Einstein–Podolsky–Rosen paradox

The EPR argument uses correlations of an entangled state and a locality assumption to challenge the completeness of the quantum description. Its Einstein–Podolsky–Rosen criterion of reality identifies quantities predictable with certainty without disturbing a distant system as elements of physical reality.

For a shared quantum entanglement resource, one party's choice of measurement can prepare different conditional ensembles for the distant party. Discarding the outcome leaves the same distant reduced density matrix, so steering does not itself permit signalling. Demonstrating steering against a local hidden-state model is stronger than merely observing a change in a postselected description.

For a Bell pair $|{\varphi }_{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$, Alice's basic measurement in the entrywise complex conjugate of any desired Bob orthonormal basis prepares each Bob pure state with probability $1/2$. Indeed, if $|u⟩={\sum }_{j=0}^1{u}_j|j⟩$, then $⟨\overline{u}|j⟩=u_j$, which proves the displayed contraction identity. The undisclosed-outcome density matrix stays $I/2$; quantum steering changes the conditional quantum state ensemble without violating the no-communication theorem.

The Einstein–Podolsky–Rosen criterion says that if one can predict a physical quantity with certainty without disturbing the system, then there is an element of physical reality corresponding to that quantity.