# Separability Criterion for Multipartite GHZ Quantum States

In "SEPARABILITY CRITERION FOR MULTIPARTITE QUANTUM STATES BASED ON THE BLOCH REPRESENTATION OF DENSITY MATRICES" by Hassan and Joag, I found this remarkable thing about entanglement of mixed GHZ states, saying:

We consider [a] $$N$$-qubit state $$\rho_{noisy}^{N} = \frac{1-p}{2^N}I + p|\psi\rangle\langle\psi|, 0\le p\le1$$ where $$|\psi\rangle$$ is a $$N$$-qubit ... GHZ state.

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Does anyone know how these value would evolve when $$N$$ grows?

Their entanglement/separability criterion is given as Theorem 1.

If a $$N$$-partite quantum state of dimension $$d_1d_2...d_N$$ with Bloch representation $$(8)$$ is fully separable, then $$||\mathcal{T}^{(N)}||_{KF}\le \sqrt{\frac1{2^N}\prod_{k=1}^N d_k(d_k-1)}$$

$$\mathcal{T}^{(N)}$$ is given as last term in $$(8)$$:

$$||\mathcal T||_{KF} = max\{||T_{(n)}^N||_{KF}\}, n=1,...,N;$$ is a Ky-Fan norm, which is the sum of the $$k$$ largest singular values of the matrix unfoldings of $$\mathcal T$$.

I'm not interested in $$|W\rangle$$ states...

• GHZ seem to have a downward trend with larger N... not sure about the W though Jul 17, 2020 at 2:19
• what entanglement criterion is being used here?
– glS
Jul 17, 2020 at 6:23
• @HasanIqbal yes, interesting isn't it? I edited my question... Jul 17, 2020 at 7:10
• @glS it is rather a separability criterion. I added the corresponding parts of the paper... Jul 17, 2020 at 7:11

There are some very simple entanglement criteria that one can apply. In particular, pick any bipartition of the system. If that bipartition contains entanglement (which might be found using the partial transpose criterion), then the state is certainly not fully separable, because it is not separable across that bipartition. For the GHZ state, this threshold occurs at $$p=\frac{1}{2^{N-1}+1}.$$ These values are lower than those stated in the table, and so give a stronger claim. It actually turns out that one can prove this is the threshold for the state becoming fully separable, so it's an exact result.
• +1 thanks, so you're saying that we are able to determine(perform a measurement) that a mixed quantum state $\rho=(1-p)I + p|GHZ_N\rangle\langle GHZ_N|$ is entangled? Jul 17, 2020 at 7:54
• hmm, but your witness $W_{x,z}$ (if we speak about this, do we?) asks for two choices of $x,z\in\{0,1\}^N$. How to pick the right ones? there are exponentially many... Jul 17, 2020 at 9:34