Quantum Multiple Access Channel (MAC)

• Quantum MACs are quantum channels where multiple independent senders exploit entanglement and non-commutative measures to exceed classical capacity limits.
• They leverage superadditivity, symmetry, and multipartite entanglement—using methods like two-mode squeezing—to unlock higher transmission rates.
• Continuous variable implementations, such as Gaussian beam-splitter and XP gate channels, demonstrate both practical advantages and sensitivity to losses and noise.

A quantum multiple access channel (MAC) is a quantum generalization of the classical MAC paradigm: multiple independent senders transmit information to a single quantum receiver through a shared channel. In contrast to the classical case, quantum MACs exhibit novel behaviors arising from quantum entanglement, non-commutative information measures, and quantum-noise-limited operations. These features fundamentally affect the classical information transmission rates achievable over such channels. The analysis in ["Schemes of transmission of classical information via quantum channels with many senders: discrete and continuous variables cases" (Czekaj et al., 2011)] uncovers core phenomena including superadditivity of capacity, the impact of multipartite entanglement on regularized capacity regions, symmetry-induced effects, and the role of squeezing and thermal noise in continuous variable (CV) Gaussian channels.

1. Superadditivity in Quantum MACs

In quantum MACs, superadditivity refers to scenarios where entangled input states across multiple uses or parallel copies of a channel allow for a total classical capacity that strictly exceeds the sum of capacities attainable with product (separable) input encodings. This phenomenon is formally expressed as

$$\mathcal{C}(\Phi_I \otimes \Phi_{II}) > \mathcal{C}(\Phi_I) + \mathcal{C}(\Phi_{II}),$$

where $\mathcal{C}$ represents a capacity figure of merit such as the Holevo or regularized classical capacity. The manifestation of capacity superadditivity in quantum MACs demonstrates that classical product-state codebooks are not in general optimal: quantum entanglement across input channels (potentially even across distinct channel copies) can "activate" additional transmissible degrees of freedom.

In the discrete-variable MACs constructed in the paper, a clear example is provided where, due to judicious use of entanglement, a sender can achieve a per-use rate exceeding the additive region defined by classical encodings—e.g., obtaining a rate of $2.5$ bits (per sender, in a balanced, fully symmetric two-sender channel structure). This shows that, even for balanced, symmetric MACs, the classically derived sum-rate bounds do not capture the full quantum potential.

2. Symmetry in Channel Architecture

The phenomenon of superadditivity is not merely a product of asymmetric channel architectures or a sender "helping" another. The construction of fully symmetric MACs, where each sender controls an equivalent set of quantum input lines and channel operations are symmetric under sender permutation (e.g., swapping the roles of A and B with equal probability), underscores that capacity enhancement via entanglement is robust to network symmetry.

Explicitly, in the "superadditivity effect in symmetric channels," both senders receive a mode label from the channel and may employ identical encoding strategies. Capacity enhancement emerges for both senders, with rate pairs achievable outside the simple classical sum of the individual channel regions. The boundary of the capacity region, e.g., $R_A \leq 1$, $R_B \leq 1$, $R_A + R_B \leq 1$ for the base channel, can be exceeded when combining with additional identity channels and entangled encoding.

3. Multipartite Entanglement and Regularized Capacity

A central result is the demonstration that bipartite entanglement does not always saturate the asymptotic transmission rate for a quantum MAC. There exist constructed MACs in which entanglement across three or more channel uses is not just beneficial but necessary to achieve the asymptotic capacity (the regularized region). Specifically, using the famous five-qubit Shor codeword spanning five MAC inputs can unlock rate regions unattainable via any strategy constrained to bipartite entanglement.

Formally, for $m$ channel copies and $n$ helper-senders (with $p = 1/n$), the rate for sender A is bounded as

$$C_A(\Phi^{\otimes m}) \leq n \sum_{i=0}^{m} p^{i} (1-p)^{m-i} \binom{m}{i} \min(2i, m).$$

A protocol based on a five-input entangled state surpasses all possible two-input entanglement-based rates, highlighting the necessity of multipartite entanglement in saturating the asymptotic rate.

