[Paper Review] Comparison of matrix norms on bipartite spaces

Christopher King, Nilufer Koldan|ArXiv.org|Apr 10, 2009

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper compares two non-commutative $L^q(L^p)$ norms on bipartite matrix spaces: the Carlen-Lieb norm and the Pisier-type norm. It proves that the Pisier-type norm is upper bounded by a constant multiple of the Carlen-Lieb norm for $1 \leq p \leq 2$, $q \geq 1$, but shows they are inequivalent in general—specifically, no lower bound exists when $p=2 < q$, implying the norms are not equivalent in this case. The authors conjecture inequivalence holds universally.

ABSTRACT

Two non-commutative versions of the classical L^q(L^p) norm on the algebra of (mn)x(mn) matrices are compared. The first norm was defined recently by Carlen and Lieb, as a byproduct of their analysis of certain convex functions on matrix spaces. The second norm was defined by Pisier and others using results from the theory of operator spaces. It is shown that the second norm is upper bounded by a constant multiple of the first for all 1 &lt;= p &lt;= 2, q &gt;= 1. In one case (2 = p &lt; q) it is also shown that there is no such lower bound, and hence that the norms are inequivalent. It is conjectured that the norms are inequivalent in all cases.

Motivation & Objective

• To compare two non-commutative $L^q(L^p)$ norms on $\mathcal{M}_n \otimes \mathcal{M}_m$: the Carlen-Lieb norm and the Pisier-type norm.
• To determine whether these norms are equivalent or if one dominates the other.
• To investigate the relationship between these norms in the context of quantum information theory, where both arise from fundamental concepts like entropy subadditivity and completely positive maps.
• To establish bounds and explore the conditions under which the norms are equivalent or inequivalent.

Proposed method

• The Carlen-Lieb norm is defined via a convexity-based infimum over positive decompositions of a block matrix representation of $Y$, using the $\Psi_{p,q}$ functional on positive semidefinite matrices.
• The Pisier-type norm is defined differently depending on the relative values of $p$ and $q$: via a supremum over operator scaling for $p \leq q$, and via an infimum over operator decompositions for $p \geq q$, with a parameter $r$ defined by $r^{-1} = |p^{-1} - q^{-1}|$.
• For $p \leq q$, the norm is expressed as $\|Y\|_{NC} = \sup_{C \geq 0, \operatorname{Tr} C = 1} \|(C^{1/2r} \otimes I_m) Y (C^{1/2r} \otimes I_m)\|_p$, leveraging concavity and unitary invariance.
• For $p=2 < q$, the paper uses a key inequality involving the trace of $\operatorname{Tr}_2(Y+A)^2 + \operatorname{Tr}_2 A^2 \geq \frac{1}{2} \operatorname{Tr}_2 Y^2$ to lower bound the Carlen-Lieb norm.
• The proof of inequivalence in the $p=2 < q$ case uses a sequence of positive semidefinite matrices $Y^{(k)}$ with $\lambda_j = c/j$, leading to $\|Y^{(k)}\|_{NC}^{-1} \Psi_{p,q}(Y^{(k)}) \to \infty$ as $k \to \infty$, showing no uniform lower bound exists.
• The analysis relies on Hölder’s inequality, trace concavity, and symmetry properties under unitary conjugation to reduce the supremum to diagonal density matrices.

Experimental results

Research questions

• RQ1 Are the Carlen-Lieb norm and the Pisier-type norm equivalent on $\mathcal{M}_n \otimes \mathcal{M}_m$ for $1 \leq p \leq 2$, $q \geq 1$?
• RQ2 Is the Pisier-type norm upper bounded by a constant multiple of the Carlen-Lieb norm in the specified parameter range?
• RQ3 Does the absence of a lower bound between the norms imply they are inequivalent in the case $p=2 < q$?
• RQ4 Can the inequivalence be extended to all $1 \leq p \leq 2$, $q \geq 1$?
• RQ5 What is the asymptotic behavior of the ratio $\Psi_{p,q}(Y)/\|Y\|_{NC}$ for specific sequences of matrices?

Key findings

• The Pisier-type norm is upper bounded by a constant multiple of the Carlen-Lieb norm for all $1 \leq p \leq 2$, $q \geq 1$, establishing a one-way domination.
• For the case $p=2 < q$, no lower bound exists: $\|Y\|_{NC}^{-1} \Psi_{p,q}(Y) \to \infty$ along a sequence of matrices, proving the norms are inequivalent.
• The ratio $\Psi_{p,q}(Y)^p / \|Y\|_{NC}^p$ is bounded below by $\|D\|_{q/p} / \|D\|_{r'}^p$, which diverges as $n \to \infty$ when $p < q$, confirming the lack of uniform lower bound.
• For $p=2 \leq q$, the Carlen-Lieb norm satisfies $\|Y\|_{CL} \geq \frac{1}{\sqrt{2}} \Psi_{2,q}(Y)$, providing a useful lower bound for that case.
• The supremum in the Pisier-type norm expression for $p \leq q$ is achieved on diagonal density matrices, simplifying the optimization problem.
• The analysis shows that the norms are not equivalent in general, and the authors conjecture they are inequivalent for all $1 \leq p \leq 2$, $q \geq 1$.