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].

A factorization property of positive maps on C*-algebras

The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.

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.

Lower bound of concurrence based on D-type positive map

In this paper, by using the D-type map, we provide an analytical lower bound of concurrence of arbitrary dimensional bipartite quantum system. It is shown that this lower bound is able to improve the existing bounds and detect entanglement better.