Precision Bounds on Continuous-Variable State Tomography Using Classical Shadows

Shadow tomography is a framework for constructing succinct descriptions of quantum states using randomized measurement bases, called “classical shadows,” with powerful methods to bound the estimators used. We recast existing experimental protocols for continuous-variable quantum state tomography in the classical-shadow framework, obtaining rigorous bounds on the number of independent measurements needed for estimating density matrices from these protocols. We analyze the efficiency of homodyne, heterodyne, photon-number-resolving, and photon-parity protocols. To reach a desired precision on the classical shadow of an $N$-photon density matrix with high probability, we show that homodyne detection requires order $\mathcal{O}(N^{4+1/3})$ measurements in the worst case, whereas photon-number-resolving and photon-parity detection require $\mathcal{O}(N^4)$ measurements in the worst case (both up to logarithmic corrections). We benchmark these results against numerical simulation as well as experimental data from optical homodyne experiments. We find that numerical and experimental analyses of homodyne tomography match closely with our theoretical predictions. We extend our single-mode results to an efficient construction of multimode shadows based on local measurements.

Popular Summary

Knowing the “state” of a quantum system tells us everything there is to know about it at that instant. Quantum state tomography is the process of estimating the state of an unknown quantum system and is an indispensable tool in quantum computing.

Classical shadows are a powerful framework for analyzing tomography of many-body quantum systems composed of qubits. However, classical shadows are not restricted to qubits if we take them not as a specific technique but as a philosophy of thinking statistically. We look at continuous-variable systems in this light.

States can be expressed as linear combinations of your favorite operators, but some combinations can be interpreted as statistical expectation values. These expectation values require not only the ingrained quantum randomness from your state but also classical randomness that you have to inject. The law of large numbers for matrices then tells you that things work out on average. Next comes the hard part: how quickly do things converge with the number of terms in your average?

Proving good convergence of variance previously relied on qubit-based techniques. We develop alternatives and apply them to conventional homodyne and photon-number-resolving schemes, certifying that they perform well with rigorous theoretical guarantees. Analysis of numerical and past experimental data shows that homodyne tomography is far more efficient than our bounds or photon-number-resolving schemes, guiding future theoretical work and the practical applications of continuous-variable shadows.

Figure 1

(a) Balanced homodyne detection. The unknown state signal interferes with a local oscillator (a coherent state $|\alpha ⟩$) and the intensities $N_1$ and $N_2$ at the output arms are measured. The difference in the intensities is proportional to the quadrature amplitude of the unknown state in the mode defined by the oscillator.(b) PNR measurement. The local oscillator is in a coherent state $|\alpha ⟩$, and detection of $N$ photons at the detector occurs with probability $p(N|\alpha )=⟨N,\alpha |\rho |N,\alpha ⟩$. BS, beam splitter.

Figure 2

Single-mode shadows. (a) Density matrix elements and (b) Wigner function representation of the original state, a homodyne shadow, and a PNR shadow of an even CSS state $|\psi ⟩\propto |\alpha ⟩+|-\alpha ⟩$ with $\alpha =\sqrt{10}$. (c),(d) Numerical sample complexity of single-mode shadows; the precision and confidence parameters are $ϵ=0.1$ and $\delta =0.1$. (c) Homodyne shadows: vacuum state ($T\propto N^{1.389}$), CSS state with amplitude $\alpha =\sqrt{10}$ ($T\propto N^{1.820}$), and random pure states ($T\propto N^{1.212}$). (d) PNR shadows: vacuum state ($T\propto N^{3.51}$) and pure random states.

Figure 3

Experimental test of single-mode shadows. (a) Density matrix elements and (b) Wigner function representation of the ideal CSS state from theory (left) and the reconstructed state from experimental homodyne measurements (right). Ideal CSS state: $|\psi ⟩\propto |\alpha ⟩+|-\alpha ⟩$, $\alpha =1.32$, $T=3\times {10}^5$. (c) Sample complexity scaling with maximum photon number $N$, from the homodyne data from the experiment. The two methods “TES 2” and “APD 1” correspond to photon detection through a TES and a SPAD, respectively. The scaling shown is for $ϵ=0.4$ and $\delta =0.1$. (d),(e) Homodyne two-mode shadows. Density matrix elements of the original states and numerically reconstructed states. The number of samples $T=5\times {10}^4$, $N=2$, and $\alpha =\sqrt{1.5}$ in both cases. (d) Two-mode separable state: $|\mathrm{\Psi }⟩=|{\psi }_{\alpha }⟩\otimes |{\psi }_{\alpha }⟩$, where $|{\psi }_{\alpha }⟩\propto |\alpha ⟩+|-\alpha ⟩$, and the infinity norm distance of the reconstructed state from the ideal state is $ϵ=0.03$. (e) Two-mode entangled state: $|\mathrm{\Psi }⟩\propto |\alpha ,\alpha ⟩+|-\alpha ,-\alpha ⟩$ and $ϵ=0.05$.