---
title: Photon-Number Teleportation Protocol
url: https://www.emergentmind.com/topics/photon-number-based-teleportation-protocol
type: topic
---

# Photon-Number Teleportation Protocol

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 [1701.06833, 2502.18204]. 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 [2511.08951, 2310.20521, 2002.03015].

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) [2310.20521, 2511.08951]. 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 [2502.18204].

## 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 [2310.20521]. 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 [1701.06833].

- **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 [2002.03015].

- **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 [2502.18204].

- **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 [2511.08951].

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: $|0,1\rangle+|1,0\rangle$| $F \geq 0.94$                    | 2310.20521     |
| Controlled teleportation         | Tripartite maximal-slice state | Varies, up to 1                  | 1701.06833     |
| Hybrid entangled resource        | Photon-displaced, SPACS     | $\bar{\mathcal F}_{pd}\geq0.90$ | 2002.03015     |
| Pixel-wise Fock teleportation    | Two-mode squeezed vacuum    | $F_j>0.9$ with $r\gtrsim1.15$   | 2502.18204     |
| Loss-robust single-photon tele   | All-optical, remotely-prep. Bell state | $F_{tel}=0.826\pm0.019$           | 2511.08951     |

## 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 [2310.20521, 2002.03015]. For multi-mode systems with $N$ modes, overall success probability scales as $(1/2)^N$ in the absence of deterministic Bell measurement technology [1302.1359].

- **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 [1701.06833].

- **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 [1701.06833]. For high-fidelity at finite resources, photon-number-based protocols outperform infinite-squeezing CV protocols [1302.1359].

- **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 [2511.08951, 2310.20521].

## 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 [2310.20521] or heralded sources based on FWM/SPDC in photonic-crystal fibers without restrictive filtering [2002.03015].
- **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 [2310.20521, 2511.08951].
- **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 [2511.08951].
- **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 [2502.18204].

## 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 [1302.1359].
- **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 [1302.1359].

## 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 [2511.08951].
- Integration with quantum memories, entanglement purification, and multimode/multichannel networks for scalable quantum communications [2511.08951, 2502.18204].
- Refined photonic resource generation leveraging quantum dots, NV centers, and high-purity SPDC/FWM sources for increased rates and fidelities [2310.20521, 2002.03015].

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.

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**References**

- "Transmission losses in optical qubits for controlled teleportation" [1701.06833]
- "Teleportation scheme for the complete state of light at the example of coherent states" [2502.18204]
- "Quantum teleportation with hybrid entangled resources prepared from heralded quantum states" [2002.03015]
- "Teleportation of a genuine single-rail vacuum-one-photon qubit generated via a quantum dot source" [2310.20521]
- "Unconditional quantum teleportational advantage of single photons" [2511.08951]
- "High Fidelity Teleportation of Continuous Variable Quantum States using Delocalized Single Photons" [1302.1359]

Source: https://www.emergentmind.com/topics/photon-number-based-teleportation-protocol