Unitary Orbit Research Articles

Preservers of restricted unitary orbits of projections

Given a pure state (rank-1 self-adjoint projection) P and a group G of unitary operators acting on a complex Hilbert space, let the restricted unitary orbit of P (relative to G) be UG(P)={UPU−1:U∈G}. We study linear transformations T which map UG(P) into pure states, or more stringently, back into UG(P). A complete characterization is obtained when G is an abelian group of unitary matrices equal to its bicommutant.

Convex hulls of unitary orbits of normal elements in C⁎-algebras with tracial rank zero

Let A be a unital separable simple C⁎-algebra with tracial rank zero and let x,y∈A be two normal elements. We show that x is in the closure of the convex hull of the unitary orbit of y if and only if there exists a sequence of unital completely positive linear maps φn from A to A such that the sequence φn(y) converges to x in norm and also approximately preserves the trace values. A purely measure theoretical description for normal elements in the closure of the convex hull of unitary orbit of y is also given. In the case that A has a unique tracial state some classical results about the closure of the convex hull of the unitary orbits in von Neumann algebras are proved to hold in the C⁎-algebraic setting.

‘The C-numerical range in infinite dimensions’

We generalize the $C$-numerical range $W_C(T)$ from trace-class to Schatten-class operators, i.e. to $C\in\mathcal B^p(\mathcal H)$ and $T\in\mathcal B^q(\mathcal H)$ with $1/p + 1/q = 1$, and show that its closure is always star-shaped with respect to the origin. For $q \in (1,\infty]$, this is equivalent to saying that the closure of the image of the unitary orbit of $T\in\mathcal B^q(\mathcal H)$ under any continous linear functional $L\in(\mathcal B^q(\mathcal H))'$ is star-shaped with respect to the origin. For $q=1$, one has star-shapedness with respect to $\operatorname{tr}(T)W_e(L)$, where $W_e(L)$ denotes the essential range of $L$. Moreover, the closure of $W_C(T)$ is convex if $C$ or $T$ is normal with collinear eigenvalues. If $C$ and $T$ are both normal, then the $C$-spectrum of $T$ is a subset of the $C$-numerical range, which itself is a subset of the closure of the convex hull of the $C$-spectrum. This closure coincides with the closure of the $C$-numerical range if, in addition, the eigenvalues of $C$ or $T$ are collinear.

Procrustes problems in Riemannian manifolds of positive definite matrices

We consider the manifold of positive definite matrices endowed with the Fisher Riemannian metric and some other distances commonly used in information theory. We show that for each of them the best approximant to A from the unitary orbit of another matrix B commutes with A.

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.

Volume of the set of locally diagonalizable bipartite states

The purpose of this article is to investigate the geometry of the set of locally diagonalizable bipartite quantum states. We have the following new results: the Hilbert–Schmidt volume of all locally diagonalizable states, and a necessary and sufficient condition for local diagonalizability in the qubit-qubit case. Besides, we partition the set of all locally diagonalizable states as local unitary orbits (or coadjoint orbits) of diagonal forms. It is well-known that the Riemannian volume of a coadjoint orbit for a regular point in a specified Weyl chamber can be calculated by Harish–Chandra’s volume formula. By modifying Harish–Chandra’s volume formula, we give, for the first time, a specific formula for the Riemannian volume of a local unitary orbit of a regular point in a specified Weyl chamber. Several open questions are presented as well.

Majorization in C*-algebras

We investigate the closed convex hull of unitary orbits of selfadjoint elements in arbitrary unital C*-algebras. Using a notion of majorization against unbounded traces, a characterization of these closed convex hulls is obtained. Furthermore, for C*-algebras satisfying Blackadar’s strict comparison of positive elements by traces or for collections of C*-algebras with a uniform bound on their nuclear dimension, an upper bound for the number of unitary conjugates in a convex combination required to approximate an element in the closed convex hull within a given error is shown to exist. This property, however, fails for certain “badly behaved” simple nuclear C*-algebras.

Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C* -algebras

AbstractIn this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C* -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

A Complete Set of Invariants for LU-Equivalence of Density Operators

We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unitary equivalence. This is done by showing that local unitary equivalence of density operators is equivalent to local ${\rm GL}$ equivalence and then using techniques from algebraic geometry and geometric invariant theory. We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of $n$-qubit pure states. Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.

Average entropy of a subsystem over a global unitary orbit of a mixed bipartite state

We investigate the average entropy of a subsystem within a global unitary orbit of a given mixed bipartite state in the finite-dimensional space. Without working out the closed-form expression of such average entropy for the mixed state case, we provide an analytical lower bound for this average entropy. In deriving this analytical lower bound, we get some useful by-products of independent interest. We also apply these results to estimate average correlation along a global unitary orbit of a given mixed bipartite state. When the notion of von Neumann entropy is replaced by linear entropy, the similar problem can be considered also, and moreover the exact average linear entropy formula is derived for a subsystem over a global unitary orbit of a mixed bipartite state.

