When Do Composed Maps Become Entanglement Breaking?

Abstract

For many completely positive maps repeated compositions will eventually become entanglement breaking. To quantify this behaviour we develop a technique based on the Schmidt number: If a completely positive map breaks the entanglement with respect to any qubit ancilla, then applying it to part of a bipartite quantum state will result in a Schmidt number bounded away from the maximum possible value. Iterating this result puts a successively decreasing upper bound on the Schmidt number arising in this way from compositions of such a map. By applying this technique to completely positive maps in dimension three that are also completely copositive we prove the so called PPT squared conjecture in this dimension. We then give more examples of completely positive maps where our technique can be applied, e.g.~maps close to the completely depolarizing map, and maps of low rank. Finally, we study the PPT squared conjecture in more detail, establishing equivalent conjectures related to other parts of quantum information theory, and we prove the conjecture for Gaussian quantum channels.

arXiv comment24 pages, no pictures

References in corpus (32)

• Quantum entanglement
• Separability of Mixed States: Necessary and Sufficient Conditions
• Separability criterion and inseparable mixed states with positive partial transposition
• General Entanglement Breaking Channels
• Entanglement Measures under Symmetry
• A Schmidt number for density matrices
• Secure key from bound entanglement
• Largest separable balls around the maximally mixed bipartite quantum state
• Schmidt number witnesses and bound entanglement
• Separability in 2xN composite quantum systems
• The asymptotic relative entropy of entanglement
• Cones of positive maps and their duality relations
• Separability from Spectrum for Qubit-Qudit States
• Limitations on Quantum Key Repeaters
• Bipartite depolarizing channels
• Activating NPPT distillation with an infinitesimal amount of bound entanglement
• Entanglement of the Antisymmetric State
• High-Dimensional Entanglement in States with Positive Partial Transposition
• Entanglement-annihilating and entanglement-breaking channels
• Extending quantum operations
• Local two-qubit entanglement-annihilating channels
• Positivity of linear maps under tensor powers
• Eventually Entanglement Breaking Maps
• Duality of cones of positive maps
• Private states, quantum data hiding and the swapping of perfect secrecy
• Quantifying the noise of a quantum channel by noise addition
• Entanglement-Breaking Indices
• Entanglement-Saving Channels
• The PPT square conjecture holds generically for some classes of independent states
• Amendable Gaussian channels:restoring entanglement via a unitary filter
• Decomposability of Linear Maps under Tensor Products
• All 2-positive linear maps from M3 to M3 are decomposable

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• The positive partial transpose conjecture for n=3
• Asymptotics of quantum channels
• Positive maps and trace polynomials from the symmetric group
• Eventually entanglement breaking Markovian dynamics: structure and characteristic times
• The PPT${}^2$ conjecture holds for all Choi-type maps
• Entanglement-breaking superchannels
• Decomposability of Linear Maps under Tensor Products
• Annihilating Entanglement Between Cones
• Mapping cone of $k$-Entanglement Breaking Maps
• Asymptotic Properties for Markovian Dynamics in Quantum Theory and General Probabilistic Theories
• Symmetric quantum states: a review of recent progress
• Separability for mixed states with operator Schmidt rank two
• All incompatible measurements on qubits lead to multiparticle Bell nonlocality
• Random covariant quantum channels
• Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps
• Entanglement in the Dicke subspace
• $k$-Positive Maps: New Characterizations and a Generation Method
• On PPT Square Conjecture
• Schoenberg Correspondence for $k$-(Super)Positive Maps on Matrix Algebras
• Some Entanglement Survives Most Measurements
• A Multiplicative Ergodic Theorem for Bistochastic Ergodic Quantum Processes with Applications to Entanglement