2017/10/03 by M. Hebenstreit, Martin Hebenstreit, M. Gachechiladze +5 · 8 citations
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Combinatorics #Discrete mathematics #Hilbert space #Mathematics #Parameterized complexity #Physics #Pure mathematics #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum and electron transport phenomena #Quantum entanglement #Quantum mechanics #State (computer science) #quant-ph
paper · pdf · doi:10.1103/physreva.97.032330
published in Physical Review A 97(3) (American Physical Society) · 29 pages, 3 figures
arxiv created 2017/10/03 · openalex created_date 2017/10/20 · openalex publication_date 2018/03/21 · arxiv updated 2018/03/28 · openalex updated_date 2026/08/06
We consider three-partite pure states in the Hilbert space ℂ2\ensuremath\bigotimesℂm\ensuremath\bigotimesℂn 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\ensuremath\bigotimesℂm\ensuremath\bigotimesℂn for n=m is the union of infinitely many SLOCC classes, which can be parameterized by m\ensuremath-3 parameters. However, for n\ensuremath≠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\ensuremath\bigotimesℂm\ensuremath\bigotimesℂn 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\ensuremath\bigotimesℂm\ensuremath\bigotimesℂ^2m\ensuremath-2 which is the optimal common resource of all states in ℂ2\ensuremath\bigotimesℂm\ensuremath\bigotimesℂm. We also show that for any n<2m it is impossible to reach all states in ℂ2\ensuremath\bigotimesℂm\ensuremath\bigotimesℂ^\stackrel\ifmmode \else \~\fin whenever \stackrel\ifmmode \else \~\fin>m.