Strict Positivity and $D$-Majorization

Abstract

Motivated by quantum thermodynamics we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of such maps to the identity channel is lower bounded by one. The notion of strict positivity comes in handy when generalizing the majorization ordering on real vectors with respect to a positive vector $d$ to majorization on square matrices with respect to a positive definite matrix $D$. For the two-dimensional case we give a characterization of this ordering via finitely many trace norm inequalities and, moreover, investigate some of its order properties. In particular it admits a unique minimal and a maximal element. The latter is unique as well if and only if minimal eigenvalue of $D$ has multiplicity one.

arXiv commentSupersedes arXiv:2003.04164

References in corpus (15)

• Fundamental limitations for quantum and nano thermodynamics
• The second laws of quantum thermodynamics
• The resource theory of informational nonequilibrium in thermodynamics
• Dividing Quantum Channels
• Contractivity of positive and trace preserving maps under $L_p$ norms
• Information preserving structures: A general framework for quantum zero-error information
• A quantum version of Wielandt's inequality
• Ergodic and Mixing Quantum Channels in Finite Dimensions
• Extending quantum operations
• Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states
• Decompositions of Hilbert Spaces, Stability Analysis and Convergence Probabilities for Discrete-Time Quantum Dynamical Semigroups
• Physical transformations between quantum states
• Unitary Dilations of Discrete-Time Quantum-Dynamical Semigroups
• Extension of the Alberti-Uhlmann criterion beyond qubit dichotomies
• The C-Numerical Range in Infinite Dimensions

Cited by in corpus (8)

• The $d$-Majorization Polytope
• Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources
• Which Bath-Hamiltonians Matter for Thermal Operations?
• Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe
• The Thermomajorization Polytope and Its Degeneracies
• Can Čencov meet Petz?
• Thermodynamic state convertibility is determined by qubit cooling and heating
• On the Alberti-Uhlmann Condition for Unital Channels