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I'm having trouble understanding the physical information one might extract from partial traces, by interpreting the partial trace of a qubit as if the qubit was averaged out of the system.

For example, imagine I have three qubits in a GHZ state $$ |GHZ> = {1 \over \sqrt{2}}(|000>_{abc}+|111>_{abc}), $$ and I give qubit $a$ to Alice, $b$ to Bob and $c$ to Charlie. If I partial trace qubit $c$, I end up with a separable mixed state. From what I understand, this means that if Alice and Bob completely ignore Charlie's qubit and any information Charlie might give, they will not be able to successfully execute a protocol that needs their qubits to be entangled, because the subsystem Alice-Bob is not entangled when the qubit $c$ is ignored.

The situation would be the opposite if the W state $$ |W> = {1 \over \sqrt{2}}(|100>_{abc}+|010>_{abc}+|001>_{abc}) $$ was considered, because the partial trace of qubit $c$ gives an entangled mixed state, pointing that the subsystem Alice-Bob is entangled.

Is this reasoning correct or not? Thank you very much for any help!

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closed as unclear what you're asking by Norbert Schuch, sammy gerbil, Martin, Cosmas Zachos, Daniel Griscom Mar 6 '18 at 2:21

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  • $\begingroup$ not sure what you are asking. Yes your reasoning is correct, the $W$ state has a more "stable" kind of entanglement in this sense, while for the GHZ the entanglement cannot be used without access to all three parties, as you correctly noted. $\endgroup$ – glS Feb 25 '18 at 1:11