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Classical capacities under physical constraints: More capacity with less entanglement

Sudipta Mondal, Pritam Halder, Saptarshi Roy, Aditi Sen De

TL;DR

The paper develops a rigorous framework for classical capacities of finite-dimensional quantum channels under physical energy-based constraints on encoding. It derives exact results for noiseless channels, analyzes energy-constrained capacities of noisy qubit dephasing channels, and introduces an energy-constrained dense coding protocol that can outperform unconstrained schemes while sometimes requiring less entanglement. A key finding is the emergence of an energy scale below which constraints significantly limit capacity, and that entanglement assistance can enhance classical capacity in energy-constrained CQ channels, contrary to the unrestricted case. These results offer a realistic, energy-aware perspective on quantum communication and suggest avenues for optimizing protocols under practical resource constraints.

Abstract

Current advancements in communication equipment demand the investigation of classical information transfer over quantum channels, by encompassing realistic scenarios in finite dimensions. To address this issue, we develop a framework for analyzing classical capacities of quantum channels where the set of states used for encoding information is restricted based on various physical properties. Specifically, we provide expressions for the classical capacities of noiseless and noisy quantum channels when the average energy of the encoded ensemble or the energy of each of the constituent states in the ensemble is bounded. In the case of qubit energy-preserving dephasing channels, we demonstrate that a nonuniform probability distribution based on the energy constraint maximizes capacity, while we derive the compact form of the capacity for equiprobable messages. We suggest an energy-constrained dense coding (DC) protocol that we prove to be optimal in the two-qubit situation and obtain a closed-form expression for the DC capacity. Additionally, we demonstrate a no-go result, which states that when the dimension of the sender and the receiver is two, no energy-preserving operation can offer any quantum advantage for energy-constrained entanglement-assisted capacity. We exhibit that, in the energy-constrained situation, classical-quantum noisy channels can show improved capabilities under entanglement assistance, a phenomenon that is unattainable in the unrestricted scenario.

Classical capacities under physical constraints: More capacity with less entanglement

TL;DR

The paper develops a rigorous framework for classical capacities of finite-dimensional quantum channels under physical energy-based constraints on encoding. It derives exact results for noiseless channels, analyzes energy-constrained capacities of noisy qubit dephasing channels, and introduces an energy-constrained dense coding protocol that can outperform unconstrained schemes while sometimes requiring less entanglement. A key finding is the emergence of an energy scale below which constraints significantly limit capacity, and that entanglement assistance can enhance classical capacity in energy-constrained CQ channels, contrary to the unrestricted case. These results offer a realistic, energy-aware perspective on quantum communication and suggest avenues for optimizing protocols under practical resource constraints.

Abstract

Current advancements in communication equipment demand the investigation of classical information transfer over quantum channels, by encompassing realistic scenarios in finite dimensions. To address this issue, we develop a framework for analyzing classical capacities of quantum channels where the set of states used for encoding information is restricted based on various physical properties. Specifically, we provide expressions for the classical capacities of noiseless and noisy quantum channels when the average energy of the encoded ensemble or the energy of each of the constituent states in the ensemble is bounded. In the case of qubit energy-preserving dephasing channels, we demonstrate that a nonuniform probability distribution based on the energy constraint maximizes capacity, while we derive the compact form of the capacity for equiprobable messages. We suggest an energy-constrained dense coding (DC) protocol that we prove to be optimal in the two-qubit situation and obtain a closed-form expression for the DC capacity. Additionally, we demonstrate a no-go result, which states that when the dimension of the sender and the receiver is two, no energy-preserving operation can offer any quantum advantage for energy-constrained entanglement-assisted capacity. We exhibit that, in the energy-constrained situation, classical-quantum noisy channels can show improved capabilities under entanglement assistance, a phenomenon that is unattainable in the unrestricted scenario.
Paper Structure (24 sections, 7 theorems, 52 equations, 4 figures)

This paper contains 24 sections, 7 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Constrained classical capacities
  3. Energy-constrained classical capacity of noiseless quantum channel
  4. Energy-constrained classical capacity of noisy quantum channels
  5. Energy-constrained Dense coding: more capacity with less entanglement
  6. Energy-constrained dense coding for two-qubits
  7. Enhancement of energy-constrained classical capacity of CQ-channels with entanglement assistance
  8. Conclusion
  9. Proof of Theorem. \ref{['theorem:eccp1']}
  10. Special cases: $d = 2,3$
  11. Dimensional advantage
  12. Proof of Lemma. \ref{['lemma:epc']}
  13. Capacity of the dephasing channel
  14. Average energy constraint
  15. Strict energy constraint
  16. ...and 9 more sections

