Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 325 / 1

For a bipartite density operator ${\rho }_{SA}$, the reduced density matrix of $S$ isIt is characterized byfor every observable $M_S$ on $S$. Equivalently, in any orthonormal basis $\{|a_k⟩\}$ of $A$, the partial trace isand the result is independent of that basis.

Let the apparatus begin in a ready state $|A_0⟩$. An ideal unitary measurement interaction is defined on the relevant subspace byThus an initial $|\varphi ⟩={\sum }_i{a}_i|e_i⟩$ evolves to the entangled stateand orthogonality of the pointer states gives

Before the interaction, the Born rule givesAfterward,The missing cross term is the lost interference between the $i$ and $j$ branches. Entanglement with orthogonal pointer records therefore explains quantum decoherence and the appearance of a classical mixture to the subsystem. It does not solve the quantum measurement problem: unitary evolution alone does not explain why one definite pointer value is observed.

The Einstein–Podolsky–Rosen criterion of reality says that if a physical quantity can be predicted with certainty without disturbing a system, then an element of physical reality corresponds to that quantity.

Let $X_i,Y_i$ be Pauli measurements on particle $i$. The GHZ state in the question obeyswith certainty. Each local outcome can therefore be predicted by spacelike-separated measurements on the other two particles. The EPR criterion assigns predetermined values $x_i,y_i\in \{-1,1\}$ satisfying those three equations. Multiplying them and using $y_i^2=1$ givesQuantum theory instead predictsso a joint $X$ measurement has product $-1$ with certainty. The EPR elements of reality therefore contradict the quantum prediction, which is the GHZ theorem form of Bell theorem.