Operator scaling with specified marginals (1801.01412v2)

Abstract: The completely positive maps, a generalization of the nonnegative matrices, are a well-studied class of maps from $n\times n$ matrices to $m\times m$ matrices. The existence of the operator analogues of doubly stochastic scalings of matrices is equivalent to a multitude of problems in computer science and mathematics, such rational identity testing in non-commuting variables, noncommutative rank of symbolic matrices, and a basic problem in invariant theory (Garg, Gurvits, Oliveira and Wigderson, FOCS, 2016). We study operator scaling with specified marginals, which is the operator analogue of scaling matrices to specified row and column sums. We characterize the operators which can be scaled to given marginals, much in the spirit of the Gurvits' algorithmic characterization of the operators that can be scaled to doubly stochastic (Gurvits, Journal of Computer and System Sciences, 2004). Our algorithm produces approximate scalings in time poly(n,m) whenever scalings exist. A central ingredient in our analysis is a reduction from the specified marginals setting to the doubly stochastic setting. Operator scaling with specified marginals arises in diverse areas of study such as the Brascamp-Lieb inequalities, communication complexity, eigenvalues of sums of Hermitian matrices, and quantum information theory. Some of the known theorems in these areas, several of which had no effective proof, are straightforward consequences of our characterization theorem. For instance, we obtain a simple algorithm to find, when they exist, a tuple of Hermitian matrices with given spectra whose sum has a given spectrum. We also prove new theorems such as a generalization of Forster's theorem (Forster, Journal of Computer and System Sciences, 2002) concerning radial isotropic position.

• The paper introduces a novel extension of operator scaling for completely positive maps under specified marginals, ensuring convergence to target distributions.
• It adapts Gurvits’ method with a Sinkhorn-Knopp style algorithm that achieves polynomial-time scaling through capacity analysis.
• The work bridges theoretical invariant properties with practical numerical bounds, impacting quantum information theory and eigenvalue computations.

Operator Scaling with Specified Marginals

Introduction

The exploration of completely positive maps seeks to extend the functionalities of nonnegative matrices to linear transformations on matrix spaces that retain positive-semidefiniteness. Operator scaling, an analogue to scaling nonnegative matrices to doubly stochastic ones, addresses various challenges such as rational identity testing in non-commuting variables, determining noncommutative rank of symbolic matrices, and issues in invariant theory. This paper studies the operator scaling problem specifically tied to given marginals, akin to scaling matrices to defined row and column sums, expanding the framework laid by Gurvits' characterization for reaching doubly stochastic states. An algorithmic approach, akin to Gurvits' method, offers a route to approximate operator scalings efficiently when existing parameters allow for polynomial-time computation.

Operator scaling, particularly under specified marginals, finds relevance across numerous domains including Brascamp-Lieb inequalities, communication complexity, eigenvalues of Hermitian matrix sums, and quantum information theory. Certain established theorems in these fields, traditionally lacking algorithmic proofs, now arise as direct corollaries of the characterization properties described. This paper bridges known results to computational practice, introducing algorithmic solutions for spectral sum problems involving Hermitian matrices and extending Forster's theorem on radial isotropic position.

Background and Capacity Analysis

Mathematical Preliminaries

The paper begins by defining completely positive maps and their associated dual transformations. A completely positive map $T:$ maps $n\times n$ matrices to $m\times m$ matrices and possesses a counterpart $T^*$. Such maps are pivotal when discussing operators that preserve positive-semidefiniteness analogously to matrix settings.

A crucial aspect is the scaling transformation $T'$ determined by invertible maps $(g,h)$, transforming $T:X \mapsto g^\dagger T(h X h^\dagger) g$, raising the question of conditions for achieving doubly stochastic states via operator manipulations.

Characterizing Scalable Operators

Operator scaling's primary task answers which completely positive maps can meet specified marginal constraints. Following Gurvits' examination, this paper extends operator scaling, proposing characterization theorems that validate algorithm feasibility for reaching targets defined by $(I_n \to Q, I_m \to P)$ conditions. The characterizations delineate properties such as $(P,Q)$-rank nondecreasingness and capacity utilization.

Reduction from Truncated Environments

Upon addressing specified marginals, reductions and truncations offer insight into how modifications affect scalability, converting operators into simpler forms that provoke equivalence in scaling potential. These reductions provide a path to apply Gurvits' conditions and verify operator compatibility in reaching the desired state with relative simplicity given integral inputs.

Algorithmic Framework

Sinkhorn-Knopp Invocation

An algorithm similarly structured to Sinkhorn-Knopp iteratively scales operators, adapting Gurvits' procedures to specific marginal constraints. Capacity under this framework undergoes examination, optimizing steps and proving theoretical bounds for operational effectiveness. This iterative approach marks significant progress in matrices meeting configurational requirements, i.e., reaching target row and column sums effectively within polynomial confines.

Capacity Bounds

Improving initial bounds from polynomial extensions of Gurvits’ estimate, utilizing variance analysis and mutual progress considerations, offers advantages. Significant numerical bounds emerge from optimizations concerning scaling integers to rational values, enabling the retention of operational capacity amidst transformation stress.

Applications and Further Research

Transformational Implications

The outcomes from specified marginal scaling algorithms verify summed eigenvalue determinations by emerging applications, proving pivotal in quantum-based computational structures and theoretical hard drives in invariant theories.

Development Prospects

The structural integrity laid by operator scaling positions further inquiry into fast-tracking polynomial-time decision problems, notably broadening spectrum position determinations and enabling algorithms that boast efficient alternatives to long-standing combinatorial solutions.

Conclusion

With a comprehensive treatment of operator scaling under specified marginals, this research enriches computational mathematics, offering verifiable techniques curbed by theoretical principles. Ultimately, this work lends credibility and computational mechanics to a domain imbued with potential, facilitating contemporary applications across diverse mathematical landscapes. The precise mapping and reduction tactics outlined herein reinforce the pursuit of practical tools amid hard computational problems.