Chapter Four

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Matrix Means

Let a and b be positive numbers. Their arithmetic, geometric, and harmonic means are the familiar objects

These have several properties that any object that is called a mean M(a, b) should have. Some of these properties are

(i)  M(a, b) > 0,

(ii)  If a ≤ b, then a ≤ M(a, b) ≤ b,

(iii)  M(a, b) = M(b, a) (symmetry),

(iv)  M(a, b) is monotone increasing in a, b,

(v)  M(αa, αb) = αM(a, b) for all positive numbers a, b, and α,

(vi)  M(a, b) is continuous in a, b.

The three of the most familiar means listed at the beginning satisfy these conditions. We have the inequality

Among other means of a, b is the logarithmic mean defined as

This has the properties (i)–(vi) listed above. Further

This is a refinement of the arithmetic-geometric mean inequality—the second part of (4.1). See Exercise 4.5.5 and Lemma 5.4.5.

Averaging operations are of interest in the context of matrices as well, and various notions of means of positive definite matrices A and B have been studied. A mean M(A, B) should have properties akin to (i)–(vi) above. The order “≤” now is the natural order X ≤ Y on Hermitian matrices. It is obvious what the analogues of properties (i)–(vi) are for the case of positive definite matrices. Property (v) has another interpretation: for positive numbers a, b and any nonzero complex number x

It is thus natural to expect any mean M(A, B) to satisfy the condition

for all A, B > O and all nonsingular X. This condition is called congruence invariance and if the equality (v^′) is true, we say that M is invariant under congruence. Restricting X to scalar matrices we see that

for all positive numbers α.

So we say that a matrix mean is a binary operation M(A, B) on the set of positive definite matrices that satisfies (the matrix versions of ) the conditions (i)–(vi), the condition (v) being replaced by (v^′).

What are good examples of such means? The arithmetic mean presents no difficulties. It is obvious that has all the six properties listed above. The harmonic mean of A and B should be the matrix . Now some of the properties (i)–(vi) are obvious, others are not. It is not clear what object should be called the geometric mean in this case. The product A^1/2B^1/2 is not Hermitian, let alone positive, unless A and B commute.

In this chapter we define a geometric mean of positive matrices and study its properties along with those of the arithmetic and the harmonic mean. We use these ideas to prove some theorems on operator monotonicity and convexity. These theorems are then used to derive important properties of the quantum entropy. A positive matrix in this chapter is assumed to be strictly positive. Extensions of some of the considerations to positive semidefinite matrices are briefly indicated.

4.1  THE HARMONIC MEAN AND THE GEOMETRIC MEAN

The parallel sum of two positive matrices A, B is defined as the matrix

This definition could be extended to positive semidefinite matrices A, B by a limit from above:

This operation was studied by Anderson and Duffin in connection with electric networks. (If two wires with resistances r₁ and r₂ are connected in parallel, then their total resistance r according to one of Kirchhoff ’s laws is given by .)

The harmonic mean of A, B is the matrix 2(A : B). To save on symbols we will not introduce a separate notation for it. Note that

By symmetry

Thus A : B is the Schur complement of A + B in either of the block matrices

Several properties of A : B can be derived from this.

4.1.1  Theorem

For any two positive matrices A, B we have

(i)   A : B ≤ A, A : B ≤ B.

(ii)  A : B is monotonically increasing and jointly concave in the arguments A, B.

(iii) 

Proof.  

(i)   The subtracted terms in (4.6) and (4.7) are positive.

(ii)  See Corollary 1.5.3.

(iii)  See Corollary 1.5.5.     ■

4.1.2  Proposition

If A ≤ B, then A ≤ 2(A : B) ≤ B.

Proof.  

A similar argument shows 2(A : B) ≤ B.     ■

Thus the harmonic mean satisfies properties (i)–(v) listed at the beginning of the chapter. (Notice one difference: for positive numbers a, b either a ≤ b or b ≤ a; this is not true for positive matrices A, B.)

How about the geometric mean of A, B? If A, B commute, then their geometric mean can be defined as A^1/2B^1/2. But this is the trivial case. In all other cases this matrix is not even Hermitian. The matrix is Hermitian but not always positive. Positivity is restored if we consider

It turns out that this is not monotone in A, B. (Exercise: construct a 2 × 2 example to show this.) One might try other candidates; e.g., e^{(log A+log B)/2}, that reduce to a^1/2b^1/2 for positive numbers. This particular one is not monotone.

Here the property (v^′)—congruence invariance—that we expect a mean to have is helpful. We noted in Exercise 1.6.1 that any two positive matrices are simultaneously congruent to diagonal matrices. The geometric mean of two positive diagonal matrices A and B, naturally, is A^1/2B^1/2.

Let us introduce a notation and state a few elementary facts that will be helpful in the ensuing discussion. Let GL(n) be the group consisting of n × n invertible matrices. Each element X of GL(n) gives a congruence transformation on . We write this as

The collection {Γ_X : X ∈ GL(n)} is a group of transformations on M_n. We have Γ_XΓ_Y = Γ_{Y X} and . This group preserves the set of positive matrices. Given a pair of matrices A, B we write Γ_X(A, B) for (Γ_X(A), Γ_X (B)) .

