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Two-level quantum systems are building blocks of quantum technologies, where the qubit is the basic unit of quantum information. The ability to design driving fields that produce prespecified evolutions of relevant physical observables is crucial to the development of such technologies. Using vector algebra and recently developed strategies for generating solvable two-level Hamiltonians, we construct the general solution to the inverse problem for a spin in a time-dependent magnetic field and its extension to any two-level system associated with fictitious spin and field. We provide a general expression for the field that drives the dynamics of the system so as to realize prescribed time evolutions of the expectation values of the Pauli operators and the autocorrelation of the Pauli vector. The analysis is applied to two-state charge transfer systems, showing that the charge transfer process can be seen as a motion of the state of the associated fictitious qubit on the Bloch sphere, and that the expectation values of the related Pauli operators describe the interference between the two differently localized electronic states and their population difference. Our formulation is proposed as a basic step towards potential uses of charge transfer in quantum computing and quantum information transfer.

In this paper, we introduce a unified method for generating ideals of Möbius-invariant Banach-valued Bloch mappings on the complex open unit disc D, through the composition with the members of a Banach operator ideal I. Using the linearization of derivatives of Banach-valued normalized Bloch mappings on D, this composition method yields the so-called ideals of I-factorizable normalized Bloch mappings I o B, where B denotes the class of normalized Bloch mappings on D. We present new examples of them as ideals of separable (Rosenthal, Asplund) normalized Bloch mappings and p-integral (strictly p-integral, p-nuclear) normalized Bloch mappings for any p ∈ [1,∞). Moreover, the Bloch dual ideal I B-dual of an operator ideal I is introduced and shown that it coincides with the composition ideal Idual o B. [ABSTRACT FROM AUTHOR]

Two-level quantum systems are building blocks of quantum technologies, where the qubit is the basic unit of quantum information. The ability to design driving fields that produce prespecified evolutions of relevant physical observables is crucial to the development of such technologies. Using vector algebra and recently developed strategies for generating solvable two-level Hamiltonians, we construct the general solution to the inverse problem for a spin in a time-dependent magnetic field and its extension to any two-level system associated with fictitious spin and field. We provide a general expression for the field that drives the dynamics of the system so as to realize prescribed time evolutions of the expectation values of the Pauli operators and the autocorrelation of the Pauli vector. The analysis is applied to two-state charge transfer systems, showing that the charge transfer process can be seen as a motion of the state of the associated fictitious qubit on the Bloch sphere, and that the expectation values of the related Pauli operators describe the interference between the two differently localized electronic states and their population difference. Our formulation is proposed as a basic step towards potential uses of charge transfer in quantum computing and quantum information transfer. [ABSTRACT FROM AUTHOR]

This work seeks to make explicit the operational connection between the preparation of two-level quantum systems with their corresponding description (as states) in a Hilbert space. This may sound outdated, but we show there is more to this connection than common sense may lead us to believe. To bridge these two separated realms—the actual laboratory and the space of states—we rely on a paradigmatic mathematical object: the Hopf fibration. We illustrate how this connection works in practice with a simple optical setup. Remarkably, this optical setup also reflects the necessity of using two charts to cover a sphere. Put another way, our experimental realization reflects the bi-dimensionality of a sphere seen as a smooth manifold. [ABSTRACT FROM AUTHOR]

For the building emergency evacuation path planning problem, existing algorithms suffer from low convergence efficiency and the problem of getting trapped in local optima. The Bloch Spherical Quantum Genetic Algorithm (BQGA) based on the least-squares principle for single-robot path planning and Bloch Spherical Quantum Bee Colony Algorithm (QABC) for multi-robots path planning are studied. Firstly, the characteristics of three-dimensional path planning are analyzed, and a linear decreasing inertia weighting approach is used to balance the global search ability of chromosomes and accelerate the search performance of the algorithm. Then, the application algorithm can generate a clear motion trajectory in the raster map. Thirdly, the least squares approach is used to fit the results, thus obtaining a progressive path. Finally, multi-robots path planning approaches based on QABC are discussed, respectively. The experimental results show that BQGA and QABC do not need to have a priori knowledge of the map, and they have strong reliability and practicality and can effectively avoid local optimum. In terms of convergence speed, BQGA improved by 3.39% and 2.41%, respectively, while QABC improved by 13.31% and 17.87%, respectively. They are more effective in sparse paths. [ABSTRACT FROM AUTHOR]

We extend Bloch sphere formalism to pure two-qubit systems. Combining insights from Geometric Algebra and the analysis of entanglement in different conjugate bases we identify two Bloch sphere geometry that is suitable for representing maximally entangled states. It turns out that the relative direction of the coordinate axes of the two Bloch spheres may be used to describe the states. Moreover, the coordinate axes of one Bloch sphere should be rignt-handed and those of the other one should be left-handed. We describe and depict separable and maximally entangled states as well as entangling and non-entangling rotations. We also offer a graphical representation of the workings of a CNOT gate for different inputs. Finally, we provide a way to also represent partially entangled states and describe entanglement measures related to the surface area of the sphere enclosing the state representation. [ABSTRACT FROM AUTHOR]

