Past exam of the quantum information course of the University of Cambridge / 2013 / Paper 1 / 3

Quantum complementarity concerns experimental arrangements or observables that cannot be combined into one sharp classical description. For a qubit, the Pauli operators $X$ and $Z$ have incompatible eigenbases: a definite $Z$ value gives equal probabilities for the two $X$ outcomes. One can measure either observable sharply on a given preparation, but a joint measurement cannot reproduce both sharp projective measurements on every state. Complementarity is not merely ignorance of two simultaneously definite quantum properties.

In a balanced two-path interferometer, which-path information reduces the interference available when the paths recombine. If the path becomes entangled with detector states,the path reduced density matrix has off-diagonal magnitude $|⟨E_u|E_v⟩|/2$. Its interference visibility is $V=|⟨E_u|E_v⟩|$. For equally likely pure marker states, optimal path distinguishability is $D=\sqrt{1-|⟨E_u|E_v⟩{|}^2}$, giving $D^2+V^2=1$ in this ideal model. Identical marker states preserve full interference; orthogonal ones permit perfect path discrimination and erase the unconditional fringes.

A concrete optical realization puts different polarizations on the two arms of a Mach-Zehnder interferometer. A detector that ignores polarization sees reduced interference because polarization records the path. Projecting the outgoing light onto a common polarization erases that information in the selected subensemble and can recover conditional fringes; summing all outcomes again removes them. This is a physical change in which alternatives remain distinguishable, not a retrospective alteration of earlier detections.

Quantum decoherence arises when different system alternatives correlate with distinguishable environmental states. For an initially uncorrelated state, a measurement-like unitary time evolution can produceTaking the environmental partial trace givesThe environmental overlap suppresses locally observable quantum coherence. Scattered photons, surrounding gas and uncontrolled apparatus variables can carry away such records. The interaction often singles out robust states forming a pointer basis. Decoherence can be much faster than thermal energy relaxation because recording a macroscopic distinction need not transfer much energy.

The overall state can remain pure and evolve unitarily even when the system alone looks like a classical mixture. Discarding environmental information is therefore not, by itself, a proof of fundamental wavefunction collapse or a solution to the definite-outcome measurement problem. A small controlled environment can return coherence, as in the single-qubit environment dephasing model; practical irreversibility arises when recovering the many dispersed records becomes inaccessible. In an interferometer, varying gas pressure or an independently controllable path marker tests the dependence of fringe contrast on this environmental coupling. Quantum computation must suppress or correct such unwanted system–environment entanglement to preserve the intended interference.

Quantum cryptography includes quantum key distribution, which uses quantum transmissions and authenticated public discussion to establish a secret random key. In BB84, the sender independently chooses a random bit and either the computational or Hadamard basis, sending one of $|0⟩,|1⟩,|+⟩,|-⟩$. A receiver independently chooses one of those two bases. They announce bases after receipt, retain matching-basis positions and reveal a random sample of those bits to estimate noise or eavesdropping. Ideal uniform independent basis choices retain about half of the detected signals before testing and further processing.^[ref]

The no-cloning theorem prevents an adversary from copying arbitrary unknown nonorthogonal signal states perfectly. In a simple intercept–resend attack, the adversary chooses a random protocol basis. On half of the sifted positions it is the wrong basis, and the resent state then yields an erroneous receiver bit with probability one half. Thus the BB84 intercept-resend error rate isif every signal is intercepted, or $Q=q/4$ when an independently selected fraction $q$ is attacked this way. This is an illustrative attack calculation, not a security proof against all possible quantum attacks.

After parameter estimation, classical error reconciliation aligns the remaining keys; privacy amplification compresses them to remove the adversary's bounded information, including information leaked during public reconciliation. The classical channel requires message authentication, otherwise an attacker could impersonate each party to the other. Real sources, losses, multiphoton pulses and detector behavior must fit the security model. A low observed error rate alone is insufficient to certify arbitrary hardware. Finally the protocol distributes a key; encrypting a later message is a separate step.