How do I disinguish between the states using a single distinguishing procedure

$$\frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$$

$$\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$$

$$\frac{1}{\sqrt{2}}(-|01\rangle + |10\rangle)$$

$$\frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)~?$$

I know how to distinguish between just the first two. But I can't find a procedure to distinguish between all of them. I know it must be possible because they all orthogonal to each other.


closed as off-topic by Norbert Schuch, Wolpertinger, Gert, Bill N, user36790 Oct 13 '16 at 16:19

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  • 1
    $\begingroup$ What, exactly, do you mean by a "distinguishing procedure"? $\endgroup$ – ACuriousMind Oct 12 '16 at 19:42
  • $\begingroup$ Yes, this is a part of my homework. However, the homework question is entirely different. I boiled down the homework question down to this problem. fyi, the question is a one-out-of-four search problem where you need to make one quantum query to a function to figure out the nature of the function and has nothing to do with bell states as such. I did like 2 pages of problem solving before I could even pose the problem in this form - which is where I got stuck. $\endgroup$ – IanDsouza Oct 17 '16 at 0:31

Typically, a Bell basis measurement is done with a CNOT and a Hadamard like so:

Bell basis measurement

It's exactly the reverse of making a Bell pair.

You can also simply do parity measurements along the X and Z axes:

Bell basis parity measurement

  • $\begingroup$ thanks. didn't understand the parity measurements but that's okay. $\endgroup$ – IanDsouza Oct 13 '16 at 21:53

Edit: As mentioned by the commenters,the method described below is not ideal because the entanglement breaks down due to the measurement. It only workes if one has an ensemble of entangled pairs all preapared in the same state. One should then do the two measurements on different pairs. The other answer is clearly better.

First you can perform a measurement on both qubits in the {|0>,|1>} basis. If these measurements give the same result, you have one of the top two states, otherwise it is one of the bottom two states.

Then you perform another measurement on both qubits, but this time in the {|+>,|->} basis, where $|+>=\frac{1}{\sqrt{2}}\left(|0>+|1>\right)$ and $|->=\frac{1}{\sqrt{2}}\left(|0>-|1>\right)$. When the outcome of this measurement is identical for both qubits, you have the second or the fourth state, otherwise it is the first or the third state.

  • $\begingroup$ Your procedure does not work: After the first measurement you have destroyed the state. You will not be able any more to carry out the second one. $\endgroup$ – Norbert Schuch Oct 13 '16 at 14:53
  • $\begingroup$ Yeah, after you measure in the single qubit computational basis, the state 'condenses' to one of the terms in the two-term sum of the bell state. You will have lost all information about the other term. In fact, measuring one qubit also confirms what the other qubit is. The other qubit is no longer in a superposition of the computational basis states. This is sort of what characterizes the entangled pair bell state. $\endgroup$ – IanDsouza Oct 13 '16 at 21:51
  • $\begingroup$ @IanDsouza ??? -- If you don't know which Bell state you have, measuring one qubit will NOT tell you what the other is. $\endgroup$ – Norbert Schuch Oct 14 '16 at 4:01
  • $\begingroup$ @NorbertSchuch Yeah, if you don't know which Bell state you have, you will now automatically know the other qubit. You will have to measure the second qubit. $\endgroup$ – IanDsouza Oct 17 '16 at 0:23
  • $\begingroup$ @NorbertSchuch If you are given the same unknown Bell state multiple times and you measure the first qubit. Given the knowledge of the first qubit, you will always get the same answer when you measure the second qubit. $\endgroup$ – IanDsouza Oct 17 '16 at 0:44

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