Entanglement.QCapacity¶
- Entanglement.QCapacity.CQClaCapacity(map_alph)¶
- \[C(\Phi) = \max S\left(\sum_{x\in\mathcal{X}} p_x \Phi(x)\right) - \sum_{x\in\mathcal{X}} p_x S(\Phi(x))\]
where \({p_x}_x\) forms probability distribution, \(S\) is the von Neumann entropy, and \(\Phi:\mathcal{X}\rightarrow \mathcal{D}(\mathcal{H})\) is some classical-quantum channel.
- Parameters:¶
map_alph (
numeric) – mapping data of the CQ-channel, should be stored in a 3D-tensor with the
:param first index representing the labelling \(x\in \mathcal{X}\) and last two indices give: :param mapped density operators.:
Note
Hayashi, M., & Nagaoka, H. (2003). General formulas for capacity of classical-quantum channels. IEEE Transactions on Information Theory, 49(7), 1753-1768.
- Entanglement.QCapacity.EntAssistedCapacity(JN, dim)¶
- \[C_{ea}(\Phi) = \max_{\rho\in\mathcal{D}(\mathcal{H})} I(\rho, \Phi)\]
where \(I(\rho, \Phi)\) is the mutual information of the channel \(\Phi\).
- Parameters:¶
JN (
numeric) – The Choi matrix of the channel.dim (
numeric) – The dimension of the input and output spaces of the channel.
- Returns:¶
The entanglement-assisted classical capacity of \(\Phi\).
- Return type:¶
numeric
Note
Bennett, C. H., Shor, P. W., Smolin, J. A., & Thapliyal, A. V. (2002). Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE transactions on Information Theory, 48(10), 2637-2655.
- Entanglement.QCapacity.QuaDegCapacity(JN, dim)¶
- \[Q(\Phi) = \max_{\rho\in\mathcal{D}(\mathcal{H})} I_c(\rho, \Phi)\]
where \(I_c(\rho, \Phi)\) is the coherent information of the (degradable) channel \(\Phi\) for input state \(\rho\).
- Entanglement.QCapacity.RelEntropyEnt(rho, dim)¶
- \[E_{\operatorname{PPT}}(\rho) = \min_{\tau\in\operatorname{PPT}} D(\rho\| \tau)\]
- Entanglement.QCapacity.RelEntropyRecovery(rhoABC, dim)¶
- \[R_{rec}(\rho_{ABC}) = \min_{\mathcal{R}:B\rightarrow BC} D(\rho_{ABC}\| (\mathcal{I}_A \otimes \mathcal{R})(\rho_{AB}))\]