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A Quantum to Classical Phase Transition in Noisy Quantum Computers

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 define the notion of entanglement length in $d$-dimensional noisy quantum computers, and show that a phase transition in entanglement occurs at a critical noise rate, where the entanglement length transforms from infinite to finite. 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 super-critical phase is shown to hold not only for quantum computers, but for any quantum system composed of macroscopically many finite state particles, with local interactions and local decoherence, subjected to some additional conditions. This phenomenon provides a possible explanation to the emergence of classical behavior in such systems. A simple formula for an upper bound on the entanglement length of any such system in the super-critical phase is given, which can be tested experimentally.