Separable Ball around any Full-Rank Multipartite Product State

Abstract

We show that around any $m$-partite product state ${\rho }_{\mathrm{prod}}={\rho }_1\otimes ...\otimes {\rho }_m$ of full rank (that is $\det ({\rho }_{\mathrm{prod}})\ne 0)$, there exists a finite-sized closed ball of separable states centered around ${\rho }_{\mathrm{prod}}$ whose radius is $\beta :=2^{1-m/2}{\lambda }_{\min }({\rho }_{\mathrm{prod}})$. Here, ${\lambda }_{\min }({\rho }_{\mathrm{prod}})$ is the smallest eigenvalue of ${\rho }_{\mathrm{prod}}$. We are assuming that the total Hilbert space is finite dimensional and we use the notion of distance induced by the Frobenius norm. Applying a scaling relation, we also give a new and simple sufficient criterion for multipartite separability based on trace: $\mathrm{Tr}[\rho {\rho }_{\mathrm{prod}}{]}^2/\mathrm{Tr}[{\rho }^2]\ge \mathrm{Tr}[{\rho }_{\mathrm{prod}}^2]-{\beta }^2$. Using the separable balls around the full-rank product states, we discuss the existence and possible sizes of separable balls around any multipartite separable states, which are important features for the set of all separable states. We discuss the implication of these separable balls on entanglement dynamics.

arXiv comment12 pages. v2: new sections, added references

References in corpus (24)

• Separability Criterion for Density Matrices
• A computable measure of entanglement
• Quantum sensing
• Separability of Mixed States: Necessary and Sufficient Conditions
• Quantum Computing
• On the volume of the set of mixed entangled states
• Bell Inequalities and the Separability Criterion
• Entanglement monotones
• Single-Ion Atomic Clock with $3\times 10^{-18}$ Systematic Uncertainty
• Robustness of entanglement
• Information-theoretic aspects of quantum inseparability of mixed states
• Separability of very noisy mixed states and implications for NMR quantum computing
• Largest separable balls around the maximally mixed bipartite quantum state
• Geometry of entangled states
• Separable balls around the maximally mixed multipartite quantum states
• Tripartite entanglement: Foundations and applications
• Quantum Separability and Entanglement Detection via Entanglement-Witness Search and Global Optimization
• Entanglement and deterministic quantum computing with one qubit
• Nonlinear entanglement witnesses, covariance matrices and the geometry of separable states
• Entanglement-Saving Channels
• Eventually entanglement breaking Markovian dynamics: structure and characteristic times
• Boundary of the set of separable states
• A bipartite separable ball and its applications
• The Transfer of Entanglement Negativity at the Onset of Interactions

Cited by in corpus (3)

• Entanglement Microscopy: Tomography and Entanglement Measures via Quantum Monte Carlo
• The Fate of Entanglement
• Separable ellipsoids around multipartite states