Key Result

Theorem 1

The energy-constrained classical capacity of a noiseless quantum channel in arbitrary dimension for both average and strict constraints is identical and can be expressed as where $\mathbb E_d = \frac{d-1}{2}$. Here $H(\{p_x\})= -\sum_x p_x \log_2 p_x$ is the Shannon entropy of the probability distribution $\{p_x\}$, where are the weights of the $d$-dimensional thermal state $\tau_E = \sum_{n=0}^

Figures (4)

  • Figure 1: Strict energy-constrained DC capacity $C^{\tt DC}_{\{E\}_S}(\Lambda_{\lambda=\frac{1}{2}})$ (red) and classical capacity $C_{\{E\}_S}(\Lambda_{\lambda=\frac{1}{2}})$ (green) in the case of complete dephasing qubit channel. In contrast to the unconstrained case Shirokov2012, entanglement provides an advantage in the classical capacity of a CQ channel in the strict energy-constrained scenario. Both axes are dimensionless.
  • Figure 2: The energy-constrained classical capacity, $C_{\{E\}}$ (vertical axis), plotted against the average energy constraint, $E$ (horizontal axis), for different dimensions $d$. As the dimension increases, the value of $C_{\{E\}}$ also increases. Dimensional advantage is significant when $E$ is high, although it saturates rapidly with increasing $d$. Both axes are dimensionless.
  • Figure 3: (a) Average energy-constrained classical capacity, $\widetilde{C}_{\{E\}_A}(\Lambda^{\mathrm{Deph}}_{\lambda})$ (vertical axis) against the dephasing noise parameter, $\lambda$ (horizontal axis), for various values of the average energy constraint, $E$. (b) Strict energy-constrained classical capacity (vertical axis) versus the dephasing noise parameter, $\lambda$ (horizontal axis), for different values of the energy bound $E$. Here, $\widetilde{C}_{\{E\}_{S}}(\Lambda^{\mathrm{Deph}}_{\lambda})$ (solid-lines) represents the capacity under an equiprobable input ensemble while $C_{\{E\}_{S}}(\Lambda^{\mathrm{Deph}}_{\lambda})$ (dashed-lines) corresponds to the optimal probability scenario. It is observed that the difference $C_{\{E\}_{S}}(\Lambda^{\mathrm{Deph}}_{\lambda}) - \widetilde{C}_{\{E\}_{S}}(\Lambda^{\mathrm{Deph}}_{\lambda})\geq$$10\sigma$ within the interval $0.4 \leq \lambda \leq 0.5$ for $E = 0.5$ and $E = 0.7$, where $\sigma = 5 \times 10^{-5}$. As $E$ increases, the curves shift from bottom to top. Both axes are dimensionless.
  • Figure 4: (a) comparison of noiseless DC capacities, $C^{\tt DC}_{\{E\}_A}, \tilde{C}^{\tt DC}_{\{E\}_A}, C^{\tt DC}_{\{E\}_S},\tilde{C}^{\tt DC}_{\{E\}_S}$ and the unassisted classical capacity $C_{\{E\}}$ (ordinate) in the region of energy bound $E\in[0,1/2]$ (abscissa). (b) DC capacities $C^{\tt DC}_{\{E\}_S}(\Lambda_{\lambda=1/2}^{\tt deph}), \tilde{C}^{\tt DC}_{\{E\}_S}(\Lambda_{\lambda=1/2}^{\tt deph})$ and entanglement-unassisted classical capacities, $C_{\{E\}_S}(\Lambda_{\lambda=1/2}^{\tt deph}),$ and $\tilde{C}_{\{E\}_S}(\Lambda_{\lambda=1/2}^{\tt deph})$ (ordinate) of complete dephasing channel against strict energy bound $E$ (abscissa). In the nontrivial energy bound region, i.e., in $E\in(0,1)$, similar to the case of $C^{\tt DC}_{\{E\}_S}(\Lambda_{\lambda=1/2}^{\tt deph})>C_{\{E\}_S}(\Lambda_{\lambda=1/2}^{\tt deph})$ as mentioned in the main text, in the case of equiprobable signal ensemble also, entanglement provides advantage in classical capacity, i.e., $\widetilde{C}^{\tt DC}_{\{E\}_S}(\Lambda_{\lambda=1/2}^{\tt deph})>\widetilde{C}_{\{E\}_S}(\Lambda_{\lambda=1/2}^{\tt deph})$. Both axes are dimensionless.

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 1
  • Proposition 1
  • proof
  • Theorem 2
  • Theorem 3
  • Proposition 2
  • Lemma 2
  • proof