The Infimum Norm of Completely Positive Maps

Let A  be a unital C* -algebra, let L: A→B(H)  be a linear map, and let ∅: A→B(H)  be a completely positive linear map. We prove the property in the following:  is completely positive}=inf {||T*T+TT*||1/2:  L= V*TπV  which is a minimal commutant representation with isometry} . Moreover, if L=L* , then  is completely positive  . In the paper we also extend the result  is completely positive}=inf{||T||: L=V*TπV}  [3 , Corollary 3.12].

The operator system of Toeplitz matrices

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of n × n n\times n Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than n n . The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the n × n n\times n complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the n × n n\times n Toeplitz matrices into the algebra of all n × n n\times n complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of n × n n\times n complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix ξ n \xi _n generates an extremal ray in the cone of all continuous n × n n\times n Toeplitz-matrix valued functions f f on the unit circle S 1 S^1 whose Fourier coefficients f ^ ( k ) \hat f(k) vanish for | k | ≥ n |k|\geq n . Lastly, it is noted that all positive Toeplitz matrices over nuclear C ∗ ^* -algebras are approximately separable.

Indecomposable positive maps on positive semidefinite matrices from Mn to Mn+1

In this paper we obtain a theorem for 2-positive linear maps from Mn(C) to Mn+1(C), where n = 2, 3, 4. In addition, we answer in the affirmative a question that asked if there exists every 2-positive linear map from M3(C) to M4(C) is indecomposable using a family of positive linear maps with Choi matrices of 2-positive maps on positive semidefinite matrices. Further it is shown that 2-positive linear map from M4(C) to M5(C) are indecomposable.

VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

Inequalities for sector matrices and positive linear maps

Ando proved that if A, B are positive definite, then for any positive linear map Φ, it holds Φ(A#λB) ≤ Φ(A)#λΦ(B), where A#λB, 0 ≤ λ ≤ 1, means the weighted geometric mean of A, B. Using the recently defined geometric mean for accretive matrices, Ando’s result is extended to sector matrices. Some norm inequalities are considered as well.

Some refinements of young type inequality for positive linear map

We obtain a refined Young type inequality in this paper. The conclusion is presented as follows: Let A, B ∈ B(𝓗) be two positive operators and p ∈ [0, 1], then $$\begin{array}{} \displaystyle A\sharp_p B+G^*(A\sharp_p B)G\le A\nabla_p B-2r(A\nabla B-A\sharp B), \end{array}$$ where r = min{p, 1 – p}, G = $\begin{array}{} \displaystyle \frac{\sqrt{L(2p)}}{2} \end{array}$ A–1S(A|B), L(t) is periodic with period one and L(t) = $\begin{array}{} \displaystyle \frac{t^2}{2}\left( \frac{1-t}{t} \right)^{2t} \end{array}$ for t ∈ [0, 1]. Moreover, we give the s-th powering of two inequalities related to the above one with s > 0 which refines Lin’s work. In the mean time, we present an inequality involving Hilbert-Schmidt norm.

The Non-m-Positive Dimension of a Positive Linear Map

We introduce a property of a matrix-valued linear map Φ that we call its ``non-m-positive dimension'' (or ``non-mP dimension'' for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of Im⊗Φ. Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map Im⊗Φ∗ can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good Φ is at detecting entanglement in quantum states. We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

Positive linear maps on normal matrices

For a positive linear map [Formula: see text] and a normal matrix [Formula: see text], we show that [Formula: see text] is bounded by some simple linear combinations in the unitary orbit of [Formula: see text]. Several elegant sharp inequalities are derived, for instance for the Schur product of two normal matrices [Formula: see text], [Formula: see text] for some unitary [Formula: see text], where the constant [Formula: see text] is optimal.

An Operator Extension of Čebyšev Inequality

Some operator inequalities for synchronous functions that are related to the čebyšev inequality are given. Among other inequalities for synchronous functions it is shown that ∥ø(f(A)g(A)) - ø(f(A))ø(g(A))∥ ≤ max{║ø(f2(A)) - ø2(f(A))║, ║ø)G2(A)) - ø2(g(A))║} where A is a self-adjoint and compact operator on B(ℋ ), f, g ∈ C (sp (A)) continuous and non-negative functions and ø: B(ℋ ) → B(ℋ ) be a n-normalized bounded positive linear map. In addition, by using the concept of quadruple D-synchronous functions which is generalizes the concept of a pair of synchronous functions, we establish an inequality similar to čebyšev inequality.