Quantum Code-Division Multiple Access

• Quantum CDMA is a family of protocols that use user-specific unitary transformations to spread quantum signals over a common channel while suppressing interference.
• It employs implementations like chaotic phase encoding and direct-sequence chip coding to multiplex and demultiplex signals in quantum networks.
• The approach underpins advanced applications in quantum key distribution and entanglement distribution, offering scalable solutions for secure quantum communications.

Searching arXiv for the cited q-CDMA papers to ground the article in current literature. Quantum code-division multiple access (q-CDMA) denotes a family of quantum networking and communication schemes in which multiple sender–receiver pairs share a common physical transmission medium while distinguishing users by code-domain operations rather than by strict separation in time, frequency, or wavelength. Across the literature, q-CDMA is realized by applying user-specific unitary spreading operations to quantum states, transmitting the encoded states through a shared quantum channel or star-coupler, and then applying matched decoding operations that recover the intended user while suppressing cross-talk from other users. The concept appears in several forms, including chaotic phase encoding with chaos synchronization for shared-channel quantum transmission (Zhang et al., 2012), direct-sequence single-photon spread-spectrum multiplexing with add–drop functionality (Garcia-Escartin et al., 2014), code-division architectures for QKD networks (Razavi, 2011), and continuous-variable quantum key distribution (CV-QKD) based on chaotic phase shifters in a two-user setting (Ali et al., 13 Feb 2025).

1. Conceptual basis and historical formulations

The central q-CDMA idea is to map the classical code-division principle into the quantum domain by replacing classical spreading sequences or spectral masks with quantum-compatible unitary transformations. In the 2012 formulation of “Quantum internet using code division multiple access,” each sender $k$ applies a chaotic phase modulation generated by a “chaotic phase shifter,” with interaction Hamiltonian

$$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$

so that an input state $\ket{\psi_k}$ acquires a broadband time-dependent phase $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$ and its spectral content is spread over a large band (Zhang et al., 2012). At the receiver, a synchronized inverse operation

$$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$

recovers the intended state (Zhang et al., 2012).

A distinct direct-sequence single-photon formulation was given in “Quantum spread spectrum multiple access,” where each user $i$ is assigned a binary code $c_i\in\{\pm1\}^S$, and spreading is implemented by unitary phase modulators acting on $S$ successive chip intervals of a single-photon wavepacket. The corresponding unitary

$$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$

applies chip-dependent phases $0$ or $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$0, and despreading is achieved by applying $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$1 again, using $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$2 and $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$3 up to an overall phase (Garcia-Escartin et al., 2014).

In QKD networking, “Multiple-Access Quantum Key Distribution Networks” treated quantum CDMA as a multi-user access method in which each raw key bit is replaced by a superposition of $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$4 sub-pulses (“chips”) of mean photon number $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$5, with Optical Orthogonal Codes (OOCs) of length $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$6 and weight $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$7 controlling chip occupancy (Razavi, 2011). This line of work emphasized medium-access behavior, collision statistics, and key-rate optimization rather than coherent phase recovery.

A later spectral formalization, “Quantum CDMA Communication Systems,” described q-CDMA in terms of continuous-mode pure states of light whose spectral amplitudes $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$8 are encoded by pseudorandom spectral phase masks $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$9, implemented by dispersing the pulse, imposing piecewise phases on each of the $\ket{\psi_k}$0 chips, and recombining (Rezai et al., 2021). This formulation unified coherent-state and number-state signaling under a common operator framework.

The 2025 two-user CV-QKD proposal explicitly specialized q-CDMA to Gaussian-modulated coherent states and reverse-reconciliation security analysis. There, two senders $\ket{\psi_k}$1 encode their states using chaotic phase shifters, combine them on a 50:50 beam splitter, send the multiplexed state through a lossy quantum channel under an entangling-cloner attack, and decode at matched receivers $\ket{\psi_k}$2 through chaos synchronization (Ali et al., 13 Feb 2025).

2. Physical realizations of spreading, multiplexing, and decoding

Two implementation families dominate the q-CDMA literature: chaotic phase encoding and direct-sequence chip coding.

In the chaotic-phase family, each sender modulates the phase of its optical mode with a broadband classical chaotic drive. In the two-user CV-QKD system, the output field $\ket{\psi_k}$3 passes through a chaotic phase shifter $\ket{\psi_k}$4, whose instantaneous classical drive $\ket{\psi_k}$5 induces a phase $\ket{\psi_k}$6. Under the Hamiltonian $\ket{\psi_k}$7, the annihilation operator transforms as

$\ket{\psi_k}$8

The encoded outputs are then mixed on a 50:50 beam splitter according to

$\ket{\psi_k}$9

and, after transmission, decoded by matched chaotic phase shifters satisfying $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$0 and $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$1, so that the imposed phase is removed at the intended receiver (Ali et al., 13 Feb 2025).

