Photon catalysis for general multimode multi-photon quantum state preparation

Abstract

Multimode multiphoton states are at the center of many photonic quantum technologies, from photonic quantum computing to quantum sensing. In this work, we derive a procedure to generate exactly, and with a predictable number of steps, any such state by using only multiport interferometers, photon number resolving detectors, photon additions and displacements. We achieve this goal by establishing a connection between photonic quantum state engineering and the algebraic problem of symmetric tensor decomposition. This connection allows us to solve the problem by using corresponding results from algebraic geometry and unveils a mechanism of photon catalysis, where photons are injected and subsequently retrieved in measurements, to generate entanglement that cannot be obtained through Gaussian operations. We also introduce a tensor decomposition, that generalizes our method and allows to construct circuits yielding perfect fidelity, using the minimum number of catalysis photons. As a benchmark, we numerically evaluate our method and compare its performance with state-of-the art results, confirming 100\% fidelity on different classes of states.

Figures (4)

• Figure 1: Circuit of the proposed method. By injecting $N$ photons, $n$ of which will be later projected-out, a state of stellar rank $d = N - n$ is obtained. The general method described in \ref{['th:main-theorem']} sets $N = dr, n = d\left(r - 1\right)$; while the special case described in \ref{['th:e2-corollary']} sets $N = M, n = M - 2$. The blue part represents the preparation of the seed state, the stellar polynomial of which factorizes into a product of linear terms, the green part represents the post-selection that allows to construct an arbitrary homogeneous state and the red part represents dehomogenization.
• Figure 2: Alternative representations of the proposed method, where approximate photon additions are implemented through injection of single photons coupled by weakly reflecting beam-splitters (\ref{['fig:main-circuit-boson']}) as in a boson sampler, or by weak single-mode squeezing operations followed by heralding of single photons (\ref{['fig:main-circuit-gaussian-boson']}) as in a Gaussian boson sampler.
• Figure 3: Probability of successfully detecting the right number of photons to obtain $\ket{\Psi_4}, \ket{\Psi_8}, \ket{\Psi_9}$ from \ref{['tbl:selection-of-states']}. The solid lines show the final PNR detection probability for the method described in \ref{['th:main-theorem']}, while the dashed lines show the PNR detection probability of the method described in \ref{['th:elementary-symmetric-decomposition']}, both maximized over 25 possible decompositions. The scaling $1$ corresponds to the Frobenius norm $\|W\| = \sqrt{N}$.
• Figure 4: Probability of success and accuracy (measured by $1 - F$, equivalently the square of the trace distance) of the boson sampling implementation, using a non-ideal photon addition with a varying beam-splitter reflectivity. For $\ket{\Psi}_{10}$, optimal circuits with two catalysis photons (red) and only one (purple) are both considered.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2: Elementary Symmetric Decomposition
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• proof
• Corollary 1
• Corollary 2
• proof