# Mathematical and actual measurements?

1. Jan 14, 2016

### Zafa Pi

Let me introduce notation so my question makes sense.

A set of orthogonal states can be distinguished by a Q-measurement, e.g. |0> and |1> (Q-computation notation) can be distinguished by measuring with the observable Z which has eigenvectors |0> and |1> with eigenvalues +1 and -1. But if those states represent horizontal and vertically polarized photons then a polarization analyzer (PA(0)) can actually pull off the measurement in the lab. Likewise cosx|0> + sinx|1> and -sinx|0> + cosx|1> can be distinguished by the observable ((cos2x)Z + (sin2x)X)/sqrt2 or using PA(x) on the appropriate polarized photons.

The set |00>, |01>, |10>, |11> can be distinguished by using Z or PA(0) on each pair of states or photons.

However, what actual PAs or other device lets me distinguish the 4 Bell states (|00> + |11>)/sqrt2,
(|00> - |11>)/sqrt2, (|01> +|10>)?sqrt2, and (|01> - |10>)?sqrt2?

2. Jan 15, 2016

### zonde

You can distinguish 2 Bell states ((|01> +|10>)/sqrt2, and (|01> - |10>)/sqrt2) out of four using Bell state analyzer (BSA):
http://arxiv.org/abs/quant-ph/9808040 look at fig.1

3. Jan 15, 2016

### jfizzix

It is actually an open research question to devise an experiment that can perform a "Bell measurement" in a reliable way.

One can mathematically construct Bell observables whose eigenstates are the four Bell states, but these observbles cannot be expressed as a simple product or sum of single qubit observables.. In short, there's no combination of polarization analyzers, beamsplitters, and linear optical devices (that I'm aware of) that can pull off a Bell measurement.

In order to perform a Bell measurement, one needs to be able to interact the two particles in a known way that will transform the set of entangled Bell states into the set of simple product states, which can be sorted reliably. Thus, the problem of Bell state discrimination is no harder than the problem of transforming uncorrelated particles into Bell-entangled particles in a reliable way (though this is no easy task either), There has been some progress on this using nonlinear optical elements, but research is still ongoing.

4. Jan 16, 2016

### Zafa Pi

Distinguishing those two is easy with a pair of PA(0)s. Measuring (|01> +|10>)/sqrt2 yields the same value for each PA(0), i.e. either both yield +1 or both measurements yield -1. Whereas measuring (|01> - |10>)/sqrt2 with the same pair of polarization analyzers yields different values for the result. This still doesn't let me distinguish all four states.
Also for some reason I was unable to access the content of your link, merely the abstract.

5. Jan 16, 2016

### Zafa Pi

Very interesting. It explains why my efforts were in vain. Thanks.
Nielsen & Chuang, (in Q-Computation & Q-Information published way back in 2000) say on page 98 that it has received partial verification in the lab, referencing Mattle, Weinfurter, Kwait, and Zeilinger. Dense coding in experimental quantum communication.

6. Jan 16, 2016

### StevieTNZ

The document is a PDF: http://arxiv.org/pdf/quant-ph/9808040v1.pdf (http://arxiv.org/abs/quant-ph/9808040 > right hand side, under Download:)

7. Jan 16, 2016

### Zafa Pi

8. Jan 18, 2016

### Zafa Pi

In spite of what you say, there are many claims of successful teleportation, so what protocol are they using. Well the following article (http://www.ijoart.org/docs/Design-a...led-NOT-gate-using-optical-implementation.pdf) purports to implement the CNOT gate in the lab which in turn would allow one to distinguish the members of the Bell basis. So I'm still confused since the other article site in this thread seems to confirm what you say.