We determine a strict-interior low-homotopy window for finite-dimensional exact-Kraus-rank CPTP channel manifolds, treat feasibility and boundary branches, identify low-stage Choi-support Chern invariants, construct a trace-preserving qubit S4 Yang-monopole channel with |c2|=1, and extend the topology to isolated full-rank Choi spectral bands under gap-preserving perturbations. We also prove finite-node non-identifiability of continuum c2 over unrestricted smooth gapped families and give conditional data-side and holdout subspace certificates. The four-dimensional numerical route retains full non-Abelian U(r) overlap information and distinguishes the trace-curvature invariant from the mathematical second Chern class, including the Chern-Weil c1^2 correction required for general U(r) bundles.
A quantum channel from a system
A of dimension
dA to a system
B of dimension
dB is a completely positive trace-preserving map from complex
dA×dA to
dB×dB matrices, and the set of all such maps with Kraus rank
r has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the
rdB×dA dimensional Stiefel manifold. Secondly, as the set of all Choi-states of a fixed rank
r. These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to extreme channels from system
A to system
B of a fixed Kraus rank
r is a smooth submanifold of dimension
2rdAdB−dA2−r2 of the set of all Choi-states of rank
r. As an application, we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from
A to
B arbitrarily well.
For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed states of fixed rank below the Hilbert-space dimension, however, are ubiquitous: reduced states of constrained subsystems, states confined to invariant sectors, and states supported on decoherence-free subspaces can have a support smaller than the Hilbert space, a support that can vary with parameters. Their geometry is far richer: the support carries genuine curvature, non-Abelian holonomy, and Chern topology. We formulate Uhlmann's theory directly on the manifold of rank-
k density matrices: minimal purifications make the fixed-rank stratum the base of a principal
U(k)-bundle whose Uhlmann connection is uniquely determined by a Sylvester equation, and in an eigenframe this connection takes a closed form reducing to the Berry, Wilczek--Zee, and faithful Uhlmann connections at
k=1, at equal weights, and at
k=N. The fixed-rank bundle inherits the topology of the Grassmannian, and non-factorizable higher Chern topology requires failure of the global eigenline splitting, which in the present spectral setting requires degeneracy, minimally realized by a rank-2 Yang monopole whose quantized second Chern number links the geometry to the four-dimensional quantum Hall response. Solvable models, from a genuinely non-Abelian holonomy to a dissipatively driven orbit whose Uhlmann holonomy is steered by the reservoir through an elliptic-integral spectral dressing, illustrate the content: the supporting subspace carries the topology, the spectral weights shape the transport, and Uhlmann holonomy provides a direct geometric probe of the resulting mixed-state structures.
Quantum process tomography (QPT) protocols are often designed by lifting quantum state tomography (QST) protocols through the Choi-Jamiolkowski isomorphism, at the cost of an extra step enforcing the trace-preserving (TP) constraint. Existing approaches fall into two classes. The folklore partial normalization, which sandwiches the density estimate between inverse square roots of its input marginal, has a closed form but is used as a heuristic, its only analyses being asymptotic and losing a factor polynomial in the dimension. Projection-based methods are principled but have no closed forms and scale poorly with dimension. We resolve this dichotomy by showing that the partial normalization is the exact projection onto the Choi states of quantum channels in fidelity (equivalently in Bures or purified distance). Consequently, any QST algorithm with guarantees in fidelity lifts to a QPT algorithm that inherits its sample complexity, through a single closed-form congruence that preserves the Choi rank and is computed in Kraus form without any eigendecomposition of Choi dimension. We lift two single-copy protocols and prove that the non-adaptive one (Fidelity Projected Least Squares) recovers the Choi rank and reaches infidelity
ε from a number of channel uses proportional to
1/ε. Numerically, for channels of low Choi rank, it improves on the state-of-the-art projected least squares in accuracy, and by orders of magnitude in the time and memory of the TP-regularization step and of the returned estimate.
A quantum channel from a system A of dimension dA to a system B of dimension dB is a completely positive trace-preserving map from complex dA × dA to dB × dB matrices, and the set of all such maps with Kraus rank r has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the rdB × dA dimensional Stiefel manifold. Second, as the set of all Choi-states of a fixed rank r. These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to extreme channels from system A to system B of a fixed Kraus rank r is a smooth submanifold of dimension 2rdAdB−dA2−r2 of the set of all Choi-states of rank r. As an application, we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from A to B arbitrarily well.
An isolated group of
k bands in an
N-level quantum system defines a rank-
k spectral projector and hence a map into the complex Grassmannian
Gr(k,N). For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential $\dd P$ as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of $\dd P$ enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal
Z2 index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.
Reduced descriptions of open quantum systems are intrinsically non-identifying: many different bath Hamiltonians, initial bath states, dilations, and microscopic couplings may generate the same reduced channel on the accessible system. This paper introduces a finite-dimensional certification protocol for this non-identifiability using the normalized Choi response object of a reduced channel. For a family of completely positive trace-preserving maps (\Phi_t), we set (B(t)=J(\Phi_t)/d_S) and study the thresholded Choi-support layer determined by (P_\tau(B)), its effective rank (r_\tau(B)), and the threshold margin (g_\tau(B)). A deterministic tolerance (\eta) defines the certification rule: if (g_\tau(B)>\eta), the threshold-rank layer is stable under operator-norm perturbations of radius (\eta); if (g_\tau(B)\le\eta), the continuation status is MARGIN_LOW, and a pathwise continuation of the previous support certificate must record CERTIFICATE_RESET before proceeding. The protocol is demonstrated on an exactly solvable two-spin central-spin dephasing model. In the selected parameter regime, the coarse Markovian envelope distance is nearly zero at the very time when the Choi threshold-support certificate fails. Thus small coarse model discrepancy does not imply stable thresholded support certification. The construction does not reconstruct the physical bath, identify a true hidden environment, equate Choi rank with physical bath dimension, or interpret certificate reset as a physical event. It provides a deterministic finite-dimensional diagnostic for reduced-channel data under declared thresholds and tolerances.https://github.com/AIDevelopersMonster/boundary-compensationRelation: isSupplementedBy / isSupplementTo, depending on Zenodo field wording