In quantummechanics andcomputing, theBloch sphere is a geometrical representation of thepure state space of atwo-level quantum mechanical system (qubit), named after the physicistFelix Bloch.^[1]

Mathematically each quantum mechanical system is associated with aseparablecomplexHilbert space${\displaystyle H}$. A pure state of a quantum system is represented by a non-zero vector${\displaystyle \psi }$ in${\displaystyle H}$. As the vectors${\displaystyle \psi }$ and${\displaystyle \lambda \psi }$ (with${\displaystyle \lambda \in {\mathbb{C}}^{\ast }}$) represent the same state, the level of the quantum system corresponds to the dimension of the Hilbert space and pure states can be represented asequivalence classes, or,rays in aprojective Hilbert space${\displaystyle \mathbf{P}(H_n)=\mathbb{C}{\mathbf{P}}^{n-1}}$.^[2] For a two-dimensional Hilbert space, the space of all such states is thecomplex projective line${\displaystyle \mathbb{C}{\mathbf{P}}^1.}$ This is the Bloch sphere, which can be mapped to theRiemann sphere.

The Bloch sphere is a unit2-sphere, withantipodal 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${\displaystyle |0⟩}$ and${\displaystyle |1⟩}$, respectively, which in turn might correspond e.g. to thespin-up andspin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to thepure states of the system, whereas the interior points correspond to themixed states.^[3][4] The Bloch sphere may be generalized to ann-level quantum system, but then the visualization is less useful.

The naturalmetric on the Bloch sphere is theFubini–Study metric. The mapping from the unit 3-sphere in the two-dimensional state space${\displaystyle {\mathbb{C}}^2}$ to the Bloch sphere is theHopf fibration, with eachray ofspinors mapping to one point on the Bloch sphere.

Given an orthonormal basis, anypure state${\displaystyle |\psi ⟩}$  of a two-level quantum system can be written as a superposition of the basis vectors${\displaystyle |0⟩}$  and${\displaystyle |1⟩}$ , where the coefficient of (or contribution from) each of the two basis vectors is acomplex number. This means that the state is described by four real numbers. However, only the relative phase between the coefficients of the two basis vectors has any physical meaning (the phase of the quantum system is not directlymeasurable), so that there is redundancy in this description. We can take the coefficient of${\displaystyle |0⟩}$  to be real and non-negative. This allows the state to be described by only three real numbers, giving rise to the three dimensions of the Bloch sphere.

We also know from quantum mechanics that the total probability of the system has to be one:

${\displaystyle ⟨\psi |\psi ⟩=1}$ , or equivalently${\displaystyle \Vert |\psi ⟩{\Vert }^2=1}$ .

Given this constraint, we can write${\displaystyle |\psi ⟩}$  using the following representation:

${\displaystyle |\psi ⟩=\cos \left(\theta /2\right)|0⟩+e^{i\varphi }\sin \left(\theta /2\right)|1⟩=\cos \left(\theta /2\right)|0⟩+(\cos \varphi +i\sin \varphi )\sin \left(\theta /2\right)|1⟩}$ , where${\displaystyle 0\le \theta \le \pi }$  and .

The representation is always unique, because, even though the value of${\displaystyle \varphi }$  is not unique when${\displaystyle |\psi ⟩}$  is one of the states (seeBra-ket notation)${\displaystyle |0⟩}$  or${\displaystyle |1⟩}$ , the point represented by${\displaystyle \theta }$  and${\displaystyle \varphi }$  is unique.

The parameters${\displaystyle \theta }$  and${\displaystyle \varphi }$ , re-interpreted inspherical coordinates as respectively thecolatitude with respect to thez-axis and thelongitude with respect to thex-axis, specify a point

${\displaystyle \overrightarrow{a}=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )=(u,v,w)}$ 

on the unit sphere in${\displaystyle {\mathbb{R}}^3}$ .

Formixed states, one considers thedensity operator. Any two-dimensional density operatorρ can be expanded using the identityI and theHermitian,tracelessPauli matrices${\displaystyle \overrightarrow{\sigma }}$ ,

 ,

where${\displaystyle \overrightarrow{a}\in {\mathbb{R}}^3}$  is called theBloch vector.

