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Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide into pure versus mixed states, or into coherent states and incoherent states. Categories with special properties include stationary states for time independence and quantum vacuum states in quantum field theory.

A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. A mixture of quantum states is again a quantum state.

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Quantum physics is most commonly formulated in terms of linear algebra, as follows. Any given system is identified with some finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1.

Multiplying a pure state by a scalar is physically inconsequential (as long as the state is considered by itself). If a vector in a complex Hilbert space H {\displaystyle H} can be obtained from another vector by multiplying by some non-zero complex number, the two vectors in H {\displaystyle H} are said to correspond to the same ray in the projective Hilbert space P ( H ) {\displaystyle \mathbf {P} (H)} of H {\displaystyle H} . Note that although the word ray is used, properly speaking, a point the projective Hilbert space corresponds to a line passing through the origin of the Hilbert space, rather than a half-line, or ray in the geometrical sense.

As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C2S+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the position in space).

A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics).

Mixed states arise in quantum mechanics in two different situations: first, when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations; and second, when one wants to describe a physical system which is entangled with another, as its state cannot be described by a pure state. In the first case, there could theoretically be another person who knows the full history of the system, and therefore describe the same system as a pure state; in this case, the density matrix is simply used to represent the limited knowledge of a quantum state. In the second case, however, the existence of quantum entanglement theoretically prevents the existence of complete knowledge about the subsystem, and it's impossible for any person to describe the subsystem of an entangled pair as a pure state.

A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of tag_hash_1172 is equal to 1 if the state is pure, and less than 1 if the state is mixed.[k][15] Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.[1]

Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space. The pure states of a quantum system correspond to the one-dimensional subspaces of the corresponding Hilbert space (and the "points" of the projective Hilbert space). For a two-dimensional Hilbert space, the space of all such states is the complex projective line C P 1 . {\displaystyle \mathbb {C} \mathbf {P} ^{1}.} This is the Bloch sphere, which can be mapped to the Riemann sphere.

For historical reasons, in optics the Bloch sphere is also known as the Poincar sphere and specifically represents different types of polarizations. Six common polarization types exist and are called Jones vectors. Indeed Henri Poincar was the first to suggest the use of this kind of geometrical representation at the end of 19th century,[4] as a three-dimensional representation of Stokes parameters.

Consider an n-level quantum mechanical system. This system is described by an n-dimensional Hilbert space Hn. The pure state space is by definition the set of 1-dimensional rays of Hn.

Formulations of quantum mechanics in terms of pure states are adequate for isolated systems; in general quantum mechanical systems need to be described in terms of density operators. The Bloch sphere parametrizes not only pure states but mixed states for 2-level systems. The density operator describing the mixed-state of a 2-level quantum system (qubit) corresponds to a point inside the Bloch sphere with the following coordinates:

When a generated beam is radially polarized, but its phase is spirally distributed as vortex phase, the beam is referred to as a radially polarized optical vortex20,21. Propagation of a beam containing a radially polarized optical vortex is different from that of a radially polarized beam12. Such a radially polarized optical vortex shows complex polarization dynamics because its polarization and phase are spatially distributed in three-dimensional free space. For this reason, vortex phase cancellation by use of an optical vortex with opposite vortex phase has been proposed as a method of creating pure vector beams3,22. This approach, however, is limited to experimental setups with complex optical alignments and low optical material throughput using polarization elements. To be best of our knowledge, radial polarization converters, which enable conversion to be the pure vector beams, have been proposed for use in a micro-structured photoconductive antenna23, a cone lens24, laser resonances1, liquid crystal modulators6,12, sub-wavelength structures15,17, nonlinear crystals25, wave plates of form birefringence fabricated by three-dimensional printing26, and meta-materials2,27 in several spectral regions from visible to terahertz. Our group also illustrated an achromatic axially-symmetric wave plate based on internal reflections and a generation of vector beams in the visible22,28, middle infrared18, and terahertz29 spectral regions. To verify details of the pure vector beam generated by use of those radial polarization converters, one must verify that the generated beam removes the vortex phase that can result from geometric phase. Although the vortex phase must be reduced in applications of pure vector beams, few studies have rigorously determined the spatial distribution of phase and its polarization of generated vector beams2,3,7,8,15,17,23, having observed only intensity distributions transmitted through a polarizer. The radial polarization converters used in these studies, moreover, are limited to narrow wavelength bands because broadband vortex phase cancellation has not been previously demonstrated.

Spatial distribution of polarization for vector beams. Spatial distribution of the polarization ellipticity (a) and azimuth (b) with respect to the azimuth angle tag_hash_125 within the beam. Ellipsometric parameters show that the vector beam is radially polarized because the ellipticity is almost zero and its azimuth varies linearly with the angle tag_hash_126. However, the phase of the vector beam was not measured.

In conclusion, we have proposed and demonstrated a pure vector beam without vortex phase by tailoring the geometric phase of the beam. This paper reports the first demonstration of a pure vector beam with radial polarization achieved by the concept for the reduction of vortex phase. We also proposed a continuous body extended to the segmented non-axially-symmetric half-wave plate. Our technique enables the vector beam to be manipulated in singular optics. Our proposed concept can also be used in conjunction with generic technologies in other different fields, such as super resolution imaging, material science, multiplexing quantum information science, high energy physics, and even astronomy.

Spatial distribution of Stokes parameters s0(tag_hash_128), s1(tag_hash_130), s2(tag_hash_132), and s3(tag_hash_134) along the angle tag_hash_135 on the input vector beam converted are modulated by the axially-symmetric wave plate. After passing through a wire-grid polarizer, the intensity distribution of the vector beam varied as a function of the angle tag_hash_136 can be expressed as

T.W. and T.H. conceived the concept of reducing the vortex phase in the vector beams. T.W. and T.H. designed the experiments. T.W. and T. H. designed, fabricated and characterized the non-axially-symmetric half-wave plate. T.W. and T.H. performed the experiments and analysed the data. T.W., T.H., K.S., M.W. and Y.O. discussed the results. T.H. supervised the project. T.W., T.H., K.S., M.W. and Y.O. contributed to writing the paper.

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We can test a gravity theory by searching for gravitational-wave (GW) polarization modes beyond general relativity. The LIGO-Virgo Collaboration analyzed several GW events in the O1 and O2 observing runs in the pure polarization framework, in which only scalar or vector polarization modes are allowed. In this paper, we reanalyze the polarizations of GW170814 (binary black hole merger) and GW170817 (binary neutron star merger) in the improved framework of pure polarizations including the angular patterns of nontensorial radiation. We find logarithms of the Bayes factors of 2.775 and 3.636 for GW170814 in favor of the pure tensor polarization against pure vector and scalar polarizations, respectively. These Bayes factors are consistent with the previous results from the LIGO-Virgo Collaboration, though the estimated parameters of the binaries should be different. For GW170817 with the priors on the location of the binary from NGC4993, we find logarithms of the Bayes factors of 21.078 and 44.544 in favor of the pure tensor polarization against pure vector and scalar polarizations, respectively. These support general relativity more strongly than the previous results by the LIGO-Virgo Collaboration due to the location prior. In addition, by utilizing the orientation information on the binary from a gamma-ray burst jet, we find logarithms of the Bayes factor of 51.043 and 60.271 in favor of the pure tensor polarization against pure vector and pure scalar polarizations, much improved from those without the jet prior. 006ab0faaa