We analyze the results of the test of π/2 qubit rotations on a public quantum computer provided by IBM. We measure a single qubit rotated by π/2 about a random axis, and we accumulate vast statistics of the results. The test performed on different devices shows systematic deviations from the theoretical predictions, which appear at level 10-3. Some of the differences, beyond 5 standard deviations, cannot be explained by simple corrections due to nonlinearities of pulse generations. The magnitude of the deviation is comparable with the randomized benchmarking of the gate, but we additionally observe a pronounced parametric dependence. We discuss other possible reasons for the deviations, including states beyond the single-qubit space. The deviations have a similar structure for various devices used at different times, so they can also serve as a diagnostic tool to eliminate imperfect gate implementations and a faithful description of the involved physical systems. [ABSTRACT FROM AUTHOR]

We analyze the results of the test of π/2 qubit rotations on a public quantum computer provided by IBM. We measure a single qubit rotated by π/2 about a random axis, and we accumulate vast statistics of the results. The test performed on different devices shows systematic deviations from the theoretical predictions, which appear at level 10–3. Some of the differences, beyond 5 standard deviations, cannot be explained by simple corrections due to nonlinearities of pulse generations. The magnitude of the deviation is comparable with the randomized benchmarking of the gate, but we additionally observe a pronounced parametric dependence. We discuss other possible reasons for the deviations, including states beyond the single-qubit space. The deviations have a similar structure for various devices used at different times, so they can also serve as a diagnostic tool to eliminate imperfect gate implementations and a faithful description of the involved physical systems.

For the building emergency evacuation path planning problem, existing algorithms suffer from low convergence efficiency and the problem of getting trapped in local optima. The Bloch Spherical Quantum Genetic Algorithm (BQGA) based on the least-squares principle for single-robot path planning and Bloch Spherical Quantum Bee Colony Algorithm (QABC) for multi-robots path planning are studied. Firstly, the characteristics of three-dimensional path planning are analyzed, and a linear decreasing inertia weighting approach is used to balance the global search ability of chromosomes and accelerate the search performance of the algorithm. Then, the application algorithm can generate a clear motion trajectory in the raster map. Thirdly, the least squares approach is used to fit the results, thus obtaining a progressive path. Finally, multi-robots path planning approaches based on QABC are discussed, respectively. The experimental results show that BQGA and QABC do not need to have a priori knowledge of the map, and they have strong reliability and practicality and can effectively avoid local optimum. In terms of convergence speed, BQGA improved by 3.39% and 2.41%, respectively, while QABC improved by 13.31% and 17.87%, respectively. They are more effective in sparse paths.

Earlier we built POVM, which is absolutely continuous relative to the scalar measure with a projector-valued density, and studied the quantum channel generated by it. In this paper, we consider the possibility of restoring states measured by the Davis–Lewis instrument. An experiment with two outcomes is considered, and we compare two cases: with post-selection and without post-selection. We study how states are transformed in these cases and whether it is possible to restore them. [ABSTRACT FROM AUTHOR]

In this paper, we discuss quantum randomness expansion using the unreliable source device and the honest measure device consisting of mutually unbiased bases in 3-dimensional Hilbert space. Then, we obtain the relationship between 3-dimensional quantum witness and random number generation rate. Furthermore, the analytic expression of the relationship is given, which is of great significance for security analysis and practical application. This work can be considered a preliminary attempt for semi-device-independent quantum random number extension protocols in 3-dimensional Hilbert space. [ABSTRACT FROM AUTHOR]

Inspired by the one-hundredth anniversary of the seminal works of Stern and Gerlach, our contribution is a proposal of how to use their famous experiment in a more contemporary perspective. Our main idea is to re-cast the experiment in the modern language of prepare-and-measure scenarios. By doing so, it is possible to connect geometric and algebraic aspects of the space of states with the physical space. We also discuss possible simulations of the SG experiment as well as some experimental properties of the experiment revealed at the statistical level. Merging a more modern perspective with a paradigmatic experiment, we hope this paper can serve as an entry door for quantum information theory and the. [ABSTRACT FROM AUTHOR]