It is this vector that indicates the point within the sphere that corresponds to a given mixed state. Specifically, as a basic feature of thePauli vector, the eigenvalues ofρ are${\displaystyle \frac{1}{2}\left(1\pm |\overrightarrow{a}|\right)}$ . Density operators must be positive-semidefinite, so it follows that${\displaystyle \left|\overrightarrow{a}\right|\le 1}$ .

For pure states, one then has

${\displaystyle \mathrm{tr}\left({\rho }^2\right)=\frac{1}{2}\left(1+{\left|\overrightarrow{a}\right|}^2\right)=1\iff \left|\overrightarrow{a}\right|=1,}$ 

in comportance with the above.^[5]

As a consequence, the surface of the Bloch sphere represents all the pure states of a two-dimensional quantum system, whereas the interior corresponds to all the mixed states.

The Bloch vector${\displaystyle \overrightarrow{a}=(u,v,w)}$  can be represented in the following basis, with reference to the density operator${\displaystyle \rho }$ :^[6]

${\displaystyle u={\rho }_{10}+{\rho }_{01}=2\mathrm{Re}({\rho }_{01})}$ 

${\displaystyle v=i({\rho }_{01}-{\rho }_{10})=2\mathrm{Im}({\rho }_{10})}$ 

${\displaystyle w={\rho }_{00}-{\rho }_{11}}$ 

where

 

This basis is often used inlaser theory, where${\displaystyle w}$  is known as thepopulation inversion.^[7] In this basis, the values${\displaystyle u,v,w}$  are the expectations of the threePauli matrices${\displaystyle X,Y,Z}$ , allowing one to identify the three coordinates with x y and z axes.

Consider ann-level quantum mechanical system. This system is described by ann-dimensionalHilbert spaceH_n. The pure state space is by definition the set of rays ofH_n.

Theorem. LetU(n) be theLie group of unitary matrices of sizen. Then the pure state space ofH_n can be identified with the compact coset space

${\displaystyle \mathrm{U}(n)/(\mathrm{U}(n-1)\times \mathrm{U}(1)).}$ 

To prove this fact, note that there is anaturalgroup action of U(n) on the set of states ofH_n. This action is continuous andtransitive on the pure states. For any state${\displaystyle |\psi ⟩}$ , theisotropy group of${\displaystyle |\psi ⟩}$ , (defined as the set of elements${\displaystyle g}$  of U(n) such that${\displaystyle g|\psi ⟩=|\psi ⟩}$ ) is isomorphic to the product group

${\displaystyle \mathrm{U}(n-1)\times \mathrm{U}(1).}$ 

In linear algebra terms, this can be justified as follows. Any${\displaystyle g}$  of U(n) that leaves${\displaystyle |\psi ⟩}$  invariant must have${\displaystyle |\psi ⟩}$  as aneigenvector. Since the corresponding eigenvalue must be a complex number of modulus 1, this gives the U(1) factor of the isotropy group. The other part of the isotropy group is parametrized by the unitary matrices on the orthogonal complement of${\displaystyle |\psi ⟩}$ , which is isomorphic to U(n − 1). From this the assertion of the theorem follows from basic facts about transitive group actions of compact groups.

The important fact to note above is that theunitary group acts transitively on pure states.

Now the (real)dimension of U(n) isn². This is easy to see since the exponential map

${\displaystyle A\mapsto e^{iA}}$ 

is a local homeomorphism from the space of self-adjoint complex matrices to U(n). The space of self-adjoint complex matrices has real dimensionn².

Corollary. The real dimension of the pure state space ofH_n is 2n − 2.

In fact,

${\displaystyle n^2-\left((n-1{)}^2+1\right)=2n-2.}$ 

Let us apply this to consider the real dimension of anm qubit quantum register. The corresponding Hilbert space has dimension 2^m.

Corollary. The real dimension of the pure state space of anm-qubitquantum register is 2^m+1 − 2.

