Almost all multipartite qubit quantum states have trivial stabilizer
arXiv:1609.01327 · doi:10.1063/1.5003015
Abstract
The stabilizer group of an n-qubit state ψis the set of all matrices of the form g=g_1\otimes\cdots\otimes g_n, with g_1,...,g_n being any 2x2 invertible complex matrices, that satisfy gψ=ψ. We show that for 5 or more qubits, except for a set of states of zero measure, the stabilizer group of multipartite entangled states is trivial; that is, containing only the identity element. We use this result to show that for 5 or more qubits, the action of deterministic local operations and classical communication (LOCC) can almost always be simulated simply by local unitary (LU) operations. This proves that almost all n-qubit states with n>4 are isolated, that is they can neither be reached nor converted into any other (n-partite entangled), LU-inequivalent state via deterministic LOCC. We also find a simple and elegant expression for the maximal probability to convert one multi-qubit entangled state to another for this generic set of states.
6 pages (main text) + 7 pages (Appendix), no figures, published version
References in corpus (4)
Cited by in corpus (26)
- Entanglement Certification From Theory to Experiment
- A resource theory of entanglement with a unique multipartite maximally entangled state
- Transformations among Pure Multipartite Entangled States via Local Operations Are Almost Never Possible
- k-stretchability of entanglement, and the duality of k-separability and k-producibility
- Tight constraints on probabilistic convertibility of quantum states
- Multipartite entanglement outperforming bipartite entanglement under limited quantum system sizes
- Mode-entanglement of Gaussian fermionic states
- Locally Maximally Entangled States of Multipart Quantum Systems
- Symmetries and entanglement of stabilizer states
- Measurement outcomes that do not occur and their role in entanglement transformations
- Local Transformations of Multiple Multipartite States
- State transformations within entanglement classes containing permutation-symmetric states
- A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems
- Classification of four qubit states and their stabilisers under SLOCC operations
- Deterministic transformations of three-qubit entangled pure states
- Metrology-assisted entanglement distribution in noisy quantum networks
- Symmetries and local transformations of translationally invariant Matrix Product States
- Entanglement Classification via Single Entanglement Measure
- Principal orbit type theorems for reductive algebraic group actions and the Kempf--Ness Theorem
- Identifying families of multipartite states with non-trivial local entanglement transformations
- Small quantum networks in the qudit stabilizer formalism
- Approximate and ensemble local entanglement transformations for multipartite states
- Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
- Multipartite Monogamy of Entanglement for Three Qubit States
- Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states
- Entanglement groups