We give a useful new characterization of the set of all completely positive, trace-preserving (i.e., stochastic) maps from 2x2 matrices to 2x2 matrices. These conditions allow one to easily check any trace-preserving map for complete positivity. We also determine explicitly all extreme points of this set, and give a useful parameterization after reduction to a certain canonical form. This allows a detailed examination of an important class of non-unital extreme points which can be characterized as having exactly two images on the Bloch sphere.
We also discuss a number of related issues about the images and the geometry of the set of stochastic maps, and show that any stochastic map on 2x2 matrices can be written as a convex combination of two "generalized" extreme points.
Positive maps are useful for detecting entanglement in quantum information theory. Any entangled state can be detected by some positive map. In this paper, the relation between positive block matrices and completely positive trace-preserving maps is characterized. Consequently, a new method for constructing decomposable maps from positive block matrices is derived. In addition, a method for constructing positive but not completely positive maps from Hermitian block matrices is also obtained.
Let $D$ be a space of $2\times n$ matrices. Then the face of the cone of all completely positive maps from $M_2$ into $M_n$ given by $D$ is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in $D$ forms a subspace of $D$ together with zero and $D^\perp$ is spanned by rank one matrices.
We introduce a generalization of the set of completely positive matrices that we call "pairwise completely positive" (PCP) matrices. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. We explore basic properties of these matrix pairs and develop several testable necessary and sufficient conditions that help determine whether or not a pair is PCP. We then establish a connection with quantum entanglement by showing that determining whether or not a pair of matrices is pai…
We present an algorithm for projecting superoperators onto the set of completely positive, trace-preserving maps. When combined with gradient descent of a cost function, the procedure results in an algorithm for quantum process tomography: finding the quantum process that best fits a set of sufficient observations. We compare the performance of our algorithm to the diluted iterative algorithm as well as second-order solvers interfaced with the popular CVX package for MATLAB, and find it to be significantly faster and more accurate while guaranteeing a physical estimate.
We show that, for SU(2) generators of arbitrary dimension $D$, there exist identities that express the completely symmetric product of $D$ matrices in terms of completely symmetric products of fewer number of matrices. We also indicate why such identities are important in characterizing electromagnetic interactions of particles.
We consider open quantum systems with factorized initial states, providing the structure of the reduced system dynamics, in terms of environment cumulants. We show that such completely positive (CP) and trace preserving (TP) maps can be unraveled by linear stochastic Schrödinger equations (SSEs) characterized by sets of colored stochastic processes (with n-th order cumulants). We obtain both the conditions such that the SSEs provide CPTP dynamics, and those for unraveling an open system dynamics. We then focus on Gaussian non-Markovian unravellings, whose known structure displays a functional …
We prove the linearity and injectivity of two maps $ϕ_1$ and $ϕ_2$ on certain subsets of $M_n$ that satisfy $\operatorname{tr}(ϕ_1(A)ϕ_2(B))=\operatorname{tr}(AB)$. We apply it to characterize maps $ϕ_i:\mathcal{S}\to \mathcal{S}$ ($i=1, \ldots, m$) satisfying $$\operatorname{tr} (ϕ_1(A_1)\cdots ϕ_m(A_m))=\operatorname{tr} (A_1\cdots A_m)$$ in which $\mathcal{S}$ is the set of $n$-by-$n$ general, Hermitian, or symmetric matrices for $m\ge 3$, or positive definite or diagonal matrices for $m\ge 2$. The real versions are also given.
We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M_N of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace-nonincreasing completely positive maps belong to the N-cube inscribed in the set of subnormalized states of size N. As a by-product we derive the measure in M_N induced by partial trace of mixed quantum states dist…
We investigate the completely positive semidefinite cone $\mathcal{CS}_+^n$, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular we study relationships between this cone and the completely positive and doubly nonnegative cones, and between its dual cone and trace positive non-commutative polynomials. We use this new cone to model quantum analogues of the classical independence and chromatic graph parameters $α(G)$ and $χ(G)$, which are roughly obtained by allowing variables to be positive semid…
We investigate structural properties of the completely positive semidefinite cone $\mathcal{CS}_+^n$, consisting of all the $n \times n$ symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set $\mathcal Q$ of bipartite quantum correlations, as projection of an affine section of it. We have two main results concerning the structure of the completely positive semidefinite cone, namely about its interior a…
Taking partial traces for computing reduced density matrices, or related functions, is a ubiquitous procedure in the quantum mechanics of composite systems. In this article, we present a thorough description of this function and analyze the number of elementary operations (ops) needed, under some possible alternative implementations, to compute it on a classical computer. As we notice, it is worthwhile doing some analytical developments in order to avoid making null multiplications and sums, what can considerably reduce the ops. For instance, for a bipartite system $\mathcal{H}_{a}\otimes\math…
Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method. We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map (the latter happens to be unitary but the former is not). In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers. In the case of the baker map it is shown that nonunitarity is restricted to a one-qubit operator (close to a Hadamard gate for some p…
We develop numerous results that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, we develop a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices. We then explore applications of our work in the theory of quantum resources. For example, we characterize exactly which quantum states have modified trace distance of coherence equal t…
The action of any lossless multilayer is described by a transfer matrix that can be factorized in terms of three basic matrices. We introduce a simple trace criterion that classifies multilayers in three classes with properties closely related with one (and only one) of these three basic matrices.
For quantum systems described by finite matrices, linear and affine maps of matrices are shown to provide equivalent descriptions of evolution of density matrices for a subsystem caused by unitary Hamiltonian evolution in a larger system; an affine map can be replaced by a linear map, and a linear map can be replaced by an affine map. There may be significant advantage in using an affine map. The linear map is generally not completely positive, but the linear part of an equivalent affine map can be chosen to be completely positive and related in the simplest possible way to the unitary Hamilto…
The completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them.
The generic linear evolution of the density matrix of a system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of hermiticity of density matrices renders an associated supermatrix hermitian and hence diagonalizable; but the positivity of the density matrix does not make this associated supermatrix positive. If it is positive, the map is called completely positive and they have a simple parametrization. This…
The completely bounded trace and spectral norms, for finite-dimensional spaces, are known to be efficiently expressible by semidefinite programs (J. Watrous, Theory of Computing 5: 11, 2009). This paper presents two new, and arguably much simpler, semidefinite programming formulations of these norms.
Linear maps of matrices describing evolution of density matrices for a quantum system initially entangled with another are identified and found to be not always completely positive. They can even map a positive matrix to a matrix that is not positive, unless we restrict the domain on which the map acts. Nevertheless, their form is similar to that of completely positive maps. Only some minus signs are inserted in the operator-sum representation. Each map is the difference of two completely positive maps. The maps are first obtained as maps of mean values and then as maps of basis matrices. Thes…
