Geometrical representation of the pure state space of a two-level quantum mechanical system
Not to be confused with Poincaré sphere (optics).
Bloch sphere
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]
Mathematically each quantum mechanical system is associated with a separable complex Hilbert space . A pure state of a quantum system is represented by a non-zero vector in . As the vectors and (with ) represent the same state, the level of the quantum system corresponds to the dimension of the Hilbert space and pure states can be represented as equivalence classes, or, rays in a projective Hilbert space .[2] For a two-dimensional Hilbert space, the space of all such states is the complex projective line This is the Bloch sphere, which can be mapped to the Riemann sphere.
The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors and , respectively, which in turn might correspond e.g. to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states.[3][4] The Bloch sphere may be generalized to an n-level quantum system, but then the visualization is less useful.
The natural metric on the Bloch sphere is the Fubini–Study metric. The mapping from the unit 3-sphere in the two-dimensional state space to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere.
In quantum mechanics and computing, the Blochsphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit)...
states. Mixed states can be represented by points inside the Blochsphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the...
other disciplines that depend on analysis and geometry, such as the Blochsphere of quantum mechanics and in other branches of physics. The extended complex...
Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss-American physicist and Nobel physics laureate who worked mainly in the U.S. He and Edward...
equate, respectively, to a rotation around the x, y and z axes of the Blochsphere by π {\displaystyle \pi } radians. The Pauli-X gate is the quantum equivalent...
(or density operators) of the quantum space C2, whose boundary is the Blochsphere. The Jones vectors correspond to the underlying space C2, that is, the...
Poincaré sphere to the real plane). A dynamical pseudospin skyrmion results from the stereographic projection of a rotating polariton Blochsphere in the...
discrete), a qubit is continuous-valued, describable by a direction on the Blochsphere. Despite being continuously valued in this way, a qubit is the smallest...
Poincaré sphere (but of radius P), as shown in the accompanying diagram. In quantum mechanics and computing, a related concept is the Blochsphere. Often...
states are represented by points in the interior. This is known as the Blochsphere picture of qubit state space. An example of pure and mixed states is...
terms of the identity matrix and the Pauli matrices also leads to the Blochsphere representation of 2 × 2 mixed states’ density matrix, (positive semidefinite...
there would be many angles through which state vectors would flip on the Blochsphere. There would be an optimal length in the Rabi setup that would give the...
Blochsphere representation of states (in blue) and optimal POVM (in red) for unambiguous quantum state discrimination on the states | ψ ⟩ = | 0 ⟩ {\displaystyle...
\Delta _{\lambda }} can be interpreted as a uniform contraction of the Blochsphere, parameterized by λ {\displaystyle \lambda } . In the case where λ =...
qubit parameters. A single qubit gate is achieved by rotation in the Blochsphere. Rotations between different energy levels of a single qubit are induced...
example, the Blochsphere to represent pure state of a two-level quantum mechanical system (qubit) is also known as the Poincaré sphere representing different...
horizontal circle (a line of latitude), or a rotation along the z-axis on the Blochsphere by φ {\displaystyle \varphi } radians. A common example is the T gate...
and has infinite order. In quantum mechanics, the Riemann sphere is known as the Blochsphere, and the Hopf fibration describes the topological structure...
unit quaternions. Versors have been used to represent rotations of the Blochsphere with quaternion multiplication. The facility of versors illustrate elliptic...
Animation showing the rotating frame. The red arrow is a spin in the Blochsphere which precesses in the laboratory frame due to a static magnetic field...
Blochsphere representation of states (in blue) and optimal POVM (in red) for unambiguous quantum state discrimination on the states | ψ ⟩ = | 0 ⟩ {\displaystyle...
single qubit gates, corresponding to 3d rotations about the axes of the Blochsphere. The Lie algebra serves to work out the representations of SU(2). The...
\mathbb {C} \mathbf {P} ^{1}} . This is known as the Blochsphere or, equivalently, the Riemann sphere. See Hopf fibration for details of the projectivization...
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