Design and Optimization of Adaptive Diversity Schemes in Quantum MIMO Channels

TL;DR

An adaptive diversity strategy for discrete-variable QuMIMO systems based on universal asymmetric cloning at the transmitter and probabilistic purification at the receiver and automatically adapts to channel symmetry and channel conditions is studied.

Abstract

As quantum networks evolve toward a full quantum Internet, reliable transmission in quantum multiple-input multiple-output (QuMIMO) settings becomes essential, yet remains difficult due to noise, crosstalk, and the mixing of quantum information across subchannels. To improve reliability in such settings, we study an adaptive diversity strategy for discrete-variable QuMIMO systems based on universal asymmetric cloning at the transmitter and probabilistic purification at the receiver. An input qubit is encoded into M approximate clones, transmitted over an N x N multi-mode quantum channel, and recovered through a purification map optimized using available channel state information (CSI). For the given cloning asymmetry parameters, we derive an eigenvalue-based expression for the decoder-optimal end-to-end fidelity in the form of a generalized Rayleigh quotient, which enables efficient tuning of the cloner without iterative optimization. As a design choice, we employ semidefinite program (SDP) to construct the purification map for only a targeted success probability p. This numerical framework is used to study fixed-noise, dimension-scaling noise, and stochastic depolarization regimes. A cloning asymmetry index is introduced to quantify the distribution of quantum information across the multiple subchannels across these operating conditions. The results show that the proposed scheme yields significant fidelity gains in crosstalk-dominated settings and automatically adapts to channel symmetry and channel conditions. This work provides design guidelines for future QuMIMO systems and establishes a robust baseline for more advanced transmission and decoding strategies.

Figures (12)

• Figure 1: A block-level representation of the QuMIMO communication system. The encoder (transmitter) employs a cloning map $\mathcal{E}_M^{\pmb{\gamma},\pmb{t}}\!\left(\cdot\right)$ to generate multiple imperfect copies of the input state $\rho = \ketbra{\psi}$. These copies propagate through the QuMIMO channel $\mathcal{H}_N^{\eta,\pmb{\lambda},\delta}\!\left(\cdot\right)$, which models crosstalk and depolarizing noise. The decoder (receiver) applies a purification map $\mathcal{D}_K^{\pmb{r},p}\!\left(\cdot\right)$ to retrieve the quantum state $\boldsymbol{\rho}'$.
• Figure 2: Sampling illustration for a three–channel configuration ($N=3$) with total depolarization budget $Z=1.0$. Left: Ternary plot of the $50$ mean depolarization allocations $\boldsymbol{\lambda}^{(m)}$ sampled from $\mathcal{A}_Z$. Middle: For a selected mean vector, ternary plots of the $50$ perturbed realizations $\mathbf{x}^{(m)}(\mu)$ generated for each noise level $\mu\in\{0.25,0.50,0.75,1.00\}$, illustrating how dispersion around the mean increases with $\mu$. Right: Kernel density estimates (KDEs) of the absolute cluster variance $v$, computed over the perturbations generated for each mean vector and each noise level, showing the shift toward higher variability as $\mu$ increases.
• Figure 3: Stochastic coupling matrices $P(\eta,\delta)$ for $N=5$ channels, illustrating the spatial coupling pattern under different crosstalk strengths $\eta = \{0.25,\,0.5,\,0.75,\,1.0\}$ with fixed decay exponent $\delta = 1$. As $\eta$ increases, the off-diagonal coupling elements become more pronounced, indicating stronger inter-channel mixing.
• Figure 4: Fidelity versus the number of clones $M$ for an $N \times N$QuMIMO channel with $M=N$, under a fixed total depolarization budget $Z$. For each value of $Z$, the system dimension is increased by generating $M$ asymmetric clones and transmitting them across $M$ parallel subchannels, each subject to independent depolarization and stochastic crosstalk. Throughout the figure, the crosstalk strength is set to $\eta=0.8$ and the purification probability is fixed at $p=0.8$. Every curve represents the average fidelity computed over $50$ mean depolarization vectors $\boldsymbol{\lambda}^{(m)}$, ensuring a statistically stable estimate of end-to-end performance. The upper row corresponds to asymmetric channels, while the lower row corresponds to symmetric channels. Each subplot reports the end-to-end average fidelity of schemes analyzed in this work: the direct baseline $\mathcal{F}_{\mathrm{dir}}$, direct transmission at $p=0.8$, $\mathcal{F}_{\mathrm{pur}}(p)$, symmetric combining $\mathcal{F}_{\mathrm{sym}}(p)$, the blind purification benchmark $\mathcal{F}_{\mathrm{blind}}(p)$, and the proposed cloning--purification strategy $\mathcal{F}_{\mathrm{div}}(p)$. The results show that the optimal choice of $M$ depends on the underlying channel structure, yet the adaptive cloning--purification framework consistently achieves the highest fidelity across all configurations, demonstrating robustness to both channel symmetry and subchannel crosstalk.
• Figure 5: Estimated probability density of the cloning asymmetry index $J$ under a fixed depolarization budget $Z$, computed for the optimal cloning asymmetry selected for each channel condition. Each panel shows the density $\widehat{\varphi}_{J}(x)$ of the cloning asymmetry index $J$ at $p=0.8$, averaged over $50$ mean depolarization vectors $\boldsymbol{\lambda}^{(m)}$. Columns correspond to $M=2,3,4$, with symmetric channels in the top row and asymmetric channels in the bottom row. Different crosstalk levels $\eta \in \{0, 0.5, 0.8\}$ are color-coded, and solid versus dashed curves represent distinct noise-budget settings $Z$. The vertical dashed line indicates the cloning asymmetry index of direct transmission, $J = 1/M$. These distributions reveal how the optimal cloning asymmetry balances redundancy and channel inequality, highlighting the strategies favored under each combination of noise, symmetry, and crosstalk.

Theorems & Definitions (8)

• Definition 1: Local depolarization
• Definition 2: Stochastic crosstalk model
• Definition 3: End-to-end fidelity
• Definition 4: Cloning asymmetry index
• Definition 5: Empirical asymmetry density
• Definition 6: Cloning isometry
• Definition 7: Fidelity boundary
• Definition 8: Haar-averaged operators