Approaches for approximate additivity of the Holevo information of quantum channels
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2018-01-25 |
| Journal | Physical review. A/Physical review, A |
| Authors | Felix Leditzky, Eneet Kaur, Nilanjana Datta, Mark M. Wilde |
| Institutions | University of Cambridge, Joint Institute for Laboratory Astrophysics |
| Citations | 108 |
| Analysis | Full AI Review Included |
Quantum Channel Capacity Analysis & Enabling Diamond Solutions
Section titled âQuantum Channel Capacity Analysis & Enabling Diamond SolutionsâExecutive Summary
Section titled âExecutive SummaryâThis paper presents a theoretical advancement in calculating the unassisted classical capacity $C(N)$ of quantum channels, a critical metric for quantum communication and computation reliability. 6CCVD, as an expert material provider for quantum hardware, highlights the necessity of high-purity, custom diamond substrates to physically realize the systems analyzed.
- Core Achievement: Derivation of tighter upper bounds on the classical capacity $C(N)$ by leveraging âapproximate additivityâ parameters, circumventing the intractable regularization required by the Holevo information formula.
- Methodology: New approximation parameters are defined based on the diamond distance ($|N - M|_\diamond$) between an arbitrary channel $N$ and channels known to have additive Holevo information ($M$, e.g., covariant, entanglement-breaking, Hadamard).
- Efficiency: The approximation parameters (e.g., covariance $\epsilon$, entanglement-breaking $\text{EB}(N)$) are efficiently computable using Semidefinite Programming (SDP).
- Qubit Relevance: The methodology is demonstrated effectively on specific qubit channels, including Amplitude Damping ($A_p$), which is highly relevant to physical solid-state quantum computing testbeds (e.g., NV centers).
- Engineering Requirement: Replication or extension of this research into physical testbeds demands ultra-low-defect, highly polished Single Crystal Diamond (SCD) wafers, which 6CCVD delivers globally.
Technical Specifications
Section titled âTechnical SpecificationsâThe core specifications derived from the analysis relate to the mathematical parameters used to define and bound the channel capacity.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Channel Dimensionality (Qubit) | A | = | |
| Maximum Qubit Count ($n$) | Up to $2^n$ | Dimension | Used for analyzing the Clifford Group ($C_n$) covariance bounds. |
| Approximation Parameters | $\epsilon$ | Unitless | Diamond distance half-difference: $1/2 |N - M|_\diamond \le \epsilon$. |
| Holevo Capacity Upper Bound | $\chi(M) + 2\epsilon \log | B | + g(\epsilon)$ |
| Error Function $g(\epsilon)$ | $(1 + \epsilon)\log(1 + \epsilon) - \epsilon\log\epsilon$ | Unitless | Function derived from continuity bounds of classical capacity. |
| Covariance Parameter ($\text{cov}_P(A_p)$) | $p/2$ | Unitless | Derived analytically for Amplitude Damping channel $A_p$ w.r.t. Pauli Group $P$. |
| Optimal Entanglement Break (EB) Bound | $\text{EB}(A_p) \ge (1-p)/2$ | Unitless | Lower bound on the entanglement-breaking distance for $A_p$. |
| Qubit Application Examples | $A_p$, $N_p$ (Mixture), $M_p$ (Composition) | N/A | Specific noise models studied for performance comparison against the strong converse bound $C_\beta(N)$. |
| Computational Method | SDP (Semidefinite Programming) | N/A | Primary method used for efficient calculation of approximation parameters $\epsilon$. |
Key Methodologies
Section titled âKey MethodologiesâThe research relies on advanced techniques in quantum information theory, primarily focusing on metric distances between channels and applying continuity bounds.
- Diamond Norm Distance: The core operational distance metric ($|N - M|_\diamond$) is used to quantify how close an arbitrary quantum channel $N$ is to an idealized channel $M$ (e.g., covariant or Hadamard). This distance is crucial as it guarantees physical relevance.
- Definition of Approximate Additivity Parameters:
- Approximate Covariance ($\text{cov}_G(N)$): Measures distance to the channelâs twirled counterpart $N_G$. Covariant channels (e.g., unital qubit channels) have known additive Holevo information.
- Approximate Entanglement-Breaking ($\text{EB}(N)$): Measures distance to an entanglement-breaking channel $M$, which is known to have strong Holevo additivity.
- Approximate Hadamard-ness ($\text{Had}(N)$ and $\text{Had}_{\text{deg}}(N)$): Measures distance to channels where the complementary channel is entanglement-breaking.
- Classical Capacity Bounding: Shirokovâs continuity bound (Theorem II.6) is adapted using the $\epsilon$-parameters derived from the diamond norm distance to provide strong upper bounds on the classical capacity $C(N)$.
- SDP Implementation: Computation of these $\epsilon$-parameters is formulated as a Semidefinite Program (SDP), allowing for efficient numerical calculation, leveraging tools like YALMIP and quantinf MATLAB packages.
