International Journal of Quantum Information

Any dielectric nanoparticle moving inside an optical cavity generates an optomechanical interaction. In this paper, we theoretically analyze the light scattering of an optomechanical cavity which strongly interacts with a dielectric nanoparticle. The cavity is driven by an external laser field. This interaction gives rise to different dynamics that can be used to cool, trap and levitate nanoparticle. We analytically calculate reflection and transmission rate of the cavity field, and study the time evolution of the intracavity field, momentum and position of the nanoparticle. We find the nanoparticle occupies a discrete position inside the cavity. This effect can be exploited to separate nanoparticle and couplings between classical particles and quantized fields.

In the circuit model of quantum computation, a desired unitary gate can be implemented deterministically, whereas in the measurement-based model the unitary gate is implemented up to a byproduct operator. In order to compensate byproducts, following measurement angles must be adjusted, or classical results must be corrected. Such a feed-forwarding requires some classical processing and tuning of the measurement device, which cause the delay of computation and the additional decoherence. We show that if we respect the no-signaling principle, which is one of the most fundamental principles in physics, byproducts cannot be avoided in measurement-based quantum computation. Furthermore, we also show by using the idea of the quantum error correcting code that due to the no-signaling principle, not all byproducts are allowed in measurement-based quantum computation.

The characterization of quantum discord (QD) and geometric discord (GD) has mostly concentrated on two-qubit states since the minimization in both discords is a daunting task for high-dimensional states. Numerical calculations of both discords are carried out for a generic bipartite state. When one-dimensional orthogonal projectors for a local measurement on n-dimensional Hilbert space are realized by the generators and the Euler angles of SU(n), the optimal measurements have a figure of merit that includes n(n - 1) Euler parameters. As an representative example, such projectors and two kinds of algorithms are used to estimate both discords for two-qutrit mixed states in recent literature. The generalized negativity as a measure of quantum entanglement is calculated for reference purposes. For those states with one parameter the discords and the negativity respectively display the nonlinear and the linear function of the parameter, with different turning points. However, they are positively correlated in the suitable ranges of the parameter for those states. The hierarchy of those quantities is discussed as well. Those shed new light on the understanding of QDs and quantum entanglement of mixed states in high-dimensions.

We theoretically investigate the quantum correlations (in terms of concurrence of indistinguishable electrons) in a prototype molecular system (hydrogen molecule). With the assistance of the standard approximations of the linear combination of atomic orbitals and the configuration interaction methods we describe the electronic wavefunction of the ground state of the H₂ molecule. Moreover, we managed to find a rather simple analytic expression for the concurrence (the most used measure of quantum entanglement) of the two electrons when the molecule is in its lowest energy. We have found that concurrence does not really show any relation to the construction of the chemical bond.

Linear entropy is widely used as a measure of operator entanglement of two-qubit gates. In this paper, we characterize the two-qubit gates of same linear entropy using the majorization criteria of the Schmidt coefficients of the gates. It is found that the gates with same linear entropy must have set of same Schmidt coefficients or set of different Schmidt coefficients which are not ordered according to the majorization relation.

Entanglement plays an important role in quantum communication, algorithms, and error correction. Schmidt coefficients are correlated to the eigenvalues of the reduced density matrix. These eigenvalues are used in von Neumann entropy to quantify the amount of the bipartite entanglement. In this paper, we map the Schmidt basis and the associated coefficients to quantum circuits to generate random quantum states. We also show that it is possible to adjust the entanglement between subsystems by changing the quantum gates corresponding to the Schmidt coefficients. In this manner, random quantum states with predefined bipartite entanglement amounts can be generated using random Schmidt basis. This provides a technique for generating equivalent quantum states for given weighted graph states, which are very useful in the study of entanglement, quantum computing, and quantum error correction.

We demonstrate the use of optimal control to design two entropy-manipulating quantum gates which are more complex than the corresponding, commonly used, gates, such as CNOT and Toffoli (CCNOT): A two-qubit gate called polarization exchange (PE) and a three-qubit gate called polarization compression (COMP) were designed using GRAPE, an optimal control algorithm. Both gates were designed for a three-spin system. Our design provided efficient and robust nuclear magnetic resonance (NMR) radio frequency (RF) pulses for ¹³C₂-trichloroethylene (TCE), our chosen three-spin system. We then experimentally applied these two quantum gates onto TCE at the NMR lab. Such design of these gates and others could be relevant for near-future applications of quantum computing devices.

Measurement of a degenerate (or non-degenerate) discrete observable is investigated in the framework of quantum measurement theory short of collapse, i.e. premeasurement theory, based on a unitary evolution operator that includes the measurement interaction between object and measuring instrument. A pointer observable with eigen-projectors of, in general, many (or even infinitely) dimensional ranges is introduced as a new approach. It leads to redundant entanglement in the final state. As the first main result, the basic dynamical relation of the approach is derived. It is shown to be equivalent to the calibration condition, which is known to define general exact measurement. The latter is given a practical form. Complete measurement (premeasurement with objectification or collapse), which is in some sense implied by the premeasurement theory, performed on a subsystem of a bipartite object in a pure state is studied with particular attention to its effect on the opposite, interactionally unaffected subsystem. The change of state of the latter is derived for exact complete subsystem measurement, and it is shown that the change is the same as for the simplest, i.e. ideal measurement (this is the second main result). It is applied to the case of twin observables and thus distant measurement obtains a new, more satisfactory, foundation (the third main result). Distant measurement is a basic concept in the EPR phenomenon. The well-known importance of the latter implies importance of the former.

We introduce new mathematical aspects of the Bell states using matrix factorizations, non-noetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial p consists of two matrices ϕ₁, ϕ₂ such that ϕ₁ϕ₂ = ϕ₂ϕ₁ = p id. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on non-noetherian singularities. Moreover, we find that the collapse of a Bell state is intimately related to the representation theory of the noncommutative blowup along its singular support. This presents an exchange in geometry: the nonlocal commutative spacetime of the entangled state emerges from an underlying local noncommutative spacetime.

Since non-permutative quantum gates have more complex rules than permutative quantum gates, it is very hard to synthesize quantum logic circuits using non-permutative quantum gates, such as controlled-square-root-of-NOT gates (CV∕CV⁺ gates). In the efficient synthesis algorithm, direct use of quantum non-permutative gates should be avoided. Rather, the key method is to use quantum gates to create new permutative quantum gates that then replace non-permutative quantum gates. This method assumes the library of quantum gate primitives are constructed so as to have the lowest possible quantum cost. In this paper, we first propose some new CV∕CV⁺-like gates, i.e. controlled-kth-root-of-NOT gates where k = 2,4,8,…, and give all corresponding matrixes. Further, we also present a novel generic method to quickly and directly construct this new optimal quantum logic gate library using CNOT and these non-permutative quantum gates. Our method introduces new means to find permutative quantum gates with lower quantum cost.