

E(A, B) = τ (∑_{j=1}^{n} ∇_{j}A^{∗}∇_{j}B)  and  E_{V} (A, B) = τ (∑_{j=1}^{m} ∇_{j}A^{∗}∇_{j}B) .

(5.17)

It will be convenient to define N_{j} = ∇_{j}^{∗}∇_{j} j = 1, …, n. Then we have

N = ∑_{j=1}^{n} N_{j}  and  N_{V} = ∑_{j=1}^{m} N_{j} ,  (5.18)

and so

D_{V} (ρ) = ∑_{j=1}^{m} τ (ln ρ, N_{j}ρ) .  (5.19)

Since N_{j}Q^{α} = \begin{cases}Q^{α} & if α(j) = 1, \\ 0 & α(j) = 0 \end{cases}, each N_{j} is an orthogonal projection, and so

(5.19) can be rewritten as

D_{V} (ρ) = ∑_{j=1}^{m} τ (N_{j}(ln ρ), N_{j}ρ) .  (5.20)

To proceed, we use a formula of Gross [18] for N_{j}f(A) where A ∈ ℭ(ℝ^{n}), and f is a continuous function. To write down Gross’s formula, first write A = B + Q_{j}C where both B and C are linear combinations of the Q^{α} with α(j) = 0. Then define Â = B − Q_{j}C. Notice that if ρ is a probability density, then ρ̂ is again a probability density. Gross’s formula is

N_{j}f(A) = \frac{1}{2} [f(A) − f(Â )] .

To prove this formula, notice that there is a unitary operator U such that Â = UAU^{∗}. (If the dimension n is odd, one can take U to be the product, in some order, of all of the Q_{k} for k ≠ j; if the dimension is even, one can add another generator.) Therefore,

{ \atop f(A)} = Uf(A)U^{∗} = f(UAU^{∗}) = f(Â ) .

Using this together with the fact that for any A ∈ 𝔄, N_{j}A = (1/2)[A − Â], we obtain Gross’s formula, which we now apply as follows:

τ (N_{j}(ln ρ)N_{j}ρ) = \frac{1}{4}τ ([ln(ρ) − ln(ρ̂)] [ρ − ρ̂])
= \frac{1}{4}τ (ln(ρ) [ρ − ρ̂]) + \frac{1}{4}τ (ln(ρ̂) [ρ̂− ρ])
= \frac{1}{4}H[ρ|ρ̂] + \frac{1}{4}H[ρ̂ |ρ]

(5.21)

where H[ρ|σ] = τρ( lnρ − ln σ) is the relative entropy of ρ with respect to σ. As is well known, (ρ, σ) ↦ H(ρ|σ) is jointly convex, and hence

ρ ↦ τ ((ln ρ)N_{j}ρ)

is convex. Furthermore, by the fundamental monotonicity property of the relative entropy under conditional expectations [37],

H(ρ_{V} |σ_{V}) ≤ H(ρ|σ)

for any two probability densities ρ and σ. It follows that τ ((ln ρ_{V})N_{j}ρ_{V}) ≤ τ ((ln ρ)N_{j}ρ). Summing on j from 1 to m, we find

D(ρ_{V}) = D_{V} (ρ_{V}) = ∑_{j=1}^{m} τ ((ln ρ_{V})N_{j}ρ_{V}) ≤ ∑_{j=1}^{m} τ ((ln ρ)N_{j}ρ) = D_{V} (ρ) .

5.2 Proof of the sufficiency

5.7 LEMMA. The condition (4.5) in Theorem 4.2 is sufficient.

Proof: For a probability density ρ in ℭ(ℝ^{n}) it is easy to see that

lim_{t→∞} S(e^{−tN} ρ) = S(1) = 0

and hence, lim_{t→∞} S(e^{−tN} (ρ_{V_{j}})) = 0 for each j = 1, …, N. Therefore, it suffices to show that

a(t) := [∑_{j=1}^{N} \frac{1}{p_{j}} S(e^{−tN} ρ_{V_{j}}) − S(e^{−tN} ρ)]

is monotone decreasing in t. Differentiating, and using (5.13), and then Lemma 5.6,

\frac{d}{dt}a(t) = [∑_{j=1}^{N} \frac{1}{p_{j}} D((e^{−tN} ρ)_{V_{j}}) − D(e^{−tN} ρ)]

≤ [∑_{j=1}^{N} \frac{1}{p_{j}} D_{V_{j}}(e^{−tN} ρ) − D(e^{−tN} ρ)]

(5.22)

Now note that by (4.12), whenever (4.5) is satisfied, ∑_{j=1}^{N} \frac{1}{p_{j}} D_{V_{j}}(σ) ≤ D(σ) for

any smooth density σ. Hence the derivative of α(t) is negative for all t > 0. Notice that the proof is almost identical, symbol for symbol, with that of the corresponding proof in the Gaussian case. The main difference of course is that the proof of the main lemma, Lemma 5.6, is considerably more intricate than that of its Gaussian counterpart. Proof of Theorem 5.1: This now follows immediately from Lemma 5.2 and 5.7.

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Eric A. Carlen Department of Mathematics Hill Center Rutgers University 110 Frelinghuysen Road Piscataway, NJ 08854-8019 USA Elliott H. Lieb Departments of Mathematics and Physics Jadwin Hall Princeton University P. O. Box 708 Princeton, NJ 08544 USA