Let A, B be positive matrices. Then

We can find a unitary matrix U such that , a diagonal matrix. So

The geometric mean of the matrices I and D is

So, if the geometric mean of two positive matrices A and B is required to satisfy the property (v^′), then it has to be the matrix

If A and B commute, then A#B = A^1/2B^1/2. The expression (4.10) does not appear to be symmetric in A and B. However, it is. This is seen readily from another description of A#B. By Exercise 1.2.13 the matrix in (4.10) is the unique positive solution of the equation

If we take inverses of both sides, then this equation is transformed to XB⁻¹X = A. This shows that

Using Theorem 1.3.3 and the relation

that we have just proved, we see that

On the other hand if X is any Hermitian matrix such that

then again by Theorem 1.3.3, we have A ≥ XB⁻¹X. Hence

Taking square roots and then applying the congruence Γ_B1/2 , we get from this

In other words A#B ≥ X for any Hermitian matrix X that satisfies the inequality (4.14).

The following theorem is a summary of our discussion so far.

4.1.3  Theorem

Let A and B be two positive matrices. Let

Then

(i)   A#B = B#A,

(ii)  A#B is the unique positive solution of the equation XA⁻¹X = B,

(iii)  A#B has an extremal property:

The properties (i)–(vi) listed at the beginning of the chapter can be verified for A#B using one of the three characterizations given in Theorem 4.1.3. Thus, for example, the symmetry (4.12) is apparent from (4.15) as well. Monotonicity in the variable B is apparent from (4.10) and Proposition 1.2.9; and then by symmetry we have monotonicity in A. This is plain from (4.15) too. From (4.15) we see that A#B is jointly concave in A and B.

Since congruence operations preserve order, the inequality (4.1) is readily carried over to operators. We have

It is easy to see either from (4.10) or from Theorem 4.1.3 (ii) that

4.1.4  Exercise

Use the characterization (4.15) and the symmetry (4.12) to give another proof of the second inequality in (4.16). Use (4.15) and (4.17) to give another proof of the first inequality in (4.16).

If A or B is not strictly positive, we can define their geometric mean by a limiting procedure, as we did in (4.5) for the parallel sum.

The next theorem describes the effect of positive linear maps on these means.

4.1.5  Theorem

Let Φ be any positive linear map on . Then for all positive matrices A, B

(i)  Φ(A : B) ≤ Φ(A) : Φ(B);

(ii)  Φ(A#B) ≤ Φ(A)#Φ(B).

Proof.   (i) By the extremal characterization (4.8)

By Exercise 3.2.2 (ii), we get from this

Again, by (4.8) this means Φ(A : B) ≤ Φ(A) : Φ(B).

The proof of (ii) is similar to this. Use the extremal characterization (4.15) for A#B, and Exercise 3.2.2 (ii).     ■

For the special map Γ_X(A) = X^∗AX, where X is any invertible matrix the two sides of (i) and (ii) in Theorem 4.1.5 are equal.This need not be the case if X is not invertible.

4.1.6  Exercise

Let A, B, and X be the 2 × 2 matrices

Show that

So, if Φ(A) = X^∗AX, then in this example we have

The inequality (4.13) and Proposition 1.3.2 imply that there exists a contraction K such that A#B = A^1/2KB^1/2. More is true as the next Exercise and Proposition show.

4.1.7  Exercise

Let U = (A^−1/2BA^−1/2)^1/2A^1/2B^−1/2. Show that U^∗U = UU^∗ = I. Thus we can write

where U is unitary.

It is an interesting fact that this property characterizes the geometric mean:

4.1.8  Proposition

Let A, B be positive matrices and suppose U is a unitary matrix such that A^1/2UB^1/2 is positive. Then A^1/2UB^1/2 = A#B.

Proof.   Let G = A^1/2UB^1/2. Then

We have another congruence

(See the proof of Theorem 1.3.3.) Note that the matrix has rank n. Since congruence preserves rank we must have A = GB⁻¹G. But then, by Theorem 4.1.3 (ii), G = A#B.     ■

Two more ways of expressing the geometric mean are given in the following propositions. We use here the fact that if X is a matrix with positive eigenvalues, then it has a unique square root Y with positive eigenvalues. A proof is given in Exercise 4.5.2.

4.1.9  Proposition

Let A, B be positive matrices and let be the the square root of A⁻¹B that has positive eigenvalues. Then

Proof.   We have the identity

Taking square roots, we get

This, in turn, shows that

4.1.10  Exercise

Show that for positive matrices A, B we have

4.1.11  Proposition

Let A, B be positive matrices. Then

(The matrix inside the square brackets has positive eigenvalues and the square root chosen is the one with positive eigenvalues.)

Proof.   Use the identity

to get

Taking square roots, we get

This gives

Using Proposition 4.1.9 and Exercise 4.1.10, we get from this

Premultiply both sides by (A + B)⁻,¹ and then take square roots, to get

This proves the proposition.     ■

On first sight, the three expressions in 4.1.9–4.1.11 do not seem to be positive matrices, nor do they seem to be symmetric in A, B.

The expression (4.10) and the ones given in Propositions 4.1.9 and 4.1.11 involve finding square roots of matrices, as should be expected in any definition of geometric mean. Calculating these square roots is not an easy task. For 2 × 2 matrices we have a formula that makes computation easier.

