Photon-Number Teleportation Protocol

• Photon-number-based teleportation is a quantum communication method using Fock-state superpositions and Bell measurements to transfer quantum states with loss resilience.
• It leverages diverse encodings—including vacuum–single photon and coherent states—to support scalable, multimode quantum operations.
• Experimental implementations demonstrate high fidelities and efficient entanglement generation, advancing robust and practical quantum networking.

A photon-number-based teleportation protocol is a quantum communication scheme in which quantum states are teleported using entanglement and measurement operations realized in the photon number (Fock) basis. Such protocols exploit single-photon states, vacuum–single-photon superpositions, or higher photon-number encodings as quantum information carriers. Teleportation is verified and conditioned exclusively on photon-number states and corresponding Bell-type measurements, with classical communication relaying the measurement outcomes to enable unitary correction. Photon-number-based approaches constitute a distinct alternative to continuous-variable quadrature teleportation, offering distinct advantages in loss tolerance, fidelity scaling, and resource requirements, with applicable encodings ranging from single-rail (vacuum–one-photon) qubits to general Fock-state superpositions.

1. Core Principles and Encodings

The photon-number-based teleportation paradigm is defined by the use of Fock-state superpositions as logical qubits, such as the vacuum–single-photon (VSP) encoding:

• Logical states: $$|\bar{0}\rangle \equiv |0\rangle,\quad |\bar{1}\rangle \equiv |1\rangle$$
• Arbitrary qubit: $$|\psi\rangle_{\rm VSP} = \alpha|0\rangle + \beta|1\rangle$$, with $|\alpha|^2 + |\beta|^2 = 1$

Coherent-state encodings utilize $|\alpha\rangle$ and $|-\alpha\rangle$ as logical basis vectors with nonzero overlap for finite $|\alpha|$:

• $|0\rangle \to |\alpha\rangle$, $|1\rangle \to |-\alpha\rangle$
• Arbitrary qubit: $$|\psi\rangle_{\rm coh} = \alpha|\alpha\rangle + \beta|-\alpha\rangle$$

For high-dimensional or multi-pixel systems (as in image teleportation), logical information is factorized across distinct modes, each teleported independently in the photon-number basis (Medina et al., 2017, Permaul et al., 25 Feb 2025). The photon-number basis supports exact orthogonality and heralded detection, providing resilience against classical strategies and certain loss models.

2. Teleportation Protocols in the Photon-Number Basis

The general photon-number teleportation process is instantiated as follows:

1. Entanglement Resource Preparation: Establishment of a maximal-entanglement state in the photon-number basis, commonly a single-photon Bell state, e.g., $$|\Phi^+\rangle_{AB} = (|0\rangle_A|1\rangle_B + |1\rangle_A|0\rangle_B)/\sqrt{2}$$, generated via spontaneous parametric down-conversion (SPDC), four-wave mixing (FWM), or direct single-photon sources and beam splitting (Peng et al., 12 Nov 2025, Polacchi et al., 2023, Dominguez-Serna et al., 2020).
2. Bell State Measurement: The input mode (carrying the unknown quantum state) is coupled to one mode of the entanglement resource (often via a 50:50 beam splitter). A joint measurement in the Fock basis—frequently partial due to the linear optics constraint—identifies certain maximally entangled basis states (most commonly the $\Phi^\pm$ states) (Polacchi et al., 2023, Peng et al., 12 Nov 2025). The measurement collapses the combined system, and success is heralded by specific photon-detection patterns.
3. Classical Communication and Unitary Correction: The Bell measurement outcomes are sent classically to the receiver. The receiver applies the appropriate Pauli rotation or photon-number shift to their part of the entangled resource to reconstruct the original quantum state.

The protocol generalizes naturally to multi-mode systems through spatial (pixel-wise) or spectral decomposition, applying the same teleportation steps independently to each mode (Permaul et al., 25 Feb 2025).

