Teleportation-Simulable Channels

• Teleportation-simulable channels are quantum channels that can be simulated via LOCC protocols using an entangled resource state (often the Choi matrix) to reproduce their action.
• They reduce complex adaptive quantum protocols to simpler block protocols, enabling computable upper bounds for quantum communication, key distribution, and metrological tasks.
• Their simulation framework spans both discrete and continuous variables, facilitating programmable quantum processing and robust error correction strategies.

Teleportation-simulable channels are quantum channels whose action can be reproduced by a local operations and classical communication (LOCC) protocol acting on an appropriate entangled resource state, often the channel’s Choi matrix. Such channels are central in quantum information theory because they enable the reduction of complex, potentially adaptive quantum protocols to much simpler block protocols, and provide computable upper bounds for quantum communication, key distribution, and metrological tasks. Their formal properties, operational significance, simulation protocols, and dualities with other quantum processes have been extensively developed for both finite-dimensional and infinite-dimensional (continuous-variable) systems.

1. Mathematical Definition and Core Properties

A channel $\mathcal{N}_{A\to B}$ is teleportation-simulable if there exist an LOCC channel $\mathcal{L}$ and an associated resource state $\omega_{RB}$ such that for any input $\rho_A$,

$$\mathcal{N}(\rho_A) = \mathcal{L}(\rho_A \otimes \omega_{RB}).$$

Frequently, $\omega_{RB}$ is chosen as the channel's Choi matrix $$\xi^{\text{CJ}}_\mathcal{N} = (\operatorname{id}_R \otimes \mathcal{N})(|\Phi^+\rangle\langle\Phi^+|)$$, where $|\Phi^+\rangle$ is the maximally entangled state.

If $\mathcal{N}$ is teleportation-covariant—i.e., for any unitary $U$ from an appropriate group, there exists $V$ such that $$\mathcal{N}(U\rho U^\dagger) = V \mathcal{N}(\rho)V^\dagger$$—then $\mathcal{N}$ is teleportation-simulable by the standard protocol. In this case, the channel action can also be written as

$$\mathcal{N}(\rho) = \mathcal{T}(\rho \otimes \xi^{\text{CJ}}_\mathcal{N}),$$

where $\mathcal{T}$ is the generalized teleportation protocol (Bell or Weyl–Heisenberg unitaries for DV, displacement for CV).

Crucially, such simulation is not limited to exact cases. Approximately teleportation-simulable channels are $\epsilon$-close in the diamond norm to a teleportation-simulable channel (Kaur et al., 2017):

$$\frac{1}{2}\|\mathcal{N} - \mathcal{M}\|_{\diamond} \leq \epsilon.$$

2. Simulation Protocols and Channel Classes

Teleportation simulation protocols differ by system dimensionality and target channel:

• Discrete variable (DV): The canonical example is the Pauli channel, where standard teleportation with a Bell-diagonal resource simulates all Pauli channels. Non-Pauli channels can be simulated by perturbing either the resource (e.g., using an amplitude-damping Choi matrix) or the classical communication step (introducing nontrivial classical noise in the correction rule), leading to more general channel classes such as Pauli-damping channels (Cope et al., 2017).
• Continuous variable (CV): The Braunstein–Kimble protocol using the two-mode squeezed vacuum resource yields heritable teleportation-simulability for all phase-insensitive Gaussian channels, including lossy, thermal, amplifier, and additive-noise channels, as reviewed with explicit uniform convergence bounds (Wilde, 2017, Pirandola et al., 2017). Port-based teleportation, generalized to CV (Pereira et al., 2023), removes the need for correction unitaries, and can simulate arbitrary energy-truncated channels.
• Probabilistic/generalized protocols: Probabilistic simulability is investigated in the context of simulating channels from the future to the past (Genkina et al., 2011). Here, the maximum probability of successful simulation is tightly linked to the channel information capacity. For instance, the identity channel on $d$-dimensional systems is simulable with maximum probability $1/d^2$, while an ideal classical channel allows probability $1/d$.
• Conditional simulation: When simulating an ensemble or mixture of channels, and joint teleportation covariance does not hold, a control system is used to "flag" the active component, with the overall simulation achieved by a conditional LOCC acting on the control and program-resource states (Pirandola et al., 2018). This enables capacity bounds for non-trivially mixed or memory channels.

3. Capacity Bounds and Resource Theory Implications

Teleportation-simulable channels admit single-letter upper bounds on two-way assisted quantum, private, and secret-key-agreement capacities:

• For channel $\mathcal{N}$ simulable via resource state $\omega_{RB}$, the two-way private (or quantum) capacity satisfies

$K(\mathcal{N}) \leq E_R(\omega_{RB}),$

where $E_R$ is the relative entropy of entanglement. For approximately simulable channels, continuity corrections proportional to the output dimension and diamond norm distance appear (Kaur et al., 2017).

The framework generalizes to other resource theories via $\nu$-freely-simulable channels, where $\nu$ is a resource state for a given theory and simulation occurs via a free channel in that theory. The amortized resourcefulness of a channel is defined as

$$V_A(\mathcal{N}) = \sup_\rho \{ V[(\operatorname{id}_R \otimes \mathcal{N})(\rho_{RA})] - V(\rho_{RA}) \},$$

leading to operational capacity bounds in the corresponding resource theory.