4.1.12  Proposition

Let A and B be 2 × 2 positive matrices each of which has determinant one. Then

Proof.   Use the formula given for A#B in Proposition 4.1.9. Let X = (A⁻¹B)^1/2. Then det X = 1 and so X has two positive eigenvalues λ and 1/λ. Further,

and hence tr . So, by the Cayley-Hamilton theorem

Multiply on the left by A and rearrange terms to get

Exercise. Let A and B be 2×2 positive matrices and let det A = α², det B = β². Then

4.2  SOME MONOTONICITY AND CONVEXITY THEOREMS

In Section 5 of Chapter 1 and Section 7 of Chapter 2 we have discussed the notions of monotonicity, convexity and concavity of operator functions. Operator means give additional information as well as more insight into these notions. Some of the theorems in this section have been proved by different arguments in Chapter 1.

4.2.1  Theorem

If A ≥ B ≥ O, then A^r ≥ B^r for all 0 ≤ r ≤ 1.

Proof.   We know that the assertion is true for r = 0, 1. Suppose r₁, r₂ are two real numbers for which A^r₁ ≥ B^r₁ and A^r₂ ≥ B^r₂. Then, by monotonicity of the geometric mean, we have A^r₁ #A^r₂ ≥ B^r₁ #B^r₂. This is the same as saying A^{(r₁+r₂)/2} ≥ B^{(r₁+r₂)/2}. Thus, the set of real numbers r for which A^r ≥ B^r is a closed convex set. Since 0 and 1 belong to this set, so does the entire interval [0, 1].     ■

4.2.2  Exercise

We know that the function f(t) = t² is not matrix monotone of order 2. Show that the function f(t) = t^r on is not matrix monotone of order 2 for any r > 1. [Hint: Prove this first for r > 2.]

It is known that a function f from into itself is operator monotone if and only if it is operator concave. For the functions f(t) = t^r, 0 ≤ r ≤ 1, operator concavity is easily proved:

4.2.3  Theorem

For 0 < r < 1, the map on positive matrices is concave.

Proof.   Use the representation

(see Theorem 1.5.8). The integrand

is concave in A by Theorem 4.1.1. Hence, so is the integral. (The integrand is also monotone in A. In Section 1.5 we used this argument to prove Theorem 4.2.1 and some other statements.)     ■

4.2.4  Exercise

For 0 < r < 1, the map on positive matrices is monotone decreasing and convex. [Use the facts that is monotone and concave, and is monotone decreasing and convex.] See Exercise 1.5.10 also.

4.2.5  Exercise

The map log A on positive matrices is monotone and concave.

[Hint: .]

4.2.6  Exercise

The map −Alog A on positive matrices is concave.

[Hint: . Use Theorem 1.5.8.]

Some results on convexity of tensor product maps can be deduced easily using the harmonic mean. The following theorem was proved by Lieb. This formulation and proof are due to Ando.

4.2.7  Theorem

For 0 < r < 1, the map is jointly concave and monotone on pairs of positive matrices A, B.

Proof.   Note that . So, by the representation (4.18) we have

The integrand can be written as

This is monotone and jointly concave in A, B. Hence, so is f (A, B).     ■

4.2.8  Exercise

Let r₁, r₂ be positive numbers with r₁ + r₂ ≤ 1. Show that f(A, B) = is jointly concave and monotone on pairs of positive matrices A, B. [Hint: Let r₁ + r₂ = r. Then .]

4.2.9  Exercise

Let r₁, r₂, . . . , r_k be positive numbers with r₁ + · · · + r_k = 1. Then the product is jointly concave on k-tuples of positive matrices A₁, . . . , A_k.

A special case of this says that for k = 1, 2, . . . , the map on positive matrices is concave. This leads to the inequality

By restricting to symmetry classes of tensors one obtains inequalities for other induced operators. For example

For k = n, this reduces to the famous Minkowski determinant inequality

It is clear that many inequalities of this kind are included in the master inequality (4.19).

4.2.10 Exercise

For 0 ≤ r ≤ 1, the map is jointly convex on pairs of positive matrices A, B. [Hint: .]

4.3  SOME INEQUALITIES FOR QUANTUM ENTROPY

Theorem 4.2.7 is equivalent to a theorem of Lieb on the concavity of one of the matrix functions arising in the study of entropy in quantum mechanics. We explain this and related results briefly.

Let (p₁, . . . , p_k) be a probability vector; i.e., p_j ≥ 0 and . The function

called the entropy function, is of fundamental importance in information theory.

In quantum mechanics, the role analogous to that of (p₁, . . . , p_k) is played by a positive matrix A with tr A = 1. Such a matrix is called a density matrix. The entropy of A is defined as

The condition tr A = 1 is not essential for some of our theorems; and we will drop it some times.

It is easy to see that the function (4.22) is jointly concave in p_j. An analogous result is true for the quantum mechanical entropy (4.23). In fact we have proved a much stronger result in Exercise 4.2.6. The proof of concavity of the scalar function (4.23) does not require the machinery of operator concave function.

4.3.1  Exercise

Let A be any Hermitian matrix and f any convex function on . Then for every unit vector x

Use this to prove that S(A) is a concave function of A. [Hint: Choose an orthonormal basis {u_i} consisting of eigenvectors of . Show that for any convex function f

4.3.2  Proposition

Let f be a convex differentiable function on and let A, B be any two Hermitian matrices. Then

Proof.   Let {u_i} be an orthonormal basis whose elements are eigenvectors of B corresponding to eigenvalues β_i. Then

The first inequality in this chain follows from (4.24) and the second from the convexity of f.     ■

Other notions of entropy have been introduced in classical information theory and in quantum mechanics. One of them is called skew entropy or entropy of A relative to K, where A is positive and K Hermitian. This is defined as

More generally we may consider for 0 < t < 1 the function

Here [X, Y ] is the Lie bracket XY − Y X. Note that

The quantity (4.26) was introduced by Wigner and Yanase, (4.27) by Dyson. These are measures of the amount of noncommutativity of A with a fixed Hermitian operator K. The Wigner-Yanase-Dyson conjecture said S_t(A, K) is concave in A for each K. Note that tr (−K²A) is linear in A. So, from the expression (4.28) we see that the conjecture says that for each K, and for 0 < t < 1, the function tr KA^tKA^1−t is concave in A. A more general result was proved by Lieb in 1973.