- Analysis of Physical Noise Models: The techniques are applied to noise models commonly found in quantum hardware, particularly the Amplitude Damping channel ($A_p$), which simulates energy loss (spontaneous emission).
6CCVD Solutions & Capabilities
Section titled â6CCVD Solutions & CapabilitiesâThe theoretical and computational work presented in this paper provides the necessary mathematical framework for future physical implementation of quantum communication testbeds. 6CCVD delivers the enabling materialsâhigh-purity MPCVD diamondârequired for realizing the physical systems (qubits, optical components) where these channels operate.
Applicable Materials
Section titled âApplicable MaterialsâThe channels analyzed are primarily qubit channels ($|A|=|B|=2$). The most promising solid-state platform for stable, low-noise qubits relevant to this work is the Nitrogen-Vacancy (NV) center in diamond.
- Optical Grade Single Crystal Diamond (SCD): Required for hosting NV centers. Our SCD offers ultra-low nitrogen content (down to < 1 ppb) and excellent crystalline quality, crucial for achieving the low noise limits (low $p$ values in $A_p$ and $N_p$) necessary for useful channel capacity.
- Custom Substrates: Since experiments require integrated optical circuits, waveguides, or specific surface chemistries, 6CCVD offers custom-cut and polished wafers.
- Dimensions: Custom plates/wafers up to 125mm (PCD) or large-area SCD for multi-qubit integration.
- Thickness Control: SCD wafers available from 0.1”m to 500”m, ensuring precise interface control for coupling.
Customization Potential for Qubit Implementation
Section titled âCustomization Potential for Qubit ImplementationâSuccessful replication of high-fidelity quantum channels relies on the physical integration of the diamond substrate into complex systems involving microwave circuitry and precise optical interfaces.
| Required Service (Based on Paper Application) | 6CCVD Capability | Technical Specification |
|---|---|---|
| High-Fidelity Qubit Hosting | Ultra-High Purity SCD | Low-defect material to minimize unwanted decoherence ($p \to 0$ in $A_p$). |
| Integration/Interface Precision | Advanced Polishing | Ra < 1nm (SCD), critical for reducing scattering losses and achieving high-quality optical interfaces (necessary for Amplitude Damping $A_p$). |
| Custom Circuitry/Contacts | Metallization Services | In-house deposition of standard quantum contacts: Au, Pt, Pd, Ti, W, Cu. Essential for microwave control (Pauli group $P$ operations) in solid-state systems. |
| Complex Geometries | Custom Dimensions & Shaping | Wafers up to 125mm, custom shapes via high-precision laser cutting for integrating diamond into cryostats or optical setups. |
Engineering Support
Section titled âEngineering SupportâThe analysis of noise models like Amplitude Damping (energy loss) and Z-Dephasing is paramount for designing robust quantum hardware. 6CCVDâs in-house PhD team provides consultative support, ensuring material properties are optimized to mitigate these noise effects.
We assist clients in selecting diamond materials tailored for projects requiring:
- Low Decoherence: Optimizing nitrogen concentration and isotopic purity for NV-based qubits.
- Thermal Management: Utilizing PCD substrates (up to 10mm thick) for applications requiring maximum thermal conductivity adjacent to quantum components.
- Optical Clarity: Providing Type IIa SCD with high transmission across the UV-visible-IR spectrum for quantum communication links.
For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.
View Original Abstract
We study quantum channels that are close to another channel with weakly\nadditive Holevo information and derive upper bounds on their classical\ncapacity. Examples of channels with weakly additive Holevo information are\nentanglement-breaking channels, unital qubit channels, and Hadamard channels.\nRelated to the method of approximate degradability, we define approximation\nparameters for each class above that measure how close an arbitrary channel is\nto satisfying the respective property. This gives us upper bounds on the\nclassical capacity in terms of functions of the approximation parameters, as\nwell as an outer bound on the dynamic capacity region of a quantum channel.\nSince these parameters are defined in terms of the diamond distance, the upper\nbounds can be computed efficiently using semidefinite programming (SDP). We\nexhibit the usefulness of our method with two example channels: a convex\nmixture of amplitude damping and depolarizing noise, and a composition of\namplitude damping and dephasing noise. For both channels, our bounds perform\nwell in certain regimes of the noise parameters in comparison to a recently\nderived SDP upper bound on the classical capacity. Along the way, we define the\nnotion of a generalized channel divergence (which includes the diamond distance\nas an example), and we prove that for jointly covariant channels these\nquantities are maximized by purifications of a state invariant under the\ncovariance group. This latter result may be of independent interest.\n
Tech Support
Section titled âTech SupportâOriginal Source
Section titled âOriginal SourceâReferences
Section titled âReferencesâ- 2006 - XIVth International Congress on Mathematical Physics