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Information storage and transmission under Markovian noise

Satvik Singh, Nilanjana Datta

TL;DR

This paper analyzes information storage and transmission under quantum Markov noise, showing that infinite-time capacities are governed by the peripheral space of the noise and are efficiently computable, with strong converse and additivity properties. It proves convergence bounds showing infinite-time capacities are reached at times of order $t\gtrsim d^2\ln(d)$, and provides explicit results for IID noise where capacities become additive per qudit and can be achieved with zero error. For data transmission over long Markovian channels, the authors derive efficiently computable bounds on all capacities in the zero- and non-zero-error regimes, along with finite-block-length bounds and strong additivity properties that extend to continuous-time Markov semigroups. The work also covers practical implications for quantum memories, illustrating how aliasing data into the peripheral space yields robust long-time storage, while non-peripheral noise leads to rapid degradation. Overall, the results unify storage and transmission under Markovian noise, yielding sharp, computable limits and providing guidance for designing memories and channels under such noise models.

Abstract

We study the information transmission capacities of quantum Markov semigroups $(Ψ^t)_{t\in \mathbb{N}}$ acting on $d-$dimensional quantum systems. We show that, in the limit of $t\to \infty$, the capacities can be efficiently computed in terms of the structure of the peripheral space of $Ψ$, are strongly additive, and satisfy the strong converse property. We also establish convergence bounds to show that the infinite-time capacities are reached after time $t\gtrsim d^2\ln (d)$. From a data storage perspective, our analysis provides tight bounds on the number of bits or qubits that can be reliably stored for long times in a quantum memory device that is experiencing Markovian noise. From a practical standpoint, we show that typically, an $n-$qubit quantum memory, with Markovian noise acting independently and identically on all qubits and a fixed time-independent global error correction mechanism, becomes useless for storage after time $t\gtrsim n2^{2n}$. In contrast, if the error correction is local, we prove that the memory becomes useless much more quickly, i.e., after time $t\gtrsim \ln(n)$. In the setting of point-to-point communication between two spatially separated parties, our analysis provides efficiently computable bounds on the optimal rate at which bits or qubits can be reliably transmitted via long Markovian communication channels $(Ψ^l)_{l\in \mathbb{N}}$ of length $l\gtrsim d^2 \ln(d)$, both in the finite block-length and asymptotic regimes.

Information storage and transmission under Markovian noise

TL;DR

This paper analyzes information storage and transmission under quantum Markov noise, showing that infinite-time capacities are governed by the peripheral space of the noise and are efficiently computable, with strong converse and additivity properties. It proves convergence bounds showing infinite-time capacities are reached at times of order , and provides explicit results for IID noise where capacities become additive per qudit and can be achieved with zero error. For data transmission over long Markovian channels, the authors derive efficiently computable bounds on all capacities in the zero- and non-zero-error regimes, along with finite-block-length bounds and strong additivity properties that extend to continuous-time Markov semigroups. The work also covers practical implications for quantum memories, illustrating how aliasing data into the peripheral space yields robust long-time storage, while non-peripheral noise leads to rapid degradation. Overall, the results unify storage and transmission under Markovian noise, yielding sharp, computable limits and providing guidance for designing memories and channels under such noise models.

Abstract

We study the information transmission capacities of quantum Markov semigroups acting on dimensional quantum systems. We show that, in the limit of , the capacities can be efficiently computed in terms of the structure of the peripheral space of , are strongly additive, and satisfy the strong converse property. We also establish convergence bounds to show that the infinite-time capacities are reached after time . From a data storage perspective, our analysis provides tight bounds on the number of bits or qubits that can be reliably stored for long times in a quantum memory device that is experiencing Markovian noise. From a practical standpoint, we show that typically, an qubit quantum memory, with Markovian noise acting independently and identically on all qubits and a fixed time-independent global error correction mechanism, becomes useless for storage after time . In contrast, if the error correction is local, we prove that the memory becomes useless much more quickly, i.e., after time . In the setting of point-to-point communication between two spatially separated parties, our analysis provides efficiently computable bounds on the optimal rate at which bits or qubits can be reliably transmitted via long Markovian communication channels of length , both in the finite block-length and asymptotic regimes.