3. Representative Protocols and Experimental Realizations

Multiple variants of photon-number-based teleportation have been demonstrated and analyzed:

• Single-Rail Qubit Teleportation: Use of vacuum–single-photon qubits generated via quantum dots or heralded photon sources. Implementation comprises on-demand generation of entangled Bell states by splitting a single photon and performing partial Bell-state measurements via beam splitters and photon counting (Polacchi et al., 2023). Practical challenges include enforcing pure single-rail qubit superpositions and ensuring indistinguishability in photon interference.
• Controlled Teleportation: Introduction of a third-party controller (Charlie's qubit) encoded via photon-number or coherent-state superpositions, distributing entangled tripartite "maximal-slice" states to guarantee teleportation can occur only with the controller's permission. Losses are treated with amplitude-damping models and closed-form formulas for both conditioned and nonconditioned teleportation fidelities are derived (Medina et al., 2017).
• Hybrid Entangled Resource Teleportation: Teleportation using entanglement generated from heralded quantum-light states, such as SPACS (single-photon–added coherent states) and photon-displaced single-photon states combined via beam splitters. Bell-state measurement and unitary corrections are performed in the photon-number basis, achieving average fidelities $\geq 0.90$ over wide bandwidths without spectral filtering (Dominguez-Serna et al., 2020).
• Pixel-by-Pixel Fock-State Teleportation: For multimode (image) teleportation, two-mode squeezed vacuum entanglement is shared per spatial "pixel." Each mode undergoes photon-number-sum and/or difference measurement, effecting Fock-basis Bell-state discrimination and conditional photon-number shift corrections (Permaul et al., 25 Feb 2025).
• Teleportation with Single-Photon Ancilla: A photon-number resource protocol demonstrated unconditional teleportation advantage over direct transmission: all-optical schemes establish long-distance heralded entanglement and perform photon-number Bell measurements, leading to a 2.95$\times$ transmission enhancement through 15 dB loss (Peng et al., 12 Nov 2025).

A summary table illustrates several key platforms and metrics, strictly based on the cited literature:

Protocol Type	Entanglement Resource	Teleportation Averaged Fidelity	Reference
Single-rail photon teleportation	Bell pair: $|0,1\rangle+|1,0\rangle$	$F \geq 0.94$	(Polacchi et al., 2023)
Controlled teleportation	Tripartite maximal-slice state	Varies, up to 1	(Medina et al., 2017)
Hybrid entangled resource	Photon-displaced, SPACS	$\bar{\mathcal F}_{pd}\geq0.90$	(Dominguez-Serna et al., 2020)
Pixel-wise Fock teleportation	Two-mode squeezed vacuum	$F_j>0.9$ with $r\gtrsim1.15$	(Permaul et al., 25 Feb 2025)
Loss-robust single-photon tele	All-optical, remotely-prep. Bell state	$F_{tel}=0.826\pm0.019$	(Peng et al., 12 Nov 2025)

4. Analytical Performance and Resource Scaling

Photon-number-based protocols are characterized by heralded fidelity and success probability metrics, both critically dependent on the properties of entanglement resources, optical loss, and detector efficiency.

• Success Probability: Limited by partial Bell-state distinguishability (linear optics resolution), with maximum two-outcome identification yielding $p_{\text{success}} \leq 0.5$ per teleporter (Polacchi et al., 2023, Dominguez-Serna et al., 2020). For multi-mode systems with $N$ modes, overall success probability scales as $(1/2)^N$ in the absence of deterministic Bell measurement technology (Andersen et al., 2013).
• Fidelity Under Loss: Optical amplitude damping is modelled via Kraus superoperators or reduction in effective squeezing. Conditioned teleportation fidelities for single rail-encoded qubits in the presence of amplitude damping are:

$$F_c = \frac{1}{6}\left[3 + 2|1 - r^2| + |1-2r^2 + 2r^4|\right],\quad r \equiv \sqrt{1-e^{-\Gamma t}}$$

Coherent-state encoding fidelities depend on the magic basis and decay with loss and reduced basis orthogonality (Medina et al., 2017).

• Efficiency Metrics: Combined metrics such as $\eta = C_p \cdot \tilde F_c$ quantify both controller authority and teleportation quality, with explicit regime dependence and non-monotonic behavior versus loss and encoding parameters (Medina et al., 2017). For high-fidelity at finite resources, photon-number-based protocols outperform infinite-squeezing CV protocols (Andersen et al., 2013).
• Experimental Outcomes: Experimental photon-number protocols consistently demonstrate performance beating the classical limit (fidelity $2/3$ or maximal classical visibility), with registered visibilities and tomographic fidelities well above classical thresholds (Peng et al., 12 Nov 2025, Polacchi et al., 2023).

