# Quantum entanglement and Bekenstein Bound

In this article Bekenstein writes that the amount of information could store in a sphere with diameter of a cm is ~$$10^{66}$$ bits. If one were to take a set of 1000 entangled electrons and put them into a sphere of about a cm in diameter, due to entanglement a complete description of there spin would be ~$$2^{1000}$$ continuous variables. How can both of these facts be true?

As expressed in the question, the Bekenstein bound is a bound on the number of bits — that is, a bound on the entropy, which is the logarithm of the number of mutually orthogonal microstates. ("Orthogonal" here refers to the inner product between state-vectors in the Hilbert space.) A collection of 1000 electron spins provides only 1000 (qu)bits, corresponding to $$2^{1000}$$ mutually orthogonal microstates. This fits comfortably within the Bekenstein bound. The Bekenstein bound says that the maximum number of mutually orthogonal microstates is $$\exp(S)$$ with $$S\sim 10^{66}$$. Our technology won't be running up against that bound anytime soon.