Quantum state manipulation of two-qubits on the local systems by special unitaries induces special orthogonal rotations on the Bloch spheres. An exact formula is given for determining the local unitaries for some given rotation on the Bloch sphere. The solution allows for easy manipulation of two-qubit quantum states with a single definition that is programmable. With this explicit formula, modifications to the correlation matrix are made simple. Using our solution, it is possible to diagonalize the correlation matrix without solving for the parameters in SU(2) that define the local unitary that induces the special orthogonal rotation in SO(3). Since diagonalization of the correlation matrix is equivalent to diagonalization of the interaction Hamiltonian, manipulating the correlation matrix is important in time-optimal control of a two-qubit state. The relationship between orthogonality conditions on SU(2) and SO(3) is given and manipulating the correlation matrix when only one qubit can be accessed is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a descendant of φ-fixed points, naturally termed as φ-coincidence points. In follow, we present a φ-fixed point theorem and a φ-coincidence point theorem, via b-simulation functions, in the turf of b-metric spaces; subsequently, we provide some interesting examples using MATLAB. As a closure, we give an application, in the field of quantum mechanics, to ensure the unique existence of a coincidence quantum state for two quantum operations on the Bloch sphere. [ABSTRACT FROM AUTHOR]

Three unit spheres were used to represent the two-qubit pure states. The three spheres are named the base sphere, entanglement sphere, and fiber sphere. The base sphere and entanglement sphere represent the reduced density matrix of the base qubit and the non-local entanglement measure, concurrence, while the fiber sphere represents the fiber qubit via a simple rotation under a local single-qubit unitary operation; however, in an entangled bipartite state, the fiber sphere has no information on the reduced density matrix of the fiber qubit. When the bipartite state becomes separable, the base and fiber spheres seamlessly become the single-qubit Bloch spheres of each qubit. Since either qubit can be chosen as the base qubit, two alternative sets of these three spheres are available, where each set fully represents the bipartite pure state, and each set has information of the reduced density matrix of its base qubit. Comparing this model to the two Bloch balls representing the reduced density matrices of the two qubits, each Bloch ball corresponds to two unit spheres in our model, namely, the base and entanglement spheres. The concurrence–coherence complementarity is explicitly shown on the entanglement sphere via a single angle.

Bipartite and multipartite entangled states are basic ingredients for constructing quantum networks and their accurate verification is crucial to the functioning of the networks, especially for untrusted networks. Here we propose a simple approach for verifying the Bell state in an untrusted network in which one party is not honest. Only local projective measurements are required for the honest party. It turns out each verification protocol is tied to a probability distribution on the Bloch sphere and its performance has an intuitive geometric meaning. This geometric picture enables us to construct the optimal and simplest verification protocols, which are also very useful to detecting entanglement in the untrusted network. Moreover, we show that our verification protocols can achieve almost the same sample efficiencies as protocols tailored to standard quantum state verification. Furthermore, we establish an intimate connection between the verification of Greenberger–Horne–Zeilinger states and the verification of the Bell state. By virtue of this connection we construct the optimal protocol for verifying Greenberger–Horne–Zeilinger states and for detecting genuine multipartite entanglement. [ABSTRACT FROM AUTHOR]

Quantum random access codes (QRACs) have attracted much attention because of their different properties from random access codes. How to explore and verify optimal QRAC strategies is an important research topic in quantum cryptography. In this paper, we focus on optimal n → 1 QRACs using single 3-level systems. Here, we present a general upper bound of maximum guess probability of n → 1 QRACs using a generalized Bloch sphere representation, and the maximum guess probability is an important indicator to verify whether it is the optimal QRAC. Furthermore, the optimal 2 → 1 and 3 → 1 QRACs are obtained through analytical methods. Finally, tight upper bounds of the maximum guess probability of 2 → 1 and 3 → 1 QRACs are acquired. These results will improve QRAC-based quantum cryptography protocols, such as quantum key distributions, quantum random number generations. [ABSTRACT FROM AUTHOR]

An arbitrarily dense discretization of the Bloch sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretized representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretized spheres) entanglement. Unlike Meyer's earlier discretization of the Bloch Sphere, there are no orthonormal triples, hence the Kocken–Specker theorem is not nullified. A physical interpretation of points on the discretized Bloch sphere is given in terms of ensembles of trajectories on a dynamically invariant fractal set in state space, where states of physical reality correspond to points on the invariant set. This deterministic construction provides a new way to understand the violation of the Bell inequality without violating statistical independence or factorization, where these conditions are defined solely from states on the invariant set. In this finite representation, there is an upper limit to the number of qubits that can be entangled, a property with potential experimental consequences. [ABSTRACT FROM AUTHOR]

The error-robust and short composite operations named ConCatenated Composite Pulses (CCCPs), developed as high-precision unitary operations in quantum information processing (QIP), are derived from composite pulses widely employed in nuclear magnetic resonance (NMR). CCCPs simultaneously compensate for two types of systematic errors, which was not possible with the known composite pulses in NMR. Our experiments demonstrate that CCCPs are powerful and versatile tools not only in QIP but also in NMR. [ABSTRACT FROM AUTHOR]