 
Summary: A Quantum to Classical Phase Transition in Noisy Quantum Computers
Dorit Aharonov
The fundamental problem of the transition from quantum to classical physics is usually
explained by decoherence, and viewed as a gradual process. The study of entanglement, or
quantum correlations, in noisy quantum computers implies that in some cases the transition
from quantum to classical is actually a phase transition. We dene the notion of entanglement
length in ddimensional noisy quantum computers, and show that a phase transition in entan
glement occurs at a critical noise rate, where the entanglement length transforms from innite
to nite. Above the critical noise rate, macroscopic classical behavior is expected, whereas
below the critical noise rate, subsystems which are macroscopically distant one from another
can be entangled.
The macroscopic classical behavior in the supercritical phase is shown to hold not only
for quantum computers, but for any quantum system composed of macroscopically many 
nite state particles, with local interactions and local decoherence, subjected to some additional
conditions. This phenomenon provides a possible explanation to the emergence of classical be
havior in such systems. A simple formula for an upper bound on the entanglement length of any
such system in the supercritical phase is given, which in principle can be tested experimentally.
I. INTRODUCTION
Quantum computation is a fascinating subject which manifests the peculiarities of quantum mechanics and
uses them in order to achieve an advantage in terms of computational power over classical computers. Shor's