The same broad structure underlies the entanglement-distribution protocol of Zhu et al., where electro-optic modulators (EOMs) driven by chaotic Colpitts circuits apply broadband chaotic phase shifts, beam splitters multiplex and demultiplex the users, and synchronized receiver-side phase shifts cancel the sender-side masking (Zhu et al., 2015). In that work, independent chaotic phase shifts are “almost uncorrelated,” enabling two pairs of users to share a single quantum channel (Zhu et al., 2015).

In the direct-sequence family, spreading is defined over chip intervals. In the Garcia-Escartin–Chamorro-Posada add–drop architecture, each photon’s wavefunction is segmented into $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$2 chips of duration $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$3, and user-specific phase modulators impose code-controlled phases on each chip (Garcia-Escartin et al., 2014). Orthogonality follows from code design: $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$4 in the ideal, lossless limit (Garcia-Escartin et al., 2014). Optical circulators and Fiber Bragg gratings (FBGs) then implement add–drop multiplexing: a user’s narrowband despread photon is reflected into or out of the shared fiber, while the spread photons of other users are largely transmitted (Garcia-Escartin et al., 2014).

Sharma and Banerjee described a related OADM architecture using two optical circulators and an FBG. Their phase modulators impose phase shifts $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$5 on the $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$6 time bins, with spreading codes drawn from maximal-length sequences generated by an $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$7-stage Linear-Feedback Shift Register, so that $\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$8 (Sharma et al., 2019). The receiver uses despreading and Gaussian-shaped FBG filtering to recover the desired photon while suppressing adjacent-channel overlap and spread noise (Sharma et al., 2019).

A more general spectral implementation appears in the continuous-mode formalism of Rezai and Salehi, in which each encoding operator is

$\theta_k(t)=\int_0^t f_k(\tau)\,d\tau$9

with inverse decoding given by its adjoint. Under encoding, $$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$0, and an input wavepacket $$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$1 is mapped to $$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$2 (Rezai et al., 2021).

3. Mathematical structure and interference suppression

The mathematical signature of q-CDMA is that intended-user components are restored by matched decoding while cross-user terms are attenuated by code orthogonality or by averaging over broadband chaotic phases.

In the 2012 chaotic q-CDMA model, the suppression of multiuser interference is quantified by a factor

$$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$3

which arises because cross-talk terms average out under chaotic spreading (Zhang et al., 2012). More generally, the 2025 CV-QKD paper defines per-user correction factors

$$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$4

and shows that, after perfect synchronization and averaging over fast chaotic phases, the two-user input–output relation becomes

$$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$5

The desired-user term scales as $$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$6, while interference and environmental terms are attenuated by $$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$7 or $$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$8 (Ali et al., 13 Feb 2025).

Expressed in quadratures, with $$U_k^{-1}=\exp\Bigl[ +\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr]$$9, Bob’s measured outcome for one quadrature is

$i$0

so that the received variable consists of the intended signal, Eve’s injected noise, interference $i$1, and beam-splitter environmental noise $i$2 (Ali et al., 13 Feb 2025).

In the direct-sequence single-photon setting, orthogonality is achieved by code inner products. For orthogonal codes of length $i$3,

$i$4

and despreading re-concentrates the intended signal while keeping interferers spread (Garcia-Escartin et al., 2014). Sharma and Banerjee instead use maximal-length sequences with correlation

$i$5

so that despreading by $i$6 yields gain $i$7 for the desired photon, whereas other users remain at level $i$8 and largely cancel out (Sharma et al., 2019).

The 2024 “chip-time interval decomposition” framework reformulates direct-sequence q-CDMA at the operator level. A creation operator $i$9 is decomposed into chip creation operators $c_i\in\{\pm1\}^S$0, yielding

$c_i\in\{\pm1\}^S$1

with orthogonal-chip commutation relations $c_i\in\{\pm1\}^S$2. A binary spreading code $c_i\in\{\pm1\}^S$3 is implemented by a phase-shifter unitary, producing

$c_i\in\{\pm1\}^S$4

(Dastgheib et al., 2024). This decomposition makes explicit that chip operators are the invariant building blocks of direct-sequence q-CDMA.

