Differentiable Geometry of Quantum Channels

• Differentiable geometric parameterization of quantum channels is a framework that represents CPTP maps as points on smooth manifolds with explicit coordinates and metrics.
• It leverages constructions like the Kraus–Stiefel quotient and Choi-state coordinates to systematically enforce complete positivity and trace preservation.
• This approach facilitates practical applications such as channel optimization, control theory, and complexity quantification using methods like gradient descent and Hamiltonian Monte Carlo.

A differentiable geometric parameterization of quantum channels provides a rigorous manifold framework in which channels—completely positive trace-preserving (CPTP) maps—can be represented, analyzed, and optimized by means of explicit coordinates, metrics, and volume elements. Such parameterizations are central to statistical analysis, optimization, control theory, and complexity quantification in quantum information science.

1. Manifold Structure and Local Coordinates

A quantum channel acting on an $n$-level system is a CPTP linear map $\Phi: M_n \rightarrow M_m$. Its space possesses a natural smooth manifold structure, realized, for instance, via the Kraus operator-sum representation. Any such channel admits a minimal decomposition $\Phi(\rho) = \sum_{i=1}^r K_i \rho K_i^\dagger$ where $K_i: \mathbb{C}^n \to \mathbb{C}^m$, satisfying $\sum_{i=1}^r K_i^\dagger K_i = I_n$ (Russkikh et al., 19 Aug 2024).

Organizing the $K_i$ as blocks yields a single $mr \times n$ complex matrix $K$ lying in the complex Stiefel manifold $$\mathrm{St}(n,\,mr) = \{K \in \mathbb{C}^{mr \times n} \mid K^\dagger K = I_n\}$$. The manifold of quantum channels is then the quotient $\mathrm{St}(n,mr)/U(r)$, factoring out unitary equivalence in Kraus representation (Iten et al., 2016).

For channels of fixed Kraus rank $r$, the corresponding smooth submanifold has real dimension $2rd_Ad_B-d_A^2-r^2$ for maps from $d_A$ to $d_B$-dimensional systems (Iten et al., 2016). In local charts, coordinates can be chosen by fixing an invertible minor and representing Stiefel elements in terms of unconstrained matrices, or equivalently by encoding Choi states.

Phase-covariant and Gaussian channels, and rank-constrained extreme channels can be parameterized globally by low-dimensional real vectors, e.g., $(\lambda_1, \lambda_3, \lambda_*)$ for qubit phase-covariant channels (Siudzińska, 2022), $(M,N,c)$ for one-mode Gaussian channels (Siudzińska et al., 2019), or symplectic invariants $(\mu_A, \mu_\sigma, \mu, \Delta)$ for Gaussian CJ states.

2. Metrics and Riemannian Geometry on Channel Manifolds

Riemannian and Finsler metrics on channel spaces are constructed either by pulling back canonical metrics from operator spaces or using the Hilbert–Schmidt metric on Choi matrices. For one-mode Gaussian channels, the CJ-isomorphism $$\Lambda \mapsto \rho_{AB} = (\Lambda \otimes \mathrm{id})(\rho_\Omega)$$ establishes a metric via the Hilbert–Schmidt line element

$$ds^2 = \frac{1}{16 \sqrt{\det{\Sigma}}} \left\{ 2\, \mathrm{Tr}[\Sigma^{-1} d\Sigma]^2 + [\mathrm{Tr}(\Sigma^{-1} d\Sigma)]^2 + 8\, d\ell^T \Sigma^{-1} d\ell \right\}$$

where $(\Sigma, \ell)$ are coordinates on the CJ Gaussian states (Siudzińska et al., 2019).

For phase-covariant qubit channels, the metric tensor in $(\lambda_1, \lambda_3, \lambda_*)$ coordinates is diagonal, $$g = \mathrm{diag}(\frac{1}{2}, \frac{1}{4}, \frac{1}{4})$$, yielding the line element

$$ds^2 = \frac{1}{4}(2\,d\lambda_1^2 + d\lambda_3^2 + d\lambda_*^2)$$

(Siudzińska, 2022).

The metric on the quotient $\mathrm{St}(n,mr)/U(r)$ descends from the ambient Euclidean metric on the Stiefel manifold, such that for tangent vectors $\Delta_1, \Delta_2$, $$\langle \Delta_1, \Delta_2 \rangle = \operatorname{Re} \operatorname{Tr}(\Delta_1^\dagger \Delta_2)$$ (Russkikh et al., 19 Aug 2024). This structure supports tangent-space projections, horizontal lifts, and retraction maps for optimization procedures.

Geometric structures for open-system dynamics and complexity measures are constructed via right-invariant metrics on Lie group manifolds, e.g., $SU(d)$, enabling the computation of geodesic lengths and Nielsen-type complexity penalties (Acevedo et al., 24 Jul 2025).

