Bloch sphere

Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system named after the physicist Felix Bloch. Alternatively, it is the pure state space of a 1 qubit quantum register. The Bloch sphere is actually geometrically a sphere and the correspondence between elements of the Bloch sphere and pure states can be explicitly given. In generalized form, the Bloch sphere may also refer to the analogous space of an n-level quantum system.

Quantum mechanics is mathematically formulated in Hilbert space or Projective Hilbert space. The space of pure states of a quantum system is given by the rays in the Hilbert space (the "points" of projective Hilbert space). The space of rays in any vector space is a projective space, and in particular, the space of rays in a two dimensional Hilbert space is the complex projective line, which is isomorphic to a sphere. Each pair of antipodal points on the Bloch sphere corresponds to a mutually exclusive pair of states of the particle, namely, spin up and spin down for a Stern-Gerlach experiment oriented along a particular axis in physical space.

The natural metric on the Bloch sphere is the Fubini-Study metric.

The qubit

To show this correspondence explicitly, consider the qubit description of the Bloch sphere; any state $|\psi \rangle$ can be written as a complex superposition of the ket vectors $|0\rangle$ and $|1\rangle$; moreover since phase factors do not affect physical state, we can take the representation so that the coefficient of $|0\rangle$ is real and non-negative. Thus $|\psi \rangle$ has a representation as

$|\psi \rangle =\cos \left({\frac {\theta }{2}}\right)|0\rangle \,+\,e^{i\phi }\sin \left({\frac {\theta }{2}}\right)|1\rangle =\cos \left({\frac {\theta }{2}}\right)|0\rangle \,+\,(\cos \phi +i\sin \phi )\,\sin \left({\frac {\theta }{2}}\right)|1\rangle$

with

$0\leq \theta \leq \pi ,\quad 0\leq \phi <2\pi .$

Except in the case $|\psi \rangle$ is one of the ket vectors $|0\rangle$ or $|1\rangle$, the representation is unique, i.e. the parameters $\phi \,$ and $\theta \,$ uniquely specify a point on the unit sphere of Euclidean space $\mathbb {R} ^{3}$, viz. the point whose coordinates $(x,y,z)$ are

${\begin{aligned}x&=\sin \theta \times \cos \phi \\y&=\sin \theta \times \sin \phi \\z&=\cos \theta .\end{aligned}}$

A generalization for pure states

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

Theorem. Let U(n) be the Lie group of unitary matrices of size n. Then the pure state space of H_n can be identified with the compact coset space

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

To prove this fact, note that there is a natural group action of U(n) on the set of states of H_n. This action is continuous and transitive on the pure states. For any state $|\psi \rangle$, the isotropy group of $|\psi \rangle$, (defined as the set of elements $g$ of U(n) such that $g|\psi \rangle =|\psi \rangle$) is isomorphic to the product group

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

In linear algebra terms, this can be justified as follows. Any $g$ of U(n) that leaves $|\psi \rangle$ invariant must have $|\psi \rangle$ as an eigenvector. 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 $|\psi \rangle$, 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 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

$A\mapsto e^{iA}$

is a local homeomorphism from the space of self-adjoint complex matrices to U(n). The space of self-adjoint complex matrices has real dimension n².

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

In fact,

$n^{2}-((n-1)^{2}+1)=2n-2.\quad$

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

Corollary. The real dimension of the pure state space of an m qubit quantum register is 2^m+1 − 2.

The geometry of density operators

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. However, while the Bloch sphere parametrizes not only pure states but mixed states for 2-level systems, 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, ..., μ_k with multiplicities n₁, ...,n_k. Then the group of unitary operators V such that V A V* = A is isomorphic (as a Lie group) to

$\operatorname {U} (n_{1})\times \cdots \times \operatorname {U} (n_{k}).$

In particular the orbit of A is isomorphic to

$\operatorname {U} (n)/(\operatorname {U} (n_{1})\times \cdots \times \operatorname {U} (n_{k})).$

We note here that, in the literature, one can find non-Bloch type parametrizations of (mixed) states that do generalize to dimensions higher than 2.

See also

• Gyrovector space

References

• Dariusz Chruściński, "Geometric Aspect of Quantum Mechanics and Quantum Entanglement", Journal of Physics Conference Series, 39 (2006) pp.9-16.
• Alain Michaud, "Rabi Flopping Oscillations" (2006). (A small animation of the bloch vector submitted to a resonant excitation.)
• Singer, Stephanie Frank (2005). Linearity, Symmetry, and Prediction in the Hydrogen Atom. New York: Springer. ISBN 0-387-24637-1.