Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 48 / 5

In the generalized measurement postulate, outcome $m$ is described by a Kraus operator $M_m$, with ${\sum }_m{M}_m^{†}{M}_m=I$. On a density operator $\rho$ its probability and conditional output, for nonzero probability, areFor a pure input the conditional vector is $M_m|\psi ⟩/\sqrt{p_m}$. The effects $E_m=M_m^{†}{M}_m$ are positive and sum to the identity, defining a positive operator-valued measure.

More generally an outcome can have several Kraus operators $M_{m\alpha }$. Then sum over $\alpha$ in both the numerator and the probability, and require ${\sum }_{m,\alpha }{M}_{m\alpha }^{†}{M}_{m\alpha }=I$.

The usual projective measurement with the Lüders rule is recovered when the measurement operators are mutually orthogonal projections, $M_m=P_m=P_m^{†}=P_m^2$, $P_m{P}_n=0$ for $m\ne n$, and ${\sum }_m{P}_m=I$. Projector-valued effects alone give projective outcome statistics; to obtain the Lüders state update as well, the instrument must implement $\rho \mapsto P_m\rho P_m$, without an additional conditional state change.

The Schmidt decomposition of a normalized bipartite pure state isThe two sets of local vectors are orthonormal. The Schmidt coefficients are the nonnegative numbers $\sqrt{{\lambda }_j}$, and the Schmidt number for this pure state, also called its Schmidt rank, is $r$, the number of nonzero coefficients. It equals the rank of either reduced density operator and of the coefficient matrix in any product bases. The general mixed-state Schmidt number specializes to this same value on a pure state.

In the computational product bases, the coefficient matrix isIts determinant is $-1/3$, so it has rank two. Therefore the Schmidt number is $\boxed{2}$. As a check, the reduced density operator $C{C}^{†}=\frac{1}{3}\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ has eigenvalues $(3+\sqrt{5})/6$ and $(3-\sqrt{5})/6$, both positive.

The two displayed amplitudes have modulus $1/\sqrt{2}$. Absorb their phases into Bob's basis rather than use complex Schmidt coefficients. Take $|u_0⟩=|0⟩$, $|u_1⟩=|1⟩$, andBoth local bases are orthonormal. A Schmidt decomposition is thereforeThe two Schmidt coefficients are $1/\sqrt{2}$, the Schmidt number is two, and both reduced density operators are $I/2$. Their equal eigenvalues make this decomposition nonunique.