CritPt / Challenge_63_main / Quantum contextuality, like Bell nonlocality, is a key signature of…

# Problem setup:
Quantum contextuality, like Bell nonlocality, is a key signature of nonclassicality in quantum information. There are two widely accepted notions of quantum contextuality: Kochen-Specker (KS) contextuality and Spekkens' generalized contextuality.

The simplest KS contextuality proof uses five three-outcome projective measurements, denoted as KCBS measurements {M_(i)}_(KCBS), where

$$ M_i=\bigl[\Pi_i,
        \Pi_{i+1},
        I -\Pi_i - \Pi_{i+1}
        \bigl],
        \qquad i = 1,\dots,5, \quad (\text{mod }5),$$

Π_(i)=|l_(i)⟩⟨l_(i)|, |l_(i)⟩=cosα|0⟩+sinα(cosφ_(i)|1⟩+sinφ_(i)|2⟩),
φ_(i)=(2π(i−1))/(5),
and α=arccos(((1)/(5))^(1/4)).

These measurements exhibit state-dependent KS contextuality: their Born-rule statistics cannot be reproduced by any deterministic noncontextual model, which can be proven by the violation of the KCBS inequality,W=∑_(i)⟨Π_(i)⟩≤2.

Though both the KS and Spekkens frameworks could conclude that such a set of measurements is useful for proving quantum contextuality,  the KS test only works for projective measurement assuming deterministic response functions; Therefore, one cannot discuss the notion of white-noise robustness for these measurements.

By comparison, in Spekken's framework,  the deterministic response function can be justified through preparational noncontextuality, and the minimal set of states for that justification are defined by the rank-1 operators in {M_(i)}_(KCBS), i.e.,

{ρ_(x)}_(KCBS)={Π_(i)}_(i=1)⁵∪{I−Π_(i)−Π_(i+1)}_(i=1)⁵,

which will be referred to as the KCBS states consisting of 10 quantum states.



# Main problem:
Within the framework of generalized contextuality (Spekkens contextuality),

(1) Determine the white-noise robustness of the measurement above, assuming one can use the set of all quantum states, and the white noise robustness η is effectively defned as the largest noise threshold such that
$$M^{\eta}_i=\bigl[\Pi^{\eta}_i,\ \Pi^{\eta}_{i+1},\ \ I -\Pi^{\eta}_i - \Pi^{\eta}_{i+1}\bigr], \qquad i=1,\dots,5, \quad (\text{mod }5)$$
        
is still useful for proving generalized contextuality (Spekkens framework) if one has access to all quantum states, where Π_(i)^(η)=ηΠ_(i)+(1−η)(I)/(3)  

Note that 0≤η≤1, and provide the result to three decimal places.

(2) Similarly, determine the white-noise robustness η of the set of effects $[\Pi^{\eta}_i \bigl], i=1,\cdots, 5$.

# Problem setup:
Quantum contextuality, like Bell nonlocality, is a key signature of nonclassicality in quantum information. There are two widely accepted notions of quantum contextuality: Kochen-Specker (KS) contextuality and Spekkens' generalized contextuality.

The simplest KS contextuality proof uses five three-outcome projective measurements, denoted as KCBS measurements $\{M_i\}_{\text{KCBS}}$, where

$$ M_i=\bigl[\Pi_i,
        \Pi_{i+1},
        I -\Pi_i - \Pi_{i+1}
        \bigl],
        \qquad i = 1,\dots,5, \quad (\text{mod }5),$$

$\Pi_i=| l_i\rangle\langle l_i|$, $
|l_i\rangle=\cos\alpha|0\rangle+\sin\alpha\left(\cos\varphi_i| 1\rangle +\sin\varphi_i| 2\rangle\right)$,
$\varphi_i =\frac{2\pi(i-1)}{5}$,
and $  \alpha =\arccos\left((\frac{1}{5})^{1/4}\right).$

These measurements exhibit state-dependent KS contextuality: their Born-rule statistics cannot be reproduced by any deterministic noncontextual model, which can be proven by the violation of the KCBS inequality,$$W=\sum_i\langle\Pi_i\rangle\le 2.$$

Though both the KS and Spekkens frameworks could conclude that such a set of measurements is useful for proving quantum contextuality,  the KS test only works for projective measurement assuming deterministic response functions; Therefore, one cannot discuss the notion of white-noise robustness for these measurements.

By comparison, in Spekken's framework,  the deterministic response function can be justified through preparational noncontextuality, and the minimal set of states for that justification are defined by the rank-1 operators in $\{M_i\}_{\text{KCBS}}$, i.e.,

$$\{\rho_x\}_{KCBS} = \{\Pi_i\}_{i=1}^5\cup\{I-\Pi_i-\Pi_{i+1}\}_{i=1}^5, $$

which will be referred to as the KCBS states consisting of 10 quantum states.



# Main problem:
Within the framework of generalized contextuality (Spekkens contextuality),

(1) Determine the white-noise robustness of the measurement above, assuming one can use the set of all quantum states, and the white noise robustness $\eta$ is effectively defned as the largest noise threshold such that
$$M^{\eta}_i=\bigl[\Pi^{\eta}_i,\ \Pi^{\eta}_{i+1},\ \ I -\Pi^{\eta}_i - \Pi^{\eta}_{i+1}\bigr], \qquad i=1,\dots,5, \quad (\text{mod }5)$$
        
is still useful for proving generalized contextuality (Spekkens framework) if one has access to all quantum states, where $\Pi^{\eta}_i=\eta \Pi_i+(1-\eta)\frac{I}{3}$  

Note that $0\le \eta \le 1$, and provide the result to three decimal places.

(2) Similarly, determine the white-noise robustness $\eta$ of the set of effects $[\Pi^{\eta}_i \bigl], i=1,\cdots, 5$.