A further distinction arises from the input state class. For coherent-state inputs, inter-user interference appears as an additive term in the output coherent-state amplitude (Rezai et al., 2021, Dastgheib et al., 2024). For number-state inputs, Rezai and Salehi show that all cross-terms vanish in the instantaneous intensity: $c_i\in\{\pm1\}^S$5 with no interference term, which they attribute to complete phase uncertainty at detection for Fock states (Rezai et al., 2021). Dastgheib, Salehi, and Rezai further identify a spread-spectrum version of the Hong–Ou–Mandel effect in the chip-time formalism (Dastgheib et al., 2024).

4. q-CDMA in quantum key distribution

QKD is one of the main application domains of q-CDMA, but the literature splits into weak-coherent-pulse / single-photon network architectures and continuous-variable schemes.

In “Multiple-Access Quantum Key Distribution Networks,” quantum CDMA QKD uses OOCs of length $c_i\in\{\pm1\}^S$6 and weight $c_i\in\{\pm1\}^S$7. In an asynchronous network, the probability that two weight-$c_i\in\{\pm1\}^S$8 codes collide in at least one chip is

$c_i\in\{\pm1\}^S$9

If $S$0 other users collide on one chip, the total background yield becomes

$S$1

and the average effective rate per user is

$S$2

In the regime where any $S$3 drives the key rate to zero, this simplifies to

$S$4

from which the paper concludes that the optimal performance is achieved at unity code weight, $S$5 (Razavi, 2011). That work also introduced a listen-before-send protocol reducing the collision probability to

$S$6

with the effective rate approaching $S$7 as $S$8 (Razavi, 2011).

The 2025 CV-QKD proposal adopts a different q-CDMA mechanism based on chaotic phase encoding. Each Alice prepares a Gaussian-modulated coherent state $S$9 with quadrature variance $$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$0, the states are multiplexed through beam splitters, and the channel is modeled by an entangling-cloner attack in which Eve injects an ancilla mode $$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$1 (Ali et al., 13 Feb 2025). For each user pair, the reverse-reconciliation secret key rate is

$$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$2

with total rate

$$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$3

The mutual information is

$$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$4

where

$$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$5

and

$$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$6

The Holevo term is computed from Eve’s covariance matrix, using symplectic eigenvalues $$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$7 and $$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$8 through entropic functions $$U_i=\exp\!\Bigl[ i\sum_{k=1}^S \phi_i(k)\,\int_{(k-1)\Delta t}^{k\Delta t}\! \hat a^\dagger(t)\hat a(t)\,dt \Bigr]$$9 (Ali et al., 13 Feb 2025).

The paper’s numerical results identify a regime in which two-user q-CDMA improves over a single-user baseline. For $0$0 and $0$1, the zero-SKR cutoff distance without q-CDMA is approximately $0$2, whereas with q-CDMA it extends beyond $0$3; at $0$4, q-CDMA yields approximately $0$5 versus approximately $0$6 for the one-user case (Ali et al., 13 Feb 2025). The same study reports that larger modulation $0$7 greatly boosts the SKR, while higher $0$8 degrades performance (Ali et al., 13 Feb 2025).

A 2026 generalization extends the chaotic-phase CV-QKD framework to an arbitrary number of users, deriving multiuser quadrature input–output relations and finite-size secret-key expressions. In that model, the asymptotic per-user SKR under collective attacks is

$0$9

with total SKR $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$00, and the finite-size rate is

$$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$01

(Ali et al., 13 Mar 2026). This suggests a broader q-CDMA QKD program in which chaotic spreading is analyzed not only asymptotically but also under estimation penalties and composable-security corrections.

5. Network architectures and representative applications

Different q-CDMA variants serve different network objectives, and the literature spans quantum internet transport, entanglement distribution, single-photon multiplexing, and multiuser QKD.

The 2012 “Quantum internet using code division multiple access” paper frames q-CDMA as a solution to the open problem of efficiently transmitting quantum data among many pairs of users via a common medium. Its architecture assumes $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$02 sender–receiver pairs, with all $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$03 senders coupling their photonic signals into a single common quantum channel, each sender using an encoder port $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$04 and each receiver a matched decoder port $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$05, synchronized through an auxiliary classical channel (Zhang et al., 2012). The paper compares q-CDMA with frequency division multiple access and states that q-CDMA can greatly increase the information rates per channel used, especially for very noisy quantum channels (Zhang et al., 2012).

