Continuous phase-space representations for finite-dimensional quantum states and their tomography

Continuous phase spaces have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states in quantum optics and beyond. A plethora of these phase-space techniques are known, however a thorough understanding of their relations is still lacking for finite-dimensional quantum states. We present a unified approach to continuous phase-space representations which highlights their relations and tomography. The infinite-dimensional case from quantum optics is then recovered in the large-spin limit.

Figure 1

(a) $s$-parametrized phase-space representations $F_{|W\rangle }(\theta ,\varphi ,s)$ for $s\in \{1,1/2,0,-1/2,-1\}$ of a (generalized) $W$ state $|W\rangle$ for a single spin with $J=10$, or equivalently the symmetric Dicke state $|J,J-1\rangle$ of $2J$ indistinguishable qubits with a single Majorana vector pointing to the south pole and $2J-1$ vectors pointing to the north pole. A decreasing $s$ (left-to-right) which smears out $F_{|W\rangle }(\theta ,\varphi ,s)$ is interpreted as a Gaussian-like convolution. Red (dark gray) and green (light gray) represent positive and negative values, respectively. The brightness reflects the absolute value of the function relative to its global maximum $\eta$. (b) Spherical Wigner functions $F_{|W\rangle }(\theta ,\varphi ,0)$ for increasing $J$ approach their planar counterpart, i.e., the single-photon state $F_{|1\rangle }(\alpha ,0)$ (see Sec. 2). Identical coordinate patches with $-1.2\le x,y\le 1.2$ have been used, where $x=Rsin\theta cos\varphi$ and $y=Rsin\theta sin\varphi$ in the first three plots and $x=\text{Re}\left(\alpha \right)$ and $y=\text{Im}\left(\alpha \right)$ in the last one. [For the plots in (b), methods from [49] to efficiently approximate phase-space representations for large $J$ have been applied.]

Figure 2

Phase-space representations $W_{\rho }$, $Q_{\rho }$, and $P_{\rho }$ of infinite- or finite-dimensional density operators $\rho$ as expectation values of the parity operator ${\mathrm{\Pi }}_s$ or $M_s$ [dashed arrows, see Eq. (1) or Eq. (4)]. Also shown is the transformation by Gaussian smoothing with $W_{|0\rangle }$ and $Q_{|0\rangle }$ or reversibly with $W_{|JJ\rangle }$ and $Q_{|JJ\rangle }$ [solid arrows, see Eq. (3) or Eq. (10)].

Figure 3

Parity-operator entries ${\left[M_s\right]}_{mm}$ corresponding to Eq. (5) [and equivalently the Stern-Gerlach reconstruction weights in Eq. (12)] for a single spin $J$ shown for the $P$ function $P_{\rho }$, the Wigner function $W_{\rho }$, and the $Q$ function $Q_{\rho }$.

Figure 4

Phase-space representations $F_{|JJ\rangle }(\theta ,s)$ of the spin-up state $|JJ\rangle$ [see Eq. (8)]. As $J$ increases, $Q_{|JJ\rangle }$ and $W_{|JJ\rangle }$ rapidly converge to the Gaussian distributions $Q_{|0\rangle }$ and $W_{|0\rangle }$ (dashed line); $P_{|JJ\rangle }$ slowly approaches the delta function $P_{|0\rangle }=\delta \left(\mathrm{\Omega }\right)$.

Figure 5

$P$, Wigner, and $Q$ functions with their corresponding convolution kernels for (a) a quantum state of a spin $J=5/2$ corresponding to the GHZ state $|\mathrm{GHZ}\rangle$ of $2J$ indistinguishable qubits, (b) a squeezed state $|\mathrm{sq}\rangle$ of a spin $J=10$, and (c) a random state $|\mathrm{rnd}\rangle$ of a spin $J=4$ [95] (see Sec. 3c). Red (dark gray) and green (light gray) represent positive and negative values, respectively. The absolute values of the spherical function relative to its global maximum $\eta$ are given by the plotted surface (left) or the brightness (right), where each variant highlights different properties of the plotted functions.

