Man-Duen Choi

Injectivity and operator spaces

Abstract Let B ( H ) be the bounded operators on a Hilbert space H. A linear subspace R ⊆ B ( H ) is said to be an operator system if 1 ϵ R and R is self-adjoint. Consider the category b of operator systems and completely positive linear maps. R ∈ C is said to be injective if given A ⊆ B, A, B ∈ C , each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C ∗ -algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting M m = B ( C m ), each map R → N ⊆ M m has approximate factorizations R → M n → N , (4) letting K be the orthogonal complement of an operator system N ⊆ M m , each map M m K → R has approximate factorizations M m K → M n → R . Analogous characterizations are found for certain classes of C ∗ -algebras.

The completely positive lifting problem for C*-algebras

(1.1)~~~~~~~~~~~~~~~~~~~1 where A, B are C*-algebras (resp., unital C*-algebras), J is a closed twosided ideal in B, 7 is the quotient map, and A, * are contractive (resp., unital) completely positive maps. The lifting problem for q is to determine whether or not one can find * so that the diagram commutes. In this paper, we will show that this is the case if A is separable, and either A, B/J, or B

Linear Maps Preserving Commutativity

Abstract Commutativity-preserving maps on the real space of all real symmetric or complex self-adjoint matrices are characterized. Related results are given for adjoint-preserving maps defined on all n × n matrices. These results are extended to infinite dimensions in the case of invertible maps.

Quantum error correcting codes from the compression formalism

We solve the fundamental quantum error correction problem for bi-unitary channels on two-qubit Hilbert space. By solving an algebraic compression problem, we construct qubit codes for such channels on arbitrary dimension Hilbert space, and identify correctable codes for Pauli-error models not obtained by the stabilizer formalism. This is accomplished through an application of a new tool for error correction in quantum computing called the “higher-rank numerical range”. We describe its basic properties and discuss possible further applications.

Geometry of higher-rank numerical ranges

We consider geometric aspects of higher-rank numerical ranges for arbitrary N u2009×u2009N matrices. Of particular interest is the issue of convexity and a possible extension of the Toeplitz–Hausdorff Theorem. We derive a number of reductions and obtain partial results for the general problem. We also conduct graphical and computational experiments. Added in proof: Following acceptance of this paper, our subject has developed rapidly. First, Hugo Woerdeman established convexity of the higher-rank numerical ranges by combining Proposition 2.4 and Theorem 2.12 with the theory of algebraic Riccati equations. See Woerdeman, H., 2007, The higher rank numerical range is convex, Linear and Multilinear Algebra, to appear. Subsequently Chi-Kwong Li and Nung-Sing Sze followed a different approach that not only yields convexity but also provides important additional insights. See Li, C.-K. and Sze, N.-S., 2007, Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations, preprint. See also Li, C.-K., Poon, Y.-T., and Sze, N.-S., 2007, Condition for the higher rank numerical range to be non-empty, preprint.

Method to find quantum noiseless subsystems.

We develop a structure theory for decoherence-free subspaces and noiseless subsystems that applies to arbitrary (not necessarily unital) quantum operations. The theory can be alternatively phrased in terms of the superoperator perspective, or the algebraic noise commutant formalism. As an application, we propose a method for finding all such subspaces and subsystems for arbitrary quantum operations. We suggest that this work brings the fundamental passive technique for error correction in quantum computing an important step closer to practical realization.

Product numerical range in a space with tensor product structure

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.

CONVEX COMBINATIONS OF PROJECTIONS

Abstract On an n -dimensional inner-product space, every operator T that satisfies O ⩽ T ⩽ I is a convex combination of as few as [log 2 n ] + 2 projections, and this number is sharp. If O ⩽ T ⩽ I and trace T is a rational number, then T is an average of projections. Further results are also obtained for the cases when the projections are required to have the same rank and⧸or to be commuting. In each case, the optimal number of projections is determined.

Numerical ranges and dilations

Denote by W(A) the numerical range of a bounded linear operator A. For two operators A and B (which may act on different Hilbert spaces), we study the relation between the inclusion relation W(A)⊆W(B) and the condition that A can be dilated to an operator of the form B⊗I. We also investigate the possibilities of dilating an operator A to operators with simple structure under the assumption that W(A) is included in a special region.

The multiplicative domain in quantum error correction

We show that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes. In the case of a unital quantum channel, these are precisely the codes that do not require a measurement as part of the recovery process, the so-called unitarily correctable codes. In the arbitrary, not necessarily unital case, they form a proper subset of unitarily correctable codes that can be computed from the properties of the channel. As part of the analysis, we derive a representation theoretic characterization of subsystem codes. We also present a number of illustrative examples.