Past exam of the quantum information course of the University of Cambridge / 2014 / Paper 1 / 3

Consistent histories assigns probabilities to suitably compatible sequences of quantum alternatives without making an external measuring apparatus fundamental. At successive times choose complete orthogonal sets of projection operators. In the Heisenberg picture, a history has class operator $C_{\alpha }=P_{{\alpha }_n}(t_n)\cdots P_{{\alpha }_1}(t_1)$. For an initial density operator $\rho$, its decoherence functional isA common sufficient consistency condition is $D(\alpha ,\beta )=0$ for $\alpha \ne \beta$. Then $p(\alpha )=D(\alpha ,\alpha )$ is nonnegative and coarse-grained alternatives have additive probabilities: the interference cross-terms vanish. The weaker requirement on real cross-terms can suffice for individual probability sum rules. Predictions must stay within one compatible framework; mutually incompatible history families cannot simply be combined into one classical account.

The many-worlds interpretation instead treats unitary quantum evolution as universally valid and interprets the correlated outcome branches as components of one global state. A measurement interaction givesWhen the environment records become nearly orthogonal, quantum decoherence suppresses observable interference between stable apparatus records. Each relative state contains an observer correlated with a definite record; no additional physical collapse is postulated. Branch structure is an emergent approximation, rather than a unique literal splitting at every arbitrary choice of basis.

Both approaches use the ordinary unitary formalism and decoherence, but their interpretations of histories and branches differ. Decoherence alone does not prove that one unique branch is selected, nor does the unitary equation by itself settle every assumption needed to justify Born rule probabilities. Neither interpretation predicts ordinary interference experiments incorrectly merely by interpreting the same state differently.

In quantum teleportation, Alice transfers an arbitrary unknown qubit $|\chi ⟩=\alpha |0⟩+\beta |1⟩$ using one shared Bell pair and two bits of classical communication. Define $|{\beta }_{mn}⟩=(I\otimes X^n{Z}^m)|{\Phi }^{+}⟩$, with $m,n\in \{0,1\}$. Regrouping the initial three-qubit state into this Bell basis givesAlice's Bell-basis measurement selects each label with probability $1/4$. She sends $(m,n)$ to Bob, who applies the inverse $Z^m{X}^n$, restoring $|\chi ⟩$. The correction depends on the measurement label, not on the unknown amplitudes. Before the message, Bob's average state is $I/2$, independent of the input; quantum no-signalling therefore prevents superluminal communication. The shared pair is consumed and Alice does not retain an independent copy, in agreement with the no-cloning theorem.

The Bouwmeester quantum teleportation experiment used photon polarization, a down-conversion entangled pair, a beam-splitter Bell-state analyzer and coincidence-conditioned output polarization measurements. Temporal indistinguishability of the two analyzer photons was essential. Its successful selected Bell outcome demonstrated the state transfer, rather than a deterministic implementation of all four outcomes.^[ref]

The general experimental tasks are to distribute entanglement, prepare and synchronize the input, perform a sufficiently discriminating joint measurement, communicate the label and apply or account for the correction. A demonstration must test nonorthogonal input states, not merely transfer a known classical basis label. Finite efficiency, accidental coincidences, distinguishability and phase drift reduce success probability or teleportation fidelity. Postselected verification, a heralded usable output and unconditional deterministic teleportation are distinct experimental claims.

Schrödinger's cat exposes the quantum measurement problem by correlating a microscopic alternative with macroscopically distinct records. Under unitary evolution a schematic state becomesThe difficulty is explaining why a definite macroscopic outcome is observed if the isolated total state is always governed by the linear Schrödinger equation. It is not a claim that one classically contradictory record has literally been observed. Environmental interactions rapidly entangle the records with different environment states, making their reduced density matrix approximately diagonal. That explains the practical loss of interference but does not, by itself, turn a global pure superposition into a selected single branch.

Wigner's friend places an observer's physical memory inside the quantum description. A friend measuring $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ can become correlated asConditioned on one friend record, the corresponding relative state contains a definite outcome. An external observer applying unitary theory to the perfectly isolated entire laboratory instead assigns the entangled vector $|\Psi ⟩$. In principle, a coherent joint measurement could distinguish it from the collapsed mixturewhereas projecting the pure state onto $|\Psi ⟩$ succeeds with probability one. Preserving that laboratory coherence, including all records and environment correlations, is an extreme idealization for macroscopic observers.

The paradox concerns how state assignment, definite records and universal unitary dynamics fit together. It does not show that an observer can send messages faster than light or that consciousness must be given a special dynamical role. The many-worlds interpretation retains the global unitary state with branch-relative records; interpretations with a physical collapse must specify when and how unitary evolution changes. Stating the chosen assumptions prevents the friend's conditional state and the external observer's unconditioned state from being treated as the same mathematical object.