Indirect Protocols via Separable Carriers

• Indirect protocols using separable carriers are communication strategies that exploit intermediary systems remaining unentangled while mediating nonclassical correlations.
• They rely on techniques like balanced beam splitters, tailored unitary operations, and rigorous separability checks to enable robust resource distribution in quantum and classical domains.
• Experimental implementations reveal dynamic switching between separability classes and effective entanglement activation in noisy channels, outperforming direct transmission methods.

Indirect protocols using separable carriers represent a class of communication, sensing, and entanglement distribution techniques where information or correlations are transmitted or activated via intermediary systems that remain unentangled (separable) with respect to the principal parties throughout the protocol. These methods exploit nonclassical correlations such as quantum discord, classical randomness, or engineered physical properties of the carrier, allowing distribution or activation of resources—entanglement, information, sensitivity, or control—without direct transmission of entangled or otherwise resourceful states. Recent research rigorously elucidates the modal structures, mathematical conditions, and experimental realizations of these protocols across quantum information, network engineering, cryptography, and biosensing domains.

1. Fundamental Principles and Definitions

Indirect protocols utilizing separable carriers are defined by the use of an ancillary system ("carrier") which, despite never becoming entangled or intrinsically correlated with the communicating parties, enables resource generation, transmission, or extraction upon appropriate local or collective operations.

Key characteristics include:

• Carrier separability: At all protocol stages, the carrier's state is separable in the relevant bipartition (carrier vs. principal nodes), often verified via Positive Partial Transpose (PPT) or convex decomposition methods (Laneve et al., 2022).
• Resource mediation: Final resource distribution (e.g., entanglement, hidden data, analyte identification) occurs without the carrier itself ever acquiring that resource.
• Activation via unitaries or classical post-processing: Nontrivial resource distribution is often realized only through specific coherent (unitary) operations or measurement/postselection, not via local measurements (LOCC) alone.

These principles have been formalized for entanglement distribution (Jr, 2016), steganography (Harb et al., 2020), random access coding (Chuan et al., 2013), and multipartite network generation (Laneve et al., 2022).

2. Quantum Entanglement Sharing via Separable States

The canonical protocol for entanglement sharing via separable carriers uses initially separable Gaussian states, balanced beam splitters, and mode exchange.

1. Preparation: Alice and Bob possess modes A and B in a two-mode separable Gaussian state with covariance matrix

$$\gamma_{AB} = \begin{pmatrix} 1 + e^{-2r}(e^{2\epsilon}-1) & 0 & e^{-2r}-1 & 0 \ 0 & e^{2r} & 0 & 0 \ e^{-2r}-1 & 0 & 2-e^{-2r} & 0 \ 0 & 0 & 0 & 1 \end{pmatrix}$$

where $r$ (squeezing) and $\epsilon$ (noise) are real parameters.

1. Splitting: Alice introduces ancillary vacuum mode A′, mixes A and A′ via a balanced beam splitter $U_{AA′}$, resulting in a three-mode state $\gamma_{AA′B}$ which is one-mode biseparable: all marginal bipartitions into one vs. two modes are separable, except $A-(A′B)$ and $A′-(AB)$, which may be entangled for $r > \epsilon$ (Jr, 2016).
2. Transmission and Entanglement Activation: Alice sends mode A′ (the separable carrier) to Bob. Bob mixes A′ and B via a balanced beam splitter $U_{BA'}$, generating entanglement between A and B if $r$ surpasses a threshold $r_e(\epsilon)$, indicated by violation of the PPT criterion for the reduced covariance matrix:

$\text{Det}\,\tilde{\gamma}_{AB} - \Delta + 1 < 0$

for specific function $\Delta$.

1. Bound entanglement signature: In intermediate states, no entanglement can be localized via Gaussian measurements and feed-forward, revealing bound entanglement features.
2. Switching separability classes: By sequential application of beam splitters, the system transitions from fully separable to one-mode biseparable to fully inseparable, demonstrating dynamic switching between entanglement classes via local coherent operations (Jr, 2016).

3. Multipartite and Network Entanglement Distribution Schemes

Scalable entanglement distribution via separable carriers (“EDSS protocols”) generalizes to $N$-node networks. The carrier can be a single qubit, remaining separable throughout controlled-phase (CPHASE) gate interactions with each node.

1. Framework: For nodes $Q_1, \dots, Q_N$ and carrier $K$, prepare combinations of separable, discorded states among selected pairs and a completely uncorrelated carrier (typically a mixture in the $\{\ket{D}, \ket{A}\}$ basis).
2. Sequence of interactions: Each node applies a CPHASE gate to the carrier:

$$\mathcal{P}_{Q_iK} = \ket{0}_{Q_i}\langle0| \otimes \mathbb{1}_K + \ket{1}_{Q_i}\langle1| \otimes \sigma_z^K$$

with the partial transpose w.r.t. $K$ remaining positive at all steps, guaranteeing separability.

