Tomographic measurements on superconducting qubit states

We propose an approach to reconstruct any superconducting charge qubit state by using quantum state tomography. This procedure requires a series of measurements on a large enough number of identically prepared copies of the quantum system. The experimental feasibility of this procedure is explained and the time scales for different quantum operations are estimated according to experimentally accessible parameters. Based on the state tomography, we also investigate the possibility of the process tomography.

Figure 1

(Color online) The black Bloch vector indicates a qubit state; the (yellow arrows) $r_x$, $r_y$, and $r_z$ represent the three components of the Bloch vector along the $x$, $y$, and $z$ axes. The 0 and 1 in the north and south poles of the Bloch sphere denote the measured states $\mid 0⟩$ and $\mid 1⟩$, respectively. The measurement instrument is attached to a pole (e.g., “1”) of the sphere. A $-\pi ∕2$ rotation of the qubit state along the $x$ direction is equivalent to a $\pi ∕2$ rotation of the measuring instrument along the $x$ direction.

Figure 2

(Color online) The Bloch vector is the same as Fig. 1. A $-\pi ∕2$ rotation of the qubit along the $y$ direction is equivalently realized by the rotation $\pi ∕2$ of the measuring instrument along the $z$ direction [from (a) to (b)], then a $\pi ∕2$ rotation along the $x$ direction [from (b) to (c)], and a $3\pi ∕2$ rotation along $z$ direction [from (c) to (d)].

Figure 3

(Color online) Graphical representation of the density matrix $\rho$ for single-qubit states, see the example explained in Sec. 2. The real ${\rho }_{ij}^{\left(R\right)}$ and imaginary ${\rho }_{ij}^{\left(I\right)}$ parts of the density matrix elements ${\rho }_{ij}=⟨i\mid \rho \mid j⟩$ ($i$, $j=0$, 1) are plotted in (a) and (b), respectively.

Figure 4

(Color online) Graphical representation of the density matrix ${\rho }^{\prime }$ for the two-qubit state described in the example given in Sec. 3. The real ${\rho }_{ij,kl}^{\prime \left(R\right)}$ and imaginary ${\rho }_{ij,kl}^{\prime \left(I\right)}$ parts of the density matrix elements for the two-qubit state ${\rho }^{\prime }$ in the basis $\mid 00⟩$, $\mid 01⟩$, $\mid 10⟩$, $\mid 11⟩$ are plotted in (a) and (b), respectively.