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.
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.
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.
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.
In this thesis, quantum channels are studied from the point of view of Mathematics. We studied the CPTP (completely positive trace preserving) and UCPTP channels, the latter being the unital channels. In both cases, the sets are compact convex, so they are the closure of their extreme points. We constructed extreme CPTP maps for every possible rank. We also constructed extreme UCPTP maps with rank 2 for each possible dimension. For ranks of at least 3, we constructed extreme UCPTP maps for many dimensions and ranks. We also proved that the tensor product of extreme CPTP maps is an extreme CPTP map. For UCPTP maps, we proved that the same is not always true, with many counterexamples. It is known that the category of CPTP maps is semicartesian. We add to this a classification of binary products. We proved that products only exist in trivial cases. For binary coproducts, we obtained the dimensions they should have, if they exist. In both cases, topological techniques were used. For this reason, we also investigated the topology of the closure of the set of extreme points of the CPTP maps. Topological invariants can be computed from decompositions of a space such as CW structures. These are a type of decomposition in which the space is expressed as a gluing of balls. In the search for a CW structure for the closure of the set of extreme CPTP maps, we obtained a partial decomposition that resembles a CW structure of the complex projective space.
Abstract Optimization on matrix manifolds, particularly on complex Stiefel manifolds, plays an important role in various tasks of modern quantum theory. The dynamics of open quantum systems are described by quantum channels, also known as Kraus maps. Properties of the geometric structure of the set of quantum channels are essential for various tasks in diverse areas of quantum theory ranging from quantum information theory to quantum computing and quantum control. This review summarizes the results on representing quantum channels as orbits of a unitary group acting on the complex Stiefel manifold and examines the corresponding quotient space geometry. The implication of this framework for Riemannian optimization is demonstrated with a particular emphasis on the absence of suboptimal extrema for broad classes of quantum control objective functionals.
A linear map L from Cn×n into Cn×n is called a quantum channel if it is completely positive and trace preserving. The set Ln of all such quantum channels is known to be a compact convex set. While the extreme points of Ln can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix Z(L) associated with the quantum channel L, we compute the maximum dimension of a proper face of Ln, and in addition the possible dimensions of faces generated by L when $rank \ Z(L)=2 $.
Constructing all extreme instances of the set of completely positive trace-preserving (CPTP) maps, i.e., quantum channels, is a challenging and valuable open problem in quantum information theory. Here we introduce a systematic approach that, despite the lack of knowledge about the full parametrization of the set of CPTP maps on arbitrary Hilbert-spaced dimension, enables us to construct exactly those extreme channels that are covariant with respect to a finite discrete group or a compact connected Lie group. Innovative labeling of quantum channels by group representations enables us to identify the subset of group-covariant channels whose elements are group-covariant generalized-extreme channels. Furthermore, we exploit essentials of group representation theory to introduce equivalence classes for the labels and also partition the set of group-covariant channels. As a result, we show that it is enough to construct one representative of each partition. We construct Kraus operators for group-covariant generalized-extreme channels by solving systems of linear and quadratic equations for all candidates satisfying the necessary condition for being group-covariant generalized-extreme channels. Deciding whether these constructed instances are extreme or quasiextreme is accomplished by solving a system of linear equations. Proper labeling and partitioning the set of group-covariant channels leads to a novel systematic, algorithmic approach for constructing the entire subset of group-covariant extreme channels. We formalize the problem of constructing and classifying group-covariant generalized extreme channels, thereby yielding an algorithmic approach to solving, which we express as pseudocode. To illustrate the application and value of our method, we solve for explicit examples of group-covariant extreme channels. With unbounded computational resources to execute our algorithm, our method always delivers a description of an extreme channel for any finite-dimensional Hilbert space and furthermore guarantees a description of a group-covariant extreme channel for any dimension and for any finite-discrete or compact connected Lie group if such an extreme channel exists.