4.3.3  Theorem (Lieb)

For any matrix X, and for 0 ≤ t ≤ 1, the function

is jointly concave on pairs of positive matrices A, B.

Proof.   We will show that Theorems 4.2.7 and 4.3.3 are equivalent.

The tensor product and the space are isomorphic. The isomorphism φ acts as . If e_i, 1 ≤ i ≤ n is the standard basis for , and E_ij the matrix units in , then

From this one can see that

Thus φ is an isometric isomorphism between the Hilbert space and the Hilbert space (the latter with the inner product ).

Let be the map induced by φ on operators; i.e., makes the diagram

commute. It is easy to see that

So

Thus for positive A, B, the concavity of tr X^∗A^tXB^1−t is equivalent to the concavity of .     ■

Other useful theorems can be derived from Theorem 4.3.3. Here is an example. The concept of relative entropy in classical information theory is defined as follows. Let p, q be two probability distributions; i.e., p = (p₁, . . . , p_k), q = (q₁, . . . , q_k), p_j ≥ 0, q_j ≥ 0, 1. Their relative entropy is defined as

The relative entropy of density matrices A, B is defined as

4.3.4  Exercise (Klein’s Inequality)

For positive matrices A, B

[Hint: Use Proposition 4.3.2. with f(x) = x log x]. Thus, if A, B are density matrices, then

Note

(I is not a density matrix.) We have seen that S(A|I) is a convex function of A.

4.3.5  Theorem

The relative entropy S(A|B) is a jointly convex function of A and B.

Proof.   For A positive and X arbitrary, let

By Lieb’s theorem I_t(A, X) is a concave function of A. Hence, so is I(A, X). Note that

Now, given the positive matrices A, B let T, X be the 2 × 2 block matrices

Then I(T, X) = −S(A|B). Since I(T, X) is concave, S(A|B) is convex.     ■

The next few results describe the behaviour of the entropy function (4.23) with respect to tensor products. Here the condition tr A = 1 will be crucial for some of the statements.

4.3.6  Proposition

Let A, B be any two density matrices. Then

Proof.   Let A, B have eigenvalues λ₁, . . . , λ_n and µ₁, . . . , µ_m, respectively. Then , and . The tensor product has eigenvalues λ_iµ_j, 1 ≤ i ≤ n, 1 ≤ j ≤ m. So

The equation (4.33) says that the entropy function S is additive with respect to tensor products.

Now let be two Hilbert spaces, and let . The operator A is called decomposable if where A₁, A₂ are operators on . Not every operator on is decomposable. We associate with every operator A on two operators A₁, A₂ on called the partial traces of A. These are defined as follows.

Let dim , dim H₂ = m and let e₁, . . . , e_m be an orthonormal basis in . For its partial trace is an operator on defined by the relation

for all .

4.3.7  Exercise

The operator A₁ above is well defined. (It is independent of the basis {e_i} chosen for .)

The partial trace is defined in an analogous way.

It is clear that if A is positive, then so are the partial traces A₁, A₂; and if A is a density matrix, then so are A₁, A₂. Further, if A₂ and A₁, A₂ are density matrices, then .

4.3.8  Exercise

Let A be an operator on with partial traces A₁, A₂. Then for every decomposable operator B of the form on we have

The next proposition is called the subadditivity property of entropy.

4.3.9  Proposition

Let A be a density matrix in with partial traces A₁, A₂. Then

Proof.   The matrix is a density matrix. So, by Exercise 4.3.4, the relative entropy is positive. By definition

By Exercise 4.3.8, this shows

Since this quantity is positive, we have the inequality in (4.36).     ■

There is another way of looking at the partial trace operation that is more transparent and makes several calculations easier:

4.3.10  Proposition

Let f₁, . . . , f_n and e₁, . . . , e_m be orthonormal bases for and . Let A be an operator on and write its matrix in the basis in the n × n partitioned form

where A_ij, 1 ≤ i, j ≤ n are m × m matrices. Then is the n × n matrix defined as

Proof.   It is not difficult to derive this relation from (4.34).     ■

4.3.11  Exercise

The map is the composition of three special kinds of maps described below.

(i)   Let ω = e^2πi/m and let U = diag(1, ω, . . . , ω^m−1). Let W be the n × n block-diagonal matrix (n copies). Let

where A is as in (4.38). Show that

(See (3.28)).

(ii)  Let V be the m × m permutation matrix defined as

Let X be the n × n block-diagonal matrix (n copies). Let

Show that

Thus the effect of Φ₂ on the block matrix (4.41) is to replace each of the diagonal matrices A_ij by the scalar matrix with the same trace as A_ij.

(iii)  Let A be as in (4.38) and let be the (1,1) entry of A_ij. Let

Note that the matrix is a principal n × n submatrix of A. We have then

(iv)  Each of the maps Φ₁, Φ₂, Φ₃ is completely positive; Φ₁, Φ₂, and Φ₃Φ₂Φ₁ are trace preserving.

