[Paper Review] The outer spectral radius and dynamics of completely positive maps

J. E. Pascoe|arXiv (Cornell University)|May 23, 2019

Algebraic structures and combinatorial models23 references4 citations

TL;DR

This paper introduces the outer spectral radius as a relaxation of the joint spectral radius for tuples of matrices, linking it to the dynamics of completely positive maps and quantum channels. It establishes a Gelfand-type formula for the outer spectral radius, proves a degenerate Quantum Perron-Frobenius theorem, and shows that the average of iterates of a completely positive map converges to a map whose Kraus operators span an ideal in the algebra generated by the original operators.

ABSTRACT

We examine a special case of an approximation of the joint spectral radius given by Blondel and Nesterov, which we call the outer spectral radius. The outer spectral radius is given by the square root of the ordinary spectral radius of the $n^2$ by $n^2$ matrix $\sum{\overline{X_i}}\otimes{X_i}.$ We give an analogue of the spectral radius formula for the outer spectral radius which can be used to quickly obtain the error bounds in methods based on the work of Blondel and Nesterov. The outer spectral radius is used to analyze the iterates of a completely postive map, including the special case of quantum channels. The average of the iterates of a completely positive map approach to a completely positive map where the Kraus operators span an ideal in the algebra generated by the Kraus operators of the original completely positive map. We also give an elementary treatment of Popescu's theorems on similarity to row contractions in the matrix case, describe connections to the Parrilo-Jadbabaie relaxation, and give a detailed analysis of the maximal spectrum of a completely positive map.

Motivation & Objective

• To unify and extend the Blondel-Nesterov approximation of the joint spectral radius via the outer spectral radius.
• To provide a Gelfand-type spectral radius formula for the outer spectral radius, enabling error bounds in approximation methods.
• To analyze the long-term dynamics of completely positive maps, especially quantum channels, using spectral properties of the associated matrix $ \sum \overline{X_i} \otimes X_i $.
• To give an elementary treatment of Popescu’s Rota-Strang theory on similarity to row contractions in the matrix case.
• To connect the outer spectral radius to the Parrilo-Jadbabaie relaxation and analyze the spectral structure of $ \sum X_i^{\otimes 2k} $.

Proposed method

• Define the outer spectral radius as $ \hat{\rho}(X_1,\ldots,X_d) = \sqrt{ \rho\left( \sum \overline{X_i} \otimes X_i \right) } $, linking it to the spectral radius of a Kronecker product matrix.
• Prove a Gelfand-type formula: $ \hat{\rho} = \lim_{k\to\infty} \sup_{\sum |a_{i_1\cdots i_k}|^2=1} \left\| \sum a_{i_1\cdots i_k} X_{i_1} \cdots X_{i_k} \right\|^{1/k} $, providing a dynamical interpretation.
• Construct a Lyapunov matrix $ L $ such that $ L - \sum X_i L X_i^* \geq 0 $, which helps analyze convergence and spectral structure.
• Use the $ \psi $-involution and partial trace to analyze the matrix $ T = \sum \overline{X_i} \otimes X_i $, showing its maximal spectrum contains a non-negative real eigenvalue.
• Apply representation theory of the symmetric group to study the invariant subspaces of $ \sum X_i^{\otimes 2k} $, linking them to symmetric polynomials.
• Reinterpret the Parrilo-Jadbabaie relaxation $ \rho_{SR,2d} $ as the spectral radius of the action of $ \sum \tau_{X_i} $ on homogeneous polynomials of degree $ 2k $.

Experimental results

Research questions

• RQ1 How does the outer spectral radius relate to the dynamics of iterated completely positive maps, especially quantum channels?
• RQ2 Can a Gelfand-type formula be established for the outer spectral radius, and what are its implications for error bounds in approximation methods?
• RQ3 What is the spectral structure of the matrix $ \sum \overline{X_i} \otimes X_i $, particularly regarding the maximal spectrum and degeneracy indices?
• RQ4 How does the outer spectral radius connect to the Parrilo-Jadbabaie relaxation and the symmetric algebra of matrices?
• RQ5 What is the role of the Lyapunov matrix in characterizing the convergence of iterates of a completely positive map?

Key findings

• The outer spectral radius satisfies a Gelfand-type formula: $ \hat{\rho}(X_1,\ldots,X_d) = \lim_{k\to\infty} \sup_{\sum |a_{i_1\cdots i_k}|^2=1} \left\| \sum a_{i_1\cdots i_k} X_{i_1} \cdots X_{i_k} \right\|^{1/k} $, providing a dynamical interpretation.
• The maximal spectrum of $ T = \sum \overline{X_i} \otimes X_i $ contains a non-negative real eigenvalue $ \lambda $ such that $ \lambda \geq |\lambda'| $ for all other eigenvalues $ \lambda' $, with equality only if $ \lambda $ has higher or equal degeneracy index.
• The average of the iterates of a completely positive map converges to a map whose Kraus operators span an ideal in the $ C^* $-algebra generated by the original Kraus operators.
• When $ \hat{\rho} = 1 $ and the $ X_i $ generate $ M_n(\mathbb{C}) $, there exist conjugations transforming the map into either a unital or trace-preserving form via $ V^{1/2} $ and $ W^{1/2} $.
• The spectral radius of the joint spectral radius can be recovered as $ \rho(X_1,\ldots,X_d) = \lim_{k\to\infty} \rho\left( \sum \tau_{X_i} \big|_{\mathbb{R}[e_1,\ldots,e_n]_{2k}} \right)^{1/2k} $, linking to polynomial actions.
• The eigenspace corresponding to the largest eigenvalue of $ \sum X_i^{\otimes 2k} $ is invariant under permutations and, in non-degenerate cases, is one-dimensional and spanned by a symmetric vector.