Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 10E / a

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- Solution a

Solution

The quantum no-signalling principle says Alice's uncommunicated local choice cannot alter Bob's reduced state. Expanding the two tensor products cancels the cross terms and gives

∣ μμ ⟩ + ∣ νν ⟩ = ∣00 ⟩ + ∣11 ⟩ .

Alice obtains either result with probability

1/2

; conditionally Bob has probability

cos^{2} θ

or

sin^{2} θ

of

0

, respectively. Averaging,

P_{B} (0) = \frac{1}{2} (cos^{2} θ + sin^{2} θ) = \frac{1}{2}, P_{B} (1) = \frac{1}{2},

independent of

θ

, exactly as no-signalling requires.

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