Mathematically the Bloch sphere for a two-spinor state can be mapped to aRiemann sphere${\displaystyle \mathbb{C}{\mathbf{P}}^1}$ , i.e., theprojective Hilbert space${\displaystyle \mathbf{P}(H_2)}$  with the 2-dimensional complex Hilbert space${\displaystyle H_2}$  arepresentation space ofSO(3).^[8]Given a pure state

${\displaystyle \alpha |\uparrow ⟩+\beta |\downarrow ⟩=|\nearrow ⟩}$ 

where${\displaystyle \alpha }$  and${\displaystyle \beta }$  are complex numbers which are normalized so that

${\displaystyle |\alpha {|}^2+|\beta {|}^2={\alpha }^{\ast }\alpha +{\beta }^{\ast }\beta =1}$ 

and such that${\displaystyle ⟨\downarrow |\uparrow ⟩=0}$  and${\displaystyle ⟨\downarrow |\downarrow ⟩=⟨\uparrow |\uparrow ⟩=1}$ ,i.e., such that${\displaystyle |\uparrow ⟩}$  and${\displaystyle |\downarrow ⟩}$  form a basis and have diametrically opposite representations on the Bloch sphere, then let

${\displaystyle u=\frac{\beta }{\alpha }=\frac{{\alpha }^{\ast }\beta }{{\alpha }^{\ast }\alpha }=\frac{{\alpha }^{\ast }\beta }{|\alpha {|}^2}=u_x+i{u}_y}$ 

be their ratio.

If the Bloch sphere is thought of as being embedded in${\displaystyle {\mathbb{R}}^3}$  with its center at the origin and with radius one, then the planez = 0 (which intersects the Bloch sphere at a great circle; the sphere's equator, as it were) can be thought of as anArgand diagram. Plot pointu in this plane — so that in${\displaystyle {\mathbb{R}}^3}$  it has coordinates${\displaystyle (u_x,u_y,0)}$ .

Draw a straight line throughu and through the point on the sphere that represents${\displaystyle |\downarrow ⟩}$ . (Let (0,0,1) represent${\displaystyle |\uparrow ⟩}$  and (0,0,−1) represent${\displaystyle |\downarrow ⟩}$ .) This line intersects the sphere at another point besides${\displaystyle |\downarrow ⟩}$ . (The only exception is when${\displaystyle u=\mathrm{\infty }}$ , i.e., when${\displaystyle \alpha =0}$  and${\displaystyle \beta \ne 0}$ .) Call this pointP. Pointu on the planez = 0 is thestereographic projection of pointP on the Bloch sphere. The vector with tail at the origin and tip atP is the direction in 3-D space corresponding to the spinor${\displaystyle |\nearrow ⟩}$ . The coordinates ofP are

${\displaystyle P_x=\frac{2u_x}{1+u_x^2+u_y^2},}$ 

${\displaystyle P_y=\frac{2u_y}{1+u_x^2+u_y^2},}$ 

${\displaystyle P_z=\frac{1-u_x^2-u_y^2}{1+u_x^2+u_y^2}.}$ 

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 ofdensity 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 pointinside the Bloch sphere with the following coordinates:

${\displaystyle \left(\sum p_i{x}_i,\sum p_i{y}_i,\sum p_i{z}_i\right),}$ 

where${\displaystyle p_i}$  is the probability of the individual states within the ensemble and${\displaystyle x_i,y_i,z_i}$  are the coordinates of the individual states (on thesurface of Bloch sphere). The set of all points on and inside the Bloch sphere is known as theBloch ball.

