Complete Positivity of Two Class of Maps Involving Depolarizing and Transpose-Depolarizing Channels

In this note, we mainly study the necessary and sufficient conditions for the complete positivity of generalizations of depolarizing and transpose-depolarizing channels. Specifically, we define [Formula: see text] and [Formula: see text], where [Formula: see text] (the set of all bounded linear operators on the finite-dimensional Hilbert space [Formula: see text] is given and [Formula: see text] is the transpose of [Formula: see text] in a fixed orthonormal basis of [Formula: see text] First, we show that [Formula: see text] is completely positive if and only if [Formula: see text] is a positive map, which is equivalent to [Formula: see text] Moreover, [Formula: see text] is a completely positive map if and only if [Formula: see text] and [Formula: see text] At last, we also get that [Formula: see text] is a completely positive map if and only if [Formula: see text] with [Formula: see text] for all [Formula: see text] where [Formula: see text] are eigenvalues of [Formula: see text].

Quantum Weak Invariants: Dynamical Evolution of Fluctuations and Correlations

Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and Rényi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in cases where the system density matrix obeys the Gorini–Kossakowski–Lindblad–Sudarshan equation.

Stinespring's construction as an adjunction

Given a representation of a unital C∗-algebra A on a Hilbert space H, together with a bounded linear map V:K→H from some other Hilbert space, one obtains a completely positive map on A via restriction using the adjoint action associated to V. We show this restriction forms a natural transformation from a functor of C∗-algebra representations to a functor of completely positive maps. We exhibit Stinespring's construction as a left adjoint of this restriction. Our Stinespring adjunction provides a universal property associated to minimal Stinespring dilations and morphisms of Stinespring dilations. We use these results to prove the purification postulate for all finite-dimensional C∗-algebras.

Weak Log-majorization of Unital Trace-preserving Completely Positive Maps

Let Φ : Mn → Mn be a unital trace preserving completely positive map and A ∈ Mn be a positive definite matrix. Weak log-majorization and weak majorization between Φ(A) and A are studied. Determinantal inequalities between Φ(A) and A are obtained as a consequence. By considering special classes of unital trace preserving completely positive map, some known matrix inequalities such as Fischer’s inequality are rediscovered. An affirmative answer to a question of Tam and Zhang in 2019 is given.

CUNTZ–KRIEGER ALGEBRAS AND ONE-SIDED CONJUGACY OF SHIFTS OF FINITE TYPE AND THEIR GROUPOIDS

A one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with a distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and a certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

Infinite dimensional generalizations of Choi’s Theorem

In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

Quantum collections

We develop the viewpoint that [Formula: see text], the opposite of the category of [Formula: see text]-algebras and unital normal ∗-homomorphisms, is analogous to the category of sets and functions. For each pair of [Formula: see text]-algebras [Formula: see text] and [Formula: see text], we construct their free exponential [Formula: see text], which in the context of this analogy corresponds to the collection of functions from [Formula: see text] to [Formula: see text]. We also show that every unital normal completely positive map [Formula: see text] arises naturally from a normal state on [Formula: see text].

When do pieces determine the whole? Extreme marginals of a completely positive map

We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extreme point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extreme, then the whole map is extreme. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.