Trace Zero Research Articles

A note on a conjecture from distillability of quantum entanglement

A conjecture from the distillability of quantum entanglement is that when A and B are 4×4 trace zero complex matrices and ‖A‖2+‖B‖2=1/4 (where ‖⋅‖ is the Frobenius norm), the sum of squares of the largest two singular values of A⊗I4+I4⊗B does not exceed 1/2. In this paper, the conjecture is proved when(i)A or B is unitarily similar to a direct sum of 2×2 trace zero matrices;(ii)A and B are unitarily similar to matrices, when partitioned into 2×2 blocks, having zero diagonal blocks.

A stronger version of Dixmier's averaging theorem and some applications

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given finite elements X1,⋯,Xn∈M, there is a finite decomposition of the identity into integer N∈N mutually orthogonal nonzero projections Ej∈M, I=∑j=1NEj, such that EjXiEj=τ(Xi)Ej for all j=1,⋯,N and i=1,⋯,n. Equivalently, there is a unitary operator U∈M such that 1N∑j=0N−1U⁎jXiUj=τ(Xi)I for i=1,⋯,n. This result is a stronger version of Dixmier's averaging theorem for type II1 factors. As the first application, we show that all elements of trace zero in a type II1 factor are single commutators and any self-adjoint elements of trace zero are single self-commutators. This result answers affirmatively Question 1.1 in [6]. As the second application, we prove that any self-adjoint element in a type II1 factor can be written a linear combination of 4 projections. This result answers affirmatively Question 6(2) in [12]. As the third application, we show that if (M,τ) is a finite factor, X∈M, then there exists a normal operator N∈M and a nilpotent operator K such that X=N+K.

Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups

We generalize the result of Gorini, Kossakowski, and Sudarshan [J. Math. Phys. 17:821, 1976] that every generator of a quantum-dynamical semigroup decomposes uniquely into a closed and a dissipative part, assuming the trace of both vanishes. More precisely, we show that given any generator [Formula: see text] of a completely positive dynamical semigroup and any matrix [Formula: see text] there exists a unique matrix [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] has trace zero, and (iii) [Formula: see text] is a real number. The key to proving this is the relation between the trace of a completely positive map, the trace of its Kraus operators, and expectation values of its Choi matrix. Moreover, we show that the above decomposition is orthogonal with respect to some [Formula: see text]-weighted inner product.

On regular Lie algebras

We study so called regular Lie algebras, i.e., Lie algebras in which each nonzero element is regular. We make a connection with an open problem whether any element of reduced trace zero in a simple associative algebra is a commutator.

Trace-free SL(2, ℂ)-representations of Montesinos links

Given a link [Formula: see text], a representation [Formula: see text] is trace-free if the image of each meridian has trace zero. We determine the conjugacy classes of trace-free representations when [Formula: see text] is a Montesinos link.

Formula omitted] norm preserving flow in matrix geometry

In this paper, we study a new L2 norm preserving heat flow in matrix geometry. We show that if the initial data has trace zero and has unit L2 norm, this flow has a global solution and enjoys the entropy stability in any finite time. We show that as the time is approaching infinity, the flow has its limit as an eigen-matrix of the Laplacian operator. Interesting operator convex property of heat equation is also derived.

Data Processing in Square Zero Preservers on Spaces of 2x2 Trace Zero Matrices

Scientific data processing "usually involves a great deal of computation (arithmetic and comparison operations) upon a relatively small amount of input data, resulting in a small volume of output." In this view, matrices play very important role.Suppose F is a Field of Characteristic Not 2, and F* is its Subset of all Nonzero Elements. Let M2 be the Space of all 2*2 full matrices over F . A matrix A M2 is called square zero if A2=0 . Let be the Subset of Consisting of all Square Zero Matrices and be the Linear Space of all Trace Zero Matrices over . we Denote by the Set of all Maps from to itself Satisfying if and only if for Every and . it was Shown that if and only if there Exists an Invertible Matrix and such that for Every .

On single commutators in II$_{1}$–factors

We investigate the question of whether all elements of trace zero in a II$_1$–factor are single commutators. We show that all nilpotent elements are single commutators, as are all normal elements of trace zero whose spectral distributions are discrete measures. Some other classes of examples are considered.

On the representation of elements of a von Neumann algebra in the form of finite sums of products of projections. III. Commutators in C*-algebras

It is proved that every skew-Hermitian element of any properly infinite von Neumann algebra can be represented in the form of a finite sum of commutators of projections in this algebra. A new commutation condition for projections in terms of their upper (lower) bound in the lattice of all projections of the algebra is obtained. For the full matrix algebra the set of operators with canonical trace zero is described in terms of finite sums of commutators of projections and the domain in which the trace is positive is described in terms of finite sums of pairwise products of projections. Applications to AF-algebras are obtained.Bibliography: 33 titles.

