Certification of linear optical quantum state preparation

TL;DR

This work tackles certifying the fidelity of linear-optical quantum state preparation under partial photon indistinguishability. It introduces an operational LOQC fidelity defined over an equivalence class of permutationally symmetric target states and develops several indistinguishability witnesses, with a comprehensive theoretical and experimental comparison. The Fourier-transform-based witness emerges as the most practical and robust approach, enabling faithful certification without strict positive partition assumptions and with favorable sampling complexity, which the authors demonstrate experimentally on a three-photon, four-mode LOQC setup. The results provide a scalable toolkit for verifying complex photonic state preparation in LOQC, with implications for quantum advantage demonstrations and future photonic quantum computing architectures. The work also clarifies the roles of partition representations and semi-device independence in photonic certification and outlines open questions for extending these methods beyond fixed-photon-number regimes.

Abstract

Certification is important to guarantee the correct functioning of quantum devices. A key certification task is verifying that a device has produced a desired output state. In this work, we study this task in the context of photonic platforms, where single photons are propagated through linear optical interferometers to create large, entangled resource states for metrology, communication, quantum advantage demonstrations and for so-called linear optical quantum computing (LOQC). This setting derives its computational power from the indistinguishability of the photons, i.e., their relative overlap. Therefore, standard fidelity witnesses developed for distinguishable particles (including qubits) do not apply directly, because they merely certify the closeness to some fixed target state. We introduce a measure of fidelity suitable for this setting and show several different ways to witness it, based on earlier proposals for measuring genuine multi-photon indistinguishability. We argue that a witness based upon the discrete Fourier transform is an optimal choice. We experimentally implement this witness and certify the fidelity of several multi-photon states.

Figures (6)