5. Practical Challenges and Physical Realizations

Photon-number teleportation faces technical constraints and has informatively addressed them in recent literature:

• Source Preparation: Direct generation of vacuum–one-photon superpositions requires single-photon sources with high coherence and indistinguishability, as achieved with semiconductor quantum dots (Polacchi et al., 2023) or heralded sources based on FWM/SPDC in photonic-crystal fibers without restrictive filtering (Dominguez-Serna et al., 2020).
• Bell-State Measurements: Linear optics schemes only partially distinguish photon-number Bell states, limiting success probability. Use of ancillary entanglement, nonlinear optics, or photon-number-resolving detectors provides potential for deterministic operation, with contemporary experiments relying on exactly resolving one or two photon events (Polacchi et al., 2023, Peng et al., 12 Nov 2025).
• Loss Management: Heralded entanglement swapping and QND heralding can remove exponential transmission loss from the logical qubit channel, dramatically improving survival probability compared to direct photon transmission (Peng et al., 12 Nov 2025).
• Scalability: Protocols teleporting multimode images scale linearly in entangled-pair resources, with physical implementation constrained by the number and quality of independent two-mode squeezed (or Bell) sources, mode sorters, and recombiners (Permaul et al., 25 Feb 2025).

6. Comparison with Other Quantum Teleportation Paradigms

Photon-number-based protocols contrast strongly with standard continuous-variable (CV) teleportation:

• Resource Requirements: Standard CV teleportation using two-mode squeezing admits unit fidelity only in the unphysical limit of infinite squeezing ($r\to\infty$), whereas photon-number-scheme protocols reach near-unit conditional fidelity with finite ensembles of maximally entangled single-photon states (Andersen et al., 2013).
• Heralding and Loss Tolerance: Photon-number approaches admit heralded, locatable failure events and are less vulnerable to unheralded noise. Channel loss in photon-number schemes affects generation rate rather than fidelity, critically differing from the CV regime, where loss directly diminishes fidelity.
• Scalability and Modality: While linear optics limits full deterministic operation, multiport implementations or near-deterministic photon-number Bell measurements (e.g., with solid-state cavity QED) are expected to provide scalable, high-fidelity quantum information transmission (Andersen et al., 2013).

7. Outlook and Generalizations

The photon-number-based teleportation framework provides a foundation for long-haul, loss-tolerant quantum networks, quantum repeaters, and distributed hybrid quantum processing. Key research directions include:

• Extension to higher photon-number (qudit) teleportation, requiring multiport beam splitters and photon-number-resolving detectors (Peng et al., 12 Nov 2025).
• Integration with quantum memories, entanglement purification, and multimode/multichannel networks for scalable quantum communications (Peng et al., 12 Nov 2025, Permaul et al., 25 Feb 2025).
• Refined photonic resource generation leveraging quantum dots, NV centers, and high-purity SPDC/FWM sources for increased rates and fidelities (Polacchi et al., 2023, Dominguez-Serna et al., 2020).

The unique features of photon-number-based teleportation—heralded success, loss resilience, and orthogonal basis encoding—confirm its central role in the architecture of quantum information and future quantum networking technologies.

---

References

• "Transmission losses in optical qubits for controlled teleportation" (Medina et al., 2017)
• "Teleportation scheme for the complete state of light at the example of coherent states" (Permaul et al., 25 Feb 2025)
• "Quantum teleportation with hybrid entangled resources prepared from heralded quantum states" (Dominguez-Serna et al., 2020)
• "Teleportation of a genuine single-rail vacuum-one-photon qubit generated via a quantum dot source" (Polacchi et al., 2023)
• "Unconditional quantum teleportational advantage of single photons" (Peng et al., 12 Nov 2025)
• "High Fidelity Teleportation of Continuous Variable Quantum States using Delocalized Single Photons" (Andersen et al., 2013)