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Length filtration of the separable states
Lin Chen1, Dragomir Ž Ðoković2
1School of Mathematics and Systems Science, Beihang University, Beijing 100191, People's Republic of China; International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, People's Republic of China.
Proceedings. Mathematical, Physical, and Engineering Sciences
|December 14, 2016
Summary
We introduce "length" for separable quantum states, defining critical and maximum lengths. For bipartite systems with a qubit, we prove critical length equals a previously defined special length.
Area of Science:
- Quantum Information Theory
- Multivariate Quantum Systems
Background:
- Investigating separable states in quantum systems is crucial for understanding quantum correlations.
- The 'length' of a separable state is defined as the minimum number of pure product states needed for its mixture.
Purpose of the Study:
- To define and analyze various length measures (maximum, critical, special) for separable states in multi-partite quantum systems.
- To investigate the relationship between critical length (L_crit) and a special length (L_c), particularly for bipartite systems.
Main Methods:
- Defining length filtration for separable states.
- Analyzing inequalities relating different length measures: d <= L_c <= L_crit <= L_max <= d^2.
- Computing the rank of a Jacobian matrix to determine L_crit = L_c for specific bipartite systems.
Main Results:
- Established bounds for length measures in multi-partite quantum systems.
- Conjectured L_crit = L_c for all finite-dimensional multi-partite systems.
- Proved L_crit = L_c for bipartite systems where one party is a qubit.
Conclusions:
- The equality L_crit = L_c holds for a significant class of bipartite quantum systems.
- This finding advances the understanding of entanglement and separability in quantum information.
- Further research is needed to prove the conjecture for all multi-partite systems.
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