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For two-qubit pure states with smaller squared Schmidt coefficients $s<t$, optimal stochastic conversion of a two-qubit pure state succeeds independently with probability $s/t$. The asymptotic pure-state entanglement conversion rate of collective LOCC is instead $h_2(s)/h_2(t)$, where $h_2$ is binary entropy. The difference is strict: differentiating gives $\frac{d}{dx}[h_2(x)/x]={\log }_2(1-x)/x^2<0$. Hence $h_2(s)/s>h_2(t)/t$. Independent filtering loses entanglement in its product-state failure branches, whereas collective concentration and dilution preserve entanglement asymptotically.

The Schmidt decomposition and local unitary operations put any entangled two-qubit pure state intoBoth Schmidt coefficients are nonzero. Alice performs a two-outcome generalized measurement with Kraus operatorsTheir effects sum to the identity, so this is a legitimate local measurement in quantum mechanics. On success,The normalized output is the Bell state ${\Phi }_{+}$. Bob's unitary matrix $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ maps it to the spin singlet state. Alice communicates her success flag; no shared quantum operation is needed. At $\lambda =1/2$ the filter is the identity and success is certain. This is the singlet target case of optimal stochastic conversion of a two-qubit pure state.

For dichotomic local outcomes $A_x,B_y\in \{-1,1\}$, the CHSH inequality isIt assumes a local hidden-variable theory with freely chosen settings independent of the hidden-variable distribution. For the spin singlet state, direct evaluation of its Pauli matrices correlations givesChoose $a_0=e_z$, $a_1=e_x$, $b_0=(e_z+e_x)/\sqrt{2}$, and $b_1=(e_z-e_x)/\sqrt{2}$. The four correlations are $-1/\sqrt{2},-1/\sqrt{2},-1/\sqrt{2},+1/\sqrt{2}$, respectively, giving

The success flag must be recorded and communicated before the subsequent independent choice of Bell-test settings. This conditioning is legitimate because setting-independent heralding preserves Bell locality: a putative local hidden-variable theory for the entire sequential experiment would, after successful filtering, still have local response functions with a setting-independent conditional hidden-variable distribution. More explicitly, if the hidden variable is $\xi$ and $h$ is the already determined success flag, conditioning changes its distribution to $\rho (\xi \mid h=1)$ but leavesClassical messages used in preparation can be included in $\xi$; none are exchanged during the spacelike Bell measurements. Such a conditioned model still obeys the CHSH inequality, contradicting the displayed singlet correlations. Therefore no local hidden-variable theory reproduces all sequential measurement correlations of any entangled two-qubit pure state. This argument does not discard outcomes on the basis of the later measurement settings.

Alice makes a two-outcome local measurement in quantum mechanics with Kraus operatorsSince $0<\tan \alpha /\tan \beta <1$, these are valid and satisfy $K_{\mathrm{s}}^{†}{K}_{\mathrm{s}}+K_{\mathrm{f}}^{†}{K}_{\mathrm{f}}=I$. Alice sends Bob the success or failure outcome. The successful unnormalized branch isTherefore the normalized success state is exactly the target andBob needs no quantum correction in this known Schmidt basis. The failure branch is proportional to $|00⟩$, a product state, and has probability $1-s/t$. The successful branch saturates the previous bound: $p_{\mathrm{succ}}{E}_2(\varphi )=E_2(\psi )$ and failure contributes zero. This establishes optimal stochastic conversion of a two-qubit pure state.