picos.expressions.exp_quantkeydist¶

Implements QuantumKeyDistribution.

Classes

class picos.expressions.exp_quantkeydist.QuantumKeyDistribution(X, subsystems=0, dimensions=2, K_list=None)[source]¶

Bases: Expression

Slice of quantum relative entropy used to compute quantum key rates.

Definition

Let $X$ be an $n \times n$ -dimensional symmetric or hermitian matrix. Let $\mathcal{Z}$ be the pinching map which maps off-diagonal blocks of a given block structure to zero, i.e., for a bipartite state

$\mathcal{Z}(X) = \sum_{i} Z_i X Z_i^\dagger,$

where $Z_i= | i \rangle \langle i | \otimes \mathbb{I}$ if we block- diagonalize over the first subsystem, and $Z_i= \mathbb{I} \otimes | i \rangle \langle i |$ if we block-diagonalize over the second subsystem. We also generalize this definition to multipartite systems where we block- diagonalize over any number of subsystems.

In general, this is the expression

$-S(\mathcal{G}(X)) + S(\mathcal{Z}(\mathcal{G}(X))),$

where $S(X)=-\operatorname{Tr}(X \log(X))$ is the quantum entropy, $mathcal{G}$ is a positive linear map given by Kraus operators

$\mathcal{G}(X) = \sum_{i} K_i X K_i^\dagger.$
If K_list=None, then $\mathcal{G}$ is assumed to be the identity map, and then this expression is simplified to

$-S(X) + S(\mathcal{Z}(X)).$

Warning

When you pose an upper bound on this expression, then PICOS enforces $X \succeq 0$ through an auxiliary constraint during solution search.

__init__(X, subsystems=0, dimensions=2, K_list=None)[source]¶

Construct an QuantumKeyDistribution.

Parameters

X (AffineExpression) – The affine expression $X$ .
subsystems (int or tuple or list) – A collection of or a single subystem number, indexed from zero, corresponding to subsystems that will be block- diagonalized over. The value $-1$ refers to the last subsystem.
dimensions (int or tuple or list) – Either an integer $d$ so that the subsystems are assumed to be all of shape $d \times d$ , or a sequence of subsystem shapes where an integer $d$ within the sequence is read as $d \times d$ . In any case, the elementwise product over all subsystem shapes must equal the expression’s shape.
K_list (None or list(numpy.ndarray)) – A list of Kraus operators representing the linear map $\mathcal{G}$ . If K_list=None, then $\mathcal{G}$ is defined as the identity map.