# For a given quantum state, is there a unique quantum state that is maximally entangled with the first?

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?

• Can you clarify your question? Dec 16, 2016 at 5:38

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.
• @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.