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The maximally entangled set of 4-qubit states

### Abstract

Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the set and on the other hand, no state in the set can be obtained from any other state via LOCC. Recently, we studied LOCC transformations among pure multipartite states and derived the MES for three and generic four qubit states. Here, we consider the non-generic four qubit states and analyze their properties regarding local transformations. As already the most coarse grained classification, due to Stochastic LOCC (SLOCC), of four qubit states is much richer than in case of three qubits, the investigation of possible LOCC transformations is correspondingly more difficult. We prove that most SLOCC classes show a similar behavior as the generic states, however we also identify here three classes with very distinct properties. The first consists of the GHZ and W class, where any state can be transformed into some other state non-trivially. In particular, there exists no isolation. On the other hand, there also exist classes where all states are isolated. Last but not least we identify an additional class of states, whose transformation properties differ drastically from all the other classes. Although the possibility of transforming states into local-unitary inequivalent states by LOCC turns out to be very rare, we identify those states (with exception of the latter class) which are in the MES and those, which can be obtained (transformed) non-trivially from (into) other states respectively. These investigations do not only identify the most relevant classes of states for LOCC entanglement manipulation, but also reveal new insight into the similarities and differences between separable and LOCC transformations and enable the investigation of LOCC transformations among arbitrary four qubit states.

Published by AIP Publishing.

Received 20 November 2015
Accepted 04 April 2016
Published online 11 May 2016

Acknowledgments:
The research of C.S. and B.K. was funded by the Austrian Science Fund (FWF): No. Y535-N16. J.I.d.V acknowledges financial support from the Spanish MINECO Grant Nos. MTM2010-21186-C02-02, MTM2011-26912, MTM2014-54692, and MTM2014-54240-P and CAM regional research consortium QUITEMAD+CM S2013/ICE-2801.

Article outline:

I. INTRODUCTION
II. GENERAL METHOD OF DETERMINING THE MES
A. Notation
B. Separable transformations and *MES*_{3} and *MES*_{4}
C. Symmetries and seed states
D. Outline of proceeding sections and results
III. THE SLOCC CLASSES *G*_{abcd}
A. Cyclic eigenvalues
1. Two cycles of size 2
2. A cycle of size 4
3. A cycle of size 3
B. A non-zero triple-degenerate eigenvalue
C. A double-degenerate non-zero eigenvalue and two different non-degenerate eigenvalues
D. Two non-zero double-degenerate eigenvalues
1. The case *a*^{2} ≠ ± *c*^{2} and *a*, *c* ≠ 0
2. The case *a*^{2} = − *c*^{2} and *a* ≠ 0
E. A double-degenerate zero eigenvalue and two different non-degenerate eigenvalues
F. A double-degenerate non-zero eigenvalue and a double-degenerate zero eigenvalue
IV. THE SLOCC CLASSES *L*_{abc2}
A. The SLOCC classes *L*_{abc2} for *c* ≠ 0
1. The case *a*^{2} ≠ *b*^{2} ≠ *c*^{2} ≠ *a*^{2} and *c* ≠ 0
2. The case *a*^{2} = *c*^{2} ≠ *b*^{2} and *c* ≠ 0
3. The case *a*^{2} = *b*^{2} ≠ *c*^{2} and *a*, *c* ≠ 0
4. The case *a*^{2} = *b*^{2} = *c*^{2} ≠ 0
5. The case *a* = *b* = 0 and *c* ≠ 0
B. The SLOCC classes *L*_{abc2} for *c* = 0
1. Case (1): *a*^{2} ≠ *b*^{2} and *a*, *b* ≠ 0
2. Case (2): *a*^{2} = *b*^{2} and *a* ≠ 0
V. THE SLOCC CLASSES *L*_{a2b2}
A. The SLOCC classes *L*_{a2b2} for *a*, *b* ≠ 0
1. Case (1): *a*^{2} ≠ ± *b*^{2}
2. Case (2): *a*^{2} = − *b*^{2} and *a*, *b* ≠ 0
3. Case (3): *a*^{2} = *b*^{2} and *a*, *b* ≠ 0
B. The SLOCC classes *L*_{a2b2} for *a* = 0 and *b* ≠ 0
C. The SLOCC class *L*_{a2b2} for *a* = *b* = 0
VI. THE SLOCC CLASSES *L*_{ba3}
A. The SLOCC classes *L*_{ba3} for *a* ≠ 0
1. The case *a*^{2} ≠ *b*^{2} and *a* ≠ 0
2. The case *a*^{2} = *b*^{2} and *a* ≠ 0
B. The SLOCC classes *L*_{ba3} with *a* = 0
1. The case *a* = 0 and *b* ≠ 0
2. The case *a* = *b* = 0
VII. THE SLOCC CLASSES *L*_{a4}
A. The SLOCC classes *L*_{a4} for *a* ≠ 0
B. The SLOCC class *L*_{a4} for *a* = 0
VIII. THE SLOCC CLASSES *L*_{a203⊕1}
IX. THE SLOCC CLASS *L*_{05⊕3}
X. THE SLOCC CLASS *L*_{07⊕1}
XI. CONCLUSIONS

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2016-09-28

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