qics.cones.QuantKeyDist¶

class qics.cones.QuantKeyDist(G_info, Z_info, iscomplex=False)[source]¶

A class representing a quantum key distribution cone

\[\mathcal{QKD}_{\mathcal{G},\mathcal{Z}} = \text{cl}\{ (t, X) \in \mathbb{R} \times \mathbb{H}^n_{++} : t \geq -S(\mathcal{G}(X)) + S(\mathcal{Z}(\mathcal{G}(X))) \},\]

where

\[S(X) = -\text{tr}[X \log(X)],\]

is the quantum (von Neumann) entropy function, \(\mathcal{G}:\mathbb{H}^n\rightarrow\mathbb{H}^{mr}\) is a positive linear map, and \(\mathcal{Z}:\mathbb{H}^{mr}\rightarrow\mathbb{H}^{mr}\) is a pinching map that maps off-diagonal blocks to zero.

Parameters:

G_infoint or list of ndarray

Defines the linear map \(\mathcal{G}\). There are two ways to specify this linear map.

If G_info is an int, then \(\mathcal{G}\) is the identity map, i.e., \(\mathcal{G}(X)=X\), and G_info specifies the dimension of \(X\).
If G_info is a list of ndarray, then G_info specifies the Kraus operators \(K_i \in \mathbb{C}^{mr \times n }\) corresponding to \(\mathcal{G}\) such that

\[\mathcal{G}(X) = \sum_{i} K_i X K_i^\dagger.\]

Z_infoint or tuple or list of ndarray

Defines the pinching map \(\mathcal{Z}\), which is of the form

\[\mathcal{Z}(Y) = \sum_{i} Z_i Y Z_i^\dagger.\]

There are three ways to specify this linear map.

If Z_info is an int, then \(Z_i=|i \rangle\langle i| \otimes\mathbb{I}\) for \(i=1,\ldots,r\), where r=Z_info.
If Z_info is a tuple of the form (dims, sys), where dims=(n0, n1) is a tuple of int and sys is an int, then
- \(Z_i=|i \rangle\langle i| \otimes\mathbb{I}_{n_1}\) for \(i=1,\ldots,n_0\) if sys=0, and
- \(Z_i=\mathbb{I}_{n_0}\otimes |i \rangle\langle i|\) for \(i=1,\ldots,n_1\) if sys=1.
We generalize this definition to the case where dims and sys are lists of any length.
If Z_info is a list of ndarray, then Z_info directly specifies the Kraus operators \(Z_i \in \mathbb{C}^{mr \times mr}\). Note that these Kraus operators must have a similar structure to those defined using the other options, i.e., must be diagonal matrices consisting of either ones or zeros, and \(Z_iZ_j=0\) for all \(i\neq j\).

iscomplexbool

Whether the matrix \(X\) is defined over \(\mathbb{H}^n\) (True), or restricted to \(\mathbb{S}^n\) (False). The default is False.