1. Entanglement realization: After tracing out or projecting the carrier, the network state exhibits genuine multipartite entanglement (e.g., noisy Bell or GHZ states) with nonzero negativity across all bipartitions, even though the carrier was separable throughout (Laneve et al., 2022).
2. Experimental considerations: Linear-optical implementation requires synchronization, high-fidelity CPHASE gates, and postselection. Noise accumulates linearly in $N$; relay schemes can reduce photon requirements at the expense of postselection probability.

4. Indirect Protocols in Quantum and Classical Cryptography

Protocols in cryptography exploit indirect transmission by embedding control or payload information into separable carriers.

Separable Steganographic Schemes:

• Employ public-key cryptosystems (Paillier) to encrypt digital image pixels as independent ciphertext pairs.
• Secret bits are embedded via homomorphic swapping of encrypted pixel components.
• Extraction and decryption are fully decoupled: hidden data is recoverable on ciphertexts without access to the image plaintext (Harb et al., 2020).
• Security: Semantic security and absence of clear-domain footprint; image PSNR remains infinite.

Microprotocols in Audio Carriers:

• Steganographic control protocols are embedded as static or dynamic headers within audio payloads.
• Embedding, extraction, error recovery, and routing can be engineered purely in the covert domain without affecting core carrier semantics (Naumann et al., 2015).
• Header overhead, embedding rate, distortion, and reliability are tunable; dynamic headers minimize repeated field visibility in long sessions.

5. Quantum Random Access and Discord-Enabled Indirect Communication

Protocols using separable but discorded quantum states outperform classical protocols under restricted randomness regimes.

• In random access codes (RACs), replacing shared classical bits with separable two-qubit (Bell-diagonal) states enables strictly higher success probability in $n\to1$ RAC tasks:

$$P_{\text{min}}^{\rm q} = \frac12\Big(1 + \frac1{\sqrt{E_1^{-2}+E_2^{-2}}}\Big)$$

for separable Bell-diagonal states with $E_i$ parameters.

• Quantum advantage arises from discorded, not entangled, correlations. For example, in $2\to1$ RAC, optimal quantum separable state yields $P_{\text{min}}^{\rm q} \approx 0.677$ vs. $P_{\text{min}}^{\rm cl}=2/3$ for classical solutions (Chuan et al., 2013).
• Methods extend generally to higher-dimensional and multipartite scenarios, contingent on discord and resource monotones yet to be fully characterized.

6. Indirect Sensing and Multiplexed Reporting via Separable Carriers

In biosensing, separable molecular carriers enable selective, multiplexed nanopore readout of complex targets.

• SE-DNA carriers: Constructed by PCR/amplification of template DNA, restriction digestion to generate sticky ends, and linker oligo ligation. Carrier length tunable from 500 bp to 10+ kbp allows multiplexed signal separation via electrophoretic dwell time and charge deficit metrics.
• Multiplex reporting: Functionalization with analyte-specific nanostructures (e.g., DNA stars) imparts secondary signature detectable in ionic blockade current profiles, readily distinguishable due to separable carrier-induced subevents (Roelen et al., 2023).
• Advantages: High selectivity, modular assembly, and avoidance of complex DNA origami or viral scaffolds. Limitations include self-circularization, PCR length efficiency, and throughput scaling with event classification reliability.

7. Carrier Separability, Noise Robustness, and Protocol Optimization

Carrier separability confers unique robustness and optimization trade-offs.

• In noisy quantum channels, entanglement can be distributed via separable carriers even when direct Bell-pair transmission fails, due to discord-mediated activation (Fedrizzi et al., 2013, Campbell et al., 2024).
• Optimal entanglement activation (negativity, concurrence) depends on noise model (dephasing, amplitude-damping, depolarizing) and memory/channel regime. EDSS approaches outperform direct distribution under certain channel/noise parameters.
• Switching between separability classes and activating bound entanglement under local restrictions (e.g., Gaussian operations unable to distill entanglement) are achievable via coherent mixing (Jr, 2016, Spedalieri et al., 2015).
• Protocol selection and resource balancing require quantitative evaluation of carrier noise, target resource, and network scaling for best-in-class operation (Campbell et al., 2024).

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In summary, indirect protocols using separable carriers constitute a rigorously defined and experimentally realized paradigm for mediation of nonclassical resources, robust information transfer, multiplexed sensing, and multipartite correlation generation. Carrier separability, far from being a limitation, is exploited as an operational mechanism, manifesting discord, structural tunability, and selective activation, with demonstrated foundational and applied relevance in quantum information, cryptography, network engineering, and single-molecule analysis.