The Garcia-Escartin–Chamorro-Posada scheme focuses on reversible add–drop multiplexing for single-photon qubits. A user’s spread photon can be inserted into a fiber carrying the qubits of other users and later extracted without disturbing the rest, up to a small unavoidable filtering loss $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$06 (Garcia-Escartin et al., 2014). The add–drop architecture combines electro-optic phase modulators, three-port circulators, and FBGs that reflect only a narrow band $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$07 corresponding to a despread photon (Garcia-Escartin et al., 2014).

The Sharma–Banerjee work likewise addresses multiuser single-photon QKD over a single fiber, emphasizing that spreading disperses channel noise and that FBG filters stop the overlapping of adjacent channels, improving signal-to-noise ratio without amplifiers and modulators in the reported architecture (Sharma et al., 2019). Their simulations examine photon-loss, crosstalk, fidelity, QBER, and key rate as functions of the spreading factor $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$08 and the number of users $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$09 (Sharma et al., 2019).

The entanglement-distribution protocol of Zhu et al. uses q-CDMA to let two sender–receiver pairs share a single quantum channel while distributing two maximally entangled states mediated by bright coherent lights (Zhu et al., 2015). Each sender and receiver contains an optical cavity and a three-level atom, beam splitters multiplex and demultiplex the optical modes, and synchronized chaotic EOMs encode and decode two independent quantum channels (Zhu et al., 2015). After dispersive atom–cavity interactions and homodyne-based post-selection, the protocol yields Bell pairs $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$10 and $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$11 with high fidelity under appropriate bandwidth and loss conditions (Zhu et al., 2015).

The continuous-mode spectral formalism of Rezai and Salehi presents q-CDMA modules more abstractly: quantum signal sources, quantum spectral encoding phase operators, an $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$12 quantum broadcasting star-coupler, spectral phase decoding operators, and quantum receivers (Rezai et al., 2021). The authors identify potential applications in point-to-point quantum communications, quantum pulse shaping, quantum radar signals and systems, quantum local-area networks, point-to-point QKD with low probability of intercept, and distributed quantum computing (Rezai et al., 2021).

6. Performance characteristics, trade-offs, and limitations

A recurring claim across q-CDMA papers is that spreading enables multiple users to share the same resource while attenuating cross-talk. The exact performance criterion, however, depends strongly on the physical model and task.

In the 2012 chaotic q-CDMA analysis, spectral efficiency is compared with FDMA. For a bosonic Gaussian loss-with-noise channel, the classical capacity per channel use in FDMA is

$$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$13

whereas in q-CDMA the full band is reused by all users and cross-talk is suppressed by a factor $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$14, leading approximately to

$$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$15

so that $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$16 (Zhang et al., 2012). The same study reports that in a hard-chaos regime with $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$17, two-user fidelity reaches $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$18, while FDMA with half-band allocation is limited to $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$19 in the cited example (Zhang et al., 2012).

For direct-sequence single-photon multiplexing, performance is commonly expressed in loss and cross-talk scaling. In the 2014 add–drop architecture, each FBG stage incurs a fraction $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$20 loss for a spread photon, and in a network of $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$21 users the worst-case loss probability is bounded by

$$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$22

Cross-talk scales as $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$23 in intensity, and losses can be made arbitrarily small by taking large $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$24 (Garcia-Escartin et al., 2014).

Sharma and Banerjee provide explicit numerical tables. For $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$25 and $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$26 users, they report $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$27 and $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$28; for the same $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$29, the average fidelity remains above $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$30 for the four BB84-type states $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$31 even up to $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$32 users (Sharma et al., 2019). They estimate

$$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$33

which for the $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$34 example gives approximately $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$35, and define the raw key-generation rate

$$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$36

(Sharma et al., 2019). Their simulations further show that increasing $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$37 from $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$38 to $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$39 progressively improves pulse recovery (Sharma et al., 2019).

In the QKD network model of 2011, the main trade-off is between code weight, frame duration, and collision probability. Increasing the code weight $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$40 raises the collision probability $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$41, and the paper concludes that unity weight is optimal (Razavi, 2011). The hybrid WDM–T/CDMA architecture considered there indicates that hundreds of users can be supported with less than $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$42 penalty per user when router isolation is at least $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$43 (Razavi, 2011).

The entanglement-distribution study quantifies performance in terms of chaotic bandwidth, correction factors, and fidelity. It reports that the chaotic correction factor $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$44 falls exponentially once the Colpitts circuit bandwidth exceeds approximately $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$45, with $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$46 at $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$47, and that for $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$48 both entanglement fidelities exceed $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$49 once each $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$50 (Zhu et al., 2015). With $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$51 and $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$52, high fidelity $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$53 is maintained up to $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$54, while beyond approximately $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$55 loss the success rate or fidelity degrades (Zhu et al., 2015).

