[Paper Review] Rank Two Bipartite Bound Entangled States Do Not Exist

Paweł Horodecki, John A. Smolin+2 more|ArXiv.org|Oct 29, 1999

Quantum Information and Cryptography4 citations

TL;DR

This paper proves that rank two bipartite quantum states cannot exhibit bound entanglement, a form of entanglement that cannot be distilled into maximally entangled states via local operations and classical communication. Using rank constraints and marginal rank analysis, the authors establish that any bound entangled state must have support on at most an n×n Hilbert space, leading to the conclusion that no such state of rank two exists, resolving a long-standing open question in quantum information theory.

ABSTRACT

We explore the relation between the rank of a bipartite density matrix and the existence of bound entanglement. We show a relation between the rank, marginal ranks, and distillability of a mixed state and use this to prove that any rank n bound entangled state must have support on no more than an n imes n Hilbert space. A direct consequence of this result is that there are no bipartite bound entangled states of rank two. We also show that a separability condition in terms of a quantum entropy inequality is associated with the above results. We explore the idea of how many pure states are needed in a mixture to cancel the distillable entanglement of a Schmidt rank n pure state and provide a lower bound of n-1. We also prove that a mixture of a non-zero amount of any pure entangled state with a pure product state is distillable.

Motivation & Objective

• To investigate the structural constraints on bipartite quantum states with low rank and their potential to exhibit bound entanglement.
• To resolve the open question of whether bound entangled states can exist with rank two.
• To establish a connection between the rank of a density matrix, its marginal ranks, and the distillability of entanglement.
• To provide a theoretical foundation for understanding the minimal rank required for bound entanglement in bipartite systems.

Proposed method

• Analyzing the relationship between the rank of a bipartite density matrix and the ranks of its reduced density matrices (marginal ranks).
• Deriving a necessary condition for distillability based on rank and marginal rank constraints.
• Applying a quantum entropy inequality to characterize separability and entanglement in low-rank states.
• Using a contradiction argument to show that a rank two state cannot support bound entanglement if it is not distillable.
• Proving that any bound entangled state must be supported on a Hilbert space of dimension at most n×n for a rank-n state.
• Demonstrating that mixing any non-zero amount of a pure entangled state with a product state results in a distillable state.

Experimental results

Research questions

• RQ1 Can a bipartite quantum state of rank two be bound entangled, i.e., entangled but not distillable?
• RQ2 What is the minimal Hilbert space dimension required for a bound entangled state of rank n?
• RQ3 How do the ranks of the density matrix and its marginals constrain the distillability of entanglement?
• RQ4 Is there a fundamental lower bound on the number of pure states needed to cancel the distillable entanglement of a Schmidt rank n pure state?
• RQ5 Does any mixture of a non-zero amount of a pure entangled state with a product state remain distillable?

Key findings

• There are no bipartite bound entangled states of rank two, as proven by rank and marginal rank constraints.
• Any bound entangled state of rank n must have support on a Hilbert space of dimension at most n×n.
• A necessary condition for distillability is derived from the interplay between the global rank and marginal ranks of a density matrix.
• The number of pure states required to cancel the distillable entanglement of a Schmidt rank n pure state is at least n−1.
• A mixture of any non-zero amount of a pure entangled state with a pure product state is always distillable.
• A quantum entropy inequality provides a separability criterion consistent with the rank-based constraints on bound entanglement.