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julia> res =separable_distance(ρ, dims); #achieve the distance to the separable space
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Iteration Primal Dual gap #Atoms
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julia> res =separable_distance(ρ, dims); #compute the distance to the separable set
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Iteration Primal Dual gap #Atoms
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11.6600e+004.0000e+001
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100003.2667e-016.1346e-0810
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200003.2667e-016.1346e-0810
@@ -68,13 +68,13 @@ Iteration Primal Dual gap #Atoms
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[ Info: Stop: maximum iteration reached
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```
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For the state ``ρ``, as the distance to the separable space `res.primal` is much larger than 0, practically, we can detect the entanglement of the state with confidence (technically speaking, ``Primal`` $\gg$ ``Dual gap``.)
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For the state ``ρ``, as the distance to the separable set `res.primal` is much larger than 0, practically, we can detect the entanglement of the state with confidence (technically speaking, ``Primal`` $\gg$ ``Dual gap``).
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## Entanglement certification
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In principle, if ``Primal`` $\geq$ ``Dual gap``, the state is outside the separable space, therefore is entangled. However, due to the heuristic method, the ``Dual gap`` is inaccuracy. In practice, we can detect the entanglement by check enlarging the factor, e.g., ``Primal`` $\geq 5 \times$ ``Dual gap``.
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In principle, if ``Primal`` $>$ ``Dual gap``, the state is outside the separable set, therefore is entangled. However, since the default method in our algorithm is heuristic, the printed value of ``Dual gap`` is a lower bound on its actual value. For practical applications, it is empirically enough (although not theoretically proven) to enlarge the factor, e.g., ``Primal`` $\geqslant 5 \times$ ``Dual gap``.
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A rigorous tool is also introduce in our package:
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For cases where this is not sufficient, a rigorous tool is also introduced in our package:
Let's consider the other case that there is more noise mixed in the state.
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Let us consider the other case, namely, when there is more noise mixed in the state.
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```julia
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julia> d =2; N =2; p =0.8; ρ = Ket.state_ghz(d, N; v =1- p) # with more white noise
@@ -95,14 +95,14 @@ julia> d = 2; N = 2; p = 0.8; ρ = Ket.state_ghz(d, N; v = 1 - p) # with more wh
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0.0-0.0im0.0-0.0im0.2+0.0im0.0+0.0im
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0.1-0.0im0.0-0.0im0.0-0.0im0.3+0.0im
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julia> res =separable_distance(ρ, dims); #achieve the distance to the separable space
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Iteration Primal Dual gap #Atoms
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julia> res =separable_distance(ρ, dims); #compute the distance to the separable set
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Iteration Primal Dual gap #Atoms
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11.3600e+004.0000e+001
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Last 7.7981e-071.0612e-0314
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[ Info: Stop: primal small enough
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```
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For this case, ``Primal`` is much smaller than ``Dual gap``, which can not be detected as an entangled state, and also can not be confirmed by entanglement witness:
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Here, ``Primal`` is much smaller than ``Dual gap``, which can not be detected as an entangled state, and also cannot be confirmed by entanglement witness:
@@ -111,7 +111,7 @@ julia> real(dot(witness.W, ρ)) < 0 # if Tr(Wρ) > 0, then the state ρ could be
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false
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```
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In order to certify separability, a geometric reconstruction procedure was introduced in Ref. [2] as the part of our package to build a unified toolbox for entanglement analysis:
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In order to certify separability, a geometric reconstruction procedure was introduced in Ref. [2] and is also provided in our package to build a unified toolbox for entanglement analysis:
The computation is based on an efficient variant of the Frank-Wolfe algorithm to iteratively find the separable state closest to the input quantum state based on correlation tensor.
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See this recent [review](https://arxiv.org/abs/2211.14103) for an introduction to the method and the package [FrankWolfe.jl](https://github.com/ZIB-IOL/FrankWolfe.jl) for the implementation on which this package relies.
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See this recent [review](https://arxiv.org/abs/2211.14103) for an introduction to the method, the [article](https://arxiv.org/abs/2506.02635) for the details on the latest improvements used in our case, and the package [FrankWolfe.jl](https://github.com/ZIB-IOL/FrankWolfe.jl) for the implementation on which this package relies.
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## Going further
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More examples can be found in the corresponding folder of the package.
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They include the application on the 10-qubit system with shortcut method and multipartite systems with different entanglement structures.
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They include the application on a 10-qubit system (with a shortcut method for early stop) and multipartite systems with different entanglement structures.
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