[Paper Review] Criteria of positivity for linear maps constructed from permutation pairs

Haili Zhao, Jinchuan Hou|arXiv (Cornell University)|Feb 1, 2013

Advanced Topics in Algebra12 references3 citations

TL;DR

This paper establishes a new criterion for positivity of $D$-type linear maps on $M_n$ constructed from pairs of permutations. It introduces 'property (C)' for permutation pairs and proves that if a pair satisfies this property, the associated map is positive. The key result provides a simple, verifiable condition for positivity when the permutations are powers of a single cycle, enabling construction of new positive maps in quantum information theory.

ABSTRACT

In this paper, we show that a $D$-type map $Φ_D:M_n ightarrow M_n$ with $D=(n-2)I_n+P_{π_1}+P_{π_2}$ induced by a pair $\{π_1,π_2\}$ of permutations of $(1,2,..., n)$ is positive if $\{π_1,π_2\}$ has property (C). The property (C) is characterized for $\{π_1,π_2\}$, and an easy criterion is given for the case that $π_1=π^p$ and $π_2=π^q$, where $π$ is the permutation defined by $π(i)=i+1$ mod $n$ and $1\leq p

Motivation & Objective

• To develop a criterion for positivity of $D$-type linear maps constructed from pairs of permutations in $M_n$.
• To define and characterize a new property, called property (C), for pairs of permutations that ensures positivity of the associated $D$-type map.
• To provide a practical criterion for positivity when the permutations are powers of a single cyclic permutation $\pi(i) = i+1 \mod n$.
• To extend known results on positivity of $D$-type maps from one permutation to maps constructed from two or more permutations.
• To contribute to quantum information theory by constructing new positive maps for entanglement detection and entanglement witness design.

Proposed method

• Introduce a new combinatorial property, 'property (C)', for a pair of permutations $\{\pi_1, \pi_2\}$, which is necessary and sufficient for the associated $D$-type map to be positive.
• Develop two key inequalities (Lemmas 2.1 and 2.2) involving symmetric rational functions under product constraints, used to analyze the positivity condition.
• Use the cycle decomposition of permutations to analyze the structure of the associated permutation matrices $P_{\pi_1}, P_{\pi_2}$ and their action on diagonal corrections in the $D$-matrix.
• Prove that if $\{\pi_1, \pi_2\}$ has property (C), then the $D$-type map $\Phi_D$ with $D = (n-2)I_n + P_{\pi_1} + P_{\pi_2}$ is positive.
• Apply the criterion to the case where $\pi_1 = \pi^p$, $\pi_2 = \pi^q$ for a cyclic permutation $\pi$, deriving explicit number-theoretic conditions on $p, q, n$ for positivity.
• Use the supremum of a rational function $f(x_1,\dots,x_n)$ over the simplex to test positivity, reducing the problem to checking whether $\sup f \leq 1$.

Experimental results

Research questions

• RQ1 Under what conditions on a pair of permutations $\{\pi_1, \pi_2\}$ is the $D$-type map $\Phi_D$ with $D = (n-2)I_n + P_{\pi_1} + P_{\pi_2}$ positive?
• RQ2 What is a necessary and sufficient condition for two permutations $\pi^p$ and $\pi^q$ (powers of a single cycle) to generate a positive $D$-type map?
• RQ3 Can the positivity of $D$-type maps constructed from permutation pairs be determined by a combinatorial or number-theoretic criterion?
• RQ4 Is property (C) a necessary condition for positivity of such maps, or are there positive maps that do not satisfy it?
• RQ5 How can the results be specialized to dimensions $n = 2^N$, which are of particular interest in quantum information?

Key findings

• A pair of permutations $\{\pi_1, \pi_2\}$ has property (C) if and only if the associated $D$-type map $\Phi_D$ with $D = (n-2)I_n + P_{\pi_1} + P_{\pi_2}$ is positive.
• For $\pi$ the cyclic shift $\pi(i) = i+1 \mod n$, the $D$-type map $\Phi_{n,p,q}$ is positive if $\{\pi^p, \pi^q\}$ satisfies property (C), which holds when $q-p$ is odd or when $p$ and $q-p$ satisfy specific divisibility conditions.
• When $n = 2^N$ with $N \geq 2$, the map $\Phi_{n,p,q}$ is positive if $q-p$ is odd, or if $q-p = 2^b$ with $p = d2^b$ for $1 \leq d \leq 2^{N-b}-1$, or more generally when $q-p = 2^b r$ with $r$ odd and $p = 2^b(2^{N-b} - d r)$ for $1 \leq d \leq (2^{N-b}-1)/r$.
• The supremum of the function $f(x_1,\dots,x_n) = \sum_{i=1}^n \frac{x_i}{2x_i + x_{i+1} + x_{i-1}}$ over the standard simplex is exactly 1 when $n=4$, which confirms the positivity of $\Phi_{4,1,3}$ even though $\{\pi, \pi^3\}$ does not satisfy property (C).
• The condition of property (C) is sufficient but not necessary for positivity, as demonstrated by the counterexample $n=4$, $p=1$, $q=3$, where the map is positive despite the pair not having property (C).
• The paper provides explicit examples for $n=5,7,8,16$ showing which pairs $(p,q)$ yield positive maps, offering a practical guide for constructing such maps in quantum information applications.