Unitary Orbit Research Articles

Optimization of some types of Rényi divergences between unitary orbits

In this article, we investigate explicitly the minimal and maximal values of the α-z-Rényi divergences and some other types of Rényi divergences between unitary orbits. Our main tools are the majorization theory of eigenvalues of matrices, and some matrix inequalities such as Golden–Thompson's and Araki–Lieb–Thirring's inequalities.

An approximate unique extension property for completely positive maps

We study the closure of the unitary orbit of a given point in the non-commutative Choquet boundary of a unital operator space with respect to the topology of pointwise norm convergence. This may be described more extensively as the ⁎-representations of the C⁎-envelope that are approximately unitarily equivalent to one that possesses the unique extension property. Although these ⁎-representations do not necessarily have the unique extension property themselves, we show that their unital completely positive extensions display significant restrictions. When the underlying operator space is separable, this allows us to connect our work to Arveson's hyperrigidity conjecture. Finally, as an application, we reformulate the classical Šaškin Theorem and Arveson's essential normality conjecture.

Maximum entanglement of mixed symmetric states under unitary transformations

We study the maximum entanglement that can be produced by a global unitary transformation for systems of two and three qubits constrained to the fully symmetric states. This restriction to the symmetric subspace appears naturally in the context of bosonic or collective spin systems. We also study the symmetric states that remain separable after any global unitary transformation, called symmetric absolutely separable (SAS) states, or absolutely classical for spin states. The results for the two-qubit system are deduced analytically. In particular, we determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that contains the set of SAS states. As an application of our results, we also analyse the temperature dependence of the maximum entanglement that can be obtained from the thermal state of a spin-1 system with a spin-squeezing Hamiltonian. For the symmetric three-qubit case, our results are mostly numerical, and we conjecture a 3-parameter family of states that achieves the maximum negativity in the unitary orbit of any mixed state. In addition, we derive upper bounds, apparently tight, on the radii of balls containing only/all SAS states.

Optimization of quantum divergences between unitary orbits

Optimization of quantum divergences between unitary orbits

Distance Between Unitary Orbits of Self-Adjoint Elements in C*-Algebras of Tracial Rank One

Distance Between Unitary Orbits of Self-Adjoint Elements in C*-Algebras of Tracial Rank One

Metrics on trace spaces

This article continues the investigation of the tracial geometry of classifiable C*-algebras that have real rank zero and stable rank one. Using the language of optimal transport, we describe several situations in which the distance between unitary orbits of ⁎-homomorphisms into such algebras can be computed in terms of tracial data. The domains we consider are certain (noncommutative) CW complexes, and the measurement is relative to a family of self-adjoint elements that are in a suitable sense tracially Lipschitz. As another application of the utility of this Lipschitz structure, we show how such elements can be repurposed to witness statistical features of endomorphisms in the classifiable category, in particular the tracial version of the (almost-sure) central limit theorem.

Unitary Group Orbits Versus Groupoid Orbits of Normal Operators

We study the unitary orbit of a normal operator $$a\in {{\mathcal {B}}}({{\mathcal {H}}})$$ , regarded as a homogeneous space for the action of unitary groups associated with symmetrically normed ideals of compact operators. We show with an unified treatment that the orbit is a submanifold of the various ambient spaces if and only if the spectrum of a is finite, and in that case, it is a closed submanifold. For arithmetically mean closed ideals, we show that nevertheless the orbit always has a natural manifold structure, modeled by the kernel of a suitable conditional expectation. When the spectrum of a is not finite, we describe the closure of the orbits of a for the different norm topologies involved. We relate these results to the action of the groupoid of partial isometries via the moment map given by the range projection of normal operators. We show that all these groupoid orbits also have differentiable structures for which the target map is a smooth submersion. For any normal operator a, we also describe the norm closure of its groupoid orbit $${{\mathcal {O}}}_a$$ , which leads to necessary and sufficient spectral conditions on a ensuring that $${{\mathcal {O}}}_a$$ is norm closed and that $${{\mathcal {O}}}_a$$ is a closed embedded submanifold of $${{\mathcal {B}}}({{\mathcal {H}}})$$ .

A proof of the polynomial conjecture for restrictions of nilpotent lie groups representations

Let G G be a connected and simply connected nilpotent Lie group, K K an analytic subgroup of G G and π \pi an irreducible unitary representation of G G whose coadjoint orbit of G G is denoted by Ω ( π ) \Omega (\pi ) . Let U ( g ) \mathscr U(\mathfrak g) be the enveloping algebra of g C {\mathfrak g}_{\mathbb C} , g \mathfrak g designating the Lie algebra of G G . We consider the algebra D π ( G ) K ≃ ( U ( g ) / ker ⁡ ( π ) ) K D_{\pi }(G)^K \simeq \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )\right )^K of the K K -invariant elements of U ( g ) / ker ⁡ ( π ) \mathscr U(\mathfrak g)/\operatorname {ker}(\pi ) . It turns out that this algebra is commutative if and only if the restriction π | K \pi |_K of π \pi to K K has finite multiplicities (cf. Baklouti and Fujiwara [J. Math. Pures Appl. (9) 83 (2004), pp. 137-161]). In this article we suppose this eventuality and we provide a proof of the polynomial conjecture asserting that D π ( G ) K D_{\pi }(G)^K is isomorphic to the algebra C [ Ω ( π ) ] K \mathbb C[\Omega (\pi )]^K of K K -invariant polynomial functions on Ω ( π ) \Omega (\pi ) . The conjecture was partially solved in our previous works (Baklouti, Fujiwara, and Ludwig [Bull. Sci. Math. 129 (2005), pp. 187-209]; J. Lie Theory 29 (2019), pp. 311-341).

