Spectral majorization for symmetric matrices

Question

In ${\mathbb R}^n$, a vector $a=(a_1,\ldots,a_n)$ is said to majorize another vector $b=(b_1,\ldots,b_n)$ if for any convex function $f\colon\mathbb R\to\mathbb R$, we have $$\sum_{i=1}^nf(a_i)\ge \sum_{i=1}^nf(b_i).$$ Note that this does not depend on the order of the coordinates. This natural pre-order has been massively studied, and has found many applications.

Now, it is natural to extend this notion to $n\times n$ symmetric matrices as follows: say that $A$ spectrally majorizes $B$ if the spectrum of $A$ majorizes that of $B$.

My question is: has this spectral notion been investigated? In particular, is there any nice characterization? Or, at least, some simple (non-spectral) sufficient conditions? Or some known contexts in which such ordered pairs (A,B) arise naturally?

Answer 1

Yes. This notion of majorization of Hermitian matrices has been investigated before in the context of quantum information theory, and there are several good characterizations of this notion of majorization. This notion of majorization is quite natural since it gives a notion of whether one density operator is 'more mixed' than another density operator.

If $A$ is a matrix, then write $\sigma(A)$ for the spectrum of $A$ (with multiplicity included). If $C,D$ are tuples of real numbers, then write $C\preceq D$ if $D$ majorizes $C$.

If $V$ is a finite dimensional complex inner product space, then let $L(V)$ denote the collection of all linear mappings from $V$ to $V$. We say that a linear mapping $\mathcal{E}:L(V)\rightarrow L(W)$ is trace preserving if $\text{Tr}(\mathcal{E}(A))=\mathcal{E}(A)$ for each $A\in L(V)$. A linear mapping $\mathcal{E}:L(V)\rightarrow L(W)$ is positive if whenever $P\in L(V)$ is positive semidefinite, then $\mathcal{E}(P)$ is also positive semidefinite. A linear mapping $\mathcal{E}:L(V)\rightarrow L(W)$ is said to be completely positive if $\mathcal{E}\otimes 1_V:L(V\otimes V)\rightarrow L(W\otimes V)$ is positive. A channel is a completely positive trace preserving mapping from some $L(V)$ to some $L(W)$. A linear mapping $\mathcal{E}:L(V)\rightarrow L(W)$ is said to be unital if $\mathcal{E}(1_V)=1_W.$ Observe that if $\mathcal{E}$ is both unital and trace preserving, then $\dim(V)=\dim(W)$.

A proof of (1-4) from the following fact can be found in the text The Theory of Quantum Information by John Watrous.

Theorem (Uhlmann): Let $A,B$ be Hermitian mappings. Then the following are equivalent.

1. $\sigma(A)\preceq\sigma(B)$.

2. There is a mixed unitary channel $\mathcal{E}$ with $A=\mathcal{E}(B)$.

3. There is a unital channel $\mathcal{E}$ with $A=\mathcal{E}(B)$.

4. There is a positive, trace preserving, and unital map $\mathcal{E}$ with $A=\mathcal{E}(B)$.

5. For every convex function $f:\mathbb{R}\rightarrow\mathbb{R}$, we have $\text{Tr}(f(A))\leq\text{Tr}(f(B))$.