CritPt / Challenge_62_main / Quantum superposition of a single particle can be demonstrated by the…

# Problem setup:
Quantum superposition of a single particle can be demonstrated by the interference pattern observed in a double-slit experiment. Although it has been shown that no genuine triple-slit interference exists for a single quantum particle, an N-slit experiment can still reveal stronger nonclassical patterns, detectable through violations of the following inequality (Note: A classical theory that allows N−1-slit inteference cannot violate such an inequality.):

p(0|0,⋯,0)+∑_(i=1)^(N)p(1|0,⋯,1_(i),⋯,0)≤N,

where p(b|a₁,⋯,a_(n)) represents the statistics of getting outcome b when each path encodes a_(i).

To see the violation, a quantum particle is prepared among N paths, such that |Ψ_(N)⟩=(1)/(√(N))∑_(i=1)^(N)|e_(i)⟩  with $|e_i\rangle=|0\rangle_{A_1}\cdots|1\rangle_{A_i}\cdots|0⟩_{A_N}$. Each party encodes a local phase on their path i given input a_(i), by applying the unitary U_(i)(ϕ_(a_(i)))=e^(iϕ_(a_(i)))|1_(i)⟩⟨1_(i)|. The resulting encoded state is given by

$$\sigma_{a_1,\cdots,a_n}= \bigotimes_i U_i(\phi_{a_i}) |\Psi_N\rangle\langle\Psi_N|\bigotimes_i U_i^{\dagger}(\phi_{a_i}),$$

which is measured using a fixed measurement {Π_(b)}_(b), and the statistics are given by

p(b|a₁,⋯,a_(n))=Tr[Π_(b)σ_(a₁,⋯,a_(n))].

We consider a specific encoding strategy for an odd number of paths, N=2k+1, as follows:
Set ϕ_(1_(i))=ϕ for i≤k; ϕ_(1_(i))=π for i=k+1; ϕ_(1_(i))=−ϕ for i>k+1; and ϕ_(i₀)=0 for all i.




# Main problem:

(1) For the case k=1(N=2k+1=3), express the violation as a function of ϕ, i.e., the violation is defined as

δ=p(0|0,⋯,0)+∑_(i=1)^(N)p(1|0,⋯,1_(i),⋯,0)−N.

(2) Determine the range of ϕ for which a quantum violation occurs, denoted as T, where T⊆[0,π] is a function of k.

(3) Determine the value of ϕ at which the maximal quantum violation occurs, denoted as ϕ_(max), as a function of k.

# Problem setup:
Quantum superposition of a single particle can be demonstrated by the interference pattern observed in a double-slit experiment. Although it has been shown that no genuine triple-slit interference exists for a single quantum particle, an $N$-slit experiment can still reveal stronger nonclassical patterns, detectable through violations of the following inequality (Note: A classical theory that allows $N-1$-slit inteference cannot violate such an inequality.):

$$
p(0|0,\cdots,0) + \sum_{i=1}^N p(1|0,\cdots,1_i,\cdots,0) \leq N,
$$

where $p(b|a_1,\cdots,a_n)$ represents the statistics of getting outcome $b$ when each path encodes $a_i$.

To see the violation, a quantum particle is prepared among $N$ paths, such that $|\Psi_N\rangle = \frac{1}{\sqrt{N}} \sum^N_{i=1} |e_i\rangle$  with $|e_i\rangle=|0\rangle_{A_1}\cdots|1\rangle_{A_i}\cdots|0⟩_{A_N}$. Each party encodes a local phase on their path $i$ given input $a_i$, by applying the unitary $U_i(\phi_{a_i}) = e^{i\phi_{a_i}}|1_i\rangle\langle1_i|$. The resulting encoded state is given by

$$\sigma_{a_1,\cdots,a_n}= \bigotimes_i U_i(\phi_{a_i}) |\Psi_N\rangle\langle\Psi_N|\bigotimes_i U_i^{\dagger}(\phi_{a_i}),$$

which is measured using a fixed measurement $\{\Pi_b\}_b$, and the statistics are given by

$$p(b|a_1,\cdots,a_n)=Tr[\Pi_b \sigma_{a_1,\cdots,a_n}].$$

We consider a specific encoding strategy for an odd number of paths, $N = 2k + 1$, as follows:
Set $\phi_{1_i} = \phi$ for $i \leq k$; $\phi_{1_i} = \pi$ for $i = k + 1$; $\phi_{1_i} = -\phi$ for $i > k + 1$; and $\phi_{i_0} = 0$ for all $i$.




# Main problem:

(1) For the case $k=1 (N=2k+1=3)$, express the violation as a function of $\phi$, i.e., the violation is defined as

$$
\delta= p(0|0,\cdots,0) + \sum_{i=1}^N p(1|0,\cdots,1_i,\cdots,0)- N.
$$

(2) Determine the range of $\phi$ for which a quantum violation occurs, denoted as $T$, where $T \subseteq [0, \pi]$ is a function of $k$.

(3) Determine the value of $\phi$ at which the maximal quantum violation occurs, denoted as $\phi_{\max} $, as a function of $k$.