The two-user CV-QKD paper identifies a different set of trade-offs. Higher modulation $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$56 improves SKR, while larger correction factor $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$57 degrades it; at fixed $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$58, the curves for $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$59 outperform those for $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$60 (Ali et al., 13 Feb 2025). This suggests that broader chaotic spectra, which produce smaller $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$61, are beneficial because they more strongly suppress user-to-user interference and Eve coupling.

The available literature also makes clear that q-CDMA is not free of implementation cost. Recurrent limitations include maintaining robust long-distance chaos synchronization across many nodes (Zhang et al., 2012), engineering sufficiently broad and stable chaotic spectra without introducing extra quantum noise (Zhang et al., 2012), hardware realization of broadband chaotic phase modulation at telecom rates over tens of kilometers (Ali et al., 13 Feb 2025), classical side-channel overhead for synchronization [(Zhang et al., 2012); (Ali et al., 13 Mar 2026)], and beam-splitter-tree overhead and vacuum noise in multiuser CV-QKD generalizations (Ali et al., 13 Mar 2026).

7. Misconceptions, open questions, and research directions

One common misconception is that q-CDMA refers to a single standardized protocol. The literature instead contains several distinct constructs: OOC-based code-domain QKD with incoherent pulse-chip recombination (Razavi, 2011), unitary single-photon spread-spectrum add–drop multiplexing (Garcia-Escartin et al., 2014), chaotic spectral spreading and chaos-synchronized recovery for generic quantum communication (Zhang et al., 2012), entanglement distribution with bright coherent probes (Zhu et al., 2015), and chaotic-phase CV-QKD with Gaussian security analysis (Ali et al., 13 Feb 2025). These share a code-division philosophy but differ in state families, hardware primitives, channel models, and performance metrics.

A second misconception is that orthogonality or interference suppression is always exact. Exact zero cross-talk appears in idealized orthogonal-code formulations such as the Garcia-Escartin–Chamorro-Posada model in the lossless limit (Garcia-Escartin et al., 2014), whereas chaotic-phase approaches typically rely on small correction factors $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$62 obtained after averaging over broadband chaotic spectra [(Zhang et al., 2012); (Ali et al., 13 Feb 2025); (Zhu et al., 2015)]. In practical systems, imperfect synchronization, limited bandwidth, channel loss, and filter nonidealities translate into residual excess noise or crosstalk.

A third misconception is that q-CDMA automatically improves all security metrics in QKD. The 2025 CV-QKD results do show higher SKR and longer secure distance than a single-user baseline in the reported two-user model (Ali et al., 13 Feb 2025), but the 2011 multiple-access QKD analysis finds that asynchronous collisions can severely reduce rates unless codes are chosen with unity weight and the listen-before-send protocol is used (Razavi, 2011). The 2026 finite-size extension further indicates that parameter-estimation penalties and finite block lengths can substantially reduce the key rate relative to the asymptotic regime (Ali et al., 13 Mar 2026). A plausible implication is that q-CDMA advantages are highly regime-dependent and must be evaluated jointly with access control, noise statistics, and security proof assumptions.

Several open directions are stated explicitly in the literature. The 2025 CV-QKD work notes that the analysis is restricted to two users and identifies extension to $$U_k=\exp\!\Bigl[-\,i\!\int_0^T f_k(t)\,a_k^\dagger a_k\,dt\Bigr],$$63 users, mutually orthogonal chaotic codes, more complex synchronization networks, and experimental validation at telecom wavelengths as future work (Ali et al., 13 Feb 2025). The 2026 multiuser extension adds finite-size analysis, arbitrary user number, and the study of correction factor, interference noise, environmental noise, and transmittance as central system parameters (Ali et al., 13 Mar 2026). Earlier works identify dynamic code assignment, free-space or satellite QKD extensions, rapid programmable phase masks, and hybrid time–wavelength–code schemes as important unresolved problems [(Razavi, 2011); (Rezai et al., 2021)].

Taken together, the q-CDMA literature defines a coherent research area rather than a single protocol: a code-domain multiplexing paradigm for quantum communication in which user-specific unitary spreading and matched decoding permit shared-medium operation while controlling multiuser interference. Its current forms range from mathematically clean chip-based single-photon constructions to experimentally motivated chaotic-phase architectures for entanglement distribution and CV-QKD, with scalability, synchronization fidelity, and security under realistic nonidealities remaining the main technical frontiers [(Zhang et al., 2012); (Garcia-Escartin et al., 2014); (Ali et al., 13 Feb 2025)].