Commutator rings

A ring is called a commutator ring if every element is a sum of additive commutators. In this note we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a nonempty set Ω, EndR(⌖ΩN) and EndR(ΠΩN) are commutator rings if and only if either Ω is infinite or EndR(N) is itself a commutator ring. We also prove that over any ring, a matrix having trace zero can be expressed as a sum of two commutators.

Galois cohomology of fields without roots of unity

There is a standard correspondence between elements of the cohomology group H 1( F, μ n ) (with the trivial action of Γ F =Gal( F s/ F) on μ n ) and cyclic extensions of dimension n over F. We extend this to a correspondence between the cohomology groups H 1( F, μ n ) where the action of Γ F on μ n varies, and the extensions of dimension n of K which are Galois over F, where K= F[ μ n ] and [ K: F] is prime to n. The cohomology groups are also related to eigenspaces of H 1(K, Z/n) with respect to the natural action of Gal( K/ F). As a result, we extend Albert's cyclicity criterion, stated in the 1930s for division algebras of prime degree, to algebras of prime-power degree over F, under the assumption stated above. We also extend the Rosset–Tate result on the corestriction of cyclic algebras in the presence of roots of unity, to extensions in which roots of unity live in an extension of dimension ⩽3 over the base field. In particular, if roots of unity live in a quadratic extension of the base field, then corestriction of a cyclic algebra along a quadratic extension is similar to a product of two cyclic algebras. Another application is that F-central algebras which are split by a certain semidirect product extension of F, are cyclic. In particular, if [ K: F]=2 then algebras over F which are split by an odd order dihedral extension, are cyclic. We also construct generic examples of algebras which become cyclic after extending scalars by roots of unity, and show the existence of elements for which most powers have reduced trace zero.

Some Trace Class Commutators of Trace Zero

It is shown that if $T$ is an operator on a separable complex Hilbert space and $X$ is a Hilbert-Schmidt operator such that $TX - XT$ is a trace class operator, then the trace of $TX - XT$ is zero provided one of the two conditions holds: (a) ${T^2}$ is normal; (b) ${T^n}$ is normal for some integer $n > 2$ and ${T^*}T - T{T^*}$ is a trace class operator. Related results involving essentially unitary operators and Cesà ro operators are also given.

Vector potential and Riemannian space

This paper uncovers the basic reason for the mysterious change of sign from plus to minus in the fourth coordinate of nature's Pythagorean law, usually accepted on empirical grounds, although it destroys the rational basis of a Riemannian geometry. Here we assume a genuine, positive-definite Riemannian space and an action principle which is quadratic in the curvature quantities (and thus scale invariant). The constant σ between the two basic invariants is equated to1/2. Then the matter tensor has the trace zero. In consequence of the constancy of the scalar curvature and the divergence identity of the matter tensor, the perturbation metric has to satisfy a scalar and a vector condition, with a negative sign in the fourth coordinate. These conditions lead to the Lorentz condition and the wave equation for the vector potential. Thus the entire Maxwell-Lorentz type of electrodynamics becomes logically derivable, making no concession to any irrationality.

Associators in simple algebras

In this paper it is shown that, with suitable hypotheses on the base field, any element of generic trace zero in an octonion algebra is a commutator and an associator, and any element of generic trace zero in a simple Jordan algebra is an associator.

Commutators and certain II 1-factors

Recently some progress has been made in the structure theory of commutators in von Neumann algebras [I]-[3]. The theory is far from complete, however, and one of the most intractable of the unsolved problems, is that of determining the commutators in a finite von Neumann algebra. A commutator in a finite von Neumann algebra must, of course, have central trace zero, and is not unreasonable to hope that the commutators in such an algebra are exactly the operators with central trace zero. However, despite considerable effort, this has been proved only in case the algebra is a finite direct sum of algebras of type I, [4]. In this note, we consider a certain class of factors of type II, discovered by Wright [13], and we show that every Hermitian operator with trace zero in such a factor is a commutator in the factor. This is accomplished by first proving that every Hermitian operator with central trace zero in an arbitrary finite von Neumann algebra of type I is a commutator in the algebra. Finally, we turn our attention to the problem of characterizing the linear manifold [GY, ac] p s anned by the commutators in an arbitrary von Neumann algebra 67 of type II, . We give three characterizations; in particular we show that [G?, ol] coincides with the set of all linear combinations C& ariEi where Cbi Qi = 0 and each E4 is equivalent in GZ to I Ei .

Matrices with Trace Zero

Matrices with Trace Zero