The next theorem says that taking partial traces of A, B reduces the relative entropy S(A|B).

4.3.12  Theorem

Let A, B be density matrices on . Then

Proof.   It is clear from the definition (4.29) that

for every unitary matrix U. Since S(A|B) is jointly convex in A, B by Theorem 4.3.5, it follows from the representations (4.40) and (4.42) that

Now note that Φ₂Φ₁(A) is a matrix of the form and Φ₃ maps it to the n × n matrix [[α_ij]]. Thus

This proves the theorem.     ■

4.3.13  Exercise

Let be any completely positive trace-preserving map. Use Stinespring’s theorem (Theorem 3.1.2) to show that

The inequality (4.46) is a special case of this.

Now we can state and prove the major result of this section: the strong subadditivity of the entropy function S(A). This is a much deeper property than the subadditivity property (4.36). It is convenient to adopt some notations. We have three Hilbert spaces ; A₁₂₃ stands for a density matrix on . A partial trace like is denoted as A₁₂, and so on for other indices. Likewise a partial trace is denoted by A₂.

4.3.14  Theorem (Lieb-Ruskai)

Let A₁₂₃ be any density matrix in . Then

Proof.   By Theorem 4.3.12, taking the partial trace gives

The equation (4.37) gives

and

Together, these three relations give (4.49).     ■

4.4  FURUTA’S INEQUALITY

We have seen that the function f(A) = A² is not order preserving on positive matrices; i.e., we may have A and B for which A ≥ B ≥ O but not A² ≥ B². Can some weaker form of monotonicity be retrieved for the square function? This question can have different meanings. For example, we can ask whether the condition A ≥ B ≥ O leads to an operator inequality implied by A² ≥ B². For positive matrices A and B consider the statements

(i)  A² ≥ B²;

(ii)  BA²B ≥ B⁴;

(iii)  (BA²B)^1/2 ≥ B².

Clearly (i) (ii) (iii). Let A ≥ B. We know that the inequality (i) does not always hold in this case. Nor does the weaker inequality (ii). A counterexample is provided by

The matrix has determinant −1.

It turns out that the inequality (iii) does follow from A ≥ B.

Note also that A² ≥ B² A⁴ ≥ AB²A A² ≥ (AB²A)^1/2. Once again, for A, B in the example given above we do not have A⁴ ≥ AB²A. But we will see that A ≥ B always implies A² ≥ (AB²A)^1/2.

The most general result inspired by these considerations was proved by T. Furuta. To put it in perspective, consider the following statements:

(i)   A ≥ B ≥ O;

(ii)  A^p ≥ B^p, 0 ≤ p ≤ 1;

(iii)  B^rA^pB^r ≥ B^p+2r, 0 ≤ p ≤ 1, r ≥ 0;

(iv)  (B^rA^pB^r)^1/q ≥ (B^p+2r)^1/q, 0 ≤ p ≤ 1, r ≥ 0, q ≥ 1.

We know that (i) (ii) (iii) (iv). When p > 1, the implication (i) (ii) breaks down. Furuta’s inequality says that (iv) is still valid for all p but with some restriction on q.

4.4.1  Theorem (Furuta’s Inequality)

Let A ≥ B ≥ O. Then

for .

Proof.   For 0 ≤ p ≤ 1, the inequality (4.50) is true even without the last restriction on q. So assume p ≥ 1. If , then . So, the inequality (4.50) holds for such q provided it holds in the special case . Thus we need to prove

for p ≥ 1, r ≥ 0. We may assume that A, B are strictly positive. Let

be the polar decomposition. Then

Hence, for any q > 0

and, therefore (using (4.52) thrice) we get

Now suppose 0 ≤ r ≤ 1/2. Then A^2r ≥ B^2r, and hence B^−2r ≥ A^−2r. Choose . Then . So, we get from (4.53)

B^−r(B^rA^pB^r)^1/qB^−r ≥ A^p/2(A^−p/2A^−2rA^−p/2)^{(p−1)/(p+2r)}A^p/2 = A.

Thus

We have thus proved the inequality (4.51) for r in the domain [0, 1/2]. This domain is extended by inductive steps as follows. Let

where r ∈ [0, 1/2] and q = (p + 2r)/(1 + 2r). We have proved that A₁ ≥ B₁. Let p₁, r₁ be any numbers with p₁ ≥ 1, r₁ ∈ [0, 1/2] and let q₁ = (p₁ + 2r₁)/(1 + 2r₁). Apply the inequality (4.54) to A₁, B₁, p₁, r₁, q₁ to get

This is true, in particular, when p₁ = q and r₁ = 1/2. So we have

Substitute the values of A₁, B₁ from (4.55) to get from this

Put 2r + 1/2 = s, and note that with the choices just made

So, (4.57) can be written as

where s ∈ [1/2, 3/2]. Thus we have enlarged the domain of validity of the inequality (4.51) from r in [0, 1/2] to r in [0, 3/2]. The process can be repeated to see that the inequality is valid for all r ≥ 0.     ■

4.4.2  Corollary

Let A, B, p, q, r be as in the Theorem. Then

Proof.   Assume A, B are strictly positive. Since B⁻¹ ≥ A⁻¹ > O, (4.50) gives us

Taking inverse on both sides reverses this inequality and gives us (4.58).     ■

4.4.3  Corollary

Let A ≥ B ≥ O, p ≥ 1, r ≥ 0. Then

4.4.4  Corollary

Let A ≥ B ≥ O. Then

These are the statements with which we began our discussion in this section. Another special consequence of Furuta’s inequality is the following.

