All Abstracts | Poster Abstracts | Talk Abstracts

Mutually unbiased measurements in finite dimensions

Amir Kalev, University of New Mexico

(Session 5 : Thursday from 5:00pm - 7:00pm)

We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long standing problem of the existence of MUB. We derive their general form, and show that in a finite, d-dimensional Hilbert space, one can construct a complete set of d+1 mutually unbiased measurements. Beside of their intrinsic link to MUB, we show, that these measurements' statistics provide complete information about the state of the system. Moreover, they capture the physical essence of unbiasedness, and in particular, they satisfy non-trivial entropic uncertainty relation similar to d+1 MUB.

Quantum process tomography of near-unitary maps

Amir Kalev, University of New Mexico

(Session 11 : Saturday from 11:15am - 11:45am)

We study the problem of quantum process tomography given the prior information that the implemented map is near to a unitary map on a d-dimensional Hilbert space. In particular, we show that a perfect unitary map is completely characterized by a minimum of d^2 + d measurement outcomes. This contrasts with the d^4 measurement outcomes required in general. To achieve this lower bound, one must probe the system with a particular set of d states in a particular order. This order exploits unitarity but does not assume any other structure of the map. We further numerically study the behaviors of two of compressed sensing estimators based on correct or faulty prior information caused by noise. The results show two important features: (1) When we have accurate prior information, one can drastically reduce the required data needed; (2) Different estimators applied to the same data are sensitive to different types of noise. The estimators could, therefore, be used as indicators of particular error models and to validate the use of prior assumptions for compressed sensing quantum process tomography. Finally, we consider the more general case of noisy quantum maps, with a low level of noise. Our study indicates that transforming to the interaction picture, where the noiseless map is represented by a diagonal operator, can provide a useful tool to identify the noise structure. This, in turn, can lead to a substantial reduction in the numerical resources needed to estimate the noisy map.