Unitary subgroups and orbits of compact self-adjoint operators

Fil: Bottazzi, Tamara Paula. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica Alberto Calderon; Argentina

Linear images of joint unitary orbits of Hermitian matrices

Let Hn be the set of all n×n Hermitian matrices and Hnm be the set of all m-tuples of n×n Hermitian matrices. For A=(A1,...,Am)∈Hnm and for any linear map L:Hnm→Rℓ, we define the L-numerical range and the L-numerical radius of A byWL(A):={L(U⁎A1U,...,U⁎AmU):U∈Cn×n,U⁎U=In} andrL(A):=max⁡{xxt:x∈WL(A)}, respectively. They can be regarded as a generalization of the C-numerical range and the C-numerical radius, respectively, of n×n complex matrices.In this paper, interplay between the geometric properties of WL(A) and algebraic properties of linear maps L:Hnm→Rℓ and A∈Hnm will be studied. We shall give a characterization of L such that rL(⋅) defines a norm on Hnm. Furthermore, we shall also characterize linear maps L:Hnm→Rℓ and A∈Hnm in which WL(A) is a singleton set. More generally, the characterization of linear maps L:Hnm→Rℓ and A∈Hnm in which WL(A) lies inside a k-dimensional affine subspace of Rℓ will be given.

Amenability and representation theory of pro-Lie groups

We develop a semigroup approach to representation theory for pro-Lie groups satisfying suitable amenability conditions. As an application of our approach, we establish a one-to-one correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits for a class of pro-Lie groups including all connected locally compact nilpotent groups and arbitrary infinite direct products of nilpotent Lie groups. The usual \(C^*\)-algebraic approach to group representation theory positively breaks down for infinite direct products of non-compact locally compact groups, hence the description of their unitary duals in terms of coadjoint orbits is particularly important whenever it is available, being the only description known so far.

Closed Convex Hulls of Unitary Orbits in $${C^{*}_{r}(\mathbb{F}_{\infty})}$$ C r ∗ ( F ∞ )

Let $${C^*_r(\mathbb{F}_{\infty})}$$ be the reduced C*-algebra of the free group on infinitely many generators. Say that $${a, b \in C^*_r(\mathbb{F}_{\infty})_{SA}}$$ . Then $${a}$$ is majorized by $${b}$$ if and only if $${a \in \overline{Conv(U(b))}.}$$ In particular, $${\tau(b)1 \in \overline{Conv(U(b))}.}$$ Moreover, in the above results, we provide uniform bounds for the number of unitary conjugates needed for a given approximation. In the above, $${Conv(U(b))}$$ is the convex hull of the unitary orbit of $${b}$$ in $${C^*_r(\mathbb{F}_{\infty})}$$ .

Quantum speed limit for mixed states using an experimentally realizable metric

Here, we introduce a new metric for non-degenerate density operator evolving along unitary orbit and show that this is experimentally realizable operation dependent metric on the quantum state space. Using this metric, we obtain the geometric uncertainty relation that leads to a new quantum speed limit (QSL). We also obtain a Margolus–Levitin bound and an improved Chau bound for mixed states. We propose how to measure this new distance and speed limit in quantum interferometry. Finally, we also generalize the QSL for completely positive trace preserving evolutions.

Minimal length curves in unitary orbits of a Hermitian compact operator

We study some examples of minimal length curves in homogeneous spaces of B(H) under a left action of a unitary group. Recent results relate these curves with the existence of minimal (with respect to a quotient norm) anti-Hermitian operators Z in the tangent space of the starting point. We show minimal curves that are not of this type but nevertheless can be approximated uniformly by those.

Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

In this paper, we study closed convex hulls of unitary orbits in various C$^*$-algebras. For unital C$^*$-algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C$^*$-algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C$^*$-algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and have a unique tracial state. Also, closed convex hulls of unitary orbits of self-adjoint operators are fully described in unital, simple, purely infinite C$^*$-algebras.

Unitary orbits of self-adjoint operators in simple Z-stable C⁎-algebras

We prove that in a simple, unital, exact, Z-stable C⁎-algebra, the distance between the unitary orbits of self-adjoint elements with connected spectrum is completely determined by spectral data. This fails without the assumption of Z-stability.

Unitary orbits in 1-dimensional NCCW complexes

Let A be a 1-dimensional NCCW complex. In this paper, it is shown that the usual distance d U defined on the approximate unitary equivalence classes (unitary orbits) of self-adjoints in A is equal to a distance d W defined on morphisms from the Cuntz semigroup of C 0 ( 0 , 1 ] to Cu ( A ) . In addition, another distance d P on approximately unitary equivalence classes related to spectral information of positive elements in the classes is proved to be equal to d U and d W on simple unital inductive limits of 1-dimensional NCCW complexes.

Pinchings and positive linear maps

We employ the pinching theorem, ensuring that some operators A admit any sequence of contractions as an operator diagonal of A, to deduce/improve two recent theorems of Kennedy–Skoufranis and Loreaux–Weiss for conditional expectations onto a masa in the algebra of operators on a Hilbert space. Similarly, we obtain a proof of a theorem of Akeman and Anderson showing that positive contractions in a continuous masa can be lifted to a projection. We also discuss a few corollaries for sums of two operators in the same unitary orbit.