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arXiv:1001.3578 · Physical Review A

Casimir invariants for systems undergoing collective motion

C. Allen Bishop, Mark Byrd, Lian-Ao Wu

Published Jun 22, 2011• 2 citations
arXivHTMLPDF

Abstract

Dicke states are an important class of states which exhibit collective behavior in many-body systems. They are interesting because (1) the decay rates of these states can be quite different from a set of independently evolving particles and (2) a particular class of these states are decoherence-free or noiseless with respect to a set of errors. These noiseless states, or more generally subsystems, avoid certain types of errors in quantum-information-processing devices. Here we provide a method for determining a set of transformations of these states which leave the states in their subsystems but still enable them to evolve in particular ways. For subsystems of particles undergoing collective motions, these transformations can be calculated by using essentially the same construction which is used to determine the famous Casimir invariants for quantum systems. Such invariants can be used to determine a complete set of commuting observables for a class of Dicke states as well as to identify possible logical operations for decoherence-free--noiseless subsystems. Our method is quite general and provides results for cases where the constituent particles have more than two internal states.

DOI
10.1103/physreva.83.062327

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Dicke states are an important class of states which exhibit collective behavior in many-body systems. They are interesting because (1) the decay rates of these states can be quite different from a set of independently evolving particles and (2) a particular class of these states are decoherence-free or noiseless with respect to a set of errors. These noiseless states, or more generally subsystems, avoid certain types of errors in quantum-information-processing devices. Here we provide a method for determining a set of transformations of these states which leave the states in their subsystems but still enable them to evolve in particular ways. For subsystems of particles undergoing collective motions, these transformations can be calculated by using essentially the same construction which is used to determine the famous Casimir invariants for quantum systems. Such invariants can be used to determine a complete set of commuting observables for a class of Dicke states as well as to identify possible logical operations for decoherence-free--noiseless subsystems. Our method is quite general and provides results for cases where the constituent particles have more than two internal states.
The interest in decoherence-free, or noiseless subsystems (DFS/NSs) of quantum systems is both of fundamental and practical interest. Understanding the invariance of a set of states under certain transformations is mutually associated with a better understanding of some fundamental aspects of quantum mechanics as well as the practical utility of invariant subsystems. For example, DFS/NSs are potentially useful for protecting quantum information in quantum cryptography and quantum computing as well as enabling universal computation. Here we discuss transformations which are compatible with a DFS/NS that is composed of d-state systems which protect against collective noise. They are compatible in the sense that they do not take the logical (encoded) states outside of the DFS/NS during the transformation. Furthermore, it is shown that the Hamiltonian evolutions derived here can be used to perform universal quantum computation on a three qudit DFS/NS. Many of the methods used in our derivations are directly applicable to a large variety of DFS/NSs. More generally, we may also state that these transformations are compatible with collective motions.
Abstract. The interest in decoherence-free, or noiseless subsystems (DFS/NSs) of quantum systems is both of fundamental and practical interest. Understanding the invariance of a set of states under certain transformations is mutually associated with a better understanding of some fundamental aspects of quantum mechanics as well as the practical utility of invariant subsystems. For example, DFS/NSs are potentially useful for protecting quantum information in quantum cryptography and quantum computing as well as enabling universal computation. Here we discuss transformations which are compatible with a DFS/NS that is composed of d-state systems which protect against collective noise. They are compatible in the sense that they do not take the logical (encoded) states outside of the DFS/NS during the transformation. Furthermore, it is shown that the Hamiltonian evolutions derived here can be used to perform universal quantum computation on a three qudit DFS/NS. Many of the methods used in our derivations are directly applicable to a large variety of DFS/NSs. More generally, we may also state that these transformations are compatible with collective motions. PACS numbers: 03.67.Pp,03.67.Lx,03.65.Yz,11.30.-j
The development of a quantum computer presents one of the greatest challenges in science and engineering to date. The promise of more efficient computing based on entangled quantum states and the superposition principle has led to a worldwide explosion of interest in the fields of quantum information and computation. Among the number of hurdles which must first be cleared before we witness a physical realization are problems associated with environment-induced decoherence and noise more generally. However, the discovery of quantum error correction and the establishment of the accuracy threshold theorem provide us with the hope of someday harnessing the potential power a functioning fault-tolerant quantum information processor has to offer. This dissertation contributes to this effort by investigating a particular class of quantum error correcting codes, namely noiseless subsystem encodings. The passive approach to error correction taken by these encodings provides an efficient means of protection from symmetrically coupled system-environment interactions. Here I will present methods for determining the subsystem-preserving evolutions for noiseless subsystem encodings supported by arbitrary-dimensional physical quantum systems. Implications for universal, collective decoherence-free quantum computation using the derived operations are discussed. Moreover, I will present a proposal for an optical device which is capable of preparing a variety of these noiseless subsystem encodings through a postselection strategy.
Noiseless subsystems offer a general and efficient method for protecting quantum information in the presence of noise that has symmetry properties. A paradigmatic class of error models displaying nontrivial symmetries emerges under collective noise behavior, which implies a permutationally invariant interaction between the system and the environment. We expand our previous investigation of the noiseless subsystem idea [L. Viola et al., Science 293, 2059 (2001)] by reporting and analyzing NMR experiments that demonstrate the preservation of a qubit encoded in a three-qubit noiseless subsystem for general collective noise. A complete set of input states is used to determine the superoperator for the implemented one-qubit process and to confirm that the fidelity of entanglement is improved for a large, noncommutative set of engineered errors. To date, this is the largest set of error operators that has been successfully corrected for by any quantum code.
When analysing quantum information processing protocols, one has to deal with large entangled systems, each consisting of many subsystems. To make this analysis feasible, it is often necessary to identify some additional structures. de Finetti theorems provide such a structure for the case where certain symmetries hold. More precisely, they relate states that are invariant under permutations of subsystems to states in which the subsystems are independent of each other. This relation plays an important role in various areas, e.g., in quantum cryptography or state tomography, where permutation invariant systems are ubiquitous. The known de Finetti theorems usually refer to the internal quantum state of a system and depend on its dimension. Here, we prove a different de Finetti theorem where systems are modelled in terms of their statistics under measurements. This is necessary for a large class of applications widely considered today, such as device independent protocols, where the underlying systems and the dimensions are unknown and the entire analysis is based on the observed correlations.
In a quantum computer, qubits are often stored in identical two-level systems separated by a distance shorter than the characteristic wavelength of the reservoirs that are responsible for decoherence. In this case the collective qubit-reservoir interaction, rather than the individual qubit-reservoir interaction, may determine the decoherence properties. We study the collective decoherence behavior in between each step in certain quantum algorithms and propose a simple alternative of implementing quantum algorithms using a quantum trajectory that is close to a decoherence-free subspace that avoids unstable Dicke's superradiant states and Schrödinger's cat state.