Quantum-process tomography: Resource analysis of different strategies

Characterization of quantum dynamics is a fundamental problem in quantum physics and quantum-information science. Several methods are known which achieve this goal, namely standard quantum-process tomography (SQPT), ancilla-assisted process tomography, and the recently proposed scheme of direct characterization of quantum dynamics (DCQD). Here, we review these schemes and analyze them with respect to some of the physical resources they require. Although a reliable figure-of-merit for process characterization is not yet available, our analysis can provide a benchmark which is necessary for choosing the scheme that is the most appropriate in a given situation, with given resources. As a result, we conclude that for quantum systems where two-body interactions are not naturally available, SQPT is the most efficient scheme. However, for quantum systems with controllable two-body interactions, the DCQD scheme is more efficient than other known quantum-process tomography schemes in terms of the total number of required elementary quantum operations.

Figure 1

Schematic of SQPT. An ensemble of states $\left\{{\rho }_j\right\}$ are prepared and each of them is subjected to the map $\mathcal{E}$, and then to the measurements $\left\{E_m\right\}$.

Figure 2

Schematic diagram of separable AAPT. An ensemble of systems is prepared in the same quantum state $\rho$. Next, they are subjected to the map $\mathcal{E}\otimes I$. Finally the operators $\left\{E_j\right\}$ are measured on both the system and the ancilla, which results in the required joint probability distributions or expectation values.

Figure 3

Schematic diagram of nonseparable AAPT. In this scheme the joint separable measurements of Fig. 2 have been replaced by (collective) mutually unbiased bases measurements on the two systems.

Figure 4

Schematic diagram of a QPT by using POVM. Here we have used the idea of “universal quantum observable” [44]. To accomplish complete process tomography, one needs two more ancillas $B_1$ and $B_2$ (in addition to the one used in AAPT, $B$) and two universal quantum observables ${\Omega }_1$ and ${\Omega }_2$.

Figure 5

Schematic diagram of the DCQD scheme. The system and the ancilla are prepared in one of the input states as in Table , and after subjecting the system to the map $\mathcal{E}$, the combined system is measured in the Bell-state basis.