Anything wrong with tomography?
“Quantum tomography”, or “state estimation”, is solidly established — or so it seemed until some months ago.
The notion is pretty simple and peacefully admitted: it’s just the quantum analog of the reconstruction of a statistical distribution from a sampling. How would you know if a die is balanced? You cast it many times and infer the probabilities. Ideally, you should cast it infinitely many times; fortunately, the whole field of statistics provides rigorous ways of assessing how much you can trust your inference from finite samples.
You can reconstruct a quantum state in a similar way. There is one main difference: the quantum state contains information about the statistics of all possible measurements and, as well known, in quantum physics not all measurements are compatible. This is solved by sampling not just for one measurement, but for several. For instance, if you want to reconstruct the state of a spin 1/2, you need to reconstruct the statistics of measurements along three orthogonal directions x,y,z. It’s like saying that you have to cast a die in three different ways, if you pass the lousy analogy.
In the lab, tomography has been used for decades, for characterization: you think you have a source that produces a given quantum state and use tomography to check how well you succeeded. Often, tomography is used to certify that the state of two or more quantum objects is “entangled”.
Theorists have been working in devising various improvements. The biggest challenge is the fact that many statistical schemes may end up reconstructing a state that is not a valid one (think of reconstructing the statistics of a die and finding out the result 5 happens with negative probability!). Also, tomography is a useful motivation to study the structure of “generalized quantum measurements” (the kind that deserve the awful acronym POVMs) and plays a crucial role even in some “interpretations” of quantum physics, notably “quantum bayesianism” (I can’t really get to the bottom of it: Chris Fuchs speaks so well that, whenever I listen to him, I get carried away by the style and forget to try and understand what he means. If you really want to make the effort, read this paper).
All is well… until, a few months ago, reports appeared that quite elementary sources of possible errors had been underestimated:
- One such source are systematic errors. Consider the example of the spin 1/2: certainly, experimentalists can’t align their devices exactly along x, y and z. They can calibrate their direction as well as the precision of their calibration devices allow. According to a paper by Gisin’s group in Geneva, the effect of the remaining error has been largely neglected. While probably not dramatic for one spin, the required corrections may become serious when it comes to estimating the state of many, possibly entangled spins.
- Another quite obvious possibility is a drift by the source. When we cast a die many times, we make the assumption that we are always casting the same die. This is not necessarily true down to ultimate principles: some tiny pieces of matter may be detached by each collision of the die with the floor, so the die may be lighter and deformed after many trials. This deterioration seems inconsequential with a die. But things may be different when it comes to quantum states that are produced by complex laboratory equipment that have the nasty tendency of not being as stable as your mobile telephone (for those who don’t know, in a research lab, the stabilization and calibration of the setup typically takes months: once it is done, the actual collection of interesting data may only take a few days or even hours). Two papers, one in December 2012 and the other posted three days ago but written earlier, explore the possibility of tomography when the source of quanta is not producing the same state in each instance.
Does all this story undermine quantum tomography? Does it even cast a gloomy shadow on science? My answer is an unambiguous NO. All the previous works on tomography were done under some assumptions. Whenever those assumptions hold, whenever there is reason to trust them, those works are correct. If the assumptions can be doubted, then obviously the conclusion should be doubted too. With these new developments, people will be able to do tomography even under more relaxed assumptions: great! The lesson to be learned is: state your assumptions (OK, you may not want to state all the assumptions in all your technical papers aimed at your knowledgeable peers: but you must be aware of them, and state them whenever you write a review paper, lecture notes or similar material).