Citation: Braunstein, Samuel L., et al. "Separability of very noisy mixed states and implications for NMR quantum computing." Physical Review Letters 83.5 (1999): 1054.
Web: https://arxiv.org/abs/quant-ph/9811018
Tags: Foundational, No-go, Approximate-structure
In this early paper, the authors demonstrate that every state in a small ball around the maximally mixed state is separable, in the sense that it is a convex combination of product states. The way they demonstrate this as follows. First, they decompose a given density matrix into Pauli matrices. Then, they split every Pauli matrix as a sum of the operators 1/2(1+sigma_i)+1/2(1-sigma_i) to write the density matrix as a combinations of tensor products of terms 1/2(1+sigma_i) and 1/2(1-sigma_i). The magic is that when you do this procedure with the identity matrix, you get the even mixture of every term. This means that if you take a linear combination with any other element, then for points sufficiently close to the maximally mixed state all of the coefficients will be positive. Moreover, this cutoff can be chosen uniformly since the coefficients are uniformly lower-bounded by general considerations.
The leading quantum computing technology at the time was NMR, which operated at high temperatures, and prepared states close to the maximally mixed state. This article shows that such states are all separable, and thus the computers were not making any real probes at the nature of entanglement (namely, you could not use them to test Bell-inequality violations).
This motivates the definition of "robustness" of a state - how much maximally mixed state one needs to add before the state becomes separable, as explored by Vidal and Tarrach:
> Vidal, Guifré, and Rolf Tarrach. "Robustness of entanglement." Physical Review A 59.1 (1999): 141.
A key takeaway from this work was summarized by Verstraete as follows: ". Mixing a barely entangled state with an amount of noise proportional to the entanglement destroys the entanglement".