Entanglement.QCapacity

Entanglement.QCapacity.CQClaCapacity(map_alph)

Compute the classical capacity of a classical-quantum (CQ) channel.

\[C(\Phi) = \max S\left(\sum_{x\in\mathcal{X}} p_x \Phi(x)\right) - \sum_{x\in\mathcal{X}} p_x S(\Phi(x))\]

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 tensor of the CQ channel. A 3D array whose first index enumerates \(x\in\mathcal{X}\), and the last two indices contain the density matrices \(\Phi(x)\).

Returns:

The capacity value of \(\Phi\) (in bits/use).

Return type:

numeric

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)

Compute the entanglement-assisted classical capacity.

\[C_{ea}(\Phi) = \max_{\rho\in\mathcal{D}(\mathcal{H})} I(\rho, \Phi)\]

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 \(\Phi\) (in bits/use).

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)

Compute the quantum capacity of a degradable channel via coherent information.

\[Q(\Phi) = \max_{\rho\in\mathcal{D}(\mathcal{H})} I_c(\rho, \Phi)\]

where \(I_c(\rho, \Phi)\) is the coherent information of the (degradable) channel \(\Phi\) for input state \(\rho\).

Parameters:

  • JN (numeric) – The Choi matrix of the channel.

  • dim (numeric) – The dimension of the input and output spaces of the channel.

Returns:

The quantum capacity of the (degradable) channel \(\Phi\) (in bits/use).

Return type:

numeric

Note

Assumes degradability and uses a degrading map to evaluate \(I_c\).

Entanglement.QCapacity.RelEntropyEnt(rho, dim)

Compute the PPT relative entropy of entanglement.

\[E_{\operatorname{PPT}}(\rho) = \min_{\tau\in\operatorname{PPT}} D(\rho\| \tau)\]
Parameters:

  • rho (numeric) – The density matrix of the target state.

  • dim (numeric) – The dimension array \([d_A,d_B]\).

Returns:

The PPT relative entropy of \(\rho\) (in bits).

Return type:

numeric

Entanglement.QCapacity.RelEntropyRecovery(rhoABC, dim)

Compute the relative entropy of recovery and the optimal recovery channel for a tripartite state.

\[R_{rec}(\rho_{ABC}) = \min_{\mathcal{R}:B\rightarrow BC} D(\rho_{ABC}\| (\mathcal{I}_A \otimes \mathcal{R})(\rho_{AB}))\]
Parameters:

  • rhoABC (numeric) – The density matrix of the tripartite target state.

  • dim (numeric) – The dimension array of the composite systems \([d_A,d_B,d_C]\).

Returns:

val: The relative entropy of recovery of \(\rho_{ABC}\) (in bits).

JR: The Choi matrix of the optimal recovery channel \(\mathcal{R}:B\to BC\).

Return type:

[numeric, numeric]