Suppose we have a particle pair (A, B) which are maximally entangled. Then A goes cavorting with some macroscopic system in a thermodynamically irreversible way, and becomes highly entangled with it. Does this entanglement with the new system reduce the entanglement between A and B? At first I want to say no, because if we were to measure a maximally entangled quanta and found it spinning one way (for example) we can immediately deduce the direction of spin for its partner. But since a measurement is just a process of sufficient interaction to produce entanglement with the measured quanta, and because only two systems may be maximally entangled with one another, I imagine that the degree of entanglement between A and B must now be something less than maximal. What am I missing?