There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0×10(-21). It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203,000 years, equivalent to a significance greater than 5.1σ. The source lies at a luminosity distance of 410(-180)(+160)  Mpc corresponding to a redshift z=0.09(-0.04)(+0.03). In the source frame, the initial black hole masses are 36(-4)(+5)M⊙ and 29(-4)(+4)M⊙, and the final black hole mass is 62(-4)(+4)M⊙, with 3.0(-0.5)(+0.5)M⊙c(2) radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.

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The Observation of Gravitational Waves from a Binary Black Hole Merger

When two single Rydberg atoms—those with electrons in highly excited states—interact, one can be used to control the quantum state of the other. Two independent experiments demonstrate such ‘Rydberg blockade’, an effect that might make long-range quantum gates between neutral atoms possible.

Collective excitation of two individual atoms in the Rydberg blockade regime

A machine equivalent to Turing machine, that is more intuitive in its working is defined. Three derivation rules are added to Elementary Arithmetic of Godel and his incompleteness theorems are proved without using any metalanguage. Two axioms are added to Zermelo-Fraenkel theory to derive the Continuum Hypothesis and to split the unit interval into infinitesimals.

Foundations of Computer Science

A theoretical scheme is proposed to implement bidirectional quantum controlled teleportation (BQCT) by using a nine-qubit entangled state as a quantum channel, where Alice may transmit an arbitrary two-qubit state called qubits $$A_1$$A1 and $$A_2$$A2 to Bob; and at the same time, Bob may also transmit an arbitrary two-qubit state called qubits $$B_1$$B1 and $$B_2$$B2 to Alice via the control of the supervisor Charlie. Based on our channel, we explicitly show how the bidirectional quantum controlled teleportation protocol works. And we show this bidirectional quantum controlled teleportation scheme may be determinate and secure. Taking the amplitude-damping noise and the phase-damping noise as typical noisy channels, we analytically derive the fidelities of the BQCT process and show that the fidelities in these two cases only depend on the amplitude parameter of the initial state and the decoherence noisy rate.

Bidirectional controlled teleportation by using nine-qubit entangled state in noisy environments

We illustrate the principle of a cryptographic switch for a quantum scenario, in which a third party (Charlie) can control to a continuously varying degree the amount of information the receiver (Bob) receives, after the sender (Alice) has sent her information through a quantum channel. Suppose Charlie transmits a Bell state to Alice and Bob. Alice uses dense coding to transmit two bits to Bob. Only if the 2-bit information corresponding to the choice of the Bell state is made available by Charlie to Bob can the latter recover Alice's information. By varying the amount of information Charlie gives, he can continuously alter the information recovered by Bob. The performance of the protocol as subjected to the squeezed generalized amplitude damping channel is considered. We also present a number of practical situations where a cryptographic switch would be of use.

The quantum cryptographic switch

Quasiprobability distributions in open quantum systems: Spin-qubit systems

Single-qubit dissipative and non-dissipative channels, set in the general scenario of a system's interaction with a squeezed thermal bath, are compared in the Choi isomorphism framework, to bring out their contrasting rank and geometric properties. The equivalence of commutativity between the signal states and the Kraus operators to that between the system and interaction Hamiltonian, and thus to non-dissipativeness, is pointed out. Two distinct unitarily equivalent Kraus representations of the dissipative channel, one based on the Choi isomorphism, and the other based on an ansatz, are used to illustrate that the orthogonality of Kraus operators under the Hilbert---Schmidt inner product is not a unitary invariant. Unlike the non-dissipative (Pauli) channels, the dissipative (squeezed generalized amplitude damping) channels do not form a convex set. Further, whereas the rank of Pauli channels can be any positive integer up to 4, that of the amplitude damping ones is either 2 or 4. In the latter case, a noise range is identified where environmental squeezing counteracts the effect of thermal decoherence.

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Dissipative and non-dissipative single-qubit channels: dynamics and geometry

We propose an all-linear-optical scheme to ballistically generate a cluster state for measurement-based topological fault-tolerant quantum computation using hybrid photonic qubits entangled in a continuous-discrete domain. Availability of near-deterministic Bell-state measurements on hybrid qubits is exploited for this purpose. In the presence of photon losses, we show that our scheme leads to a significant enhancement in both tolerable photon-loss rate and resource overheads. More specifically, we report a photon-loss threshold of ∼3.3×10^{-3}, which is higher than those of known optical schemes under a reasonable error model. Furthermore, resource overheads to achieve logical error rate of 10^{-6}(10^{-15}) is estimated to be ∼8.5×10^{5}(1.7×10^{7}), which is significantly less by multiple orders of magnitude compared to other reported values in the literature.

Resource-Efficient Topological Fault-Tolerant Quantum Computation with Hybrid Entanglement of Light.

Characterizing noisy quantum processes is important to quantum computation and communication (QCC), since quantum systems are generally open. To date, all methods of characterization of quantum dynamics (CQD), typically implemented by quantum process tomography, are \textit{off-line}, i.e., QCC and CQD are not concurrent, as they require distinct state preparations. Here we introduce a method, "quantum error correction based characterization of dynamics", in which the initial state is any element from the code space of a quantum error correcting code that can protect the state from arbitrary errors acting on the subsystem subjected to the unknown dynamics. The statistics of stabilizer measurements, with possible unitary pre-processing operations, are used to characterize the noise, while the observed syndrome can be used to correct the noisy state. Our method requires at most $2(4^n-1)$ configurations to characterize arbitrary noise acting on $n$ qubits.

S. Omkar

Papers

The quantum cryptographic switch

Quasiprobability distributions in open quantum systems: Spin-qubit systems

Dissipative and non-dissipative single-qubit channels: dynamics and geometry

Resource-Efficient Topological Fault-Tolerant Quantum Computation with Hybrid Entanglement of Light.

Characterization of quantum dynamics using quantum error correction