Fragile entanglement statistics

TL;DR

This paper demonstrates that certain highly entangled quantum states can exhibit seemingly independent measurement outcomes for subsets of particles, yet are fragile in their entanglement, as removing one particle destroys the entanglement.

Contribution

It constructs explicit examples of N-particle GHZ states with observables showing subset independence despite overall entanglement, highlighting the fragility of such entangled states.

Findings

Identified observables with independent outcomes for subsets of particles

Showed that the entanglement is fragile and easily destroyed by tracing out a single particle

Provided explicit construction of such states in the GHZ framework

Abstract

If X and Y are independent, Y and Z are independent, and so are X and Z, one might be tempted to conclude that X, Y, and Z are independent. But it has long been known in classical probability theory that, intuitive as it may seem, this is not true in general. In quantum mechanics one can ask whether analogous statistics can emerge for configurations of particles in certain types of entangled states. The explicit construction of such states, along with the specification of suitable sets of observables that have the purported statistical properties, is not entirely straightforward. We show that an example of such a configuration arises in the case of an N-particle GHZ state, and we are able to identify a family of observables with the property that the associated measurement outcomes are independent for any choice of 2, 3, ..., N-1 of the particles, even though the measurement outcomes for all N particles are not independent. Although such states are highly entangled, the entanglement turns out to be 'fragile', i.e. the associated density matrix has the property that if one traces out the freedom associated with even a single particle, the resulting reduced density matrix is separable.