Separability Criteria of Quantum States based on Generalized Bloch Representation

TL;DR

The paper introduces a unified framework for entanglement detection across bipartite and multipartite quantum systems based on a generalized Bloch representation. It constructs a parameterized extended correlation tensor and a novel mixed mode matrix unfolding to extend bipartite criteria to N-partite settings, yielding practical bounds on trace norms that certify separability. The main bipartite result unifies and subsumes several existing criteria under specific parameter and basis choices, while the multipartite extension provides biseparability and full separability criteria with an explicit genuine $N$-partite entanglement test. Numerical examples demonstrate improved entanglement detection relative to several established criteria, highlighting the approach’s potential as a versatile diagnostic toolbox for complex quantum states.

Abstract

Quantum entanglement serves as a fundamental resource in quantum information theory. This paper presents a comprehensive framework of separability criteria for detecting entanglement across quantum systems, from bipartite to multipartite states. We propose a novel unified parameterized extended correlation tensor, constructed via the generalized Bloch representation under an arbitrary orthogonal basis, which bridges our bipartite criterion with several existing ones. Moreover, we develop a specialized tensor unfolding technique -- termed mixed mode matrix unfolding -- that naturally generalizes the conventional $k$-mode matrix unfolding and enables the generalization of the extended correlation tensor construction to multipartite systems. And we derive several separability criteria for multipartite states. Numerical examples demonstrate that our separability criteria exhibit enhanced capability in detecting entanglement.

Figures (6)

• Figure 1: The relation between the entanglement detection of $\rho_p$ and $p,x$ in Example \ref{['example: bipartite ppt state']} with setting of $\bm{u}=\left(x,x,x,x\right)^{\rm T}$, $\bm{v}=\left(\sqrt{x},\sqrt{x},\sqrt{x},\sqrt{x}\right)^{\rm T}$, $\bm{\alpha}=0.063782$, and $\bm{\beta}=0.0786454$.
• Figure 2: The relation between the entanglement detection of $\rho_p$ and $p,x$ in Example \ref{['example: bipartite bounded entangled state']} with setting of $a=0.9$, $\bm{u}=\left(x,x\right)^{\rm T}$, $\bm{v}=\left(\sqrt{x},\sqrt{x}\right)^{\rm T}$, $\bm{\alpha}=\left(-3.23405, 1.35293\right)^{\rm T}$, and $\bm{\beta}=\left(-1.83346, -0.969888\right)^{\rm T}$.
• Figure 3: The mixed $\left(\left\{1\right\},1;\left\{2,3\right\},1\right)$-mode matrix unfolding of $\mathcal{A}$ in Example \ref{['example: mixed mode matrix unfolding of A']}.
• Figure 4: The mixed $\left([3],1;[4,6],2\right)$-mode matrix unfolding of $\mathcal{B}$ in Example \ref{['example: mixed mode matrix unfolding of B']}.
• Figure 5: The relation between the entanglement detection of $\rho_p$ and $p,x$ in Example 5 with setting of $\bm{u}=\left(x,x\right)^{\rm T}$, $\bm{v}=\left(\sqrt{x},\sqrt{x}\right)^{\rm T}$, $\bm{\alpha}=\left(-0.516382, -0.165015\right)^{\rm T}$, and $\bm{\beta}=\left(-0.259148, 0.485242\right)^{\rm T}$.

Theorems & Definitions (28)

• Lemma 1
• Theorem 1
• Example 1
• Example 2
• Definition 1: Mixed Mode Matrix Unfolding of Tensors
• Example 3
• Example 4
• Theorem 2
• Theorem 3
• Lemma 2