Bloch sphere should not be confused with Poincaré sphere (optics).
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.
H
\psi
H
\psi
λ\psi
λ\inC
P(Hn)=CPn-1
CP1.
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
|0\rangle
|1\rangle
C2
|\psi\rangle
|0\rangle
|1\rangle
|0\rangle
We also know from quantum mechanics that the total probability of the system has to be one:
\langle\psi|\psi\rangle=1
\||\psi\rangle\|2=1
Given this constraint, we can write
|\psi\rangle
|\psi\rangle= \cos\left(\theta/2\right)|0\rangle+ei\phi\sin\left(\theta/2\right)|1\rangle= \cos\left(\theta/2\right)|0\rangle+(\cos\phi+i\sin\phi)\sin\left(\theta/2\right)|1\rangle
0\leq\theta\leq\pi
0\leq\phi<2\pi
The representation is always unique, because, even though the value of
\phi
|\psi\rangle
|0\rangle
|1\rangle
\theta
\phi
The parameters
\theta
\phi
\vec{a}=(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)=(u,v,w)
on the unit sphere in
R3
\vec{\sigma}
\begin{align} \rho&=
1 | |
2 |
\left(I+\vec{a} ⋅ \vec{\sigma}\right)\\ &=
1 | |
2 |
\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}+
ax | |
2 |
\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}+
ay | |
2 |
\begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}+
az | |
2 |
\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}\\ &=
1 | |
2 |
\begin{pmatrix} 1+az&ax-iay\\ ax+iay&1-az \end{pmatrix} \end{align}
\vec{a}\inR3
It is this vector that indicates the point within the sphere that corresponds to a given mixed state. Specifically, as a basic feature of the Pauli vector, the eigenvalues of are
1 | |
2 |
\left(1\pm|\vec{a}|\right)
\left|\vec{a}\right|\le1
For pure states, one then has
\operatorname{tr}\left(\rho2\right)=
1 | |
2 |
\left(1+\left|\vec{a}\right|2\right)=1 \Leftrightarrow \left|\vec{a}\right|=1~,
in comportance with the above.[2]
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
\vec{a}=(u,v,w)
\rho
u=\rho10+\rho01=2\operatorname{Re}(\rho01)
v=i(\rho01-\rho10)=2\operatorname{Im}(\rho10)
w=\rho00-\rho11
where
\rho= \begin{pmatrix}\rho00&\rho01\ \rho10&\rho11\end{pmatrix}=
1 | |
2 |
\begin{pmatrix}1+w&u-iv\ u+iv&1-w\end{pmatrix}.
This basis is often used in laser theory, where
w
u,v,w
X,Y,Z
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 rays of Hn.
Theorem. Let U(n) be the Lie group of unitary matrices of size n. Then the pure state space of Hn can be identified with the compact coset space
\operatorname{U}(n)/(\operatorname{U}(n-1) x \operatorname{U}(1)).
To prove this fact, note that there is a natural group action of U(n) on the set of states of Hn. This action is continuous and transitive on the pure states. For any state
|\psi\rangle
|\psi\rangle
g
g|\psi\rangle=|\psi\rangle
\operatorname{U}(n-1) x \operatorname{U}(1).
In linear algebra terms, this can be justified as follows. Any
g
|\psi\rangle
|\psi\rangle
|\psi\rangle
The important fact to note above is that the unitary group acts transitively on pure states.
Now the (real) dimension of U(n) is n2. This is easy to see since the exponential map
A\mapstoei
Corollary. The real dimension of the pure state space of Hn is 2n − 2.
In fact,
n2-\left((n-1)2+1\right)=2n-2.
Let us apply this to consider the real dimension of an m qubit quantum register. The corresponding Hilbert space has dimension 2m.
Corollary. The real dimension of the pure state space of an m-qubit quantum register is 2m+1 − 2.
