Joint Seminar on Quantum Information and Technologies

We present an error analysis of various tomographic protocolsbased on the linear inversion for the reconstruction of an unknowntwo-qubit state. We solve the problem of finding a tomographicprotocol which is the most robust against errors in terms of thelowest value (i.e., equal to 1) of a condition number, as requiredby the Gastinel–Kahan theorem. In contrast, standard tomographicprotocols, including those based on mutually unbiased bases, arenonoptimal for determining all the 16 elements of an unknowntwo-qubit density matrix. Our method is based on the measurementsof the 16 generalized Pauli operators, where twelve of them can belocally measured, and the other four require nonlocal Bellmeasurements. Our method corresponds to selectively measuring, oneby one, all the real and imaginary elements of an unknowntwo-qubit density matrix. We describe two experimentally feasiblesetups of this protocol for the optimal reconstruction of twophotons in an unknown polarization state using conventionaldetectors and linear-optical elements. Moreover, we define theoperators for the optimal reconstruction of the states ofmultiqubit or multilevel (qudit) systems.