Pinching

Let $A$ be a Hermitian matrix, and $A = \sum_j \lambda_j P_j$ be the spectral decomposition of $A$. The pinching map defined by $A$ is given by

$$\mathcal{P}_A (X) = \sum_j P_j X P_j ,$$

for any Hermitian matrix $X$.

Theorem 1. Let $A$ be a positive semi-definite matrix and $B$ be a Hermitian matrix. The following statements hold.

1. $\mathcal{P}_B (A)$ commutes with $B$.
2. $\mathrm{Tr} ( \mathcal{P}_B (A) B ) = \mathrm{Tr} ( A B )$.
3. (Pinching inequality) $\vert \mathrm{spec} (B) \vert \, \mathcal{P}_B (A) \geq A$, where $\mathrm{spec} (B)$ denotes the set of eigenvalues of $B$.

The first two statements are easy to check. The earliest reference on the pinching inequality I can find is the classic book by Jacques Dixmier. A simple proof of the pinching inequality can be found in the textbook by Masahito Hayashi.

One main issue in matrix analysis is non-commutativity. The first statement in Theorem 1 hints that pinching can be an useful tool to deal with this issue. In the next section, the pinching trick is illustrated using the Golden-Thompson inequality as an example.

A proof of the Golden-Thompson Inequality

The Golden-Thompson inequality says that

$$\mathrm{Tr} ( \exp ( A + B ) ) \leq \mathrm{Tr} ( \exp (A) \exp (B) ) ,$$

for any two Hermitian matrices $A$ and $B$. Obviously, if $A$ commutes with $B$, the Golden-Thompson inequality holds with an equality; however, in general one needs to take non-commutativity into consideration. Below we present a very elegant proof using the pinching trick from a recent paper by D. Sutter et al.

The key observation is that $\vert \mathrm{spec} ( A^{\otimes n} ) \vert$ does not grow rapidly with $n$ for any Hermitian matrix $A$.

Lemma 1. One has $\vert \mathrm{spec} ( A^{\otimes n} ) \vert = O ( \mathsf{poly} (n) )$ for any Hermitian matrix $A$.

Proof (Golden-Thompson inequality).

Let $X$ and $Y$ be two positive definite matrices. Then one can write

$$\log \mathrm{Tr} ( \exp ( \log X + \log Y ) ) = \frac{1}{n} \log \mathrm{Tr} ( \exp ( \log X^{\otimes n} + \log Y^{\otimes n} ) ),$$

for any positive integer $n$. By the pinching inequality, one has

$$\frac{1}{n} \log \mathrm{Tr} ( \exp ( \log X^{\otimes n} + \log Y^{\otimes n} ) ) \leq \frac{1}{n} \log \mathrm{Tr} \{ \exp [ \log ( \vert \mathrm{spec} ( Y^{\otimes n} ) \vert \, \mathcal{P}_{Y^{\otimes n}} ( X^{\otimes n} ) ] + \log Y^{\otimes n} ) \}$$

By the first two statements in Theorem 1 and Lemma 1, one has

$$\begin{align} \text{RHS} & = \frac{1}{n} \log \mathrm{Tr}\, ( \mathcal{P}_{Y^{\otimes n}} ( X^{\otimes n} ) Y^{\otimes n} ) + \frac{\log \mathsf{poly} (n)}{n} \notag \\ & = \frac{1}{n} \log \mathrm{Tr} ( X^{\otimes n} Y^{\otimes n} ) + \frac{\log \mathsf{poly} (n)}{n} \notag \\ & = \log \mathrm{Tr}\, ( X Y ) + \frac{\log \mathsf{poly} (n)}{n} . \end{align}$$

Then one obtains the Golden-Thompson Inequality by letting $n \to \infty$.