state_props.sk_vec_norm

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

Functions

sk_vector_norm(rho[, k, dim])

Compute the S(k)-norm of a vector [1].

Module Contents

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

Compute the S(k)-norm of a vector [1].

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

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

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 entangled 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)
np.float64(0.5)

References

[1] (1,2)

Nathaniel Johnston and David W. Kribs. A family of norms with applications in quantum information theory. Journal of Mathematical Physics, Aug 2010. URL: http://dx.doi.org/10.1063/1.3459068, doi:10.1063/1.3459068.

Parameters:

  • rho (numpy.ndarray) – A vector.

  • k (int) – An int.

  • dim (int | list[int]) – The dimension of the two sub-systems. By default it’s assumed to be equal.

Returns:

The S(k)-norm of rho.

Return type:

float