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Let's say we want to analyze the entanglement property of the two-qubit maximally entangled state with white noise.
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We demonstrate how to analyze the entanglement property of the two-qubit maximally entangled state with white noise.
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Using `EntanglementDetection.jl`, here is what the code looks like.
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```julia
@@ -68,13 +68,40 @@ julia> res = separable_distance(ρ, dims); # compute the distance to the separab
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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 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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For the state ``ρ``, as the distance to the separable set `res.primal` is significantly greater 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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### Rapid entanglement detection
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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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In principle, if ``Primal`` > ``Dual gap``, the state lies outside the separable set and is therefore entangled. However, since the default method in our algorithm is heuristic, the reported ``Dual gap`` is only a lower bound on the true value.
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Empirically, requiring ``Primal`` ≥ 5 × ``Dual gap`` is often sufficient for robust detection, especially in noisy but clearly experimental entangled states.
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For cases where this is not sufficient, a rigorous tool is also introduced in our package:
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To support this, we provide a shortcut mode that speeds up the detection process.
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The package also accepts raw experimental input in the form of a correlation tensor from standard full state tomography.
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```julia
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# the tensor of the density matrix; can be replaced by real experimental data
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julia> C =correlation_tensor(ρ, dims)
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4×4 Matrix{Float64}:
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1.00.00.00.0
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0.00.20.00.0
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0.00.0-0.20.0
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0.00.00.00.2
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# the default basis is the generalized Gell-Mann basis (Pauli basis for qubits)
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# any informationally complete basis with normalization 2 can be used
@@ -83,9 +110,9 @@ julia> real(dot(witness.W, ρ)) < 0 # if Tr(Wρ) < 0, then the state ρ is entan
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true
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```
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## Separability certification
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### Separability certification
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Let us consider the other case, namely, when there is more noise mixed in the state.
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Now consider a noisier version of the same 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
@@ -102,7 +129,7 @@ julia> res = separable_distance(ρ, dims); # compute the distance to the separab
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[ Info: Stop: primal small enough
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```
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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:
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In this case, ``Primal`` is much smaller than ``Dual gap``, which can not be detected as an entangled state, and also cannot be confirmed by entanglement witness:
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