4.4.5  Corollary

Let A ≥ B ≥ O. Then for 0 < p < ∞

Proof.   Choose r = p/2 and q = 2 in (4.58) to get

This is equivalent to the inequality

Using the relation (4.17) we get (4.63).     ■

For 0 ≤ p ≤ 1, the inequality (4.63) follows from Theorem 4.2.1. While the theorem does need this restriction on p, the inequality (4.63) exhibits a weaker monotonicity of the powers A^p for p > 1.

4.5  SUPPLEMENTARY RESULTS AND EXERCISES

The matrix equation

is called the Sylvester equation. If no eigenvalue of A is an eigenvalue of B, then for every Y this equation has a unique solution X. The following exercise outlines a proof of this.

4.5.1  Exercise

(i)   Let . This is a linear operator on . Each eigenvalue of A is an eigenvalue of with multiplicity n times as much. Likewise the eigenvalues of the operator are the eigenvalues of B.

(ii)  The operators and commute. So the spectrum of is contained in the difference , where stands for the spectrum of .

(iii)  Thus if σ(A) and σ(B) are disjoint, then does not contain the point 0. Hence the operator is invertible. This is the same as saying that for each Y, there exists a unique X satisfying the equation (4.64).

The Lyapunov equation (1.14) is a special type of Sylvester equation.

There are various ways in which functions of an arbitrary matrix may be defined. One standard approach via the Jordan canonical form tells us how to explicitly write down a formula for f(A) for every function that is n − 1 times differentiable on an open set containing σ(A). Using this one can see that if A is a matrix whose spectrum is in (0, ∞), then it has a square root whose spectrum is also in (0, ∞). Another standard approach using power series expansions is equally useful.

4.5.2  Exercise

Let and . Then

Suppose all eigenvalues of B₁ and B₂ are positive. Use the (uniqueness part of ) Exercise 4.5.1 to show that B₁ = B₂. This shows that for every matrix A whose spectrum is contained in (0, ∞) there is a unique matrix B for which B² = A and σ(B) is contained in (0, ∞).

The same argument shows that if σ(A) is contained in the open right half plane, then there is a unique matrix B with the same property that satisfies the equation B² = A.

4.5.3  Exercise

Let Φ be a positive unital linear map on . Use Theorem 4.1.5(i) to show that if A and B are strictly positive, then

Thus the map is monotone and concave on the set of positive matrices.

4.5.4  Exercise

Let Φ be a positive unital linear map. Show that for all positive matrices A

(See Proposition 2.7.1.)

The Schur product A ◦ B is a principal submatrix of . So, there is a positive unital linear map Φ from into such that . This observation is useful in deriving convexity and concavity results about Schur products from those about tensor products. We leave it to the reader to obtain such results from what we have done in Chapters 2 and 4.

The arithmetic, geometric, and harmonic means are the best-known examples of means. We have briefly alluded to the logarithmic mean in (4.2). Several other means arise in various contexts. We mention two families of such means.

For 0 ≤ ν ≤ 1 let

We call these the Heinz means. Notice that H_ν = H_1−ν, H_1/2 is the geometric mean, and H₀ = H₁ is the arithmetic mean. Thus, the family H_ν interpolates between the geometric and the arithmetic mean. Each H_ν satisfies conditions (i)–(vi) for means.

4.5.5  Exercise

(i)   For fixed positive numbers a and b, H_ν(a, b) is a convex function of ν in the interval [0, 1], and attains its minimum at ν = 1/2.

Thus

(ii)  Show that

and use this to prove the inequality (4.3).

For −∞ ≤ p ≤ ∞ let

These are called the power means or the binomial means. Here it is understood that

The arithmetic and the harmonic means are included in this family. Properties (i)–(vi) of means may readily be verified for this family.

In Section 4.1 we defined the geometric mean A#B by using the congruence Γ_A−1/2 to reduce the pair (A, B) to the commuting pair (I, A^−1/2BA^−1/2). A similar procedure may be used for the other means. Given a mean M on positive numbers, let

4.5.6  Exercise

From properties (i)–(vi) of M deduce that the function f on has the following properties:

(i)  f(1) = 1,

(ii)  xf(x⁻¹) = f(x),

(iii)  f is monotone increasing,

(iv)  f is continuous,

(v)  f(x) ≤ 1 for 0 < x ≤ 1, and f(x) ≥ 1 for x ≥ 1.

4.5.7  Exercise

Let f be a map of into itself satisfying the properties (i)–(v) given in Exercise 4.5.6. For positive numbers a and b let

Show that M is a mean.

Given a mean M(a, b) on positive numbers let f(x) be the function associated with it by the relation (4.69). For positive matrices A and B let

When this formula gives the geometric mean A#B defined in (4.10). Does this procedure always lead to an operator mean satisfying conditions (i)–(vi)? For the geometric mean we verified its symmetry by an indirect argument. The next proposition says that such symmetry is a general fact.

4.5.8  Proposition

Let M(A, B) be defined by (4.71). Then for all A and B

Proof.   We have to show that

If A^−1/2B^1/2 = UP is the polar decomposition, then B^1/2A^−1/2 = PU^∗ and B^−1/2A^1/2 = P⁻¹U^∗. The left-hand side of (4.73) is, therefore, equal to

The right-hand side of (4.73) is equal to

So, (4.73) will be proved if we show that f(P²) = Pf(P⁻²)P. This follows from the fact that for every x we have f(x²) = x²f(x⁻²) as shown in Exercise 4.5.6.     ■

4.5.9  Exercise

Show that M(A, B) defined by (4.71) is invariant under congruence; i.e.,

for every invertible matrix X. [Hint: Use the polar decomposition.]

