Rank-1 Operator Research Articles

Optimal discrimination of geometrically uniform qubit and qutrit states

We consider discrimination of geometrically uniform states, and analytically solve this problem for qubits. For qutrits, we obtain the exact solution to the optimal measurement when the defining unitary matrix for the geometrically uniform states is degenerate. We also show that if the unitary is nondegenerate then the optimal measurement for discriminating geometrically uniform qubits or qutrits can always be expressed as a rank-1 operator, which converts the original discrimination problem into an optimization of a sum of trigonometric functions with two real variables in the case of qutrits. Additionally, a geometrical interpretation of the optimal measurement for discriminating geometrically uniform qubits is given via the Bloch sphere representation.

Joint Schmidt-type decomposition for two bipartite pure quantum states

It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we derive several generally valid purity-type formulae for one-party reductions of rank-1 operators, and we point out relevant relations between the Schmidt decomposition and the Bloch representation of bipartite pure states.

Decomposability of Finite Rank Operators in Lie Ideals of Nest Algebras

A set of operators is said to be decomposable if each finite rank operator in it can be written as a sum of finitely many rank-1 operators in it. In this note, we establish a sufficient and necessary condition on the nest such that each closed Lie ideal in the associated nest algebra is decomposable.

G-narrow operators and G-rich subspaces

Abstract Let X and Y be Banach spaces. An operator G: X → Y is a Daugavet center if ‖G +T‖ = ‖G‖+‖T‖ for every rank-1 operator T. For every Daugavet center G we consider a certain set of operators acting from X, so-called G-narrow operators. We prove that if J is the natural embedding of Y into a Banach space E, then E can be equivalently renormed so that an operator T is (J ○ G)-narrow if and only if T is G-narrow. We study G-rich subspaces of X: Z ⊂ X is called G-rich if the quotient map q: X → X/Z is G-narrow.

Local quantum protocols for separable measurements with many parties

In a recent paper [S. M. Cohen, Phys. Rev. A 84, 052322 (2011)], we showed how to construct a protocol for implementing a bipartite, separable quantum measurement using only local operations on subsystems and classical communication between parties (LOCC) within any fixed number of rounds of communication, whenever such a protocol exists. Here, we generalize that construction to one that applies for any number of parties. One important observation is that the construction automatically determines the ordering of the parties' measurements, overcoming a significant apparent difficulty in designing protocols for more than two parties. We also present various other results about LOCC, including showing that if, in any given measurement operator of the separable measurement under consideration, the local parts for two different parties are rank-1 operators that are not repeated in any other measurement operator of the measurement, then this separable measurement cannot be exactly implemented by LOCC in any finite number of rounds.

NUMERICAL RADIUS OF RANK-1 OPERATORS ON BANACH SPACES

We study the rank-1 numerical index of a Banach space, namely the infimum of the numerical radii of those rank-1 operators on the space which have norm 1. We show that the rank-1 numerical index is always greater than or equal to 1/ e. We also present properties of this index and some examples.

Almost Daugavet centers

An operator G : X → Y is an almost Daugavet center if there exists a norming subspace Z ⊂ Y ⁎ such that ‖ G + T ‖ = ‖ G ‖ + ‖ T ‖ for every rank-1 operator T : X → Y of the form T = x ⁎ ⊗ y where y ∈ Y and x ⁎ ∈ W = G ⁎ ( Z ) ¯ . This notion is both a generalization of the almost Daugavet property when G = I and X = Y , and a generalization of the notion of Daugavet centers when W = X ⁎ . We give a characterization of the almost Daugavet centers in terms of the thickness of an operator and in terms of canonical ℓ 1 -type sequences. We show that, for a separable space Y , an operator G : X → Y is similar to an almost Daugavet center if and only if G fixes an isomorphic copy of ℓ 1 . We also give some geometric characterizations of this property.

Characterizing operations preserving separability measures via linear preserver problems

We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that send separable pure states to separable pure states. We also provide a new proof of an analogous statement in the multipartite setting. We use these results to develop a bipartite version of a classical result about the structure of maps that preserve rank-1 operators and then characterize the isometries for two families of norms that have recently been studied in quantum information theory. We see, in particular, that for k ≥ 2 the operator norms induced by states with Schmidt rank k are invariant only under local unitaries, the swap operator and the transpose map. However, in the k = 1 case there is an additional isometry: the partial transpose map.

Daugavet centers and direct sums of Banach spaces

Abstract A linear continuous nonzero operator G: X → Y is a Daugavet center if every rank-1 operator T: X → Y satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X 1⊕F X 2 of two Banach spaces X 1 and X 2 by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X 1 and X 2 there exists a Daugavet center acting from X 1⊕F X 2, and the class of those F such that for some pair of spaces X 1 and X 2 there is a Daugavet center acting into X 1⊕F X 2. We also present several examples of such Daugavet centers.

Rank-1 perturbations of cosine functions and semigroups

Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A + B generates a cosine function for each B ∈ L ( D ( ( ω − A ) 1 / 2 ) , X ) . If A is unbounded and 1 2 < γ ⩽ 1 , then we show that there exists a rank-1 operator B ∈ L ( D ( ( ω − A ) γ ) , X ) such that A + B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A + B generates a distribution semigroup for each operator B ∈ L ( D ( A ) , X ) of rank-1, then A generates a holomorphic C 0 -semigroup. If A + B generates a C 0 -semigroup for each operator B ∈ L ( D ( ( ω − A ) γ ) , X ) of rank-1 where 0 < γ < 1 , then the semigroup T generated by A is differentiable and ‖ T ′ ( t ) ‖ = O ( t − α ) as t ↓ 0 for any α > 1 / γ . This is an approximate converse of a perturbation theorem for this class of semigroups.

Certain structure of Similarity-preserving linear maps

It is proved that if the range of a bounded similarity-preserving linear map on contains the set of all compact operators, then it is surjective, and therefore is a nonzero scalar multiple of either an automorphism or an anti-automorphism. We also give the structure of a weakly continuous similarity-preserving map which maps one rank-1 operator to rank-1.

Denseness for norm attaining operator-valued functions

In this note we offer a short, constructive proof for Hilbert spaces of Lindenstrauss' famous result on the denseness of norm attaining operators. Specifically, we show given any A∈ L( H) there is a sequence of rank-1 operators K n such that A+K n is norm attaining for each n and K n converges in norm to zero. We then apply our construction to establish denseness results for norm attaining operator-valued functions.

Multiplicative maps on ℬ(X)

A structural feature of multiplicative maps on ℬ(X) which sends some rank-1 operator to an operator of rank not greater than 1 is given, and based on it, some characterizations of automorphism of ℬ(X) are obtained and some multiplicative preserver problems are answered.