Wigner process tomography: Visualization of spin propagators and their spinor properties

We study the tomography of propagators for spin systems in the context of finite-dimensional Wigner representations, which completely characterize and visualize operators using shapes assembled from linear combinations of spherical harmonics. The Wigner representation of a propagator can be experimentally recovered by measuring expectation values of rotated axial spherical tensor operators in an augmented system with an additional ancilla qubit. The methodology is experimentally demonstrated for standard one-qubit quantum gates using nuclear magnetic resonance spectroscopy. In particular, this approach provides a direct and compelling visualization of the spinor property of the propagators corresponding to the rotation of a spin-1/2 particle.

Figure 1

(a) The two-spin operator $A=\frac{1}{2}\mathbf{1}+I_{1x}+I_{2z}+I_{1x}{I}_{2x}+I_{1x}{I}_{2y}+I_{1x}{I}_{2z}$ [28, 29] is represented using multiple spherical functions $f^{\left(\ell \right)}=f^{\left(\ell \right)}(\theta ,\varphi )$, and individual components $A^{\left(\ell \right)}$ of $A$ mapped to $f^{\left(\ell \right)}$ and graphically visualized. (b) $f^{\left\{12\right\}}$ (box) decomposed into its $2^j$-multipole contributions $f_j^{\left\{12\right\}}$ with $j\in \{0,1,2\}$. (c) $f_1^{\left\{12\right\}}$ (circle) decomposed into spherical harmonics of order $m\in \{-1,0,1\}$; $Y_{1,-1}$ and $Y_{1,1}$ are rainbow colored [30].

Figure 2

For a system of interest consisting of a single ($N=1$) spin 1/2, the figure illustrates the tomographic reconstruction of a spherical function $f^{\left(\ell \right)}(\beta ,\alpha )$ representing the Hadamard gate based on Result 1. (a) Samples (crosses) with different polar angles $\beta$ in the range $[0,\pi ]$ and phases $\alpha$ in the range $[0,2\pi ]$ acquired by expectation values ${\langle I_a\otimes T_{j,\alpha \beta }^{\left(\ell \right)\left[N\right]}\rangle }_{{\rho }_U^{[N+1]}}$ with $a\in \{x,y\}$ and $j\in \{0,1\}$. The colors (shades of gray) of the circles of latitude are defined as a function of the polar angle $\beta$ by the given color bar (gray scale). (b) Predicted expectation values (line) and experimentally measured expectation values $\langle I_x\otimes T_{1,\alpha \beta }\rangle :={\langle I_x\otimes T_{1,\alpha \beta }^{\left(\ell \right)\left[N\right]}\rangle }_{{\rho }_U^{[N+1]}}$ (crosses) for a simple sampling scheme with an equidistant discrete set of polar angles $\beta \in \{0,\frac{\pi }{12},\frac{2\pi }{12},\cdots ,\pi \}$ and azimuthal angles $\alpha \in \{0,\frac{\pi }{12},\frac{2\pi }{12},\cdots ,2\pi \}$. For each discrete value ${\beta }_n=n\pi /12$ with $n\in \{0,1,2,\cdots ,12\}$, the azimuthal angle $\alpha$ is incremented in steps of $\pi /12$ from 0 to $2\pi$. This scheme results in $13\times 25=325$ measurement points acquired in 13 cycles to fully scan the sphere. Note that the remaining expectation values ${\langle I_x\otimes T_{0,\alpha \beta }^{\left(\ell \right)\left[N\right]}\rangle }_{{\rho }_U^{[N+1]}}$, ${\langle I_y\otimes T_{0,\alpha \beta }^{\left(\ell \right)\left[N\right]}\rangle }_{{\rho }_U^{[N+1]}}$, and ${\langle I_y\otimes T_{1,\alpha \beta }^{\left(\ell \right)\left[N\right]}\rangle }_{{\rho }_U^{[N+1]}}$ are zero and not shown here. (c) Smooth surface interpolated from individual samples with distance from the origin given by $f^{\left(\ell \right)}(\beta ,\alpha )$, the phase of which determines the color (gray scale) of the surface (see Fig. 4).

