94 research outputs found
A 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.Comment: 5 pages, to appear in Physical Review Letter
A Unified and Generalized Approach to Quantum Error Correction
We present a unified approach to quantum error correction, called operator
quantum error correction. This scheme relies on a generalized notion of
noiseless subsystems that is not restricted to the commutant of the interaction
algebra. We arrive at the 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, by combining active error correction with this generalized noiseless
subsystem method. 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.Comment: 5 page
A quantum computing primer for operator theorists
This is an exposition of some of the aspects of quantum computation and
quantum information that have connections with operator theory. After a brief
introduction, we discuss quantum algorithms. We outline basic properties of
quantum channels, or equivalently, completely positive trace preserving maps.
The main theorems for quantum error detection and correction are presented and
we conclude with a description of a particular passive method of quantum error
correction.Comment: 24 pages, to appear in Lin. Alg. App
Quantum Error Correction of Observables
A formalism for quantum error correction based on operator algebras was
introduced in [1] via consideration of the Heisenberg picture for quantum
dynamics. The resulting theory allows for the correction of hybrid
quantum-classical information and does not require an encoded state to be
entirely in one of the corresponding subspaces or subsystems. Here, we provide
detailed proofs for the results of [1], derive a number of new results, and we
elucidate key points with expanded discussions. We also present several
examples and indicate how the theory can be extended to operator spaces and
general positive operator-valued measures.Comment: 22 pages, 1 figure, preprint versio
Generalization of Quantum Error Correction via the Heisenberg Picture
We show that the theory of operator quantum error correction can be naturally
generalized by allowing constraints not only on states but also on observables.
The resulting theory describes the correction of algebras of observables (and
may therefore suitably be called ``operator algebra quantum error
correction''). In particular, the approach provides a framework for the
correction of hybrid quantum-classical information and it does not require the
state to be entirely in one of the corresponding subspaces or subsystems. We
discuss applications to quantum teleportation and to the study of information
flows in quantum interactions.Comment: 5 pages, preprint versio
The structure of preserved information in quantum processes
We introduce a general operational characterization of information-preserving
structures (IPS) -- encompassing noiseless subsystems, decoherence-free
subspaces, pointer bases, and error-correcting codes -- by demonstrating that
they are isometric to fixed points of unital quantum processes. Using this, we
show that every IPS is a matrix algebra. We further establish a structure
theorem for the fixed states and observables of an arbitrary process, which
unifies the Schrodinger and Heisenberg pictures, places restrictions on
physically allowed kinds of information, and provides an efficient algorithm
for finding all noiseless and unitarily noiseless subsystems of the process
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