The <Schmidt decomposition> of an entangled pure two-qubit state has two nonzero terms. Absorb both complex phases into the local basis vectors and interchange the labels of the second qubit to obtain
$$
|\psi\rangle=c_0|\uparrow\rangle_1|\downarrow\rangle_2
+c_1|\downarrow\rangle_1|\uparrow\rangle_2,
\qquad
c_0,c_1>0,
\qquad
c_0^2+c_1^2=1.
$$
In this state the only nonzero same-axis two-qubit <Pauli correlators> are
$$
\langle X\otimes X\rangle
=\langle Y\otimes Y\rangle=2c_0c_1,
\qquad
\langle Z\otimes Z\rangle=-1.
$$
All mixed-axis correlators vanish. Expanding $(\mathbf a\cdot\boldsymbol\sigma)\otimes(\mathbf b\cdot\boldsymbol\sigma)$ therefore gives
$$
\boxed{E(\mathbf a,\mathbf b)
=2c_0c_1(a_xb_x+a_yb_y)-a_zb_z}.
$$

For the four stated vectors this becomes
$$
E(\mathbf a,\mathbf b)=-\cos\beta,
\quad
E(\mathbf a,\mathbf b')=-\cos\beta',
\quad
E(\mathbf a',\mathbf b)=-2c_0c_1\sin\beta,
\quad
E(\mathbf a',\mathbf b')=-2c_0c_1\sin\beta'.
$$
It follows immediately that
$$
|E(\mathbf a,\mathbf b)-E(\mathbf a,\mathbf b')|
+|E(\mathbf a',\mathbf b)+E(\mathbf a',\mathbf b')|
=|\cos\beta-\cos\beta'|
+2c_0c_1|\sin\beta+\sin\beta'|.
$$
Every <separable quantum state> obeys the corresponding <CHSH inequality> with upper bound two. Set $s=2c_0c_1>0$, choose $\beta'=\pi-\beta$, and take $\tan\beta=s$ with $0<\beta<\pi/2$. The displayed expression is then
$$
2(\cos\beta+s\sin\beta)=2\sqrt{1+s^2}>2.
$$
Thus every entangled pure two-qubit state has local measurement correlations that no separable state can reproduce, which is <Gisin's theorem>. In an ideal <Bose--Marletto--Vedral experiment>, optimized local measurements can therefore witness any nonzero pure-state entanglement generated during the gravitational interaction. If gravity is the only interaction between the masses, such a violation shows that the mediator cannot be described by a purely classical local variable under the assumptions of the proposal; experimentally, control of <decoherence> and nongravitational forces is essential to that inference.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 325 / 4 / Solution

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Every entangled pure state of two qubits violates a <CHSH inequality> for a suitable choice of local measurement axes. If its two <Schmidt coefficients> are $c_0,c_1>0$, the optimal construction already gives a value $2\sqrt{1+(2c_0c_1)^2}>2$.

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 Incoming links: Gisin's theorem