In a paper arXiv:1312.0158 the authors proved that any complex vector (thus also the quantum state vector) can be uniquely determined (up to a global phase factor) by 4N-4 generic intensity measurements, where N is the dimension of the complex space. Thus if one can from an experiment obtain the absolute values of inner products of the state vector with 4N-4 other vectors that constitute a generic frame, then he is able to find the state vector. My question is: is it really experimentally doable to find such a number of projections of the state vector?
http://arxiv.org/abs/1312.0158
Dear Artur,
My question is: is it really experimentally doable to find such a number of projections of the state vector?
You can do this. At least for qubits (N =2) the solution is simple. Please see "Minimal qubit tomography" by Jaroslav Řeháček et al. (see the link below). There, the set of projectors corresponds to the vertices of a tetrahedron (4*2 -4 = 4) embedded within the Bloch sphere.
You may also want to look for other paper on minimal quantum tomography.
http://arxiv.org/pdf/quant-ph/0405084v2.pdf
I see that nobody has replied so far so maybe I shall add a bit of comment to encourage others to this discussion. Generally, I believe that it is easier to repeat the same kind of measurement a few times than to devise a few distinct measurements and perform each of them once. Thus, I developed an approach in which you can reconstruct a complex state vector by a number of projectors significantly lower than 4N-4. The difference is that in my case the same projective measurement is performed at different time instants. I believe my approach can be beneficial. But I just wanted to ask other whether for others that seems also like a good idea.
Dear Artur,
My question is: is it really experimentally doable to find such a number of projections of the state vector?
You can do this. At least for qubits (N =2) the solution is simple. Please see "Minimal qubit tomography" by Jaroslav Řeháček et al. (see the link below). There, the set of projectors corresponds to the vertices of a tetrahedron (4*2 -4 = 4) embedded within the Bloch sphere.
You may also want to look for other paper on minimal quantum tomography.
http://arxiv.org/pdf/quant-ph/0405084v2.pdf
Dear Karol!
Thank you very much for your comment and for the paper you recommended. I will definitely read it and compare this approach with mine.
My approach to quantum tomography is dynamic, which in case of qubits allows me to reconstruct the complex state vector on the basis of two projectors but each of them is measured in two time instants t1 = 0 and t2>0. Thus, in my approach you also perform four measurements so the difference is not so significant... It might be more advantageous to use my approach in higher dimensional cases, but I'm not 100% sure about it.
Anyway, soon I will post my newest article on arxiv so and I will ask you some feedback.
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