TY - JOUR

T1 - Coarse graining of entanglement classes in 2×m×n systems

AU - Hebenstreit, M.

AU - Gachechiladze, M.

AU - Gühne, O.

AU - Kraus, B.

N1 - Publisher Copyright:
© 2018 American Physical Society.

PY - 2018/3/21

Y1 - 2018/3/21

N2 - We consider three-partite pure states in the Hilbert space C2 - Cm - Cn and investigate to which states a given state can be locally transformed with a nonvanishing probability. Whenever the initial and final states are elements of the same Hilbert space, the problem can be solved via the characterization of the entanglement classes which are determined via stochastic local operations and classical communication (SLOCC). In the particular case considered here, the matrix pencil theory can be utilized to address this point. In general, there are infinitely many SLOCC classes. However, when considering transformations from higher to lower dimensional Hilbert spaces, an additional hierarchy among the classes can be found. This hierarchy of SLOCC classes coarse grains SLOCC classes which can be reached from a common resource state of higher dimension. We first show that a generic set of states in C2 - Cm - Cn for n=m is the union of infinitely many SLOCC classes, which can be parameterized by m-3 parameters. However, for n≠m there exists a single SLOCC class which is generic. Using this result, we then show that there is a full-measure set of states in C2 - Cm - Cn such that any state within this set can be transformed locally to a full measure set of states in any lower dimensional Hilbert space. We also investigate resource states, which can be transformed to any state (not excluding any zero-measure set) in the smaller dimensional Hilbert space. We explicitly derive a state in C2 - Cm - C2m-2 which is the optimal common resource of all states in C2 - Cm - Cm. We also show that for any n<2m it is impossible to reach all states in C2 - Cm - Cn whenever n>m.

AB - We consider three-partite pure states in the Hilbert space C2 - Cm - Cn and investigate to which states a given state can be locally transformed with a nonvanishing probability. Whenever the initial and final states are elements of the same Hilbert space, the problem can be solved via the characterization of the entanglement classes which are determined via stochastic local operations and classical communication (SLOCC). In the particular case considered here, the matrix pencil theory can be utilized to address this point. In general, there are infinitely many SLOCC classes. However, when considering transformations from higher to lower dimensional Hilbert spaces, an additional hierarchy among the classes can be found. This hierarchy of SLOCC classes coarse grains SLOCC classes which can be reached from a common resource state of higher dimension. We first show that a generic set of states in C2 - Cm - Cn for n=m is the union of infinitely many SLOCC classes, which can be parameterized by m-3 parameters. However, for n≠m there exists a single SLOCC class which is generic. Using this result, we then show that there is a full-measure set of states in C2 - Cm - Cn such that any state within this set can be transformed locally to a full measure set of states in any lower dimensional Hilbert space. We also investigate resource states, which can be transformed to any state (not excluding any zero-measure set) in the smaller dimensional Hilbert space. We explicitly derive a state in C2 - Cm - C2m-2 which is the optimal common resource of all states in C2 - Cm - Cm. We also show that for any n<2m it is impossible to reach all states in C2 - Cm - Cn whenever n>m.

UR - http://www.scopus.com/inward/record.url?scp=85044405290&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.97.032330

DO - 10.1103/PhysRevA.97.032330

M3 - Article

AN - SCOPUS:85044405290

SN - 2469-9926

VL - 97

JO - Physical Review A

JF - Physical Review A

IS - 3

M1 - 032330

ER -