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For quantum key distribution with Bell pairs, a source distributes one photon to Alice and its partner to Bob. One implementation is the Ekert protocol, using the spin singlet state $|{\psi }^{-}⟩=(|01⟩-|10⟩)/\sqrt{2}$. Each party independently chooses a polarization measurement setting after distribution. Equal settings give opposite outcomes; Bob reverses his outcome-to-bit assignment so these rounds give matching random bits. Other setting pairs test a Bell inequality, with ideal spin singlet state correlation $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$ for the corresponding Bloch sphere axes. The optical polarizer angles correspond to twice those angles on the appropriate Bloch sphere great circle.

Over an authenticated public classical channel, Alice and Bob announce settings and detection times, select coincident detection rounds, and perform key sifting. They reveal a test sample to estimate the quantum bit error rate and the chosen Bell inequality signal, discarding all revealed bits. Excess disturbance causes an abort. Otherwise information reconciliation makes the retained strings agree and privacy amplification shortens them to remove an adversary's residual information, including information disclosed during reconciliation. The tested quantum entanglement supplies correlations unavailable to an independently prepared classical mixture; a security claim also requires stated detector and sampling assumptions.^[ref]

Each photon polarization analyzer uses a setting-dependent waveplate followed by a polarizing beam splitter. A single-photon detector on each output converts the routed photon into an electrical click, recording the outcome and its time. For example, a single-photon avalanche diode is a reverse-biased photodiode operated above breakdown: absorption creates a charge carrier, impact ionization triggers an avalanche, and the resulting pulse is registered. A quenching circuit lowers the bias to stop the avalanche and then resets the device. The detector dead time during recovery limits the count rate; dark counts produce false clicks, and imperfect detection loses rounds. The polarization information comes from the preceding analyzer, while the detector registers which output fired. Coincidence timing associates Alice's and Bob's records with the same emitted pair. ^[ref]

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]

The disagreement probability of Alice's and Bob's sifted bits under their agreed outcome-to-bit maps. A finite test sample estimates this probability; the estimate and its uncertainty enter parameter estimation before information reconciliation and privacy amplification.