Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 34 / 5

The generalized measurement postulate specifies Kraus operators $M_{i\alpha }$, where $i$ is the recorded outcome and $\alpha$ permits unobserved alternatives for that outcome, satisfyingFor a density operator $\rho$, outcome $i$ has probability and conditional stateWith one operator per outcome, this is the usual formula $p_i=\mathrm{Tr}(M_i\rho M_i^{†})$, ${\rho }_i=M_i\rho M_i^{†}/p_i$. The positive operators $E_i={\sum }_{\alpha }{M}_{i\alpha }^{†}{M}_{i\alpha }$ form a POVM, while the state-update maps form a quantum instrument.

The full postulate reduces to the usual projective measurement with the Lüders update when there is one operator per outcome and $M_i=P_i$, with $P_i=P_i^{†}=P_i^2$, $P_i{P}_j=0$ for $i\ne j$, and ${\sum }_i{P}_i=I$. Then $p_i=\mathrm{Tr}(P_i\rho )$ and ${\rho }_i=P_i\rho P_i/p_i$. Projective effects $E_i=P_i$ alone ensure these probabilities but do not fix the state update: an additional outcome-dependent unitary could still be applied. The choice $M_i=P_i$ makes the required projective state-update condition explicit.

The unread measurement is the pinching map $\mathcal{P}(\rho )={\rho }^{\prime }={\sum }_i{P}_i\rho P_i$, a nonselective projective measurement. Its output is block diagonal and commutes with each $P_i$, so $\log {\rho }^{\prime }$ also commutes with the projectors on its support.

First the quantum relative entropy $D(\rho \Vert {\rho }^{\prime })$ is finite. If $v$ lies in the kernel of ${\rho }^{\prime }$, positivity givesEvery summand is zero; summing ${\rho }^{1/2}{P}_{i}v=0$ gives ${\rho }^{1/2}v=0$. Thus $\ker {\rho }^{\prime }\subseteq \ker \rho$, equivalently $\mathrm{supp}\rho \subseteq \mathrm{supp}{\rho }^{\prime }$.

The block-diagonal logarithm obeys ${\sum }_i{P}_i(\log {\rho }^{\prime })P_i=\log {\rho }^{\prime }$ on this support. Cyclicity of the trace therefore givesConsequently the relative entropy of a pinched state is exactly the entropy increase:By the proved nonnegativity of quantum relative entropy and its equality condition,Equivalently, equality holds exactly when all off-diagonal blocks $P_i\rho P_j$ for $i\ne j$ already vanish, or when $\rho$ commutes with every measurement projector. This argument proves the equality condition even when the projectors are degenerate and the states singular.

The orthogonal projection onto the $+1$ eigenspace of the Pauli matrix ${\sigma }_z$ is $P_{+}=(I+{\sigma }_z)/2$. By the Born rule and the Bloch vector representation,Since $s_z=1/5$,The given vector has length $19/30<1$, so it lies inside the Bloch ball and indeed defines a valid mixed qubit state. For the physical spin operator $S_z=(\hslash /2){\sigma }_z$, the corresponding outcome is $+\hslash /2$ with the same probability.