Abstract. We construct explicit bistochastic entanglement-breaking maps on Mn whose completely bounded idempotence defects tend to zero, while their distances from all unital completely positive idempotents on Mn have lower bounds tending to one. This answers negatively Kitaev’s question on dimension-independent approximation by idempotent unital completely positive maps. The construction uses a regular simplex with n + 1 vertices, and the defect is exactly 2(n2 − n− 1)/n3. The obstruction persists under arbitrary finite matrix amplifications, with an asymptotic lower bound of 1/4 uniform in the auxiliary dimension. The proof combines a rank obstruction with the Choi–Effros theorem and an elementary rigidity result: two finite-dimensional C∗-algebras admitting unital completely positive approximate inverses with completely bounded error below 1/4 are ∗-isomorphic.
Abstract. We construct explicit bistochastic entanglement-breaking maps on Mn whose completely bounded idempotence defects tend to zero, while their distances from all unital completely positive idempotents on Mn have lower bounds tending to one. This answers negatively Kitaev’s question on dimension-independent approximation by idempotent unital completely positive maps. The construction uses a regular simplex with n + 1 vertices, and the defect is exactly 2(n2 − n− 1)/n3. The obstruction persists under arbitrary finite matrix amplifications, with an asymptotic lower bound of 1/4 uniform in the auxiliary dimension. The proof combines a rank obstruction with the Choi–Effros theorem and an elementary rigidity result: two finite-dimensional C∗-algebras admitting unital completely positive approximate inverses with completely bounded error below 1/4 are ∗-isomorphic.
Almost-idempotent quantum channels and approximate C∗-algebras
Let Φ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that Φ is η-idempotent, namely, ∥Φ2−Φ∥cb≤η, and construct an associated ε-C∗ algebra (of almost-invariant observables) for ε=O(η). This type of structure has the axioms of a unital C∗ algebra but the associativity and other axioms involving the multiplication and the unit hold up to ε. We prove that any finite-dimensional ε-C∗ algebra A is O(ε)-isomorphic to some genuine C∗ algebra B. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When A comes from a finite-dimensional η-idempotent UCP map Φ, the O(η)-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel Φ∗ into a decoding channel, producing a state on B, and an encoding channel.
Preprint. This manuscript has not undergone external peer review. We characterize exact projection channels on all tensor powers of a finite-dimensional system. Fix d ≥ 2. Suppose the channels are unital, completely positive, trace preserving and idempotent, covariant under collective SU(d) transformations and tensor permutations, consistent with identity embeddings, and completely depolarizing on one site. Then the entire family is either complete depolarization or collective Haar twirling. The proof combines standard conditional-expectation theory, the two-site Schur–Weyl commutant and an elementary consequence of infinite exchangeability. It requires no general classification of invariant subalgebras. For qubits, any nontrivial family necessarily preserves a logical qubit on three spins, excluding a nontrivial purely classical retained algebra under these assumptions. We illustrate the obstruction with a total-spin measurement channel and give sharp three-spin disturbance bounds. The entanglement-breaking bound is an application of the established trine state-reconstruction benchmark. An explicit example separates small normalized-Choi consistency defects from uniform operational closeness, clarifying a limitation of approximate extensions.
We prove dimension-independent Hyers--Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here ε≥0 is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most 9ε/2 in the trace metric and 22ε in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to 3ε even under the hypothesis of δ-surjectivity, independently of δ. In infinite dimensions, we construct spectrum-preserving homeomorphisms of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as ε→0, but no such modulus can be chosen independently of dimension.
We analyze linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. If a linear map becomes entanglement breaking after finitely many iterations, we say that the map has a finite index of separability. In particular, we show that every unital positive partial transpose (PPT) channel has a finite index of separability and that the class of unital channels that have a finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have a finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. Our work is motivated by Christandl’s PPT-squared conjecture. This conjecture states that every PPT channel, when composed with itself, becomes entanglement breaking.
In this paper, we study the [Formula: see text]-convex set of unital entanglement breaking (EB-)maps on matrix algebras. General properties and an abstract characterization of [Formula: see text]-extreme points are discussed. By establishing a Radon–Nikodym-type theorem for a class of EB-maps we give a complete description of the [Formula: see text]-extreme points. It is shown that a unital EB-map [Formula: see text] is [Formula: see text]-extreme if and only if it has Choi-rank equal to [Formula: see text]. Finally, as a direct consequence of the Holevo form of EB-maps, we derive a non-commutative analog of the Krein–Milman theorem for [Formula: see text]-convexity of the set of unital EB-maps.
A quantum channel can always be implemented by a unitary interaction with an environment that starts in a pure state. With a finite, maximally mixed environment, the implementable channels are the noisy operations. A tracial von Neumann algebra as environment gives the factorizable channels, and by a theorem of Haagerup and Musat combined with MIP* = RE, some of these lie outside the closure of the noisy operations. Version 1.0 of this record wrote down explicit rational channels of this kind, with no value for their distance from the closure. This version gives explicit lower bounds. Using Wang and Zhi's dimension-uniform reflection estimate, it constructs a factorizable unital channel on twelve qubits with Kraus entries in {0, 1, ±3/5, 4/5}, whose normalized Choi and diamond distances from the closure are at least 2^-(2J+50), with J = 2^(2^1122000). The bound comes from a compiler theorem: group relations that are rigid uniformly in the matrix dimension give a rational channel with an explicit distance bound. For the channels of version 1.0, a new spectral certificate for the root groups of their presentation gives explicit gaps: 2^-(2J+41) for the fifteen-qubit channel and its two variants and, by the bound of version 1.0, 2^-(2J+36) in Choi distance and 2^-(2J+21) in diamond distance for the fully referenced channel. The numbers are tiny because Wang and Zhi's tolerance is 2^-J. The results depend on recent preprints of Wang and Zhi, Thom, and Alekseev, Liu and Thom, which are not yet refereed. This archive includes the paper, the channel data, the certificates, and programs that rebuild and check them; python verify_all.py ends with OK ALL.