CP1
P(H2)
H2
\alpha\left|\uparrow\right\rangle+\beta\left|\downarrow\right\rangle=\left|\nearrow\right\rangle
\alpha
\beta
|\alpha|2+|\beta|2=\alpha*\alpha+\beta*\beta=1
\langle\downarrow|\uparrow\rangle=0
\langle\downarrow|\downarrow\rangle=\langle\uparrow|\uparrow\rangle=1
\left|\uparrow\right\rangle
\left|\downarrow\right\rangle
u={\beta\over\alpha}={\alpha*\beta\over\alpha*\alpha}={\alpha*\beta\over|\alpha|2}=ux+iuy
If the Bloch sphere is thought of as being embedded in
R3
R3
(ux,uy,0)
Draw a straight line through u and through the point on the sphere that represents
\left|\downarrow\right\rangle
\left|\uparrow\right\rangle
\left|\downarrow\right\rangle
\left|\downarrow\right\rangle
u=infty
\alpha=0
\beta\ne0
\left|\nearrow\right\rangle
Px={2ux\over1+
2 | |
u | |
x |
+
2}, | |
u | |
y |
Py={2uy\over1+
2 | |
u | |
x |
+
2}, | |
u | |
y |
Pz={1-
2 | |
u | |
x |
-
2 | |
u | |
y |
\over1+
2 | |
u | |
x |
+
2}. | |
u | |
y |
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:
\left(\sumpixi,\sumpiyi,\sumpizi\right),
where
pi
xi,yi,zi
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. Suppose A is a density operator on an n level quantum mechanical system whose distinct eigenvalues are μ1, ..., μk with multiplicities n1, ..., nk. Then the group of unitary operators V such that V A V* = A is isomorphic (as a Lie group) to
\operatorname{U}(n1) x … x \operatorname{U}(nk).
In particular the orbit of A is isomorphic to
\operatorname{U}(n)/\left(\operatorname{U}(n1) x … x \operatorname{U}(nk)\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.
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 the Lie algebra for the group of unitary and hermitian matrices
SU(2)
SO(3)
The rotations of the Bloch sphere about the Cartesian axes in the Bloch basis are given by[4]
\begin{align} Rx(\theta)&=e(-i=\cos(\theta/2)I-i\sin(\theta/2)X= \begin{bmatrix} \cos\theta/2&-i\sin\theta/2\\ -i\sin\theta/2&\cos\theta/2 \end{bmatrix}\\ Ry(\theta)&=e(-i=\cos(\theta/2)I-i\sin(\theta/2)Y= \begin{bmatrix} \cos\theta/2&-\sin\theta/2\\ \sin\theta/2&\cos\theta/2 \end{bmatrix}\\ Rz(\theta)&=e(-i=\cos(\theta/2)I-i\sin(\theta/2)Z= \begin{bmatrix} e-i&0\\ 0&ei\end{bmatrix} \end{align}
If
\hat{n}=(nx,ny,nz)
R\hat{n
An interesting thing to note is that this expression is identical under relabelling to the extended Euler formula for quaternions.
q=
| |||||
e |
= \cos
\theta | |
2 |
+(uxi+uyj+uzk)\sin
\theta | |
2 |
Ballentine[5] 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 found here.
Consider a family of unitary operators
U
S
U(0)=I
U(s1+s2)=U(s1)U(s2)
0,s1,s2,\inS
We define the infinitesimal unitary as the Taylor expansion truncated at second order.
U(s)=I+
dU | |
ds |
|s=0s+O\left(s2\right)
By the unitary condition:
U\daggerU=I
Hence
U\daggerU=I+s\left(
dU | |
ds |
|s=0+
dU\dagger | |
ds |
|s=0\right)+O\left(s2\right)=I
For this equality to hold true (assuming
O\left(s2\right)
dU | |
ds |
|s=0+
dU\dagger | |
ds |
|s=0=0
dU | |
ds |
|s=0=iK
K
U(s)=eiKs
(\sigmax,\sigmay,\sigmaz)
(\hat{x},\hat{y},\hat{z})
\hat{n}
R\hat{n
K=\hat{n} ⋅ \vec{\sigma}/2.
1 | |
2 |
(11+\veca ⋅ \vec\sigma)= \begin{pmatrix} \cos2\theta/2&\sin\theta/2~\cos\theta/2~e-i\phi\\ \sin\theta/2~\cos\theta/2~ei\phi&\sin2\theta/2 \end{pmatrix}
acts on the state eigenvector
(\cos\theta/2,ei\phi\sin\theta/2)