Optimized tomography of continuous variable systems using excitation counting

We propose a systematic procedure to optimize quantum state tomography protocols for continuous variable systems based on excitation counting preceded by a displacement operation. Compared with conventional tomography based on Husimi or Wigner function measurement, the excitation counting approach can significantly reduce the number of measurement settings. We investigate both informational completeness and robustness, and provide a bound of reconstruction error involving the condition number of the sensing map. We also identify the measurement settings that optimize this error bound, and demonstrate that the improved reconstruction robustness can lead to an order-of-magnitude reduction of estimation error with given resources. This optimization procedure is general and can incorporate prior information of the unknown state to further simplify the protocol.

Figure 1

Procedure of estimating the ${\alpha }_i$ via Husimi Q function. (a) shows the true Q function of the state; (b) shows the estimated Q function via iMLE after measuring $Q_n^{\beta }\left(\rho \right)$ at three ${\beta }^{\prime }\mathrm{s}$ shown as the crosses; (c) and (d) are estimations after measuring at four and five ${\beta }^{\prime }\mathrm{s}$, respectively. Apparently the estimate in (c) already converges to the true Q function shown in (a).

Figure 2

Condition number of the sensing map as a function of $\beta$ for cat states with number of components $p=2,3,4$. Upper panels: Numerical results for CN. Lower panels: A simple estimate of the CN using the expression $\kappa \left(\beta \right)\sim {\sum }_{i,j}exp\left[{\left(d_i-d_j\right)}^2/2\right]$ where $d_i\equiv \left|{\alpha }_i-\beta \right|$. We also included the white lines on which the sensing map is strictly informationally incomplete (see main text). Blue stars indicate the positions of the coherent components $\left|{\alpha }_i\right.〉$. For visual clarity, values beyond 100 are all mapped to white. The minimum CNs achievable for the three cases are 1.74, 6.81, and 38.64 (numerical results), respectively. Here the maximal resolved excitation number $n_c$ is taken sufficiently large. If $n_c$ decreases, CN for large $\left|\beta \right|$ gets worse.

Figure 3

Determinant of the Fisher information $\mathcal{I}\left(\vec{\rho }\right)$ as a function of $\beta$ for four different states. (a) Two-component maximally mixed cat state, ${\rho }_{ij}\propto {\delta }_{ij}$. In other words, the Bloch vector for the effective two level system is $\vec{0}$. (b) A two-component cat state, with Bloch vector $0.9·(1,1,0)/\sqrt{2}$. (c) Three-component maximally mixed cat state, ${\rho }_{ij}\propto {\delta }_{ij}$. (d) A mixture $\rho =(1-\lambda )I/3+\lambda \left|\psi \right.〉\left.〈\psi \right|$ where $I$ is the identity and $\left|\psi \right.〉={(1,1,1)}^{†}\sqrt{3}$. The shape of the good detection region for maximally mixed states is very similar to that predicted by the condition number while for higher purity states additional “interference fringes” appear. The worst case of Fisher information over all true states appears to be that of the maximally mixed states. The good regions for $\beta$ predicted by worst-case Fisher information agree well with that given by condition number.

Figure 4

Main panel: Condition numbers of full-ring configuration (FRC) and half-ring configuration (HRC) as a function of the ring radius ($m_c=4$ case). Top two insets: FRC and HRC in phase space. For both schemes, ${\beta }_j=\left|\beta \right|e^{i{\varphi }_j}$. FRC: ${\varphi }_j=\frac{2\pi }{2m_c+1}j,j=0,1,2,...,2m_c$. HRC: ${\varphi }_j=\frac{\pi }{2m_c+1}j,j=0,1,2,...,m_c$. The condition number of HRC approaches that of FRC as $\left|\beta \right|$ gets large, as predicted by theory. Bottom inset: Figure of merit $\kappa \sqrt{N_{\beta }}$ for HRC and FRC.

Figure 5

Optimal condition number for $Q_n$ measurements as a function of $m_c$. Vertical axis shows $\kappa {\left(A\right)}^2$. Red solid line shows a linear fit with equation ${\kappa }^2=3.28m_c-0.07769$.

Figure 6

Comparison of the figures of merits (assuming shot noise only) $\kappa \sqrt{N_{\beta }}$ for optimized $Q_n$ tomography with large enough $\left|\beta \right|$ and optimized Wigner tomography obtained from gradient-based optimization.

Figure 7

Comparison of performances of Wigner measurements where ${\beta }^{\prime }\mathrm{s}$ form a square lattice (yellow triangles), Wigner measurements with optimized measurement settings obtained from gradient search (red squares), and $Q_n$ measurements with optimized measurement settings (blue circles). Left and right panels correspond to $m_c=2$ and 5. The true state $\rho$ is a randomly generated density matrix with excitation number cutoff $m_c=5$. Each scatter point corresponds to one reconstruction via semidefinite programming based on a set of simulated measurement records containing only shot noise. The $y$ axis shows the reconstruction infidelity $\delta F=1-F(\rho ,{\rho }^{\prime })$ and the $x$ axis shows the total number of measurements performed, i.e., total number of copies of unknown states consumed.

Figure 8

Greedy optimization of the set of ${\beta }^{\prime }\mathrm{s}$. Crosses show the position of ${\alpha }^{\prime }\mathrm{s}$ and stars indicate all the ${\beta }^{\prime }\mathrm{s}$ added to the set $S$. At each step, the optimal $\beta$ is added to the set $S$. When the condition number is low enough (smaller than a preset threshold), $m_c$ is increased by one and the optimization goes on.