Adaptive Compressive Tomography with No a priori Information

Quantum state tomography is both a crucial component in the field of quantum information and computation and a formidable task that requires an incogitable number of measurement configurations as the system dimension grows. We propose and experimentally carry out an intuitive adaptive compressive tomography scheme, inspired by the traditional compressed-sensing protocol in signal recovery, that tremendously reduces the number of configurations needed to uniquely reconstruct any given quantum state without any additional a priori assumption whatsoever (such as rank information, purity, etc.) about the state, apart from its dimension.

Figure 1

Schematic diagram of a particular adaptive step in ACT tomography. In a clockwise flow, ACT first performs ICC to check whether data (blue) collected from measuring $\mathcal{B}$ are IC or not. If not, it proceeds to choose a good basis to measure in the next step.

Figure 2

Schematic of the OAM-based experimental setup. A 16-dimensional OAM state is generated at SLM-A using a holographic technique that allows the tailoring of the intensity and phase profile of the incoming beam. The modulated first-order of diffraction is filtered out using an iris ($I$) and a pair of lenses ($f_1$ and $f_2$). A similar holographic technique is used at the second SLM-B to measure the state in a given basis. The first measurement basis, ${\mathcal{B}}_1$, is given by the OAM computational basis. In the case of the rank 1 state shown on SLM-A, the corresponding eigenbasis is achieved after the fourth iteration.

Figure 3

Plots of simulation (noiseless) and experimental values of $s_{\mathrm{CVX}}$ and ${\widehat{\rho }}_k$ fidelity against the measured basis number $k$ for $d=16$, where ${\widehat{\rho }}_k≔{\rho }_{\min }$ for the RP scheme. Estimated experimental error bars reflect the propagated Poissonian-source standard deviations. All plot markers are averaged over five ${\rho }_r\mathrm{s}$. The filled markers represent results for rank-1 ${\rho }_r\mathrm{s}$ whereas the unfilled ones represent those for rank-3 ${\rho }_r\mathrm{s}$. The insets showcase simulation performances of rank-6 states as a demonstration of high-rank $(r\approx D/2)$ compressive tomography, with $1\le k\le (D+1)=17$ restricted to the minimal bases number for arbitrary-state tomography. The lower experimental fidelities for RP and PACT are due to a technical bias of the OAM setup for finite $N$, where bases close to the eigenbasis of ${\rho }_r$ tend to give estimated Born probabilities that are relatively more accurate than those that are not. So for OAM sources, ACT is the most favorable option, as both PACT and RP correspond to measurement bases that are never close to the eigenbasis of ${\rho }_r$. Even with noisy data, ICC can still validate whether the resulting ML probabilities obtained from data are IC (left panels), which is the point of ACT.

Figure 4

Plots of simulation (noiseless) and experimental values of $k_{\mathrm{IC}}$ and the ${\widehat{\rho }}_{k=k_{\mathrm{IC}}}$ fidelity against the rank $1\le r\le 3$ of ${\rho }_r$. All $k_{\mathrm{IC}}$ values ascertained using ICC are averaged over five ${\rho }_t\mathrm{s}$ per rank. Otherwise all specifications are the same as Fig. 3. Although, for real data, positivity modifies the $k_{\mathrm{IC}}$ performances with ML, and PACT achieves informational completeness much quicker than RP as far as local bases are concerned. A comparison with random IC orthonormal bases shows that ACT gives a much lower value owing to the additional assessment of and optimization over $\mathcal{C}$.