Figure 3

Schematic representation of the tomography scheme proposed by Result 2 to scan the spherical functions $f^{\left(\ell \right)}(\beta ,\alpha )$ for the case $N=1$ with $\ell \in \{\varnothing ,1\}$ by measuring the expectation values ${\langle {\left(M_{j,n}^{\left(\ell \right)}\right)}_a^{\left[2\right]}\rangle }_{{\widetilde{\stackrel{̃}{\rho }}}_U^{\left[2\right]}}$ for $j\in \{0,1\}$, $n=1$, and $a\in \{x,y\}$ as described in Sec. 5b. See also Fig. 2.

Figure 4

Experimentally reconstructed (a) and theoretical (b) spherical functions $f(\theta ,\varphi )$ representing propagators for the Id, not, $\sqrt{\text{not}}$, and the Hadamard gate. The colors red (dark gray), yellow (light gray), green (gray), and blue (black) correspond to phase factors $exp\left(i\phi \right)$ of 1, $i$, $-1$, and $-i$ [9]. See the color bar (gray scale) for $0\le \phi \le 2\pi$.

Figure 5

Experimentally reconstructed (a) and theoretical (b) spherical functions $f(\theta ,\varphi )$ representing propagators of phase-shift gates for 0, $\pi /2$, $\pi$, $3\pi /2$, and $2\pi$.

Figure 6

Reconstructed spherical functions $f(\theta ,\varphi )$ representing propagators for rotations around the $x$ axis for rotation angles $0,\pi /2,\pi ,3\pi /2,2\pi ,5\pi /2,3\pi ,7\pi /2$, and $4\pi$. The experimentally sampled shapes are positioned along the inner circle, whereas the theoretical functions are positioned along the outer circle. The colors red (dark gray), yellow (light gray), green (gray), and blue (black) correspond to phase factors $exp\left(i\phi \right)$ of 1, $i$, $-1$, and $-i$; see the color bar (gray scale) in Fig. 4. Note the sign change of the experimentally measured and theoretical DROPS representation when a given rotation angle is increased by $2\pi$, nicely illustrating the spinor property of propagators corresponding to rotations of spin-1/2 particles.

Figure 7

The Wigner representation $f(\theta ,\varphi )$ of a ${\left[\frac{\pi }{2}\right]}_x$ rotation propagator is decomposed into its contributions $f_0^{\left\{\varnothing \right\}}(\theta ,\varphi )$ and $f_1^{\left\{1\right\}}(\theta ,\varphi )$.

Figure 8

Theoretical Wigner function $f(\theta ,\varphi )$ representing various propagators realizing $x$ rotations decomposed into its contributions $f_0^{\left\{\varnothing \right\}}(\theta ,\varphi )$ (positioned along the inner circle) and $f_1^{\left\{1\right\}}(\theta ,\varphi )$ (positioned along the outer circle).

Figure 9

Experimentally reconstructed Wigner function $f(\theta ,\varphi )$ representing various propagators realizing $x$ rotations decomposed into its contributions $f_0^{\left\{\varnothing \right\}}(\theta ,\varphi )$ (positioned along the inner circle) and $f_1^{\left\{1\right\}}(\theta ,\varphi )$ (positioned along the outer circle).

Figure 10

For the case of a system consisting only of a single ($N=1$) spin 1/2, the simulations show how different sources of propagator errors can be identified in the DROPS representation: (a) Ideal case of a rotation with rotation angle $\mathrm{\Psi }=\pi$ and rotation axis $\vec{n}$ with $n_x=1$ and $n_y=n_z=0$, (b) rotation angle increased by 10%, (c) rotation axis deviating by an angle of $\pi /10$ from the $x$ axis resulting in $n_x=cos(\pi /10)$ and $n_y=sin(\pi /10)$ and $n_z=0$, and (d) simultaneous error of the flip angle as in case (b) and of the rotation axis as in case (c).