Mixed-state twin observables
  Twin observables, i.e. opposite subsystem observables A+ and A- that are indistinguishable in measurement in a given mixed or pure state W, are investigated in detail algebraicly and geometrically. It is shown that there is a far-reaching correspondence between the detectable (in W) spectral entities of the two operators. Twin observables are state-dependently quantum-logically equivalent, and direct subsystem measurement of one of them ipso facto gives rise to the indirect (i.e. distant) measurement of the other. Existence of nontrivial twins requires singularity of W. Systems in thermodynamic equilibrium do not admit subsystem twins. These observables may enable one to simplify the matrix representing W. 
F. Herbut and M. Damnjanovic