Thermodynamic Interpretation of the Quantum Error Correcting Criterion

Shannon's fundamental coding theorems relate classical information theory to thermodynamics. More recent theoretical work has been successful in relating quantum information theory to thermodynamics. For example, Schumacher proved a quantum version of Shannon's 1948 classical noiseless coding theorem. In this note, we extend the connection between quantum information theory and thermodynamics to include quantum error correction. There is a standard mechanism for describing errors that may occur during the transmission, storage, and manipulation of quantum information. One can formulate a criterion of necessary and sufficient conditions for the errors to be detectable and correctable. We show that this criterion has a thermodynamical interpretation. PACS: 03.67; 05.30; 63.10     

Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction

The 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     

Approximate Quantum Error Correction

The errors that arise in a quantum channel can be corrected perfectly if and only if the channel does not decrease the coherent information of the input state. We show that, if the loss of coherent information is small, then approximate quantum error correction is possible. PACS: 03.67.H, 03.65.U     

Quantum Computing and Information Extraction for Dynamical Quantum Systems

We discuss the simulation of complex dynamical systems on a quantum computer. We show that a quantum computer can be used to efficiently extract relevant physical information. It is possible to simulate the dynamical localization of classical chaos and extract the localization length with quadratic speed up with respect to any known classical computation. We can also compute with algebraic speed up the diffusion coefficient and the diffusion exponent, both in the regimes of Brownian and anomalous diffusion. Finally, we show that it is possible to extract the fidelity of the quantum motion, which measures the stability of the system under perturbations, with exponential speed up. The so-called quantum sawtooth map model is used as a test bench to illustrate these results. PACS: 03.67.Lx, 05.45.Mt     

Geometric Approach to Digital Quantum Information

We present geometric methods for uniformly discretizing the continuous N-qubit Hilbert space H_N. When considered as the vertices of a geometrical figure, the resulting states form the equivalent of a Platonic solid. The discretization technique inherently describes a class of /2 rotations that connect neighboring states in the set, i.e., that leave the geometrical figures invariant. These rotations are shown to generate the Clifford group, a general group of discrete transformations on N qubits. Discretizing H_N allows us to define its digital quantum information content, and we show that this information content grows as N². While we believe the discrete sets are interesting because they allow extra-classical behavior—such as quantum entanglement and quantum parallelism—to be explored while circumventing the continuity of Hilbert space, we also show how they may be a useful tool for problems in traditional quantum computation. We describe in detail the discrete sets for one and two qubits.PACS: 03.67.Lx; 03.67.pp; 03.67.-a; 03.67.Mn.PACS: 03.67.Lx; 03.67.pp; 03.67.-a; 03.67.Mn.     

Quantum Information Processing in Cavity-QED

We give a brief overview of cavity-QED and its roles in quantum information science. In particular, we discuss setups in optical cavity-QED, where either atoms serve as stationary qubits, or photons serve as flying qubits. PACS: 42.50.Pq, 03.67.Lx, 03.67.Hk, 32.80.Pj     

Ion Trap Quantum Computing with Ca+ Ions

The scheme of an ion trap quantum computer is described and the implementation of quantum gate operations with trapped Ca⁺ ions is discussed. Quantum information processing with Ca⁺ ions is exemplified with several recent experiments investigating entanglement of ions. PACS: 03.67.Lx, 03.67.Mn, 32.80.Pj     

Towards Scalable Linear-Optical Quantum Computers

Scalable quantum computation with linear optics was considered to be impossible due to the lack of efficient two-qubit logic gates, despite the ease of implementation of one-qubit gates. Two-qubit gates necessarily need a non-linear interaction between the two photons, and the efficiency of this non-linear interaction is typically very small in bulk materials. However, it has recently been shown that this barrier can be circumvented with effective non-linearities produced by projective measurements, and with this work linear-optical quantum computing becomes a new avenue towards scalable quantum computation. We review several issues concerning the principles and requirements of this scheme. PACS: 03.67.Lx, 03.67.Pp, 42.50.Dv, 42.65.Lm     

How Do Two Observers Pool Their Knowledge About a Quantum System?

In the theory of classical statistical inference one can derive a simple rule by which two or more observers may combine independently obtained states of knowledge together to form a new state of knowledge, which is the state which would be possessed by someone having the combined information of both observers. Moreover, this combined state of knowledge can be found without reference to the manner in which the respective observers obtained their information. However, we show that in general this is not possible for quantum states of knowledge; in order to combine two quantum states of knowledge to obtain the state resulting from the combined information of both observers, these observers must also possess information about how their respective states of knowledge were obtained. Nevertheless, we emphasize this does not preclude the possibility that a unique, well motivated rule for combining quantum states of knowledge without reference to a measurement history could be found. We examine both the direct quantum analog of the classical problem, and that of quantum state-estimation, which corresponds to a variant in which the observers share a specific kind of prior information. PACS: 03.67.-a, 02.50.-r, 03.65.Bz     

