Heralded Operator in Quantum Optics

• Heralded operator is a quantum operation that couples target transformations with auxiliary detection events, enabling real-time certification of successful quantum processes.
• It is implemented in quantum optics through heralded state preparation using parametric down-conversion and heralded quantum gates for controlled photonic interactions.
• Experimental realizations leverage nonclassicality criteria and efficient photon detection to enhance gate fidelity and scalability in quantum computing architectures.

A heralded operator is a quantum operation whose successful application is flagged by a correlated detection event—commonly termed a "herald"—that enables post-selection or enables conditional processes in quantum optical experiments. Heralded operators are crucial in photonic quantum information due to the probabilistic nature of photon generation and interaction, enabling scalable protocols contingent on detectable success signals. Two principal domains of heralded operator implementation are heralded quantum state preparation (notably in single-mode parametric down-conversion) and heralded quantum gates (notably in linear optics). The concept is realized by coupling quantum operations with auxiliary detection events that certify, in real time, whether the desired transformation has occurred, thus separating successful from unsuccessful trials without destroying target quantum information (Horoshko et al., 2019, Li et al., 2020).

1. Heralded Quantum State Preparation

Heralded quantum state preparation exploits correlated quantum processes to probabilistically generate non-classical states, with the heralding event filtering out unsuccessful realizations. In single-mode parametric down-conversion (PDC), detection of an idler photon “heralds” the generation of a correlated signal photon whose quantum state is thus conditionally prepared. The resulting signal-mode state for on/off heralding is precisely characterized as a thermal-difference state (TDS), a non-Gaussian bosonic state described by a weighted difference of two thermal states (Horoshko et al., 2019). The explicit density operator is

$$ρ_{\text{heralded}} = C \left[ \frac{ρ_{\text{th}}(q)}{1-q} - d\,\frac{ρ_{\text{th}}(qp)}{1-qp} \right]$$

where $q$, $d$, $p$ and normalization $C$ are analytic functions of system parameters: PDC gain ($\xi$), signal transmittance ($\eta_s$), idler transmittance ($\eta_i$). In the single-photon limit ($\xi\to0$, $\eta_s\to1$, $d\to1$), this operator reduces to the Fock state $|1⟩⟨1|$.

2. Nonclassicality Criteria and Measures

Heralded state operators can be categorized by rigorous nonclassicality criteria. For the TDS family, closed-form witnesses include:

• Glauber–Sudarshan $P$-function negativity: The analytic expression

$P(0) = \frac{C}{q\pi}(1 - d/p)$

implies nonclassicality when $d > p$, invariably satisfied for $\eta_i > 0$ in heralded PDC (Horoshko et al., 2019).

• Negative volume of the Wigner function ($N_W$): Quantifies phase-space negativity, computed analytically for all TDS members; in the single-photon limit, $N_W(|1⟩) \approx 0.213$.
• Sum of quantum Fisher informations (QFI): For all classical states, $M_{\text{QFI}}(ρ) \leq 1$. TDS states exceed this bound, signifying quantum-enhanced parameter sensitivity.
• Ordering sensitivity (OS): Given by

$$S(ρ) = \frac{\sum_n (p_n - p_{n+1})^2 (n+1)}{\sum_n p_n^2}$$

with $S(ρ) > 1$ signaling nonclassicality. All such measures decrease as initial brightness $\bar n$ or signal channel loss increases, evidencing a nonclassicality–brightness trade-off.

3. Heralded Quantum Entangling Gates

Heralded operators extend directly to quantum gates, such as the heralded controlled-NOT (CNOT) implemented between photonic qubits. The theoretical map is framed as a Kraus operator decomposition; successful trials correspond to a heralded projection of ancillary photon states (e.g., onto Bell states in linear optical networks). The measurement event constitutes the "herald," conditioning the acceptance of gate output. For the heralded CNOT (Li et al., 2020), the operational Kraus operator for the success channel is

$$E_{\text{success}} = \langle Φ^+ |_{2,3} U_{\text{tot}} |a_1, a_2⟩_{2,3} = \frac{1}{2} U_{\text{CNOT}},$$

with heralding probability $P_{\text{herald}} = \frac{1}{8}$ in ideal linear optics.

4. Experimental Realizations and Performance Metrics

In physical systems, heralded operators manifest through photon detection networks, single-photon sources, and partial Bell state analyzers, with performance constrained by transmission and detection efficiencies:

• Single-photon source: Resonantly driven quantum dots in micropillar cavities deliver on-demand photons with brightness $\eta_s$, e.g. $\eta_s \approx 0.175$ (Li et al., 2020).
• Detection efficiencies: Superconducting nanowire detectors (SNSPDs) yield $\eta_d \approx 0.8$; the observed four-fold coincidence to herald signal ratio is $\eta_h \approx 0.008$.
• Gate fidelities: Experimental gate operation yields average gate fidelity $F_{\text{gate}} = (87.8 \pm 1.2)\%$ and Bell state fidelity $F_{\text{Bell}} = (83.4 \pm 2.4)\%$.

5. Advantage and Scalability via Heralding

Heralded operators provide immediate "go/no–go" signals, critical in quantum architectures requiring feed-forward correction and loss tolerance, such as cluster-state computation and teleported gates within the KLM protocol (Li et al., 2020). Unlike unheralded probabilistic gates, which provide only post-selected correlations or destroy output photons, heralded gates retain output photons for sequential modular operations, offering a functional path toward scalable, all-optical quantum computation.

6. Benchmark States and Testbed Applications

The TDS family defines benchmark single-mode bosonic states for nonclassicality analysis due to analytic access to $P$, Wigner, Q-, and QFI distributions (Horoshko et al., 2019). Their parameter space encompasses vacuum, thermal, photon-added/subtracted thermal, Fock, and mixed states $(|0⟩,|1⟩)$. The simplicity and physical relevance of heralded operator–derived states make them ideal for comparative evaluation of quantum state criteria and for calibrating experimental quantum optics platforms. A plausible implication is that future enhancements in single-photon source brightness ($\eta_s \to 1$) and photon-number resolving detection efficiency ($\eta_d \to 1$) will allow heralded operators to approach deterministic operation, further impacting quantum processing and information transmission.