Internal Boundary between Entanglement and Separability Within a Quantum State
arXiv:2003.00607 · doi:10.1109/TIT.2022.3204712
Abstract
Quantum states are the key mathematical objects in quantum mechanics, and entanglement lies at the heart of the nascent fields of quantum information processing and computation. However, there has not been a general, necessary and sufficient, and operational separability condition to determine whether an arbitrary quantum state is entangled or separable. In this paper, we show that whether a quantum state is entangled or not is determined by a threshold within the quantum state. We first introduce the concept of \emph{finer} and \emph{optimal} separable states based on the properties of separable states in the role of higher-level witnesses. Then we show that any bipartite quantum state can be decomposed into a convex mixture of its optimal entangled state and its optimal separable state. Furthermore, we show that whether an arbitrary quantum state is entangled or separable, as well as positive partial transposition (PPT) or not, is determined by the robustness of its optimal entangled state to its optimal separable state with reference to a crucial threshold. Moreover, for an arbitrary quantum state, we provide operational algorithms to obtain its optimal entangled state, its optimal separable state, its best separable approximation (BSA) decomposition, and its best PPT approximation decomposition while it was an open question that how to calculate the BSA in high-dimension systems.
12 pages, 4 figures, close to the published version
References in corpus (36)
- Quantum entanglement
- Supplementary information for "Quantum supremacy using a programmable superconducting processor"
- Entanglement detection
- Efficient classical simulation of slightly entangled quantum computations
- Quantum Resource Theories
- On the reality of the quantum state
- Quantum entanglement in photosynthetic light harvesting complexes
- Classification of mixed three-qubit states
- Graph States for Quantum Secret Sharing
- A complete family of separability criteria
- Separable states are more disordered globally than locally
- Quantifying Entanglement with Witness Operators
- Some Properties of the Computable Cross Norm Criterion for Separability
- Entanglement witnesses: construction, analysis and classification
- Reflections upon separability and distillability
- Covariance matrices and the separability problem
- How robust is a quantum gate in the presence of noise?
- On the quantification of entanglement in infinite-dimensional quantum systems
- Steering bound entangled states: A counterexample to the stronger Peres conjecture
- Separable Multipartite Mixed States - Operational Asymptotically Necessary and Sufficient Conditions
- On the geometry of entangled states
- Geometric entanglement witnesses and bound entanglement
- Operational interpretation of weight-based resource quantifiers in convex quantum resource theories of states
- Representation of entanglement by negative quasi-probabilities
- Separable approximation for mixed states of composite quantum systems
- On structural physical approximations and entanglement breaking maps
- Best Separable Approximation of multipartite diagonal symmetric states
- Sufficient separability criteria and linear maps
- Optimal Lewenstein-Sanpera Decomposition for some Biparatite Systems
- Entangled states in the role of witnesses
- The hierarchies of "witnesses" and the properties and characterization of entangled states as super entanglement witnesses
- Optimal indecomposable witnesses without extremality as well as spanning property
- Optimal Lewenstein-Sanpera decomposition of two-qubit states using Semidefinite Programming
- Degree of separability of bipartite quantum states
- Entanglement Verification, with or without tomography
- Universal methods for extending any entanglement witness from the bipartite to the multipartite case