Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 325 (Quantum Information, Foundations and Gravity)

For a subsystem $A$ of a multipartite system with density operator ${\rho }_{A{E}_1\cdots E_n}$, its reduced density matrix is the unique operatorsuch that $\mathrm{Tr}(M_A{\rho }_A)=\mathrm{Tr}[(M_A\otimes I_E){\rho }_{AE}]$ for every observable $M_A$ on $A$. In an orthonormal basis $\{|e_j⟩\}$ of the other subsystems, the partial trace iswhich is independent of the chosen basis.

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Write $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. Before any interaction the system is in the pure state $|{\psi }_S⟩=\alpha |0⟩+\beta |1⟩$, soOne interaction leaves the environmental state $|+⟩$ unchanged on the system branch $|0⟩$, while on the branch $|1⟩$ it producesAfter interactions with $n$ distinct environment qubits, the joint state isand the decoherence factor is the product $(\cos ϵ{)}^n$. Taking the partial trace givesFor small $ϵ$, $(\cos ϵ{)}^n\simeq e^{-n{ϵ}^2/2}$: the populations remain fixed while the phase coherence decays. This is environmental decoherence, caused by entanglement with unobserved records rather than by a nonunitary evolution of the complete state.

A measurement in the computational basis cannot reveal the decay because its probabilities remain $|\alpha {|}^2$ and $|\beta {|}^2$. An interference measurement can. For example, measuring in the Hadamard basis givesChanging the measurement phase similarly accesses the imaginary part, so the shrinking interference visibility directly displays decoherence.

If the same environment qubit is reused, the conditional rotation accumulates coherently. After $n$ interactions the overlap is $\cos (nϵ)$ rather than $(\cos ϵ{)}^n$, and the off-diagonal entries oscillate. Coherence vanishes at some times but returns periodically: this finite environment exhibits quantum recoherence rather than effectively irreversible decay.

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The Einstein–Podolsky–Rosen criterion of reality says that if, without disturbing a system, one can predict a physical quantity with certainty, then an element of physical reality corresponds to that quantity.

For two settings on each wing and outcomes in $\{-1,1\}$, the CHSH inequality for every local hidden-variable theory is

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Choose all four axes in one plane, at anglesTheir relevant unsigned separations areThe specified correlation function therefore gives $E(a,b)=E(a,b^{\prime })=E(a^{\prime },b)=1$ and $E(a^{\prime },b^{\prime })=-1$. Hencethe algebraic maximum and thus a maximal violation of the CHSH inequality.

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Suppose local elements of reality predetermined outputs $x(a,\lambda ),x(a^{\prime },\lambda ),y(b,\lambda ),y(b^{\prime },\lambda )\in \{-1,1\}$. For every hidden state $\lambda$,equals $2$ or $-2$, because exactly one bracket vanishes and the other equals $\pm 2$. Averaging over $\lambda$ proves the CHSH inequality. Part (b) instead gives $4$, so no such locally predetermined response functions can reproduce the device. This is the standard Bell theorem obstruction.

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For each pair of inputs, define the joint probabilitiesThey are nonnegative because $|f|\le 1$, sum to one, and have the required correlation:Both marginals are uniform,independently of the remote input. The device is thus a no-signalling box: its superquantum correlation does not by itself transmit a message, so it is compatible with relativistic causality.

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For equal masses with positions ${\mathbf{r}}_1,{\mathbf{r}}_2$, the two-body Schrodinger equation isNeglecting wave-packet spreading and overlap, write $|a{⟩}_i$ for the packet ${\psi }_{ai}$. The initial state and its branchwise gravitational evolution areup to local kinetic phases. The branch dependence of the Newtonian gravitational potential energy is the source of gravitationally induced entanglement.

Under the approximation stated in the question, only the matching pairs $a=b$ acquire an appreciable common phaseThe coefficient matrix of the resulting bipartite state iswhere $J$ is the all-ones matrix. ThereforeA maximally entangled state would require ${\rho }_1=I/n$. This demandsThat equation has no solution for $n>4$. Thus the requested large-$n$ conclusion does not follow from the assumptions printed in the paper: within the stated approximation, the system never becomes maximally entangled in the large-$n$ limit. For completeness, maximal entanglement is possible for $n=3$ at $\varphi =2\pi /3$ and for $n=4$ at $\varphi =\pi$; for $n=2$ it occurs at $\varphi =\pi /2$ or $3\pi /2$. The $n=4$ time would bebut it is not a large-$n$ answer.

The system also does not remain entangled for every $t>0$. Whenever $\varphi =2\pi k$, the phase matrix again factorizes and the state returns to its initial product state. The revival period isFor $n>2$ it is entangled at all intervening times; for $n=2$ there is the additional product-state revival at $\varphi =\pi$.

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Assume an objective-collapse theory in which collapses occur at localized spacetime events. At an event $x$, define the local quantum state under objective collapse by taking the reduced density operator on spacelike hypersurfaces that approach the past light cone of $x$. A quantum state readout device at $x$ prints a complete classical description of this local density operator, in a fixed basis and here with infinite precision, without disturbing it.

For a localized pure state whose relevant preparation lies in the past light cone, the output is its rank-one projector $|\psi ⟩⟨\psi |$. For an improper mixed state produced by entanglement with an uncollapsed exterior system, it is the reduced density matrix. For a proper mixed state, once the selecting preparation or collapse event is in the past light cone, it is the actually selected component; before that causal information arrives, it is the corresponding local mixture.

A spacelike-separated collapse is excluded from the hypersurface limit and cannot change the output. Its effect appears only on and inside its future light cone. The definition therefore prevents superluminal signalling while allowing light-speed changes in the readout.

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One obtains a logically consistent extension by specifying three rules: ordinary quantum evolution and localized objective collapses determine the global state; the past-light-cone prescription determines a unique local density operator at every readout event; and a readout copies a classical description of that operator into a classical register without changing the quantum state. These rules assign an output to every finite-dimensional input and every spacetime history.

The device is outside the class of ordinary quantum measurements and permits tasks such as learning and then preparing a copy of an unknown pure state, but violating the operational no-cloning theorem for two pure states does not make the enlarged axioms contradictory. Feed-forward operations can be defined from earlier readouts in the same causal region. Because every output depends only on collapse events in its past light cone, no closed causal dependence or spacelike signal is introduced. The extension is therefore mathematically coherent, although it is hypothetical and goes beyond standard quantum theory.

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Let $z_A,z_B\in \{+1,-1\}$ denote the two $Z$ outcomes and $x_C\in \{+1,-1\}$ the $X$ outcome. Measuring $Z_A$ in the GHZ state projects $B$ onto the same $Z$ eigenstate, while the later $X_C$ outcome is unbiased. Hence the possible triples, in the order $(z_A,x_C,z_B)$, areeach with probability $1/4$.

Choose $(z_A,x_C,z_B)=(+,+,+)$. Immediately before $t=0$, each local readout is $I/2$. The collapse at $(t,x)=(0,0)$ is immediately in $A$'s past light cone, reaches $B$ at $t=1$, and reaches $C$ at $t=2$. ThusHere $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. The $X_C$ collapse at $(3,2)$ reaches $B$ at $t=4$ and $A$ at $t=5$, but the state is already a product after the first collapse, so it does not change their local states. Likewise, the $Z_B$ collapse at $(5,1)$ reaches $A$ and $C$ at $t=6$ without changing their states. For another allowed outcome, replace $|0⟩$ by $|1⟩$ according to $z_A=z_B$ and replace $|+⟩$ by $|-⟩$ according to $x_C$.

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