toqito.state_props.sk_vector_norm

toqito.state_props.sk_vector_norm(rho, k=1, dim=None)

Compute the S(k)-norm of a vector [NJDK09].

The \(S(k)\)-norm of of a vector \(|v \rangle\) is defined as:

\[\big|\big| |v\rangle \big|\big|_{s(k)} := sup_{|w\rangle} \Big\{ |\langle w | v \rangle| : \text{Schmidt - rank}(|w\rangle) \leq k \Big\}\]

It’s also equal to the Euclidean norm of the vector of \(|v\rangle\)’s k largest Schmidt coefficients.

This function was adapted from QETLAB.

Examples

The smallest possible value of the \(S(k)\)-norm of a pure state is \(\sqrt{\frac{k}{n}}\), and is attained exactly by the “maximally entagled states”.

>>> from toqito.states import max_entangled
>>> from toqito.state_props import sk_vector_norm
>>> import numpy as np
>>>
>>> # Maximally entagled state.
>>> v = max_entangled(4)
>>> sk_vector_norm(v)
0.5

References

[NJDK09]

“A Family of Norms With Applications In Quantum Information Theory” Nathaniel Johnston, David W. Kribs https://arxiv.org/abs/0909.3907

Parameters:

• rho – A vector.

• k – An int.

• dim – The dimension of the two sub-systems. By default it’s assumed to be equal.

Returns:

The S(k)-norm of rho.