Data-pattern tomography of entangled states

We discuss the data-pattern tomography for reconstruction of entangled states of light. We show that for a moderate number of probe coherent states it is possible to achieve high accuracy of representation not only for single-mode states but also for two-mode entangled states. We analyze the stability of these representations to the noise and demonstrate the conservation of the purity and entanglement. Simulating the probe and signal measurements, we show that systematic error inherent for representation of realistic signal response with finite sets of responses from probe states still allows one to infer reliably the signal states preserving entanglement.

Figure 1

Measurement scheme of data-pattern approach.

Figure 2

The fidelities of the representation of mixed states $\rho =p|0\rangle \langle 0|+(1-p)|1\rangle \langle 1|$ ($p\in [0,1]$) using coherent projectors with amplitudes selected in (a) the square lattice on the phase plain with $N\times N=6\times 6, d=0.1$ and (b) the helical grid with $N=17, \mathrm{\Delta }r=0.016, \mathrm{\Delta }\phi =\pi /4$.

Figure 3

The fidelity of the representation of entangled double-mode states against the grid pitch $d$ of the square lattice of coherent projectors: (a) the number of nodes along each axis $N=6$; (b) the number of nodes along each axis $N=7$. The predefined double-mode states are $\psi =\text{const}\times \left({|\alpha \rangle }_1{|-\alpha \rangle }_2+{|-\alpha \rangle }_1{|\alpha \rangle }_2\right)$, with $\alpha =0.5$ (black bars), $\psi =\left({|0\rangle }_1{|1\rangle }_2+{|1\rangle }_1{|0\rangle }_2\right)/\sqrt{2}$ (dark gray bars), $\psi =\left({|0\rangle }_1{|0\rangle }_2+{|1\rangle }_1{|1\rangle }_2\right)/\sqrt{2}$ (light gray bars).

Figure 4

The fidelities of the representation of entangled mixed states $\rho =\frac{1-p}{2}\left({|0\rangle }_1{|1\rangle }_2+{|1\rangle }_1{|0\rangle }_2\right)\left({\langle 0|}_1{\langle 1|}_2+{\langle 1|}_1{\langle 0|}_2\right)+\frac{p}{2}\left({|0\rangle }_1{|0\rangle }_2+{|1\rangle }_1{|1\rangle }_2\right)\left({\langle 0|}_1{\langle 0|}_2+{\langle 1|}_1{\langle 1|}_2\right)$ ($p\in [0,1]$) using coherent projectors with amplitudes selected in the square lattice on the phase plain with $N=7, d=0.05$.

Figure 5

The fidelities of the representation (5) against the rms amplitude error. (a) The fidelities of the representation of the coherent state with amplitude $\alpha =0.5$ for the square lattice of coherent projectors with $N=6, d=0.15$. (b) The fidelities of the representation of coherent state with amplitude $\alpha =0.5$ for the helical grid with $N=17, \mathrm{\Delta }r=0.016, \mathrm{\Delta }\phi =\pi /4$. (c) The fidelities of the representation of the state $\psi =\left({|0\rangle }_1{|1\rangle }_2+{|1\rangle }_1{|0\rangle }_2\right)/\sqrt{2}$ using coherent projectors in the pitches of the square lattice with $N=7, d=0.15$.

Figure 6

Plots of the trace $Tr\left(W\rho \right)$ against the grid pitch $d$ of the square lattice on the phase plane: (a) the number of nodes along each axis $N=6$ (fidelity level of representation ${\rho }^{\mathrm{Appr}}$ is 0.994–0.999); (b) the number of nodes along each axis $N=7$ (fidelity level of representation ${\rho }^{\mathrm{Appr}}$ is 0.999–0.99999). The entanglement witnesses $W$ are obtained solving the optimization problem for the precise density matrices. Filled bars represent the results for the density operators ${\rho }^{\mathrm{Appr}}$ being the representation of the following states: $\psi =\text{const}\times \left({|\alpha \rangle }_1{|-\alpha \rangle }_2+{|-\alpha \rangle }_1{|\alpha \rangle }_2\right)$, with $\alpha =0.5$ (black bars), $\psi =\left({|0\rangle }_1{|1\rangle }_2+{|1\rangle }_1{|0\rangle }_2\right)/\sqrt{2}$ (dark gray bars) and $\psi =\left({|0\rangle }_1{|0\rangle }_2+{|1\rangle }_1{|1\rangle }_2\right)/\sqrt{2}$ (light gray bars). White bars represent the trace $Tr\left(W\rho \right)$ for the precise density matrices of the corresponding states.