For states of higher dimensions there is difficulty in extending this to mixed states. The topological description is complicated by the fact that the unitary group does not act transitively on density operators. The orbits moreover are extremely diverse as follows from the following observation:

Theorem. SupposeA is a density operator on ann level quantum mechanical system whose distinct eigenvalues are μ₁, ..., μ_k with multiplicitiesn₁, ...,n_k. Then the group of unitary operatorsV such thatV A V* =A is isomorphic (as a Lie group) to

${\displaystyle \mathrm{U}(n_1)\times \cdots \times \mathrm{U}(n_k).}$ 

In particular the orbit ofA is isomorphic to

${\displaystyle \mathrm{U}(n)/\left(\mathrm{U}(n_1)\times \cdots \times \mathrm{U}(n_k)\right).}$ 

It is possible to generalize the construction of the Bloch ball to dimensions larger than 2, but the geometry of such a "Bloch body" is more complicated than that of a ball.^[9]

A useful advantage of the Bloch sphere representation is that the evolution of the qubit state is describable by rotations of the Bloch sphere. The most concise explanation for why this is the case is that theLie algebra for the group of unitary and hermitian matrices${\displaystyle SU(2)}$  is isomorphic to the Lie algebra of the group of three dimensional rotations${\displaystyle SO(3)}$ .^[10]

The rotations of the Bloch sphere about the Cartesian axes in the Bloch basis are given by^[11]

 

If${\displaystyle \hat{n}=(n_x,n_y,n_z)}$  is a real unit vector in three dimensions, the rotation of the Bloch sphere about this axis is given by:

${\displaystyle R_{\hat{n}}(\theta )=\exp \left(-i\theta \hat{n}\cdot \frac{1}{2}\overrightarrow{\sigma }\right)}$ 

An interesting thing to note is that this expression is identical under relabelling to the extended Euler formula forquaternions.

${\displaystyle \mathbf{q}=e^{\frac{1}{2}\theta (u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})}=\cos \frac{\theta }{2}+(u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\sin \frac{\theta }{2}}$ 

Ballentine^[12] presents an intuitive derivation for the infinitesimal unitary transformation. This is important for understanding why the rotations of Bloch spheres are exponentials of linear combinations of Pauli matrices. Hence a brief treatment on this is given here. A more complete description in a quantum mechanical context can be foundhere.

Consider a family of unitary operators${\displaystyle U}$  representing a rotation about some axis. Since the rotation has one degree of freedom, the operator acts on a field of scalars${\displaystyle S}$  such that:

${\displaystyle U(0)=I}$ 

${\displaystyle U(s_1+s_2)=U(s_1)U(s_2)}$ 

where${\displaystyle 0,s_1,s_2,\in S}$ 

We define the infinitesimal unitary as the Taylor expansion truncated at second order.

${\displaystyle U(s)=I+\frac{dU}{ds}{|}_{s=0}s+O\left(s^2\right)}$ 

By the unitary condition:

${\displaystyle U^{†}U=I}$ 

Hence

${\displaystyle U^{†}U=I+s\left(\frac{dU}{ds}{|}_{s=0}+\frac{d{U}^{†}}{ds}{|}_{s=0}\right)+O\left(s^2\right)=I}$ 

For this equality to hold true (assuming${\displaystyle O\left(s^2\right)}$  is negligible) we require

${\displaystyle \frac{dU}{ds}{|}_{s=0}+\frac{d{U}^{†}}{ds}{|}_{s=0}=0}$ .

This results in a solution of the form:

${\displaystyle \frac{dU}{ds}{|}_{s=0}=iK}$ 

where${\displaystyle K}$  is any Hermitian transformation, and is called the generator of the unitary family.Hence

${\displaystyle U(s)=e^{iKs}}$ 

Since the Pauli matrices${\displaystyle ({\sigma }_x,{\sigma }_y,{\sigma }_z)}$  are unitary Hermitian matrices and have eigenvectors corresponding to the Bloch basis,${\displaystyle (\hat{x},\hat{y},\hat{z})}$ , we can naturally see how a rotation of the Bloch sphere about an arbitrary axis${\displaystyle \hat{n}}$  is described by

${\displaystyle R_{\hat{n}}(\theta )=\exp (-i\theta \hat{n}\cdot \overrightarrow{\sigma }/2)}$ 

with the rotation generator given by${\displaystyle K=\hat{n}\cdot \overrightarrow{\sigma }/2.}$ 

• Online Bloch sphere visualization by Konstantin Herb

• Atomic electron transition
• Gyrovector space
• Versors
• Specific implementations of the Bloch sphere are enumerated under thequbit article.

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