Paper Structure

This paper contains 41 sections, 57 theorems, 288 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Data storage
  4. Data transmission
  5. Outline of the paper
  6. Preliminaries
  7. Channel capacities
  8. One-shot capacities
  9. Asymptotic capacities
  10. Entropic quantities
  11. Entropic bounds on channel capacities
  12. Zero-error communication
  13. Spectral properties
  14. The peripheral space
  15. Storage perspective
  16. ...and 26 more sections

Key Result

Theorem 1.2

Let $\Psi : \mathcal{L}({\mathcal{H}})\to \mathcal{L}({\mathcal{H}})$ be a quantum channel and $\varepsilon\in [0,1)$. There exist positive integers $K, d_1, \dots d_K$ that can be efficiently computed from $\Psi$ such that These integers arise from the peripheral space $\mathscr{X}(\Psi) = \text{span}\{X\in \mathcal{L}({\mathcal{H}}) : \exists \, \theta\in \mathbb{R} \text{ with } \Psi(X)= e^{i\

Figures (7)

  • Figure 1: Schematic for a physical memory device experiencing Markovian noise in time modelled by a dQMS $(\Psi^t)_{t\in\mathbb{N}}$, where $\Psi:\mathcal{L}({\mathcal{H}})\to \mathcal{L}({\mathcal{H}})$ is a quantum channel. Logical data is encoded in the memory at time $t=0$ using an encoding channel $\mathcal{E}:\mathcal{L}({\mathcal{H}_{\operatorname{data}}})\to \mathcal{L}({\mathcal{H}})$. After some time $t$, a decoding channel $\mathcal{D}:\mathcal{L}({\mathcal{H}})\to \mathcal{L}({\mathcal{H}_{\operatorname{data}}})$ is applied to recover the data, so that $\mathcal{D}\circ \Psi^t \circ \mathcal{E}\simeq_{\varepsilon} \operatorname{id}$ approximately simulates the identity channel on the data system, where $\varepsilon\in [0,1)$ is the error allowed in the recovery process.
  • Figure 2: The error-rate curve for data storage in a quantum memory comprised of $n$ physical qudits (Hilbert space $\mathcal{H}\simeq (\mathbb{C}^{q})^{\otimes n}$) experiencing IID Markovian noise modelled by a dQMS $(\Psi^t)_{t\in \mathbb{N}}$, where $\Psi = \Gamma^{\otimes n}$ for some quantum channel $\Gamma:\mathcal{L}({\mathbb{C}^{q}})\to \mathcal{L}({\mathbb{C}^{q}})$. One can store $\log\left( \max_k d_k \right)$ many logical qubits per physical qudit inside the largest block in the peripheral space of the local channel $\mathscr{X}(\Gamma)$ (see Eq. \ref{['eq:phasespace-intro']}) perfectly with zero data recovery error. Moreover, any attempt to store data at a higher rate $R>\log \left(\max_k d_k\right)$ fails as $t\to \infty$ with certainty, since the data recovery error $\varepsilon_n\to 1$ exponentially as $n\to \infty$.
  • Figure 3: Schematic for a physical memory device experiencing Markovian noise modelled by a dQMS $(\Psi^t)_{t\in\mathbb{N}}$, where $\Psi:\mathcal{L}({\mathcal{H}})\to \mathcal{L}({\mathcal{H}})$ is a quantum channel. Logical data is encoded in the memory at time $t=0$ using an encoding channel $\mathcal{E}:\mathcal{L}({\mathcal{H}_{\operatorname{data}}})\to \mathcal{L}({\mathcal{H}})$. After some time $t$, a decoding channel $\mathcal{D}:\mathcal{L}({\mathcal{H}})\to \mathcal{L}({\mathcal{H}_{\operatorname{data}}})$ is applied to recover the data, so that $\mathcal{D}\circ \Psi^t \circ \mathcal{E}\simeq_{\varepsilon} \operatorname{id}$ approximately simulates the identity channel on the data system, where $\varepsilon\in [0,1)$ is the error allowed in the recovery process.
  • Figure 4: The error-rate curve for data storage in a quantum memory with $n$ qudits experiencing IID Markovian noise modelled by a dQMS $((\Gamma^{\otimes n})^t)_{t\in \mathbb{N}}$, where $\Gamma:\mathcal{L}({\mathbb{C}^{q}})\to \mathcal{L}({\mathbb{C}^{q}})$ is a quantum channel. One can store $n(\log\max_k d_k)$ many logical qubits inside the largest block in the peripheral space $\mathscr{X}(\Gamma)$ (see Eq. \ref{['eq:phasespace-storage']}) perfectly with no recovery error $\varepsilon_n=0$. Moreover, any attempt to store data at a higher rate $R>\log\max_k d_k$ fails with certainty as $t\to \infty$, since the data recovery error $\varepsilon_n\to 1$ exponentially as $n\to \infty$.
  • Figure 5: Quantum memory with $n$ physical qubits undergoin IID depolarizing noise with parameter $p\in (0,1)$. An error-correction procedure $\Psi_{\operatorname{ecc}}$ is designed to periodically detect and correct errors induced by the noise. We show that the memory becomes useless for data storage after time $t\gtrsim n2^{2n}$ scaling exponentially with the number of qubits.
  • ...and 2 more figures

Theorems & Definitions (127)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 117 more