• Figure 1: Certification target class for linear optics. The class $\mathcal{C}_{\mathrm{LO}}$ contains all of the $m$-mode states consisting of the first $n$ modes containing a completely permutationally invariant state with exactly 1 photon in each mode and the remaining $m-n$ modes in the vacuum state (we denote such a state $\rho^\mathrm{sym}_{n,m}$) that is evolved through a linear optical unitary $U_\mathrm{LO}$. We provide some examples of states in $\mathcal{C}_{\mathrm{LO}}$ for the case ($n=m=2$) to highlight that as well as the canonical case of two identical (blue) photons entering the unitary we may also have a statistical mixture all photons being in orthogonal internal modes (all blue or all green) or even an even superposition of each photon being in one internal mode or the other. As explained in the main text, so long as the input state is symmetric under all permutations then all these states are functionally equivalent in the sense that they realize a 'perfect' LOQC experiment.
• Figure 2: Certification setup and assumptions. (a) States with an incoherent, discrete distinguishability structure can be written as a linear combination of partition states which are those where all photons are partitioned into groups of identical or orthogonal internal modes, illustrated here for the case of $n=3$. Any state, $\rho$, that satisfies a property called orbit invariance (see Section \ref{['sec:LOQC_fid']}) is indistinguishable from some state of this form for all LOQC experiments and is said to have a positive partition representation which we denote by $\rho\sim \rho_{\underline{\Lambda}}$. We say that $\rho$ has a negative partition representation is we can find an equivalence $\rho\sim \sum_{\underline{\Lambda}} p_{\underline{\Lambda}} \ket{{\underline{\Lambda}}}\bra{{\underline{\Lambda}}}$ where $\sum_{\underline{\Lambda}} p_{\underline{\Lambda}} = 1$ but not all $p_{\underline{\Lambda}}$ are non-negative. (b) The certification setup considered in Thm. \ref{['thm_wit']} for witnessing the LOQC fidelity takes an unknown state preparation device (indicated by the solid black box) and evaluates its LOQC fidelity with a target state in $\rho^U_{\mathbf{1}_{n,m}} \in \mathcal{C}_\mathrm{LO}$, thereby simultaneously verifying the quality of the photonic source and interferometric processing. The unknown state is fed into a programmable, trusted interferometer although these trust requirements can be somewhat relaxed for some witnesses indicated by the grey dashed enclosing box. The interferometer will be programmed to implement the inverse of the unitary defined by the target state or one of the $\nu = \mathrm{rev}$ of the inverse followed by one of the indistinguishability witnesses $\nu \in \{\mathrm{Four},\mathrm{cyc},\mathrm{2cor},\mathrm{sHOM} \}$. The output of each measurement setting is detected using detectors that resolve the number of photons in each external mode, but not their internal degrees of freedom as indicated by the colourless circle in each detector generating outputs $\mathbf{s}^\nu$. These outputs are then classically processed to obtain a witness for the LOQC fidelity. Depending upon the witness, obtaining a bound will require additional information in the form of assumptions on the unknown state represented by the input labelled $\rho_?$ and we indicate this assumption dependence by writing $F_\mathrm{LO}^*$ for the certified LOQC fidelity. Depending upon the witness the assumption will typically be whether the the state $P_{\mathbf{1}_{n,m}}\rho P_{\mathbf{1}_{n,m}}$ has a (positive) partition representation. In Thm. \ref{['thm_exp']} we present an alternative protocol where the black box of the device is opened and the indistinguishability witness directly photonic source under the stronger assumptions on the source and interferometric map $M$.
• Figure 3: Interferometers for the four fidelity witnesses.(a)Superposed HOM dips: A network of balanced beam splitters implementing multiple Hong-Ou-Mandel-type interference experiments between a reference photon, that was evenly split by a discrete Fourier transform, and the remaining photons. The witness is based on the measured bunching probability $p_{\text{b}}$. (b)Two-mode correlator interferometer: A programmable multiport unitary optimized to maximize two-mode number correlations $C_{i,j}$, providing an estimate of pairwise interference contributions. (c)Cyclic interferometer: A shallow two-layer beam-splitter network with a tunable phase $\alpha$, arranged such that photons interfere along a cyclic pattern. The resulting interference fringe yields a visibility $V(\alpha)$ that probes cyclic permutation symmetry. (d)Fourier interferometer: A discrete Fourier transform acting on the external modes, exploiting exact many-particle suppression laws. The witness is obtained from the total probability $p_\mathrm{f}$ of strictly suppressed output events.
• Figure 4: Witness comparison.(a) Geometric depiction of an $n=4$ photon states and their respective angles. This is an example of an n-photon state where one photon pair is completely orthogonal, yet it remains undetected by the Cyclic Interferometer witness. (b) Analytic dependence of the five partition weights (for $n=3$) on a distinguishability parameter (time delay $\tau$). For the time delay model discussed in the text and depicted in the center of the graph, one of the partition weights (the green one) is negative for all $\tau>0$. This can lead to the failure of some of the witnesses, which is shown in the next graph. (c) This graph shows the $c_n$ values in the noiseless case, $\epsilon=0$ (see (d) for noise model), for the negative partition representation state. We can see that in this case, the superposed HOM dips witness overshoots the true value of $c_n$. (d) Monte-Carlo analysis of device independence for the four witnesses. Each data point corresponds to a random perturbation of the ideal interferometer $U_{\mathrm{ideal}}$ of the form $U(\epsilon) = U_{\mathrm{ideal}} \cdot e^{\epsilon \log(U_{\mathrm{Haar}})},$ where $U_{\mathrm{Haar}}$ is a Haar-random unitary and $\epsilon \geq 0$ controls the perturbation strength. The big dots mark the ideal (noiseless) values for each interferometer. The shaded region is the value range above the true $c_n$ value. A semi-device-independent witness never overestimates $c_n$, corresponding to points lying below the upper boundary. Note, that the Fourier Interferometer witness is the only one that is tight and does not overshoot. The cyclic interferometer is tight but frequently overshoots, and the other witnesses do not even reach the boundary in the noiseless case (meaning that they are never tight). (e) This graph shows numerical simulations of the $c_n$ value as a function of the cyclic overlap (a distinguishability parameter, in this case $x_{12}x_{13}x_{23}$) for $n=3$, $m=6$, $\epsilon=0.1$, $N=500$, and HOM dip values $\textbf{x}=(0.98,0.94,0.91)$. An optimal witness would be linear in the cyclic overlap and thus follow the boundary of the gray shaded region. We can see that both the Fourier transform and the Cyclic Interferometer witness perform well in this sense.
• Figure 5: Experimental setup. (a) A Ti:Sapph pump laser at 775 nm is split by a beamsplitter into the two arms of the single photon source, and then the polarization is controlled with a half-wave plate ($\frac{\lambda}{2}$) to match the ppKTP crystal. The pumped ppKTP crystal generates photon pairs by type-II spontaneous down conversion. The pump is removed by a high-pass filter (HPF). The photon pairs are split by polarizing beam splitters. The polarization is aligned with the fiber by another half-wave plate. Then, a band pass filter (BPF) is used to improve the spectral purity of the photons of three of the photons that are used as the input state. The arrival time of the single photons in the input state is synchronized with the linear stages. The other photon is used to herald the single photon state. (b) The three single photons are used as the input state. The input is $\ket{1,1,1,0}$ (green box) for the superposed HOM dips, Fourier transform, and two-mode correlator witness. For the cyclic interferometer, the $\ket{1,0,1,0,1,0}$ (blue box) input state vector is used. The Clements decomposition is used to determine the voltages on the heater of the photonic processor to implement the necessary unitary transformation ($V_{LO}^{\nu}$) for the four different interferometers. The output is measured with a bank of superconducting nanowire single-photon detectors. For the superposed HOM, Fourier transform, and two-mode sorrelator, four three-photon pseudo-photon-number-resolving detectors (green detectors) are used, and for the cyclic interferometer, six two-photon pseudo-photon-number-resolving detectors (blue detectors) are used.

Theorems & Definitions (17)

• Definition 1: Certification protocols
• Definition 2: LOQC equivalence
• Definition 3: Class of LOQC target states $\mathcal{C}_\mathrm{LO}$
• Definition 4: LOQC certification metrics
• Theorem 1: LOQC fidelity witness
• Definition 5: Partition representation
• Theorem 2: LOQC fidelity via indistinguishability witnesses
• Theorem 3: LOQC witness via source measurement
• Theorem 4: Relationship between LOQC metrics
• Lemma 1: Fourier-twirling equivalence