CritPt / Challenge_67_main / We study the fundamental limit of a quantum channel called the private channel.…

# Problem setup:
We study the fundamental limit of a quantum channel called the private channel. To be more specific, the Choi operator of the channel is given by the private state, with the input space given by the private system and shield system. Our channel is determined by the following Werner state: q|ψ_(+)⟩⟨ψ_(+)|^(a₀b₀)⊗(1)/(d_(sym))P_(sym)^(A₀B₀)+(1−q)|ψ_(−)⟩⟨ψ_(−)|^(a₀b₀)⊗(1)/(d_(asym))P_(asym)^(A₀B₀), q∈[0,1],
where a₀,b₀ have dimension 2 and A₀,B₀ have dimension d. P_(sym) is the projection onto the symmetric subspace.


# Main problem:

What is the quantum capacity of this channel when q=(d+1)/(2d)?

# Problem setup:
We study the fundamental limit of a quantum channel called the private channel. To be more specific, the Choi operator of the channel is given by the private state, with the input space given by the private system and shield system. Our channel is determined by the following Werner state: $$ q |\psi_+\rangle \langle \psi_+|^{a_0b_0} \otimes \frac{1}{d_{\text{sym}}} P_{\mathrm{sym}}^{A_0B_0} + (1-q) |\psi_-\rangle \langle \psi_-|^{a_0b_0} \otimes \frac{1}{d_{\mathrm{asym}}} P_{\mathrm{asym}}^{A_0B_0},\ q\in [0,1],$$
where $a_0,b_0$ have dimension 2 and $A_0,B_0$ have dimension $d$. $P_{\mathrm{sym}}$ is the projection onto the symmetric subspace.


# Main problem:

What is the quantum capacity of this channel when $q = \frac{d+1}{2d}$?