Figure 7

The purity of the representation of entangled double-mode states against the grid pitch $d$ of the square lattice of coherent projectors: (a) the number of nodes along each axis $N=6$ (fidelity level of representation ${\rho }^{\mathrm{Appr}}$ is 0.994–0.999); (b) the number of nodes along each axis $N=7$ (fidelity level of representation ${\rho }^{\mathrm{Appr}}$ is 0.999–0.99999). The states considered are $\psi =\text{const}\times \left({|\alpha \rangle }_1{|-\alpha \rangle }_2+{|-\alpha \rangle }_1{|\alpha \rangle }_2\right)$, with $\alpha =0.5$ (black bars), $\psi =\left({|0\rangle }_1{|1\rangle }_2+{|1\rangle }_1{|0\rangle }_2\right)/\sqrt{2}$ (dark gray bars), $\psi =\left({|0\rangle }_1{|0\rangle }_2+{|1\rangle }_1{|1\rangle }_2\right)/\sqrt{2}$ (light gray bars).

Figure 8

The fidelities of the reconstruction of single-mode states against the number of state copies $N_{\mathrm{rep}}$. The amplitudes of the probe state and projectors $\left\{{\mathrm{\Pi }}_j\right\}$ are chosen in the nodes of the square lattice with the optimal parameters $N=6, d=0.15$. The reconstruction process is simulated for the single-photon state (rhombuses), the coherent state with amplitude $\alpha =0.5$ (squares), the even coherent state with amplitude $\alpha =0.5$ (triangles), and the superposition of the vacuum and the single-photon states (circles).

Figure 9

The fidelities of the reconstruction of entangled double-mode states against the number of state copies $N_{\mathrm{rep}}$. The amplitudes of the probe state and projectors $\left\{{\mathrm{\Pi }}_j\right\}$ are chosen in the nodes of the square lattice with the optimal parameters $N=7, d=0.15$. The reconstruction is done for $\psi =\text{const}\times \left({|\alpha \rangle }_1{|-\alpha \rangle }_2+{|-\alpha \rangle }_1{|\alpha \rangle }_2\right)$, with $\alpha =0.5$ (circles), $\psi =\left({|0\rangle }_1{|1\rangle }_2+{|1\rangle }_1{|0\rangle }_2\right)/\sqrt{2}$ (squares), $\psi =\left({|0\rangle }_1{|0\rangle }_2+{|1\rangle }_1{|1\rangle }_2\right)/\sqrt{2}$ (triangles).

Figure 10

Plot of the trace $Tr\left(W^{\mathrm{Appr}}{\rho }^{\mathrm{Appr}}\right)$ against the number of state copies $N_{\mathrm{rep}}$ for the following signal states: $\psi =\text{const}\times \left({|\alpha \rangle }_1{|-\alpha \rangle }_2+{|-\alpha \rangle }_1{|\alpha \rangle }_2\right)$, with $\alpha =0.5$ (black bars), $\psi =\left({|0\rangle }_1{|1\rangle }_2+{|1\rangle }_1{|0\rangle }_2\right)/\sqrt{2}$ (dark gray bars), and $\psi =\left({|0\rangle }_1{|0\rangle }_2+{|1\rangle }_1{|1\rangle }_2\right)/\sqrt{2}$ (light gray bars). The entanglement witnesses $W^{\mathrm{Appr}}$ are obtained by solving the optimization problem for the reconstructed density matrices. The values of $Tr\left(W\rho \right)$ obtained by solving the optimization problem for the precise density matrices are approximately equal to $-0.00089$ for the signal state $\psi =\text{const}\times \left({|\alpha \rangle }_1{|-\alpha \rangle }_2+{|-\alpha \rangle }_1{|\alpha \rangle }_2\right)$ and $-0.00249$ for the signal states $\psi =\left({|0\rangle }_1{|1\rangle }_2+{|1\rangle }_1{|0\rangle }_2\right)/\sqrt{2}$ and $\psi =\left({|0\rangle }_1{|0\rangle }_2+{|1\rangle }_1{|1\rangle }_2\right)/\sqrt{2}$. The fidelity level for the reconstructed states is 0.86–0.999.