Marco Piani course @ UFF – Lecture 3Posted: August 30, 2013
Lecture 3 – Wed 28 Aug
Today Marco started by wrapping up some topics from the previous lecture: He first noted that there is a connection between the witness and PnCP-map methods for detecting entanglement: for any given entangled states, given a map that detects its entanglement allows us to construct a witness, and vice-versa. Nevertheless, there is a definite practical distinction between the two techniques: in order to apply the witness method in the lab, one only needs to measure a single (but well-chosen) expectation value of the state. On the other hand, applying the PnCP map method requires first obtaining (eg tomographing) the full state.
He then mentioned that the entanglement-detection criterion based on the partial transposition PnCP map is necessary and sufficient in the cases of 2×2 or 2×3-dimensional systems.
Finally, he noted that the degree of violation of this criterion, measured by a quantity called the Negativity, is non-increasing under LOCC, and may be used as an easily computed measure of entanglement. He cautioned, however, that no single measure is sufficient to capture the richness of entanglement.
Marco then moved on to studying nonlocality. After recalling the definition of a local joint probability distribution from lecture 1, he proved the celebrated CHSH inequality, and showed that it may be violated by the results of local measurements on an entangled quantum state.
Geometrically, this means that, within the space formed by all possible probability distributions , the subset of local distributions is strictly contained within a wider set composed of the distributions that can be obtained from quantum measurements. Marco then showed that this set is, in turn, part of an even wider set of ‘nonsignalling’ (NS) distributions, which do not allow communication from A to B or vice – versa. More technically: these are the distributions for which a change of (say) Bob’s input y does not affect the marginal statistics of Alice’s measurement results.
Interestingly, is is now known that there exist nonsignalling distributions which are not achievable by QM. Marco discussed briefly the convexity properties of the sets of Local, QM and NS distributions. Finally, he raised the point of why it is interesting to study such supra-quantum correlations. Apart from the sheer fun of it, three reasons were mentioned: first, even though we currently believe QM is the correct framework for physics, it is conceivable that QM may have to be modified or extended to account for phenomena such as gravity. In this case, it is important to know if and how this could be done without breaking fundamental principles such as the impossibility of instantaneous signalling, which is incompatible with relativity. Second, if such an extension does turn out to be needed for the correct description of physics, this will have implications for the security of cryptographic protocols that assume the validity of QM. Designing protocols that are secure even if certain supra-quantum correlations exist serves as an insurance policy against such an eventuality. Finally, the fact that the nonsignalling condition does not, by itself, imply the validity of QM leads to the question of which further principles would need to be invoked in order to constrain the set of compatible correlations precisely to the quantum ones.