4. Continuous Variable (CV) Quantum MACs and Gaussian Channels

In the CV regime, the paper studies Gaussian MACs such as the beam-splitter channel ($\Phi_\theta$) and the XP "non-demolition" gate channel. Key features include:

• Beam-splitter channel ($\Phi_\theta$): with transmittance $T = \cos^2\theta$, inputs from two senders (e.g., encoded as displaced or squeezed states) are mixed on a beam splitter. Sender B can transmit a two-mode squeezed state (one half through the main channel, the other through an identity channel), allowing sender A to access superadditive rates otherwise blocked for product or coherent state encodings.
• XP gate channel: realized via a measurement-induced, continuous-variable interaction, allowing sender A's dual-mode input to modulate the output state via interaction with sender B's input.

Sensitivity to thermal noise is pronounced: inserting a thermal-noise channel with mean photon number $N_t$ and transmissivity $T$ between input and output can degrade or destroy the superadditivity window. Specifically, capacity superadditivity in the beam-splitter channel is only observable within an experimentally favorable noise regime—e.g., exceeding about 15% loss can eliminate the advantage.

5. Two-Mode Squeezing as a Resource

Two-mode squeezing is analytically and operationally crucial for enabling capacity superadditivity. In the entanglement-assisted scheme, sender B transmits half of a two-mode squeezed vacuum of squeezing parameter $r$ through the MAC, while the other half traverses an identity channel. The rate achievable by sender A is

$$R_A^{\text{ent}} = \log \left[ 1 + \frac{\sigma^2 \sin^2\theta}{2 (\cosh r - \cos\theta \sinh r)^2} \right].$$

Optimization yields a maximal rate $R_A^{\text{ent,opt}} = \log(1 + N_A)$ when $\cos\theta = \tanh r$, matching the rate of an ideal lossless channel. Importantly, the required two-mode squeezing to realize this enhancement is within experimentally realistic bounds (e.g., $4.37$–$4.55$ dB), notably less demanding than the levels required by alternative single-mode squeezed schemes.

This two-mode squeezed protocol allows the loss introduced by the channel to be completely counteracted, as long as the squeezing parameter is chosen appropriately.

6. Implications and Experimental Prospects

The ability of entanglement (including multipartite entanglement across channel uses) to achieve rates beyond classical product-state limits demonstrates an essential distinction between quantum and classical MACs. Channel symmetry does not preclude superadditivity, and, in fact, protocols exploiting multipartite entanglement (as in the five-qubit example) can be necessary to maximize transmission rates.

In CV Gaussian MACs, the beneficial impact of entanglement and squeezing is counterbalanced by tangible sensitivity to losses and thermal noise, imposing practical constraints for implementation—especially in optical systems. Nevertheless, the paper's squeezing requirements are considerably more optimistic than previously reported, suggesting that these superadditive effects are within reach of current quantum optics technology.

7. Summary Table: Principal Phenomena in Quantum MACs

Phenomenon	Discrete MACs	Continuous Variable (CV) MACs
Superadditivity via entanglement	Yes; explicit rate boost	Yes; via two-mode squeezing protocols
Symmetry and equivalence	Symmetric channels enhance both senders	Equally valid (through symmetric mixing)
Multipartite entanglement required	Yes (e.g., five-qubit code)	Not directly, but multipartite squeezing possible
Sensitivity to imperfections	Channel construction dependent	High: thermal noise and losses critical
Achievable rate bounds	Surpassing product-state sum	Achieve single-user lossless bound
Resource requirements	Entangled codewords (multi-copy)	Two-mode squeezing ($\sim$4.5 dB)

• "Schemes of transmission of classical information via quantum channels with many senders: discrete and continuous variables cases" (Czekaj et al., 2011)
• Specific equations, sections, and derived rates reference the notation and constructions defined therein.

The findings establish the fundamental role of quantum entanglement, channel symmetry, and quantum-optical resources in shaping and surpassing the fundamentally classical limits for multi-user quantum communication.