4. Simulation of Gaussian Channels: Finite-Energy Constraints and Error Correction

Simulating bosonic Gaussian channels requires finite-energy resources, as infinite-energy Choi states are unphysical. For any phase-insensitive Gaussian channel $\mathcal{G}_{\tau, v}$, one can construct a pure, finite-energy resource state whose entanglement of formation matches that of the Choi state (Tserkis et al., 2018). This resource enables exact simulation of the channel via LOCC, provided the channel is not the ideal identity (which demands infinite entanglement). Error correction protocols are constructed via distillation (e.g., noiseless linear amplification) on the resource, allowing noise reduction below that of the original channel.

Recent advances enable heralded teleportation schemes—blending measurement-based noiseless linear amplification (MBNLA) with non-unity gain teleporters—permitting probabilistic simulation of Gaussian channels with either very low added noise or even in the non-physical regime inaccessible to deterministic approaches (Shajilal et al., 16 Aug 2024). This expands the class of simulable channels to include Gaussian noise-suppressed channels, and enables enhanced entanglement transmission.

5. Metrological Consequences and Limitations

When a channel is teleportation covariant, its adaptive metrology protocols reduce to block protocols in which the quantum Fisher information (QFI) per use is bounded by that of the Choi matrix of the channel (or, equivalently, the program state for simulation) (Laurenza et al., 2017, Cope et al., 2018). Thus, the quantum Cramér–Rao bound is additive:

$$\delta\theta^2 \geq \frac{1}{n \cdot \text{QFI}(\xi^{\text{CJ}}_\mathcal{N})},$$

implying no adaptive protocol can beat the standard quantum limit (SQL, scaling as $1/n$) for such channels. The search for feasible channel simulation protocols attaining Heisenberg scaling (i.e., error scaling as $1/n^2$) remains open in this class, with the lack of such simulations tied to the absorption of all parameter dependence by a single-copy program state.

6. Channel Classes, Trade-Offs, and Duality Structures

A variety of explicit channel classes have been analyzed:

• Pauli and non-Pauli qubit channels: Standard teleportation can simulate all Pauli channels; with noisy classical communication or non-Bell-diagonal resources, a larger class of affine qubit channels or Pauli-damping channels is accessible (Cope et al., 2017, Pereira et al., 2019).
• Cloning and partial trace channels: A time-reversal duality links partial trace simulation (fewer outputs than inputs) and universal cloning (more outputs than inputs). The maximum simulation probability for cloning from $N$ to $M > N$ copies equals that of simulating the dual trace channel from $M$ to $N$ copies (Genkina et al., 2011).

Channel Type	Max Simulation Probability	Remarks
Identity ($d$-dim.)	$p_q = 1/d^2$	Full quantum info, via standard teleportation
Classical (basis)	$p_{cl} = 1/d$	Only classical info preserved
Measure&Prepare	$p_{mp} \ge 1/d_{out}$	Estimation, unitarily covariant
Partial Trace ($N\to M$)	$p_{q,N\to M}^+ = \frac{d_+^{|N-M|}}{d_+^N d_+^M}$	$d_+^k = \binom{d+k-1}{k}$

The inherent trade-offs are between the informativeness of the channel and the simulation probability or resource requirements. Channels transmitting more quantum information require lower probabilities or higher entanglement.

7. Practical Implications and Operational Applications

Teleportation-simulable channels yield considerable operational benefits:

• Tight capacity bounds: Used to set or numerically evaluate strong converse bounds for private communication and entanglement distribution over lossy bosonic (and other teleportation-covariant) channels (Namiki, 2016, Wilde, 2017, Pirandola et al., 2017).
• Simplified analysis of adaptive protocols: Through "teleportation stretching" and "peeling arguments," adaptive protocols (e.g., QKD, metrology, channel discrimination) can be reduced to block-LOCC forms involving only the resource Choi state (Wilde, 2017, Laurenza et al., 2017, Cope et al., 2018).
• Programmable quantum processing: Port-based teleportation, and its CV analog, provide universal simulation of qubit and CV channels via appropriately engineered resource (“program”) states (Pereira et al., 2019, Pereira et al., 2023).
• Robust error correction: Resource engineering combined with distillation or weak measurement reversal enables noise suppression and improved fidelity, with practical protocols involving environment-assisted and weak measurement strategies (Harraz et al., 2022, Shajilal et al., 16 Aug 2024).

8. Open Problems and Future Directions

• Heisenberg scaling: The absence of simulation schemes achieving quadratic precision scaling for parameter estimation marks a key theoretical limitation (Laurenza et al., 2017).
• Resource-efficient simulation: Minimizing energy and entanglement requirements for high-fidelity simulation of continuous-variable channels (finite-energy Choi-state constructions) remains critical for implementable protocols (Tserkis et al., 2018, Shajilal et al., 16 Aug 2024).
• Extension to non-covariant and memory channels: Conditional simulation using explicit control systems generalizes to structurally complex scenarios, including mixtures and memory effects (Pirandola et al., 2018), but automated extraction and resource state optimization for arbitrary channel sets is an open direction.
• Quantum-gravity-inspired simulation: Recent wormhole teleportation protocols, in which classical communication and engineered interaction terms enable effective quantum teleportation through SYK-like models, invite cross-disciplinary efforts with high-energy physics and holography (Lykken et al., 13 May 2024).

---

Teleportation-simulable channels form a foundational class underpinning a broad range of quantum information protocols, marrying rigorous capacity analyses with explicit simulation constructions. Their theory connects resource measures, channel capacities, metrological limitations, and implementation tradeoffs, and is a central pillar in both the mathematical structure of quantum channels and the design of practical quantum communication systems.