Parametrizing Quantum States and Channels

This work describes one parametrization of quantum states and channels and several of its possible applications. This parametrization works in any dimension and there is an explicit algorithm which produces it. Included in the list of applications are a simple characterization of pure states, an explicit formula for one additive entropic quantity which does not require knowledge of eigenvalues, and an algorithm which finds one Kraus operator representation for a quantum operation without recourse to eigenvalue and eigenvector calculations. PACS: 03.67a, 03.67-Hk, 03.67-Lx     

Experimental Concatenation of Quantum Error Correction with Decoupling

We experimentally explore the reduction of decoherence via concatenating quantum error correction (QEC) with decoupling in liquid-state NMR quantum information processing. Decoupling provides an efficient means of suppressing decoherence from noise sources with long correlation times, and then QEC can be used more profitably for the remaining noise sources. PACS: 03.67.Lx, 03.65.Bz     

Controlling Spin Qubits in Quantum Dots

We review progress on the spintronics proposal for quantum computing where the quantum bits (qubits) are implemented with electron spins. We calculate the exchange interaction of coupled quantum dots and present experiments, where the exchange coupling is measured via transport. Then, experiments on single spins on dots are described, where long spin relaxation times, on the order of a millisecond, are observed. We consider spin-orbit interaction as sources of spin decoherence and find theoretically that also long decoherence times are expected. Further, we describe the concept of spin filtering using quantum dots and show data of successful experiments. We also show an implementation of a read out scheme for spin qubits and define how qubits can be measured with high precision. Then, we propose new experiments, where the spin decoherence time and the Rabi oscillations of single electrons can be measured via charge transport through quantum dots. Finally, all these achievements have promising applications both in conventional and quantum information processing. PACS: 03.67.Lx, 03.67.Mn, 73.23.Hk, 85.35.Be     

Quantum Comparison Machines with One-Sided Error

It is always possible to decide, with one-sided error, whether two quantum states are the same under a specific unitary transformation. However we show here that it is impossible to do so if the transformation is anti-linear and non-singular. This result implies that unitary and anti-unitary operations exist on an unequal footing in quantum information theory. PACS: 03.67.-a     

Locality and Information Transfer in Quantum Operations

We investigate the situation in which no information can be transferred from a quantum system B to a quantum system A, even though both interact with a common system C.PACS:03.67.-a, 03.65.Ud     

Estimation of Quantum Channels with Finite Resources

We examine the problem of determining the parameters that describe a quantum channel. It is assumed that the users of the channel have at best only partial knowledge of it and make use of a finite amount of resources to estimate it. We discuss simple protocols for the estimation of the parameters of several classes of channels that are studied in the current literature. A quantitative measures of the quality of the estimation schemes can be given on the basis of the standard deviation or of the fidelity. Protocols that employ entangled particles are also discussed. The use of entangled particles as a nonclassical resource enhances the estimation quality of some classes of quantum channel. Further, the methods presented here can be extended to higher dimensional quantum systems. PACS: 03.67.Hk     

Spin-Based Quantum Dot Quantum Computing in Silicon

The spins of localized electrons in silicon are strong candidates for quantum information processing because of their extremely long coherence times and the integrability of Si within the present microelectronics infrastructure. This paper reviews a strategy for fabricating single electron spin qubits in gated quantum dots in Si/SiGe heterostructures. We discuss the pros and cons of using silicon, present recent advances, and outline challenges. PACS: 03.67.Pp, 03.67.Lx, 85.35.Be, 73.21.La     

Securing QKD Links in the Full Hilbert Space

Many quantum key distribution QKD analyses examine the link security in a subset of the full Hilbert space that is available to describe the system. In reality, information about the photon state can be embedded in correlations between the polarization space and other dimensions of the full Hilbert space in such a way that Eve can determine the polarization of a photon without affecting it. This paper uses the concept of suitability Hockney et al. Suitability versus Fidelity for Rating Single Photon Guns to quantify the available information for Eve to exploit, and demonstrate how it is possible for Alice and Bob to fool themselves into thinking they have a highly secure link.PACS:03.67.Dd; 03.67.HK; 42.50.-p.     

Quantum Information Processing with Trapped Neutral Atoms

Quantum information can be processed using large ensembles of ultracold and trapped neutral atoms, building naturally on the techniques developed for high-precision spectroscopy and metrology. This article reviews some of the most important protocols for universal quantum logic with trapped neutrals, as well as the history and state-of-the-art of experimental work to implement these in the laboratory. Some general observations are made concerning the different strategies for qubit encoding, transport and interaction, including trade-offs between decoherence rates and the likelihood of two-qubit gate errors. These trade-offs must be addressed through further refinements of logic protocols and trapping technologies before one can undertake the design of a general-purpose neutral-atom quantum processor. PACS: 03.67.Lx, 32.80.Pj, 34.50.-s