David W. Kribs

Unified and generalized approach to quantum error correction.

We present a unified approach to quantum error correction, called operator quantum error correction. Our scheme relies on a generalized notion of a noiseless subsystem that is investigated here. By combining the active error correction with this generalized noiseless subsystems method, we arrive at a unified approach which incorporates the known techniques--i.e., the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method--as special cases. Moreover, we demonstrate that the quantum error correction condition from the standard model is a necessary condition for all known methods of quantum error correction.

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.

Isometric Dilations of Non-Commuting Finite Rank n-Tuples

A contractive n-tuple A = (A1, . . . , An) has a minimal joint isometric dilation S = (S1, . . . , Sn) where the Sis are isometries with pairwise orthogonal ranges. This determines a rep- resentation of the Cuntz-Toeplitz algebra. When A acts on a finite dimensional space, the wot-closed nonself-adjoint algebra S generated by S is completely described in terms of the properties of A. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra S is always hyper-reflexive. In the last section, we describe similarity invariants. In partic- ular, an n-tuple B of d × d matrices is similar to an irreducible n-tuple A if and only if a certain finite set of polynomials vanish on B.

Geometry of higher-rank numerical ranges

We consider geometric aspects of higher-rank numerical ranges for arbitrary N  × N 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.

Decoherence-Insensitive Quantum Communication by Optimal

The central issue in this paper is to transmit a quantum state in such a way that after some decoherence occurs, most of the information can be restored by a suitable decoding operation. For this purpose, we incorporate redundancy by mapping a given initial quantum state to a messenger state on a larger dimensional Hilbert space via a C* -algebra embedding. Our noise model for the transmission is a phase damping channel which admits a noiseless subsystem or decoherence-free subspace. More precisely, the transmission channel is obtained from convex combinations of a set of lowest rank yes/no measurements that leave a component of the messenger state unchanged. The objective of our encoding is to distribute quantum information optimally across the noise-susceptible component of the transmission when the noiseless component is not large enough to contain all the quantum information to be transmitted. We derive simple geometric conditions for optimal encoding and construct examples of such encodings.

C^{\ast }

We consider the problem of computing the family of operator norms recently introducedin [1]. We develop a family of semidefinite programs that can be used to exactly computethem in small dimensions and bound them in general. Some theoretical consequencesfollow from the duality theory of semidefinite programming, including a new constructiveproof that for all r there are non-positive partial transpose Werner states that are r-undistillable. Several examples are considered via a MATLAB implementation of thesemidefinite program, including the case of Werner states and randomly generated statesvia the Bures measure, and approximate distributions of the norms are provided. Weextend these norms to arbitrary convex mapping cones and explore their implicationswith positive partial transpose states.

-Encoding

Abstract We examine k -minimal and k -maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k -minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k -positive linear maps and bound entanglement. Similarly, we investigate the k -super minimal and k -super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k -block positive operators and (unnormalized) states with Schmidt number no greater than k , respectively. We characterize a class of norms on the k -super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.

A family of norms with applications in quantum information theory II

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.

Minimal and maximal operator spaces and operator systems in entanglement theory

AbstractThe effect of noise on a quantum system can be described by a set of operators obtained from the interaction Hamiltonian. Recently it has been shown that generalized quantum error correcting codes can be derived by studying the algebra of this set of operators. This led to the discovery of noiseless subsystems. They are described by a set of operators obtained from the commutant of the noise generators. In this paper we derive a general method to compute the structure of this commutant in the case of unital noise. PACS: 03.67.–a, 03.67.Pp

Method to find quantum noiseless subsystems.

We show that every correctable subsystem for an arbitrary noise operation can be recovered by a unitary operation, where the notion of recovery is more relaxed than the notion of correction insofar as it does not protect the subsystem from subsequent iterations of the noise. We also demonstrate that in the case of unital noise operations one can identify a subset of all correctable subsystems\char22{}those that can be corrected by a single unitary operation\char22{}as the noiseless subsystems for the composition of the noise operation with its dual. Using the recently developed structure theory for noiseless subsystems, the identification of such unitarily correctable subsystems is reduced to an algebraic exercise.