3. Constraints: Complete Positivity and Trace Preservation

Differentiable parameterizations encode CPTP constraints either via geometric embeddings or explicit frame conditions. In Kraus-vector constructions, each channel is encoded by mutually constrained unit vectors subject to Euclidean orthogonality and symplectic orthogonality constraints, enforcing completeness and positivity:

$$\|\mathbf{v}_i\|^2 = 1, \quad \mathbf{v}_i \cdot \mathbf{v}_j = 0, \quad \mathbf{v}_i^T S \mathbf{v}_j = 0,$$

where $S$ is block-diagonal with symplectic form factors (Ateeq et al., 29 Nov 2025). These guarantee that every point on the "Kraus sphere" yields a valid CPTP map.

In Choi-state parameterizations, positivity and trace-preservation translate to $C \geq 0$ and $\mathrm{Tr}_B C = I_{d_A}$. Spherical coordinate constructions for the block columns of the Choi matrix enforce orthonormality and positivity automatically (Sim et al., 2019).

4. Volume Elements and Integration over Channel Manifolds

The Riemannian metric enables computation of the volume form on the channel manifold, essential for statistical inference, typicality analysis, and integration over submanifolds. For one-mode Gaussian channels in symplectic invariants, the Hilbert–Schmidt volume element is

$$dV = \frac{\mu^{11/2}}{64 \sqrt{2} \mu_A^3 \mu_\sigma^2} \, d\mu_A\, d\mu_\sigma\, d\mu\, d\Delta,$$

up to gauge-group normalization (Siudzińska et al., 2019).

For phase-covariant qubit channels,

$$dV = \frac{\sqrt{2}}{8} d\lambda_1\,d\lambda_3\,d\lambda_*,$$

with integrals over constrained domains yielding volumes for CPTP, entanglement-breaking, and time-local generator maps (Siudzińska, 2022).

Hamiltonian Monte Carlo sampling leverages these coordinate parameterizations and associated Jacobians to produce high-quality channel samples, as the parameter-space is unconstrained and differentiable (Sim et al., 2019).

5. Optimization and Control on Channel Manifolds

Gradient-based optimization, control, and learning tasks on quantum channels are rendered tractable by explicit manifold models. Fidelity-based loss functions, evaluated over channel manifolds (e.g., the Kraus sphere), permit efficient gradient descent for quasi-inverse synthesis:

$$\mathcal{L}(\boldsymbol\theta) = 1 - \bar{F}(\boldsymbol\theta)$$

with optimality enforced by projection steps onto the constraint manifold at each iteration (Ateeq et al., 29 Nov 2025).

Reachable set approximation in coherently controlled quantum channels utilizes Lie semigroup parameterization and tangent cones ("Lie wedges"), which serve as differentiable charts for finite product integrals approximating all accessible channels (O'Meara et al., 2011).

On Stiefel quotients, gradients and Hessians of control objectives descend via horizontal lifts, supporting steepest-descent and trust-region methodologies. For convex kinematic costs, all local extrema on channel manifolds are global (Russkikh et al., 19 Aug 2024).

6. Special Cases and Applications

In Gaussian settings, the geometry enables closed-form volume ratios for entanglement-breaking and incompatibility-breaking subspaces, facilitating statistical typicality analysis (Siudzińska et al., 2019). For depolarizing channels, affine contraction induces a deformation of the Uhlmann metric on the Bloch ball, quantifying distinguishability and randomness (Cafaro et al., 2011).

Circuit synthesis and quantum algorithm complexity for open and closed systems are quantified by Riemannian lengths of curves (geodesic actions) in the associated group manifold (e.g., $SU(d_S d_E)$), with precise formulas for channel complexity as Nielsen-length differences (Acevedo et al., 24 Jul 2025).

The smooth submanifold of extreme CPTP maps gives the minimal number of real parameters required for circuit architectures to approximate all such channels, setting lower bounds for control resources (Iten et al., 2016).

7. Summary and Outlook

A differentiable geometric parameterization of quantum channels is established via several equivalent constructions: Kraus–Stiefel quotients, Choi-state coordinates, symmetry-adapted parameters, or Lie semigroup charts. These models encode CPTP constraints, admit explicit Riemannian metrics, and provide volume elements for integration. They underpin optimization algorithms, control theory, statistical inference, and complexity measures for quantum channels across bosonic, fermionic, and multi-qubit systems. Relative volumes, typicality questions, and reachable-set characterization are tractable as finite-dimensional integrals over the corresponding smooth manifolds (Siudzińska et al., 2019, Ateeq et al., 29 Nov 2025, Russkikh et al., 19 Aug 2024, Iten et al., 2016).