Figure 6

Simulated tomography of Wigner functions $W_{\rho }(\theta ,\varphi )$ for a random ensemble of $N_{\rho }=2200$ spin-$5/2$ states $\rho$. (a) Pointwise tomography at the phase-space point $(\theta ,\varphi )=(0,0)$ as discussed in Sec. 4a. The relative reconstruction errors (relative to the global maximum of the ideal phase-space function) empirically follow a Gaussian distribution. Its mean $\mu$ and standard deviation $\sigma$ are obtained from a fitted Gaussian distribution. And $\sigma$ empirically scales as $N_r^{-1/2}$ with the number $N_r$ of repetitions. (b) Pointwise tomographies from (a) evaluated at ${22}^2=484$ phase-space points (as discussed in Sec. 4b) for points $({\theta }_k,{\varphi }_q)$ from an equiangular grid (refer to Sec. 4c). The relative reconstruction errors (relative to the global maximum of the ideal phase-space function) are averaged over the grid points $({\theta }_k,{\varphi }_q)$. The mean $\mu$ and standard deviation $\sigma$ of the average error are obtained from a fitted Gaussian distribution; $\sigma$ empirically scales as $N_r^{-1/2}$. (c) Full tomography as discussed in Sec. 4c using an equiangular grid of ${22}^2=484$ phase-space points. The relative $L^2$-norm errors (relative to the global maximum of the ideal phase-space function) empirically follow a log-normal distribution. The mean and standard deviation are determined from a fitted log-normal distribution and the mean also scales as $N_r^{-1/2}$. (d) Examples of reconstructed Wigner functions from (c) with their relative $L^2$-norm errors.

Figure 7

Relative reconstruction errors (relative to the global maximum of the ideal phase-space function) of simulated full tomographies of $P$, $W$, and $Q$ functions evaluated at the phase-space point $(\theta ,\varphi )=(0,0)$ for a random ensemble of $N_{\rho }=2200$ spin-$5/2$ states using $N_r=1000$ Stern-Gerlach repetitions. This is similar as discussed in Sec. 4c but the reconstruction errors are only evaluated at $(\theta ,\varphi )=(0,0)$. The directly reconstructed phase-space functions (see Result 4) for the green histograms on the diagonal are in a second step transformed with a spherical convolution (see Result 2) to $P$, $W$, and $Q$ functions for the red, off-diagonal histograms. The direct reconstruction is usually preferable. The mean $\mu$ and standard deviation $\sigma$ are obtained from a fitted Gaussian distribution.

Figure 8

Similar as in Fig. 7, the relative reconstruction errors are given here, however, for the full phase-space function as relative $L^2$ errors (and not only for a single phase-space point). The directly reconstructed phase-space functions for the green histograms on the diagonal are usually preferable to the red, off-diagonal histograms that are obtained from the green ones by applying an additional spherical convolution (see Result 2).

Figure 9

(a), (b) Stern-Gerlach reconstruction weights ${\left[M_0^R\right]}_{mm}$ in Eq. (13) for the Radon transform of a Wigner function applicable to a single spin $J$ (see Fig. 3). (c), (d) Wigner function (see Fig. 5), its Radon transform, and its point-symmetric part reconstructed by inverse Radon transformation. (c) Quantum state of a single spin with $J=5/2$ corresponding to the GHZ state of $2J$ qubits [see Fig. 5]. (d) Squeezed state $|\mathrm{sq}\rangle$ of a single spin with $J=10$ [see Fig. 5] approximately localized on the upper hemisphere of its Wigner function $W_{|\mathrm{sq}\rangle }$; the corresponding Radon transform $R_{|\mathrm{sq}\rangle }$ is highly localized around the equator and can be reconstructed using few measurements. Red (dark gray) and green (light gray) represent positive and negative values, respectively.