Our next concern is whether M(A, B) defined by (4.71) is monotone in the variables A and B. This is so provided the function f is operator monotone. In this case monotonicity in B is evident from the formula (4.71). By symmetry it is monotone in A as well.

For the means that we have considered in this chapter the function f is given by

The first five functions in this list are operator monotone. The last enjoys this property only for some values of p.

4.5.10  Exercise

Let f be an operator monotone function on (0, ∞). Then the function g(x) = [f(x^p)]^1/p is operator monotone for 0 < p ≤ 1. [It may be helpful, in proving this, to use a theorem of Loewner that says f is operator monotone if and only if it has an analytic continuation mapping the upper half-plane into itself. See MA Chapter V.]

4.5.11  Exercise

Show that the function

is operator monotone if and only if −1 ≤ p ≤ 1.

Thus the binomial means B_p(a, b) defined by (4.68) lead to matrix means satisfying all our requirements if and only if −1 ≤ p ≤ 1. The mean B₀(a, b) leads to the geometric mean A#B.

The logarithmic mean is important in different contexts, one of them being heat flow. We explain this briefly. Heat transfer by steady unidirectional conduction is governed by Fourier’s law. If the direction of heat flow is along the x-axis, this law says

where q is the rate of heat flow along the x-axis across an area A normal to the x-axis, dT/dx is the temperature gradient along the x direction, and k is a constant called thermal conductivity of the material through which the heat is flowing.

The cross-sectional area A may be constant, as for example in a cube. More often, as in the case of a fluid traveling in a pipe, it is a variable. In such cases it is convenient for engineering calculations to write (4.74) as

where A_m is the mean cross section of the body between two points at distance ∆x along the x-axis and ∆T is the difference of temperatures at these two points. For example, in the case of a body with uniformly tapering rectangular cross section, A_m is the arithmetic mean of the two boundary areas A₁ and A₂.

A very common situation is that of a liquid flowing through a long hollow cylinder (like a pipe). Here heat flows through the sides of the cylinder in a radial direction perpendicular to the axis of the cylinder. The cross sectional area in this case is proportional to the distance from the centre.

Consider the section of the cylinder bounded by two concentric cylinders at distances x₁ and x₂ from the centre. Total heat flow across this section given by (4.74) is

where A = 2πxL, L being the length of the cylinder. This shows that

If we wish to write it in the form (4.75) with ∆x = x₂ − x₁, then we must have

In other words,

the logarithmic mean of the two areas bounding the section under consideration. In the chemical engineering literature this is called the logarithmic mean area.

If instead of a hollow cylinder we consider a hollow sphere, then the cross-sectional area is proportional to the square of the distance from the center. In this case we get from (4.76)

The reader can check that in this case

the geometric mean of the two areas bounding the annular section under consideration.

In Chapter 6 we will see that the inequality between the geometric and the logarithmic mean plays a fundamental role in differential geometry.

4.6  NOTES AND REFERENCES

The parallel sum (4.4) was introduced by W. N. Anderson and R. J. Duffin, Series and parallel addition of matrices, J. Math. Anal. Appl., 26 (1969) 576–594. They also proved some of the basic properties of this object like monotonicity and concavity, and the arithmetic-harmonic mean inequality. Several other operations corresponding to different kinds of electrical networks were defined following the parallel sum. See W. N. Anderson and G. E. Trapp, Matrix operations induced by electrical network connections—a survey, in Constructive Approaches to Mathematical Models, Academic Press, 1979, pp. 53–73.

The geometric mean formula (4.10) and some of its properties like (4.13) are given in W. Pusz and S. L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys., 8 (1975) 159–170. The notations and the language of this paper are different from ours. It was in the beautiful paper T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl., 26 (1979) 203–241, that several concepts were clearly stated, many basic properties demonstrated, and the power of the idea illustrated through several applications. The interplay between means and positive linear maps clearly comes out in this paper, concavity and convexity of various maps are studied, a new proof of Lieb’s theorem 4.27 is given and many new kinds of inequalities for the Schur product are obtained. This paper gave a lot of impetus to the study of matrix means, and of matrix inequalities in general.

W. N. Anderson and G. E. Trapp, Operator means and electrical networks, in Proc. 1980 IEEE International Symposium on Circuits and Systems, pointed out that A#B is the positive solution of the Riccati equation (4.11). They also drew attention to the matrix in Proposition 4.1.9. This had been studied in H. J. Carlin and G. A. Noble, Circuit properties of coupled dispersive lines with applications to wave guide modelling, in Proc. Network and Signal Theory, J. K. Skwirzynski and J. O. Scanlan, eds., Peter Pergrinus, 1973, pp. 258–269. Note that this paper predates the one by Pusz and Woronowicz.

An axiomatic theory of matrix means was developed in F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246 (1980) 205–224. Here it is shown that there is a one-to-one correspondence between matrix monotone functions and matrix means. This is implicit in some of our discussion in Section 4.5.

A notion of geometric mean different from ours is introduced and studied by M. Fiedler and V. Pták, A new positive definite geometric mean of two positive definite matrices, Linear Algebra Appl., 251 (1997) 1–20. This paper contains a discussion of the mean A#B as well.

