Past exam of the laser science and quantum information processing course of the University of Oxford / 2009

Only the supplied question extract is held; Oxford's C2 support-page record attributes it to 2009. The page does not establish its historical examination stage or sitting date.

Source extract: users.physics.ox.ac.uk/~Foot/C2/pdf/C22009Q8.pdf

The year is attributed from the preserved Oxford support-page record; it is not printed on the held page.

TODO: The remainder of this historical paper is not held. Only the supplied question is covered.

Normalize the supplied Bell state as $|{\Psi }^{+}⟩=(|01⟩+|10⟩)/\sqrt{2}$, with $|V⟩=|0⟩$ and $|H⟩=|1⟩$. For a measurement of linear photon polarization, the two orthogonal analyzer states can be taken asAssigning outcomes $+1,-1$ gives the observablewhere $X,Z$ are the Pauli operators. The angle of the Bloch vector is twice the physical analyzer angle. HenceEach has eigenvalues $\pm 1$ and square equal to the identity operator.

Use an Ekert protocol. Alice and Bob choose their analyzer settings independently and randomly for each received pair, record the outcomes, and confirm reception before announcing settings over an authenticated public classical channel. The authentication prevents an adversary from impersonating either party; it does not conceal the setting announcements. Announcing settings after reception also avoids the premature basis announcement in quantum key distribution attack.

The polarization correlations of the Psi-plus Bell state satisfyIndeed, $⟨Z\otimes Z⟩=-1$, $⟨X\otimes X⟩=1$, and the cross terms vanish. In the specified setting order,

For key sifting, retain the setting pairs $(i,j)=(1,1)$ and $(3,2)$. Their outcomes are perfectly anticorrelated. Alice assigns bit $0$ to outcome $+1$ and bit $1$ to outcome $-1$; Bob uses the opposite assignment. Their raw keys then agree. With uniform independent choices of the three settings, these pairs supply a raw-key fraction $2/9$, before sacrificing a test sample. They must keep the remaining raw-key outcomes secret. The four pairs $(1,2),(1,3),(2,2),(2,3)$ supply Bell-test data, and the other three setting pairs can be discarded.

Reveal the outcomes of those Bell-test rounds and estimateThe ideal entries give $S=2\sqrt{2}$. The CHSH inequality gives $|S|\le 2$ for local hidden-variable theories: for predetermined outcomes $a_1,a_2,b_2,b_3\in \{\pm 1\}$, the signed combination is $-a_1(b_2+b_3)-a_2(b_2-b_3)$. One bracket is zero and the other is $\pm 2$, so averaging cannot exceed two in magnitude. The Tsirelson bound limits quantum correlations to $2\sqrt{2}$.

For the specified, trusted Pauli operators, the maximal CHSH witness for a fixed Bell state makes the ideal security statement precise. Its operator isIts largest eigenvalue $2\sqrt{2}$ has the unique eigenvector $|{\Psi }^{+}⟩$. Exact attainment therefore fixes the shared density matrix to this pure state. Since purity decouples a subsystem from its purification, Eve cannot hold information correlated with that pair. This rules out information-gaining eavesdropping in the ideal model, rather than proving that no observer exists. More generally, CHSH self-testing identifies the entangled qubit pair up to local changes of representation under its own assumptions.

A quantum measurement that reads Eve's chosen polarization generally disturbs correlations in an incompatible basis. The no-cloning theorem also prevents copying arbitrary unknown photon polarizations and retaining a perfect copy while forwarding an unchanged original. The Bell test checks several incompatible settings, so observing perfect correlation in only one key basis is insufficient.

Alice and Bob also reveal a randomly chosen sample of their raw keys to estimate the quantum bit error rate, then discard that sample. Information reconciliation corrects discrepancies, and privacy amplification removes residual information, including leakage from the public transcript. With finite data they need statistical bounds and a security analysis appropriate to their devices and noise. A measured violation $S>2$ alone does not assert zero information for Eve; the exact ideal argument above uses the maximal value.^[ref]

Eve's computational-basis intercept-resend attack applies a projective measurement of $Z$ to Bob's photon and discards her outcome from Alice and Bob's description. Their resulting density matrix isThis separable state keeps $⟨Z\otimes Z⟩=-1$ but removes $⟨X\otimes X⟩$. HenceUsing the same signed Bell-test statistic givesThe ideal value is halved, and the attack no longer violates the CHSH inequality.

The key data show why several settings matter. For $(1,1)$, Eve's measurement leaves perfect anticorrelation, so this subset has zero bit errors after Bob flips his assignment. For $(3,2)$, $E=-1/2$, so correct anticorrelation has probability $(1-E)/2=3/4$: its quantum bit error rate is $1/4$. If both key-setting pairs occur equally often, their combined raw-key error rate is $1/8$. The Bell-test change detects this attack even though the vertical/horizontal key subset alone would not.