Past exam of the quantum information course of the University of Cambridge / 2015 / Paper 1 / 1

Linearity of the detector's unitary time evolution maps the two input branches separately, giving the quantum premeasurement stateIn the ordered tensor product basis $(|\uparrow d_{\uparrow }⟩,|\uparrow d_{\downarrow }⟩,|\downarrow d_{\uparrow }⟩,|\downarrow d_{\downarrow }⟩)$, its pure density matrix isA nonselective projective measurement in the indicated detector basis applies $\rho \mapsto {\sum }_j(I_S\otimes {\Lambda }_j)\rho (I_S\otimes {\Lambda }_j)$, removing the two off-diagonal terms:This is the usual nonselective meaning of von Neumann process 1. It is not equal to the pure projector $|{\Phi }^C⟩⟨{\Phi }^C|$. If “the detector has clicked” means conditioning on an actual up outcome, the normalized postselected state is instead $|\uparrow d_{\uparrow }⟩$, with probability $1/2$. The correlated superposition describes the coherent premeasurement before that selection.

Put $c=\cos (\theta /2)$ and $s=\sin (\theta /2)$. The detector rotation is real and orthogonal, so its inverse is the same transformation. Substituting it givesThus the requested orthonormal system states areThe pure density operator has not changed: this is only a different Schmidt decomposition of a maximally entangled state. But choosing a nonselective projective measurement in this new detector basis gives a different operation and generally a different state:For an explicit matrix in the original ordered basis, write $q=c^2{s}^2$ and $h=cs(c^2-s^2)/2$. ThenAt $\theta =\pi /2$ the entries $(1,4)$ and $(2,3)$ are $1/4$, whereas they vanish in ${\rho }_{up/down}$. Nevertheless, the system and detector individually each have reduced density matrix $I/2$ for either mixture.

This is the preferred-basis problem: the correlated state alone does not specify which detector observable becomes a stable classical record. Arbitrarily declaring collapse in a Schmidt basis changes the physical prediction. The actual detector interaction and environment must supply the basis selection, rather than a mere choice of notation.

Write $H=g{I}_S\otimes Z_D\otimes X_E$, with $Z_D={\Lambda }_{\uparrow }-{\Lambda }_{\downarrow }$. The orthogonal projection operators obey ${\Lambda }_{\uparrow }{\Lambda }_{\downarrow }=0$ and ${\Lambda }_j^2={\Lambda }_j$. Hence $[Z_D,{\Lambda }_j]=0$, and the remaining tensor factors commute with detector operators. ThereforeThe environment interaction preserves these detector populations, making their basis a candidate pointer basis. It can change their relative coherence without flipping the stored labels.

Since $(Z_D\otimes X_E{)}^2=I$, the solution of the Time-dependent Schrödinger equation isThe two detector eigenvalues generate opposite rotations of the environment. Acting on the initial state givesThese are the indicated $|E_{-}⟩$ and $|E_{+}⟩$. Their conditional environment overlap is $⟨E_{+}|E_{-}⟩=\cos 2A$. The partial trace therefore multiplies the coherence between $|\uparrow d_{\uparrow }⟩$ and $|\downarrow d_{\downarrow }⟩$ by $\cos 2A$. This is an exactly solvable single-qubit environment dephasing model, with fully reversible global evolution.

At $A=\pi /4$, defineThey are normalized and orthogonal. The full state is the pure superposition $(|\uparrow d_{\uparrow }{E}_{\leftarrow }⟩+|\downarrow d_{\downarrow }{E}_{\to }⟩)/\sqrt{2}$. A nonselective von Neumann process 1 selecting the original detector records givesWithout imposing any projection on the full state, tracing out the orthogonal environment states already gives ${\rho }_{SD}={\rho }_{up/down}$ from part (a). These two global descriptions share that reduced state but differ in their global coherence. Quantum decoherence does not make a pure global state literally equal to the corresponding statistical mixture.

Inverting the detector rotation at this time givesFor a representation with one system factor and one environment factor per displayed detector state, each parenthesized system–environment vector would have to be a product state. But the first has coefficient matrix $\mathrm{diag}(c,s)$ in the indicated orthonormal bases; the second has matrix $\mathrm{diag}(s,-c)$. Their matrix ranks are two whenever $cs\ne 0$. Equivalently, their two nonzero Schmidt coefficients are $|c|,|s|$, which no independent basis changes can remove. Thus the proposed factorization is impossible when $\sin \theta \ne 0$.

The exception matters: at $\theta =m\pi$, the detector basis is the original basis up to phases or interchange, and a factorization does exist. In particular, the literal condition $\theta \ne 0$ is insufficient if it includes $\theta =\pi$. The intended obstruction concerns genuinely mixed detector bases.

The environment has selected robust detector records: its interaction commutes with the original projectors, and orthogonal environmental records suppress their interference in the reduced density matrix. A rotated detector branch instead leaves the system and environment entangled. This is environment-induced pointer-basis selection, the issue analyzed by W. H. Zurek in his apparatus-environment model.^[ref]

This addresses the basis ambiguity but does not by itself select one realized outcome from the global superposition. Moreover, a two-state environment has recurrences: $\cos 2A$ returns to unit modulus, so the toy model does not establish irreversible macroscopic collapse. A large environment can make records effectively persistent over experimental times, but that requires further dynamical assumptions. The distinction is part of the measurement problem, not an algebraic error cured by a partial trace.