Random Unitaries and Random States in Quantum Information Theory - beyond Uniform Randomness
ABSTRACT :
We study quantum tomography from continuous measurement records obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. We consider application of a random diagonal unitary at each time step and quantify the information gain in tomography using Fisher information of the measurement record. We then compare this with the information generated and fidelities obtained by repeated application of a single unitary operator as well as application of a different Haar random unitary at each time step. We surprisingly find that we get very high fidelity reconstruction using random diagonal unitaries even though the measurement record is not informationally complete. We quantify this information incompleteness by finding the number of directions in the operator space where we have no knowledge​.
Moreover, fidelities obtained in the case of repeated application of a single random unitary at each time step are very close to those obtained from application of random diagonals and, in some cases, even exceed them.
We see in our study of tomography, that an observable under unitary evolution traces a trajectory in the operator space spanning all of the space or most of it depending on the unitary used. That occurs because unitary dynamics can generate incompatible observables. We study the “amount of incompatibility" generated by various classes of random unitaries by using entropic uncertainty relations.
Lastly, we extend and generalize Levy's Lemma which is a statement about the concentration of measure on a high dimensional hypersphere.
In particular, we obtain a bound on the probability that a function, at a point picked at random, on a high dimensional hypersphere, from a Lipschitz continuous probability density, lies in a small neighborhood around its expectation value. We briefly discuss its application to quantum information theory.