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Case I: Alice and Bob each have a quantum state. Their quantum states are maximally entangled.

Case II: Alice and Charlie each have a quantum state. Their quantum states are maximally entangled.

Cases I and II are not happening simultaneously.

Say each person's state was stored in a (separate) quantum memory. Alice used the same quantum memory in both cases. Is it guaranteed that given the condition of maximal entanglement, that Bob and Charlie had idenitcal quantum states in their respective quantum memories?

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  • $\begingroup$ Can you clarify your question? $\endgroup$ – Norbert Schuch Dec 16 '16 at 5:38
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Saying that the state is maximally entangled by definition is equivalent to saying that the state of each subsystem is maximally mixed. I.e. the pure state in $\mathcal{H}_A\times \mathcal{H}_B$ is maximally entangled if the reduced density matrix of each subsystem is proportional to identity matrix $\frac{1}{\mathrm{dim}\mathcal{H}_A}\mathbb{I}_A$ and $\frac{1}{\mathrm{dim}\mathcal{H}_B}\mathbb{I}_B$. If those two subsystems are identical you can say that they for maximally entangled state they are in identical states but there's nothing nontrivial in it.

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  • $\begingroup$ I'm having a little trouble understanding the second paragraph. Should Bob and Charlie had idenitcal quantum states in their respective quantum memories? I don't know if this is critical to understand it or not.. But given any arbitrary pure state that I have with me, can I always find another state that I can be maximally entangled with? And if so, will that state be unique for a given state that I have? $\endgroup$ – IanDsouza Dec 16 '16 at 23:35
  • $\begingroup$ @IanDsouza Ok, do you actually know the meaning of the words "pure state", "mixed state", "maximally mixed state", "entangled state"? Because it seems to me that you don't at all and that's the crux of the problem. If you have with you some pure state it's definitely not maximally entangled with anything because again by definition of maximal entanglement state of subsystem should be maximally mixed and it's like a contrary of pure... $\endgroup$ – OON Dec 17 '16 at 0:18
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    $\begingroup$ @IanDsouza I don't know,maybe you'll get it with example. For two qubits you can have maximally entangled state in the form $\frac{1}{\sqrt{2}}(|0\rangle|1\rangle+|1\rangle|0\rangle)$. If you calculate reduced density matrix for both systems it will be $\frac{1}{2}I$ where $I$ is identity matrix. This is obviously not pure state but a maximally mixed state. And for maximally entangled state this is the only possibility you can get. $\endgroup$ – OON Dec 17 '16 at 4:18
  • $\begingroup$ Thanks. I'm just learning this stuff. How about asking whether the reduced density matrix for Bob and Charlie in the two cases would be the same if it is known that the reduced density matrix for Alice is the same in both the cases? $\endgroup$ – IanDsouza Dec 18 '16 at 2:48

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