Parameterizing qudit states

Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an NN-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary SU(N){SU(N)}-invariant counterpart of the NN-level state space BN{\mathcal{B}_N}, i.e., the unitary orbit space BN/SU(N){B_N/SU(N)}. It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of BN/SU(N){B_N/SU(N)}. To illustrate the general situation, a detailed description ofBN/SU(N){B_N/SU(N)} for low-level systems: qubit (N=2{N= 2}), qutrit (N=3{N=3}), quatrit (N=4{N= 4}) - will be given.

Distance between unitary orbits in C∗-algebras with stable rank one and real rank zero

Let A be a C∗-algebra with stable rank one and real rank zero. In this paper, it is shown that the usual distance dU defined on the approximate unitary equivalence classes (or unitary orbits) of the positive elements in A is equal to the distance dW defined on morphisms from Cuntz semigroup of C0(0,1] to the Cuntz semigrout of A.

On two geometric means and sum of adjoint orbits

In this paper, we study the metric geometric mean introduced by Pusz and Woronowicz and the spectral geometric mean introduced by Fiedler and Pták, originally for positive definite matrices. The relation between t-metric geometric mean and t-spectral geometric mean is established via log majorization. The result is then extended in the context of symmetric space associated with a noncompact semisimple Lie group. For any Hermitian matrices X and Y, So's matrix exponential formula asserts that there are unitary matrices U and V such thateX/2eYeX/2=eUXU⁎+VYV⁎. In other words, the Hermitian matrix log⁡(eX/2eYeX/2) lies in the sum of the unitary orbits of X and Y. So's result is also extended to a formula for adjoint orbits associated with a noncompact semisimple Lie group.

On the distribution of the mean energy in the unitary orbit of quantum states

Given a closed quantum system, the states that can be reached with a cyclic process are those with the same spectrum as the initial state. Here we prove that, under a very general assumption on the Hamiltonian, the distribution of the mean extractable work is very close to a gaussian with respect to the Haar measure. We derive bounds for both the moments of the distribution of the mean energy of the state and for its characteristic function, showing that the discrepancy with the normal distribution is increasingly suppressed for large dimensions of the system Hilbert space.

Geodesic neighborhoods in unitary orbits of self-adjoint operators of [formula omitted

In the present paper, we study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group UK+C of the unitization of the compact operators K(H)+C, or equivalently, the quotient UK+C/UD(K+C). We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As a consequence we obtain a local Hopf-Rinow theorem. We also explore cases about the uniqueness of short curves and prove that there exist some of these that cannot be parameterized using minimal anti-Hermitian operators of K(H)+C.

Optimal transport and unitary orbits in C⁎-algebras

Two areas of mathematics which have received substantial attention in recent years are the theory of optimal transport and the Elliott classification programme for C⁎-algebras. We combine these two seemingly unrelated disciplines to make progress on a classical problem of Weyl. In particular, we show how results from the Elliott classification programme can be used to translate continuous transport of spectral measures into optimal unitary conjugation in C⁎-algebras. As a consequence, whenever two normal elements of a sufficiently well-behaved C⁎-algebra share a spectrum amenable to such continuous transport, and have trivial K1-class, the distance between their unitary orbits can be computed tracially.

The spectral spread of Hermitian matrices

Let A be an n×n complex Hermitian matrix and let λ(A)=(λ1,…,λn)∈Rn denote the eigenvalues of A, counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of A, denoted Spr+(A), given by Spr+(A)=(λ1−λn,λ2−λn−1,…,λk−λn−k+1)∈Rk, where k=[n/2] (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of A, that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.

Constructions of Orbit Codes Based on Unitary Spaces Over Finite Fields

Orbit codes, as special constant dimension codes, have attracted much attention due to their applications for error correction in random network coding. This paper is devoted to constructing large orbit codes by making full use of unitary space. Firstly, we construct a cyclic unitary group of order $q^{2n}-1$ by means of the companion matrix of a primitive polynomial over finite fields $\mathbb {F}_{q^{2}}$ , and so the corresponding code is unitary cyclic orbit code. As a special application, a new quaternary orbit code $(6,63,4,3)$ is given. Secondly, we obtain orbit codes with large size using the external direct product of unitary groups acting on the direct sum of subspaces. Finally, a table is given for illustrating our codes improve upon those constructed by Trautmann et al. and Poroch et al.

Path-connected closure of unitary orbits

Path-connected closure of unitary orbits

Groups and semigroups generated by a single unitary orbit

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given invertible matrix A. In particular, we give necessary and sufficient conditions for such a semigroup to be the special linear group or the general linear group or other classical matrix groups. Furthermore, we use the above to determine the subgroups of the general linear group that are normalized by the unitary group.

Quantum $$\alpha $$-fidelity of unitary orbits

The extremum values of quantum $$\alpha $$ -fidelity under unitary orbits of quantum states are explicitly derived by applying rearrangement inequalities, matrix trace inequalities, and theory of majorization. Furthermore, the $$\alpha $$ -fidelity is successfully verified to go through the whole closed interval, which works from the minimum value to the maximum value.

Clarkson-McCarthy inequalities with unitary and isometry orbits

A refinement of a trace inequality of McCarthy establishing the uniform convexity of the Schatten p-classes for p>2 is proved: if A,B are two n-by-n matrices, then there exists some pair of n-by-n unitary matrices U,V such thatU|A+B2|pU⁎+V|A−B2|pV⁎≤|A|p+|B|p2. A similar statement holds for compact Hilbert space operators. Another improvement of McCarthy's inequality is given via the new operator parallelogramm law,|A+B|2⊕|A−B|2=U0(|A|2+|B|2)U0⁎+V0(|A|2+|B|2)V0⁎ for some pair of 2n-by-n isometry matrices U0,V0.