Past exam of the quantum information course of the University of Cambridge / 2012 / Paper 1 / 3

A positive operator-valued measure consists of positive semidefinite effects $E_r$ with ${\sum }_r{E}_r=I$. Its outcome probabilities are $p_r=\mathrm{Tr}(\rho E_r)$, which are nonnegative and sum to one. Unlike a projective measurement, its effects need not be mutually orthogonal projections, and there can be more outcomes than the system's Hilbert-space dimension.

For example, choose three equatorial Bloch vectors ${\mathbf{n}}_r$ separated by $120^{\circ }$ and set $E_r=(I+{\mathbf{n}}_r\cdot \mathbf{\sigma })/3$. Each effect has eigenvalues zero and $2/3$, and the vectors sum to zero, so the three effects sum to $I$. This trine qubit POVM gives $p_r=(1+\mathbf{s}\cdot {\mathbf{n}}_r)/3$ for input Bloch vector $\mathbf{s}$. It cannot be a three-outcome orthogonal projective measurement on that qubit.

A physical implementation can couple the system to an ancilla and projectively measure the enlarged system. More generally, operators $M_{r\alpha }$ implement an outcome with $E_r={\sum }_{\alpha }{M}_{r\alpha }^{†}{M}_{r\alpha }$; the normalized conditional state is ${\sum }_{\alpha }{M}_{r\alpha }\rho M_{r\alpha }^{†}/p_r$. Thus a POVM does not determine the post-measurement state: different measurement instruments can have the same outcome probabilities and different disturbance. Generalized measurements are useful for state discrimination and realistic detection, including explicitly modeled inconclusive outcomes.

Quantum cryptography includes quantum key distribution, which establishes a shared secret key rather than transmitting an arbitrary plaintext directly. In BB84, Alice randomly prepares one of the two computational-basis or two complementary-basis qubit states; Bob randomly measures in either basis. They announce bases over an authenticated classical channel and retain the positions where their choices agree, about half in the ideal unbiased case. They test a subset for errors, reconcile the remaining strings, and apply privacy amplification to remove a bounded eavesdropper's information. A protocol can abort when the observed data do not support a secret key.^[ref]

Nonorthogonal states cannot be perfectly copied by a state-independent device, by the no-cloning theorem. An intercept-resend attacker who measures each signal in a random BB84 basis chooses the wrong basis half the time. On those sifted positions Bob has a one-half error probability, giving the ideal BB84 intercept-resend error rate $1/4$. This simple calculation demonstrates detectable disturbance for that attack; it is not a complete security proof for arbitrary attacks.

Authentication is essential: otherwise an adversary can impersonate both participants. Error correction reveals information that must be included in the privacy budget, and losses, finite sample sizes, multiphoton emission and detector behavior belong in the security model. Entanglement-based variants can use Bell correlations, but observing some entanglement or a Bell violation alone does not replace a quantitatively specified key-distillation protocol. The final key can subsequently be used in a suitable classical encryption scheme.

For quantum teleportation, Alice holds an unknown input $|\psi {⟩}_1$ and one member of a shared $|{\Phi }^{+}{⟩}_{23}$ pair; Bob holds qubit 3. Alice applies a CNOT gate from 1 to 2 and a Hadamard gate on 1, then measures both in the computational basis. Expansion gives the Bell-basis teleportation identityEvery outcome pair $(m,n)$ has probability $1/4$, independently of the input. Alice sends these two classical bits, and Bob appliesThe order of the two corrections matters only by a global sign for the doubly nonzero outcome. The transformation also transfers entanglement with an untouched reference, because the identity is linear in the input amplitudes.

Before receiving the bits, Bob must average over all four Pauli-related states, obtaining $I/2$ for any input. Thus teleportation requires classical communication and does not send information faster than light. The original input is consumed by Alice's measurement, so no second copy remains and the no-cloning theorem is respected. Ideal qubit teleportation consumes one shared Bell pair and two classical bits, as well as local gates and measurements. Imperfect entanglement, gate error, detection error or lost classical records reduce practical fidelity; teleportation transfers the quantum state, not the underlying material particle.