Length filtration of the separable states

TL;DR

This paper studies the structure of separable quantum states by analyzing their length filtration, defining key length parameters, and proving that for bipartite systems with a qubit, the critical length equals a previously defined special length.

Contribution

It introduces the length filtration concept for separable states, defines critical and special lengths, and proves the equality of these lengths for bipartite qubit systems.

Findings

Established bounds for the maximum and critical lengths of separable states.

Proved that for bipartite systems with a qubit, the critical length equals the special length.

Computed the rank of the Jacobian matrix to support the main result.

Abstract

We investigate the separable states $\r$ of an arbitrary multipartite quantum system with Hilbert space $\cH$ of dimensionin $d$. The length $L(\r)$ of $\r$ is defined as the smallest number of pure product states having $\r$ as their mixture. The length filtration of the set of separable states, $\cS$, is the increasing chain $\emptyset\subset\cS'_1\subseteq\cS'_2\subseteq\cdots$, where $\cS'_i=\{\r\in\cS:L(\r)\le i\}$. We define the maximum length, $L_{\rm max}=\max_{\r\in\cS} L(\r)$, critical length, $L_{\rm crit}$, and yet another special length, $L_c$, which was defined by a simple formula in one of our previous papers. The critical length indicates the first term in the length filtrartion whose dimension is equal to $\dim\cS$. We show that in general $d\le L_c\le L_{\rm crit}\le L_{\rm max}\le d^2$. We conjecture that the equality $L_{\rm crit}=L_c$ holds for all finite-dimensional multipartite quantum systems. Our main result is that $L_{\rm crit}=L_c$ for the bipartite systems having a single qubit as one of the parties. This is accomplished by computing the rank of the Jacobian matrix of a suitable map having $\cS$ as its range.