Entropy is an important notion in statistical mechanics and in information theory. Both of these subjects have their classical and quantum versions. Eminently readable introductions are given by E. H. Lieb and J. Yngvason, A guide to entropy and the second law of thermodynamics, Notices Am. Math. Soc., 45 (1998) 571–581, and in other articles by these two authors such as The mathematical structure of the second law of thermodynamics, in Current Developments in Mathematics, 2001, International Press, 2002. The two articles by A. Wehrl, General properties of entropy, Rev. Mod. Phys., 50 (1978) 221–260, and The many facets of entropy, Rep. Math. Phys., 30 (1991) 119–129, are comprehensive surveys of various aspects of the subject. The text by M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, 1993, is another resource. The book by M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000, reflects the renewed vigorous interest in this topic because of the possibility of new significant applications to information technology.

Let (p₁, . . . , p_n) be a probability vector; i.e., p_i ≥ 0 and = 1. C. Shannon, Mathematical theory of communication, Bell Syst. Tech. J., 27 (1948) 379–423, introduced the function S(p₁, . . . , p_n) = as a measure of “average lack of information” in a statistical experiment with n possible outcomes occurring with probabilities p₁, . . . , p_n. The quantum analogue of a probability vector is a density matrix A; i.e., A ≥ O and tr A = 1. The quantum entropy function S(A) = −tr A log A was defined by J. von Neumann, Thermodynamik quantenmechanischer Gesamtheiten, Göttingen Nachr., 1927, pp. 273–291; see also Chapter 5 of his book Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. Thus von Neumann’s definition preceded Shannon’s. There were strong motivations for the former because of the work of nineteenth-century physicists, especially L. Boltzmann.

Theorem 4.3.3 was proved in E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. Math., 11 (1973) 267–288. Because of its fundamental interest and importance, several different proofs appeared soon after Lieb’s paper. One immediate major application of this theorem was made in the proof of Theorem 4.3.14 by E. H. Lieb and M. B. Ruskai, A fundamental property of quantum-mechanical entropy, Phys. Rev. Lett., 30 (1973) 434–436, and Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys., 14 (1973) 1938–1941. Several papers of Lieb are conveniently collected in Inequalities, Selecta of Elliot H. Lieb, M. Loss and M. B. Ruskai eds., Springer, 2002. The matrix-friendly proof of Theorem 4.3.14 is adopted from R. Bhatia, Partial traces and entropy inequalities, Linear Algebra Appl., 375 (2003) 211–220.

Three papers of G. Lindblad, Entropy, information and quantum measurements, Commun. Math. Phys., 33 (1973) 305–322, Expectations and entropy inequalities for finite quantum systems, ibid. 39 (1974) 111–119, and Completely positive maps and entropy inequalities, ibid. 40 (1975) 147–151, explore various convexity properties of entropy and their interrelations. For example, the equivalence of Theorems 4.3.5 and 4.3.14 is noted in the second paper and the inequality (4.48) is proved in the third.

In the context of quantum statistical mechanics the tensor product operation represents the physical notion of putting a system in a larger system. In quantum information theory it is used to represent the notion of entanglement of states. These considerations have led to several problems very similar to the ones discussed in Section 4.3. We mention one of these as an illustrative example. Let Φ be a completely positive trace-preserving (CPTP) map on M_n. The minimal entropy of the “quantum channel” Φ is defined as

It is conjectured that

for any two CPTP maps Φ₁ and Φ₂. See P. W. Shor, Equivalence of additivity questions in quantum information theory, Commun. Math. Phys., 246 (2004) 453–472 for a statement of several problems of this type and their importance.

Furuta’s inequality was proved by T. Furuta, A ≥ B ≥ O assures (B^rA^pB^r)^1/q ≥ B^(p+2r)/q for r ≥ 0, p ≥ 0, q ≥ 1 with (1 + 2r)q ≥ p + 2r, Proc. Am. Math. Soc. 101 (1987) 85–88. This paper sparked off several others giving different proofs, extensions, and applications. For example, T. Ando, On some operator inequalities, Math. Ann., 279 (1987) 157–159, showed that if A ≥ B, then e^−tA#e^tB ≤ I for all t ≥ 0. It was pointed out at the beginning of Section 4.4 that A ≥ B ≥ O does not imply A² ≥ B² but it does imply the weaker inequality (BA²B)^1/2 ≥ B². This can be restated as A⁻²#B² ≤ I. In a similar vein, A ≥ B does not imply e^A ≥ e^B but it does imply e^−A#e^B ≤ I. It is not surprising that Furuta’s inequality is related to the theory of means and to properties of matrix exponential and logarithm functions.

The name “Heinz means” for (4.65) is not standard usage. We have called them so because of the famous inequalities of E. Heinz (proved in Chapter 5). The means (4.68) have been studied extensively. See for example, G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Second Edition, Cambridge University Press, 1952. The matrix analogues

are analysed in K. V. Bhagwat and R. Subramanian, Inequalities between means of positive operators, Math. Proc. Cambridge Philos. Soc., 83 (1978) 393–401. It is noted there that the limiting value . We have seen that this is not a suitable definition of the geometric mean of A and B.

A very interesting article on the matrix geometric mean is J. D. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Am. Math. Monthly 108 (2001) 797–812. The importance of the logarithmic mean in engineering problems is discussed, for example, in W. H. McAdams, Heat Transmission, Third Edition, McGraw Hill, 1954.