Catalytic majorization and ${\ell }_p$ norms

Abstract

An important problem in quantum information theory is the mathematical characterization of the phenomenon of quantum catalysis: when can the surrounding entanglement be used to perform transformations of a jointly held quantum state under LOCC (local operations and classical communication) ? Mathematically, the question amounts to describe, for a fixed vector $y$, the set $T(y)$ of vectors $x$ such that we have $x\otimes z\prec y\otimes z$ for some $z$, where $\prec$ denotes the standard majorization relation. Our main result is that the closure of $T(y)$ in the ${\ell }_1$ norm can be fully described by inequalities on the ${\ell }_p$ norms: $\Vert x{\Vert }_p\le \Vert y{\Vert }_p$ for all $p\ge 1$. This is a first step towards a complete description of $T(y)$ itself. It can also be seen as a ${\ell }_p$-norm analogue of Ky Fan dominance theorem about unitarily invariant norms. The proofs exploits links with another quantum phenomenon: the possibiliy of multiple-copy transformations ($x^{\otimes n}\prec y^{\otimes n}$ for given $n$). The main new tool is a variant of Cramér$ theorem on large deviations for sums of i.i.d. random variables.

References in corpus (3)

• Multiple-copy entanglement transformation and entanglement catalysis
• Inequalities that Collectively Completely Characterize the Catalytic Majorization Relation
• Stochastic domination for iterated convolutions and catalytic majorization

Cited by in corpus (30)

• Catalytic quantum teleportation and beyond
• Quantum thermodynamics of correlated-catalytic state conversion at small-scale
• The structure of Renyi entropic inequalities
• Resource convertibility and ordered commutative monoids
• Catalytic Transformations of Pure Entangled States
• Catalysis of entanglement and other quantum resources
• Correlation in Catalysts Enables Arbitrary Manipulation of Quantum Coherence
• Entropy and reversible catalysis
• Random quantum channels II: Entanglement of random subspaces, Renyi entropy estimates and additivity problems
• Fundamental Limits on Correlated Catalytic State Transformations
• Majorization relations and entanglement generation in a beam splitter
• Extending a characterization of majorization to infinite dimensions
• Qubit channels with small correlations
• Arbitrary Amplification of Quantum Coherence in Asymptotic and Catalytic Transformation
• Local convertibility of the ground state of the perturbed Toric code
• Catalytic Transformations in Coherence Theory
• Two constructive proofs on d-majorization and thermo-majorization
• Stochastic domination for iterated convolutions and catalytic majorization
• Entanglement catalysis for quantum states and noisy channels
• Catalysis always degrades external quantum correlations
• The multiplicative property characterizes ${\ell }_p$ and $L_p$ norms
• Monotone additive statistics
• Matrix majorization in large samples
• Dirichlet polynomials, Majorization, and Trumping
• Revisiting Pure State Transformations with Zero Communication
• Quantum thermodynamics with coherence: Covariant Gibbs-preserving operation is characterized by the free energy
• Matrix Majorization in Large Samples with Varying Support Restrictions
• A finite sufficient set of conditions for catalytic majorization
• Refining Ky Fan's majorization relation with linear programming
• Symmetric polynomials and $l^p$ inequalities for certain intervals of $p$